Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Vanbergen, Sean
Doctor of Philosophy - PhD
University of British Columbia
The TRIUMF UltraCold Advanced Neutron (TUCAN) collaboration is developing a new measurement of the neutron electric dipole moment (nEDM) with the goal of improving the current best limit of dn ≤ 1.8 × 10-²⁶ ecm (90% C.L.) to the level of dn ≲ 2 × 10-²⁷ ecm (90% C.L.). A non-zero nEDM requires the violation of charge-parity (CP) symmetry, and so the measurement of the nEDM can shed light on unanswered questions of fundamental physics, such as baryon asymmetry, the strong CP problem, and extensions of the Standard Model.The first part of this dissertation relates to the development of a new source of ultracold neutrons (UCNs). This source is based on the conversion of cold neutrons produced by a dedicated spallation source to UCNs by downscattering in superfluid ⁴He. The new source will enable the desired statistical reach of the experiment. Prior to the construction of the new source, a prototype source was operated at TRIUMF from 2017 to 2019, and was used to perform experiments on UCN production, storage, and transport to assist in the design of the new source. This dissertation describes the results of a subset of those experiments, relating to the production and lifetime of UCNs in superfluid 4He. The second part of this dissertation relates to the design of the TUCAN EDM spectrometer, which implements Ramsey’s method of separated oscillating magnetic fields to search for a shift in precession frequency associated with the presence of a non-zero nEDM in a large electric field. To achieve the desired sensitivity, the spectrometer must be designed to minimize systematic uncertainties. The calculations and simulations described here demonstrate the requirements to achieve this goal. This work is largely focused on the design of the central region of the spectrometer, which contains the UCNs and applies the electric field. The work done on the central region includes the development of a prototype cell and characterization of its storage properties, simulation studies of the electric fields and impact of magnetic properties, and measurements of ferromagnetic contamination in components of the central region.
Studies of Ultracold Neutron Dynamics andSystematic Effects for the TUCAN EDMExperimentbySean VanbergenB.A.Sc., The University of British Columbia, 2015B.C.S., The University of British Columbia, 2018A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2025c© Sean Vanbergen 2025The following individuals certify that they have read, and recommend to the Faculty of Graduateand Postdoctoral Studies for acceptance, the thesis entitled:Studies of Ultracold Neutron Dynamics and Systematic Effects for the TUCAN EDMExperimentsubmitted by Sean Vanbergen in partial fulfillment of the requirements forthe degree of Doctor of Philosophyin PhysicsExamining Committee:Ruediger Picker, Research Scientist, TRIUMFCo-supervisorKirk Madison, Professor, Physics and Astronomy, UBCCo-supervisorScott Oser, Professor, Physics and Astronomy, UBCUniversity ExaminerWilliam Andrew MacFarlane, Chemistry, UBCUniversity ExaminerBrad Plaster, Professor, Physics and Astronomy, University of KentuckyExternal ExaminerAdditional Supervisory Committee Members:Douglas Bryman, Professor, Physics and Astronomy, UBCSupervisory Committee MemberDouglas Scott, Professor, Physics and Astronomy, UBCSupervisory Committee MemberReiner Kruecken, Nuclear Science Division Director, Lawrence Berkeley National LaboratorySupervisory Committee MemberiiAbstractThe TRIUMF UltraCold Advanced Neutron (TUCAN) collaboration is developing a new mea-surement of the neutron electric dipole moment (nEDM) with the goal of improving the currentbest limit of dn ≤ 1.8×10−26 ecm (90% C.L.) to the level of dn . 2×10−27 ecm (90% C.L.). Anon-zero nEDM requires the violation of charge-parity (CP) symmetry, and so the measurementof the nEDM can shed light on unanswered questions of fundamental physics, such as baryonasymmetry, the strong CP problem, and extensions of the Standard Model.The first part of this dissertation relates to the development of a new source of ultracoldneutrons (UCNs). This source is based on the conversion of cold neutrons produced by adedicated spallation source to UCNs by downscattering in superfluid 4He. The new sourcewill enable the desired statistical reach of the experiment. Prior to the construction of thenew source, a prototype source was operated at TRIUMF from 2017 to 2019, and was used toperform experiments on UCN production, storage, and transport to assist in the design of thenew source. This dissertation describes the results of a subset of those experiments, relating tothe production and lifetime of UCNs in superfluid 4He.The second part of this dissertation relates to the design of the TUCAN EDM spectrometer,which implements Ramsey’s method of separated oscillating magnetic fields to search for a shiftin precession frequency associated with the presence of a non-zero nEDM in a large electric field.To achieve the desired sensitivity, the spectrometer must be designed to minimize systematicuncertainties. The calculations and simulations described here demonstrate the requirementsto achieve this goal. This work is largely focused on the design of the central region of thespectrometer, which contains the UCNs and applies the electric field. The work done on thecentral region includes the development of a prototype cell and characterization of its storageproperties, simulation studies of the electric fields and impact of magnetic properties, andmeasurements of ferromagnetic contamination in components of the central region.iiiLay SummaryThe TUCAN collaboration is building an experiment to measure how neutrons behave in electricfields, which will advance our understanding of the laws of physics and could help to answerquestions on topics such as the origin of matter in our Universe. This experiment will useultracold neutrons, which have such low energies that they can be stored in special vessels forhundreds of seconds. The first part of this work is about how to make these neutrons and howthey behave. The second part is about designing and building the vessel in which the neutronswill be stored during the experiment. This vessel must meet strict requirements to be able tostore the neutrons, to maintain the electric field, and to minimize effects that would cause errorsin the measurement. The work presented here will ensure that the TUCAN collaboration canbuild this experiment with unprecedented measurement precision.ivPrefaceAll of the writing in this dissertation is my own. All of the figures that do not cite a sourceI have created myself. The figures borrowed from other publications are acceptable use underthe corresponding copyright policies and licenses. Figure 2.1 appeared in Ref. [1], Figure 2.2appeared in Ref. [2], and Figure 2.4 appeared in Ref. [3]. As an author of all three of thesepapers the replication of these figures is permitted under the corresponding journal’s policiesand licenses, and is permitted by the TUCAN collaboration which forms the larger authorlists. Figure 5.4 appeared in Ref. [4], published under a Creative Commons Attribution 4.0license (http://creativecommons.org/licenses/by/4.0/), and the replication here is permittedunder that license.The prototype source described in Chapter 2, and the experimental campaigns, involveda large number of people in the TUCAN collaboration who participated in the installation ofthe source, took shifts during the campaigns, and participated in data collection. A full listof contributors to the source development can be found in the author lists of Refs. [3] and[5]. For my own part, I was involved in the physical installation of the source in 2017 as anundergraduate, and did much of the programming of the slow control for the cryostat and gashandling systems. I was involved in the general planning and preparation, shift taking, anddata collection in both the 2018 and 2019 UCN runs, as well as the first cooldown of the systemin 2017, although I was not present for the 2017 UCN production. A large part of my workon the source was on analyzing experiments from the 2018 and 2019 runs, and much of thatultimately did not make it into this dissertation.Chapter 3 is based on work contributed to by a large number of members of the TUCANcollaboration. The actual data collection was spread across many members of the collaboration,including myself, who participated in shift taking during the 2018 experimental campaigns.The initial analysis of the raw data to obtain the storage lifetimes was performed using codeoriginally developed by W. Schreyer and T. Andalib. A simplified version of the CAD modelfor the source, used in PENTrack, was developed by W. Schreyer for the analysis in Ref. [3]. Imodified that model to be more detailed and to describe the geometry for the specific simulationsthat make up the bulk of Chapter 3. The setting up and running of the simulations, and all ofthe subsequent analysis and interpretation of them that is included in this chapter is my ownwork.The calculations in Chapter 4 are entirely my own. I wrote the code to carry out all ofthe analytical calculations that are included in this chapter, and all of the interpretation ofthose results is my own. The PENTrack simulations of UCN correlations for the depolarizationcalculations were also my own. For some calculations here, and in subsequent chapters, I usedas input the simulation results of S. Sidhu from his dissertation [6], and when that is the caseI have cited that appropriately.The design and construction of the prototype cell described in Chapter 5 was a majorcollaborative effort. R. Picker suggested the concept of the cell valve, and the detailed design ofthe internals was developed by co-op student V. G. Tiepo under the supervision of I. Nikonov.I modelled the rest of the conceptual design of the cell, and the detailed drawings were madevPrefaceby S. Horn. The first assembly of the cell internals was done primarily by M. Good, with someinput from me on modifications that had to be made. The full assembly and testing of themechanics of the cell was done by me. The coating machine used to produce the insulator ringwas initially worked on by F. Kuchler, with design work by I. Nikonov and N. Massacret. Itook over the development of these coatings after the initial construction was finished. I didall of the test coatings and final dPS coating of the insulator ring. I also modified the coatingmachine for dPE coatings and did all of the development of those, although it ultimately didnot lead to a final coating of a 0.5 m ring.I proposed and planned the beamtimes of Chapter 5 myself, in consultation with K. Mishimaat J-PARC. A number of people other than myself participated in the setup and data-takingat J-PARC in 2022 and 2023: A. Brossard, K. Mishima, T. Higuchi, S. Imajo, S. Kawasaki,R. Picker, J. W. Martin, I. Press, H. Akatsuka, and I. Ide. The processing of the raw datafor background subtraction and normalization used code provided by S. Imajo. All other partsof the analysis for both the TOF and storage lifetime measurements I developed myself. Thedisassembly, cleaning, and reassembly of the cell between the 2022 and 2023 beamtimes wasdone by me, with some help from A. Brossard as a second pair of hands for the assembly.The Opera simulation described in Chapters 6 and 7 was initially developed in a simplifiedform, which included only the electrodes, by students R. Gaur and S. Weitz for magneticfield simulations under the supervision of R. Picker. I took over this model and rebuilt itto more closely match the full design of the geometry of the central region, including the finaldimensions, details of the vacuum chamber, HV feedthrough, detailed features of the electrodes,and introducing magnetic coils. All of the field simulations using the Opera model that arepresented in this dissertation I ran myself, and the interpretation and analysis of the results ismy own.S. Weitz developed first PENTrack simulations of the B1 field, and this was a helpful basisfor some of the studies presented in Chapter 7. The PENTrack simulations using the Operamaps that are presented in this dissertation I set up and ran on my own, and all of the analysisof the results was also my own.The alternative FEMM simulation used in Chapter 6 was originally based on a simulationby Y. Li from 2015, which I adapted as an undergraduate in 2017. This early simulation was ofa completely different geometry from that currently planned for the TUCAN EDM experiment,so at this point the FEMM simulation has been almost entirely rebuilt from scratch, but Y. Li’ssimulation was a helpful starting point. All simulations using this model and all calculationsbased on them that are presented in this dissertation I performed myself.Other results that are presented in Chapters 6 and 7, and which did not use the Opera andFEMM simulations, were also entirely my own work. This includes the calculations related toleakage currents, HV instability, and magnetic dipoles.The tests at PTB described in Chapter 7 involved a few people. The first batch of ∼20samples in 2021 were gathered and brought to PTB by F. Kuchler, and the measurements weredone by A. Schnabel and K. Rolfs of PTB. The second batch of ∼100 samples in 2023 I collectedand prepared. The measurements at PTB were done by myself, R. Picker, and W. Klassen,with the guidance of J. Voigt. The analysis of these results in Chapter 7 is entirely my own.Many of the equations used in this dissertation can be found in other publications onnEDM measurements. During the course of my research I found that said publications neverincluded or referred to truly complete derivations, and so as part of my work I carried outcomplete re-derivations of all of the needed results for the TUCAN EDM measurement fromfirst principles. The goal of this was to fully elucidate where these equations were comingfrom, what assumptions were involved, and the validity of applying those equations to a higherviPrefaceprecision experiment. These derivations, which are included in Appendices A, B, C, and E aretherefore my own work in the sense that I carried out the derivations myself, but the ideasoriginated in prior works — I have cited those works accordingly. Occasionally, the directionof a derivation relied on a step that had been written in a previous publication, and when thatis the case I have explicitly cited the relevant source.PENTrack simulations that are described throughout this dissertation were run on theDigital Research Alliance of Canada (DRAC) computing clusters.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixSymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Physical Symmetries and Violations . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Electric Dipole Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1 Non-neutron EDM Searches . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.2 Searches for the Neutron EDM . . . . . . . . . . . . . . . . . . . . . . . . 121.2.3 Measurement Technique for the nEDM . . . . . . . . . . . . . . . . . . . 141.3 Ultracold Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.3.1 Properties of Ultracold Neutrons . . . . . . . . . . . . . . . . . . . . . . . 171.3.2 Production of Ultracold Neutrons . . . . . . . . . . . . . . . . . . . . . . 221.3.3 Simulating Ultracold Neutrons . . . . . . . . . . . . . . . . . . . . . . . . 251.4 Outline of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27I Ultracold Neutron Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 Ultracold Neutrons at TRIUMF . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 The TUCAN Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2.1 Beamline 1U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2.2 Moderators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2.3 UCN Production Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3 The RCNP Source at TRIUMF . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.4 Experimental Campaigns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.4.2 Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38viiiTable of Contents2.4.3 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.4.4 Raw Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 Lifetime of Ultracold Neutrons in He-II . . . . . . . . . . . . . . . . . . . . . . 413.1 UCN Production and Storage Lifetime in He-II . . . . . . . . . . . . . . . . . . . 413.2 Experiment Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3 Storage Lifetime Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.3.1 Experiment Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.3.2 Analysis Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.3.3 Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.4 Production Rate in He-II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.4.1 Experiment Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . 623.4.2 Analysis Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.4.3 Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.5 Long-term Behaviour of Source Lifetime . . . . . . . . . . . . . . . . . . . . . . . 683.5.1 Source Storage Lifetime vs. Time . . . . . . . . . . . . . . . . . . . . . . 683.5.2 Source Spoiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73II The TUCAN EDM Central Region . . . . . . . . . . . . . . . . . . . . . . . 764 The nEDM Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2 The TUCAN EDM Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2.1 Magnetic Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.2.2 Ambient Compensation Coils . . . . . . . . . . . . . . . . . . . . . . . . . 794.2.3 Spin Polarization and Transport . . . . . . . . . . . . . . . . . . . . . . . 804.2.4 The Central Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.2.5 Magnetic Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.2.6 Magnetometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.2.7 Spin Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.3 Overview of Sensitivity Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.3.1 General Features of the UCN Population . . . . . . . . . . . . . . . . . . 884.3.2 Magnetic Field Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 894.3.3 Single Cell vs Double Cell . . . . . . . . . . . . . . . . . . . . . . . . . . 904.3.4 Redfield Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.3.5 Magnetic Depolarization . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.3.6 BE Frequency Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.3.7 B2 Frequency Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.3.8 E2 Frequency Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.3.9 Spatially-Localized Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.3.10 Net Motional Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.3.11 AC Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.3.12 Co-Magnetometer Systematics . . . . . . . . . . . . . . . . . . . . . . . . 1214.3.13 Miscellanea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.3.14 The TUCAN EDM Systematics Budget and Requirements . . . . . . . . 122ixTable of Contents5 Development of a Prototype Storage Cell . . . . . . . . . . . . . . . . . . . . . 1255.1 A Prototype Storage Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.1.1 Cell Valve Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.1.2 Insulator Coatings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.2 J-PARC BL05 UCN Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305.3 Energy Spectrum Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1325.3.1 Experiment Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1325.3.2 Analysis Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1325.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.4 Storage Lifetime Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.4.1 Experiment Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.4.2 Pressure Dependent Measurements . . . . . . . . . . . . . . . . . . . . . 1415.4.3 Analysis Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1445.4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1506 Electrical Fields in the Central Region Design . . . . . . . . . . . . . . . . . . 1596.1 High Voltage Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1596.2 Finite Element Simulations of the Electric Field . . . . . . . . . . . . . . . . . . 1636.2.1 Comparison of 2D and 3D Simulation Results . . . . . . . . . . . . . . . 1696.3 Impact of Electric Field Inhomogeneity . . . . . . . . . . . . . . . . . . . . . . . 1696.3.1 Rotational v × E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1696.3.2 Linear vxE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1856.4 High-Voltage Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1886.4.1 Leakage Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1896.4.2 High Voltage Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1926.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1957 Magnetic Considerations in the Central Region Design . . . . . . . . . . . . . 1977.1 Materials and Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1977.1.1 Eddy Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1997.1.2 Thermal Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1997.2 Para- and Diamagnetic Distortion of the B0 Field . . . . . . . . . . . . . . . . . 2007.2.1 Material Choice of Other Components . . . . . . . . . . . . . . . . . . . 2067.3 Eddy Currents and the B1 Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 2077.4 Ferromagnetism and Magnetic Dipoles . . . . . . . . . . . . . . . . . . . . . . . 2147.4.1 Systematic Contributions from Magnetic Dipoles . . . . . . . . . . . . . . 2147.4.2 Measurements of Ferromagnetic Dipoles . . . . . . . . . . . . . . . . . . . 2177.4.3 Fully Reversing Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2287.5 Thermal Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2327.5.1 Observations at PTB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2327.5.2 Heating of the Central Region . . . . . . . . . . . . . . . . . . . . . . . . 2347.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2358 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2378.0.1 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242xTable of ContentsAppendicesA Derivations of Ramsey’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 275A.1 Probability of Spin Flip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275A.1.1 Rotation Transformation Results . . . . . . . . . . . . . . . . . . . . . . . 275A.1.2 Hamiltonian in Rotating Frame . . . . . . . . . . . . . . . . . . . . . . . 276A.1.3 Solving the Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277A.1.4 A Linearly Oscillating Field . . . . . . . . . . . . . . . . . . . . . . . . . 280A.2 The Central Fringe and Asymmetry Equation . . . . . . . . . . . . . . . . . . . 282A.2.1 The Central Fringe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282A.2.2 The Asymmetry Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 284A.3 Statistical Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285A.3.1 Asymmetry Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285A.3.2 Hessian Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287A.3.3 Including the Co-Magnetometer . . . . . . . . . . . . . . . . . . . . . . . 292B Redfield Theory in an nEDM Experiment . . . . . . . . . . . . . . . . . . . . . 293B.1 Relaxation Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293B.2 Relaxation Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297B.3 Frequency Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302B.4 Harmonic Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307C Derivations of the Crossing Point Method . . . . . . . . . . . . . . . . . . . . . 312C.1 Single Cell Crossing Point Equations . . . . . . . . . . . . . . . . . . . . . . . . 312C.2 Single Cell Crossing Point Uncertainty . . . . . . . . . . . . . . . . . . . . . . . 313C.3 Double Cell Crossing Point Equations . . . . . . . . . . . . . . . . . . . . . . . . 315C.4 Double Cell Crossing Point Uncertainty . . . . . . . . . . . . . . . . . . . . . . . 316D UCN Cleaning Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317D.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317D.2 Cleaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317D.2.1 Small Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317D.2.2 Large Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319D.2.3 O-Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322D.3 Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323E Derivation of the Frequency Shift from AC Ripple . . . . . . . . . . . . . . . 325F Additional Material on Ferromagnetic Dipoles . . . . . . . . . . . . . . . . . . 327F.1 Single-Angle Dipole Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327F.2 PTB Samples Detailed List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327xiList of Tables1.1 Summary of current measured EDM limits and experiments under developmentin the leptonic sector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2 Summary of the current measured EDM limits and experiments under develop-ment for hadronic systems, not including neutrons. . . . . . . . . . . . . . . . . . 111.3 Summary of the history of nEDM measurements and experiments under devel-opment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.4 Commonly referenced neutron energy ranges and their properties. . . . . . . . . 181.5 Fermi potentials of some commonly-used materials in UCN experiments, calcu-lated using data from Ref. [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.6 Summary of superthermal UCN sources and their main features. . . . . . . . . . 262.1 Summary of the differences between the vertical source and the planned TUCANsource at TRIUMF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.1 Summary of past experimental measurements of He-II upscattering coefficients. . 463.2 Measured storage lifetimes and uncertainties for the temperature-dependent stor-age lifetime experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.3 Summary of material properties used in the PENTrack simulation. The valuesand ranges are explained in the main text. . . . . . . . . . . . . . . . . . . . . . . 523.4 Best-fit values of Ws for each of the simulated values of BHe. . . . . . . . . . . . 623.5 Best-fit values of Ws for each of the simulated values of BHe in the analysis ofthe production experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.1 Table of the harmonic polynomial basis functions for magnetic fields. . . . . . . . 904.2 Summary of systematic uncertainty contributions in the TUCAN EDM experi-ment, with expected size of the contributions at the 1σ level. . . . . . . . . . . . 1245.1 Weighting functions used in the simulation analysis of the total energy spectrumfrom the TOF experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355.2 Parameter values used in the simulation analysis of the total energy spectrum. . 1355.3 List of storage experiments performed in the 2022 and 2023 beamtimes. . . . . . 1395.4 Measured storage lifetimes in all cell experiments. . . . . . . . . . . . . . . . . . . 1505.5 Values of WNiP obtained from analysis of the three storage lifetime measurementsTCN22-010-SL, TCN22-012-SL, and TCN23-010-SL. . . . . . . . . . . . . . . . . 1525.6 Values of WdPS obtained from analysis of the two storage lifetime measurementsTCN22-020-SL and TCN23-020-SL. . . . . . . . . . . . . . . . . . . . . . . . . . . 1546.1 Properties of select dielectric materials of relevance to the TUCAN EDM exper-iment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1626.2 Summary of systematic uncertainties calculated in Chapter 6. . . . . . . . . . . . 196xiiList of Tables7.1 Magnetic susceptibilities of materials of interest for the TUCAN EDM experiment.1987.2 Electrical conductivities of materials of interest for the TUCAN EDM experiment.1997.3 False EDM signal resulting from BE frequency shifts calculated from simulationsof the distortion of a uniform 1µT field due to magnetic susceptibility of thecentral region materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2037.4 False EDM signal resulting from BE frequency shifts calculated from simulationsof the distortion of a uniform 1µT field due to magnetic susceptibility of thecentral region materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2047.5 Summary of systematic uncertainties calculated in Chapter 7. . . . . . . . . . . . 236B.1 Free-space magnetic field polynomial basis up to order l = 3. . . . . . . . . . . . 309B.2 Free space magnetic field polynomial basis functions of order l = 4 . . . . . . . . 310B.3 Free space magnetic field polynomial basis functions of order l = 5 . . . . . . . . 311F.1 Details of the samples tested at PTB. . . . . . . . . . . . . . . . . . . . . . . . . 333xiiiList of Figures1.1 Illustration of the connections between different theoretical sources of CP viola-tion and measurable systems, based on the description of Ref. [8]. . . . . . . . . 61.2 An illustration of Ramsey’s method of separated oscillating magnetic fields. . . . 151.3 Plot of the calculated probability that a neutron spin in the state ψ↑ will bemeasured to be in the ψ↓ state following the sequence of Ramsey’s method. . . . 162.1 Schematics of the BL1 facility at TRIUMF, focused on BL1U. These diagramswere previously published in Ref. [1], and their use here is permitted by theTUCAN collaboration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2 Photograph of the spallation target used for the TUCAN source. . . . . . . . . . 322.3 Geometry of the TUCAN source cut through the central plane. . . . . . . . . . . 332.4 Geometry of the prototype UCN source at TRIUMF. . . . . . . . . . . . . . . . . 353.1 Plot of the dispersion curves of free neutrons and He-II. . . . . . . . . . . . . . . 423.2 Feynman diagrams of the three upscattering processes in He-II relevant for UCNlosses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3 Schematic of the geometry of the vertical source and experimental configurationfor experiments relating to the behaviour of UCNs in He-II. The top view showsthe layout from above, and the bottom view shows the layout from the side. Alldimensions given in mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.4 Histogram showing the distribution of fHe−II, the fraction of time spent by UCNsin the He-II volume, for different storage times in the source. . . . . . . . . . . . 533.5 Histogram showing the distribution of fHe−II, the fraction of time spent by UCNsin the He-II volume, for different conditions during storage in the source. . . . . 533.6 Picture of the Solidworks model of the vertical source showing the helium vapourfilled up to IV1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.7 Cut-through picture of the Solidworks model of IV2, indicating the internal sur-faces and geometry in the open position. . . . . . . . . . . . . . . . . . . . . . . . 573.8 Plot of experimental storage lifetime results vs. temperature compared to thetheoretical He-II upscattering model. . . . . . . . . . . . . . . . . . . . . . . . . . 583.9 Heatmap plots showing the value of ∆χ2 for different simulation values of (BHe,W )compared to the experimental storage lifetime results. . . . . . . . . . . . . . . . 593.10 Heatmap plot of ∆χ2 for different simulated values of (BHe,W ) for the3He detector. 613.11 Measured UCN count rate in the 6Li detector versus temperature as the sourcewas warmed up to 1.8 K and then back down to 1.1 K. . . . . . . . . . . . . . . . 633.12 Heatmap of ∆χ2 for different simulated values of (BHe,W ) for the productionexperiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.13 Plot of the storage lifetime in the source measured over the duration of the 2018experimental run. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69xivList of Figures3.14 Plot of the temperature of the He-II measured over the duration of the 2018experimental run. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.15 Simulated storage lifetime vs. temperature for different reductions in the levelof the He-II volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.16 Measured “daily” storage lifetimes in the source vs. the measured temperatureof the He-II in the source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.17 Plot of measured source storage lifetimes versus temperature while spoiling thesource by injecting nitrogen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.1 Illustration showing the layout of the major components of the TUCAN EDMspectrometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.2 Concept model of the central region of the TUCAN EDM spectrometer. . . . . . 814.3 Schematic showing the geometry of the cells assumed for systematic calculationsand PENTrack simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.4 Plot of the energy spectra of UCNs used for PENTrack simulations and calculations. 884.5 Plot of the height distribution of UCNs seen in PENTrack simulations. . . . . . . 894.6 Simulated value of the autocorrelation of the magnetic field basis function Π1,0,z =z for UCNs versus time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.7 Simulated values of T2,m versus energy for a magnetic field of G1,0 = 1 nT/m. . . 974.8 Plot of visibility after 150 s due to intrinsic depolarization, for different magneticfields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.9 Plot of visibility after 150 s due to gravitational depolarization, for different mag-netic fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.10 Calculated false EDM signals caused by BE frequency shifts for different orderand magnitude of magnetic fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.11 Calculated false EDM signals from BE frequency shifts for different orders andmagnitudes of magnetic field gradient versus cell radius. . . . . . . . . . . . . . . 1064.12 Calculated false EDM signals from BE frequency shifts for m 6= 0 modes in acell that is offset from the coordinate centre of the magnetic field. . . . . . . . . . 1074.13 Calculated false EDM signals from BE frequency shifts for l-even modes whenthe cells are vertically offset from the coordinate centre of the magnetic field. . . 1084.14 Calculated residual false EDM signals from BE frequency shifts for G3,0 and G5,0.1134.15 Calculated false EDM signal resulting from the gravitational shift between UCNsand Hg for a polarity-dependent gradient in a double cell configuration. . . . . . 1144.16 Calculated false EDM signals from B2 frequencies shifts in a single cell whenthere is a difference in B2 with HV polarity reversal. . . . . . . . . . . . . . . . . 1174.17 Calculated false EDM signals caused by B2 frequency shifts in a double cell whenthere is a difference in B2 with HV polarity. . . . . . . . . . . . . . . . . . . . . . 1185.1 Diagrams of the prototype cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.2 Diagram of the cell valve mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . 1275.3 Diagram of the deuterated-plastic coating machine. . . . . . . . . . . . . . . . . . 1295.4 Diagram of the J-PARC MLF BL05 Doppler shifter UCN source. . . . . . . . . . 1315.5 Diagrams of the geometry of the TOF measurement. . . . . . . . . . . . . . . . . 1335.6 Plot of normalized counts vs TOF measured out of the DS. . . . . . . . . . . . . 1375.7 Example fits of the simulated TOF spectrum to match the measured spectrum. . 1395.8 Simulated initial total energy spectra of neutrons from the DS. . . . . . . . . . . 1405.9 Diagrams of the geometry of the storage lifetime measurement with cell valve. . . 142xvList of Figures5.10 Diagram of the modified experiment layout for TCN22-030-SL. . . . . . . . . . . 1435.11 Measured UCN counts after 15 s storage in the cell versus filling time. . . . . . . 1435.12 A sample of pressure data versus time from the auxiliary pressure measurementused to calibrate pressure inside the cell during storage. . . . . . . . . . . . . . . 1455.13 Photograph of a contaminated guide taken following TCN22-011-SL. . . . . . . . 1515.14 Simulated storage lifetime of the cell versus WNiP for the TCN2X-01Y-SL con-figuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1535.15 Simulated storage lifetime of the cell versus WdPS for the TCN2X-020-SL config-uration using a uniform angular distribution. . . . . . . . . . . . . . . . . . . . . 1555.16 Simulated storage lifetime of the cell versus WdPS for the TCN2X-020-SL config-uration using a collimated angular distribution. . . . . . . . . . . . . . . . . . . . 1566.1 Electric field strength at breakdown versus pressure for different gases, calculatedfrom Paschen’s law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1606.2 Schematic showing the key geometric features and dimensions of the cell for HVsimulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1636.3 Schematic showing the geometric features of the central region around the HVfeedthrough. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1646.4 Electric field map inside the top cell in the xz plane, simulated using Opera . . . 1656.5 Electric field maps in the vacuum region of the central region, focused aroundthe feedthrough, simulated using Opera. . . . . . . . . . . . . . . . . . . . . . . . 1666.6 Electric field maps inside the central region around the HV feedthrough, with a15.24 cm insulator height, simulated using Opera. . . . . . . . . . . . . . . . . . . 1686.7 Map of the radial electric field component in the top cell, simulated using FEMMand Opera. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1706.8 Illustration of the path geometry of a neutron in the horizontal plane, beingreflected off the perimeter wall of a cylindrical cell. . . . . . . . . . . . . . . . . . 1736.9 Simulated decay of the azimuthal velocity vφ of a UCN population over time. . . 1756.10 Mean τr of the net motion vφ versus diffuse reflection probability PL, calculatedusing different methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1766.11 Magnitude of the simulated false signal from net rotational motion vs. the stepsize of the electric field map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1786.12 Magnitude of the simulated false EDM signal from net rotational motion vs.allowed angular deviation when exporting the STL geometry from Solidworks. . . 1796.13 Distributions of |qn| for different allowed angular deviations of the STL mesh ofthe cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1816.14 Visual comparison of the mesh at the inner edge of the insulator trench of thecell, with different allowed angular deviations. . . . . . . . . . . . . . . . . . . . . 1816.15 Calculated magnitude of the false EDM signal caused by a rotational v × Eeffect in the top cell, assuming maximum initial vφ, for different diffuse reflectionprobabilities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1826.16 Magnitude of the false EDM signal caused by a rotational v×E effect in the topand bottom cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1836.17 Distribution of time tf spent in the cell prior to the end of the filling period,simulated in PENTrack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1846.18 Horizontal electric field magnitude |Ex| vs. position x inside the top cell, takenfrom Opera simulations of the central region. . . . . . . . . . . . . . . . . . . . . 1866.19 Mean y coordinate of UCNs vs. time from PENTrack simulations. . . . . . . . . 187xviList of Figures6.20 Two example leakage current paths in the bottom cell. . . . . . . . . . . . . . . . 1896.21 Calculated magnitude of the false EDM signal produced by a 1 nA leakage currenttravelling a distance φ around the insulator at a height of h from the bottom ofthe cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1916.22 Calculated false EDM signal produced by a leakage current in the bottom cell,with no leakage current in the top cell, when combining data from the top andbottom cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1926.23 Diagram of an AC circuit for a cell. . . . . . . . . . . . . . . . . . . . . . . . . . . 1936.24 Calculated false EDM signal resulting from different amounts of AC ripple in thehigh voltage supply for a single cell vs. frequency of the ripple. . . . . . . . . . . 1946.25 False EDM signal resulting from an AC ripple in the high voltage supply for asingle cell vs. frequency of the ripple, for different inline resistances. . . . . . . . 1957.1 Maps of paramagnetic distortion of an applied Bz = 1 µT field in the top cellwith a vacuum chamber made of Al6061. . . . . . . . . . . . . . . . . . . . . . . . 2027.2 Picture of the geometry as implemented in the Opera model, indicating thesimplified model of the titanium bellows for the cell valve actuation. . . . . . . . 2067.3 Calculated asymmetry of simulated Ramsey working points versus step size ofthe map of the B1 field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2087.4 Simulated final polarization of the UCN population in the top cell versus applied~B1 magnitude, for vacuum chambers of different materials. . . . . . . . . . . . . 2097.5 Simulated polarization of the UCN population in the top cell versus applied ~B1magnitude, for vacuum chambers of different materials. . . . . . . . . . . . . . . 2107.6 Picture of the Opera model with three different possible B1 coil geometriesaround the central region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2117.7 Simulated polarization of the UCN population in the top cell versus coil currentfor three possible coil configurations . . . . . . . . . . . . . . . . . . . . . . . . . 2127.8 Histograms of the final Sz component of the spin vectors of simulated UCNs for~B1 fields using different coils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2137.9 Map of the calculated magnitude of a magnetic dipole, located at each point inthe map, that will cause a false EDM signal of 1.26× 10−29 ecm . . . . . . . . . 2167.10 Pictures showing the measurement setup at PTB . . . . . . . . . . . . . . . . . . 2187.11 Calculated maximum field strength that would be measured for dipoles of differ-ent strength, at different distances from the dipole centre. . . . . . . . . . . . . . 2197.12 Measured magnetic fields of aluminium samples at a nominal distance of 27.5 mm.2217.13 Measured magnetic fields of titanium samples at a nominal distance of 27.5 mm. 2227.14 Measured magnetic fields of brass samples at a nominal distance of 27.5 mm. . . 2237.15 Measured magnetic fields of copper alloy samples at a nominal distance of 27.5 mm.2247.16 Measured magnetic fields of plastic samples at a nominal distance of 27.5 mm. . . 2257.17 Measured magnetic fields from samples not in any of the broader metal or plasticcategories, measured at a nominal distance of 27.5 mm. . . . . . . . . . . . . . . . 2267.18 Calculated magnitude of the false EDM signal produced by B2 frequency shiftsfrom a magnetic dipole of 1 nAm2 that reverses with the electric field . . . . . . . 2297.19 Calculated magnitude of the false EDM signal from uncompensated gravitationalfrequency shifts produced by a magnetic dipole of 1 nAm2 that reverses withelectric field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2317.20 Measured magnetic fields of samples that were exposed to some heat source ofunknown temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233xviiList of Figures7.21 Calculated temperature change of the central region induced by an optimized ~B1pulse using a saddle coil mounted on the vacuum chamber at a distance of 1.5 cm.235F.1 Map of the calculated magnitude of a magnetic dipole, located at each point inthe map, that will cause a false EDM signal of 1.26 × 10−29 ecm, with dipoleorientations of θ=0◦, 30◦, 60◦, and 90◦, with φ = 0◦. . . . . . . . . . . . . . . . . 328F.2 Map of the calculated magnitude of a magnetic dipole, located at each point inthe map, that will cause a false EDM signal of 1.26 × 10−29 ecm, with dipoleorientations of θ=-30◦, -60◦, 120◦, and 150◦, with φ = 0◦. . . . . . . . . . . . . . 329xviiiList of Acronyms2C two cell2C200 two cells, 200 kVABS acryolonitrile butadiene styreneAC alternating currentADC analog-to-digital converterAl6061 aluminium alloy 6061ASA alternating spin analyzerASF adiabatic spin flipperBL05 Beamline 05 Neutron Optics and Fundamen-tal PhysicsBL1U beamline 1UBMSR-2 Berlin Magnetically Shielded Room 2BMT Bargmann-Michel-TelegdiBSM beyond-the-standard-modelC chargeCBS cold beryllium sourceCEDM chromoelectric dipole momentCGS centimetre-gram-seconds system of unitsCKM Cabibbo-Kobayashi-MaskawaCP charge parityCPT charge parity and time-reversalCsM cesium magnetometerCuBe copper berylliumD2O heavy waterDLC diamond-like carbonDN double normal functiondPE deuterated polyethylenedPS deuterated polystyreneDRAC Digital Research Alliance of CanadaDS Doppler shifterecm electron charge centimetresEDM electric dipole momentEPR ethylene propylene rubberxixList of AcronymsFEM finite element methodFRD fully reversing dipoleFS full simulationGN generalized normal functionGS20 6Li-enriched glassGS30 6Li-depleted glassGW generalized Weibull functionh.c. Hermitian conjugateHe-II superfluid helium-4HEX1 primary heat exchanger in TUCAN sourceHV high voltageILL Institut Laue-LangevinIV1 isolation valve 1IV2 isolation valve 2J-PARC Japan Proton Accelerator Research ComplexKT kinetic theoryLANL Los Alamos National LaboratoryLD2 liquid deuteriumLF low frequencyLFU lepton flavour universalityLN2 liquid nitrogenMIDAS maximum integrated data acquisition systemMLF Material and Life Science Experimental Facil-ityMSR magnetically shielded roomN single normal functionnEDM neutron electric dipole momentNiP electroless nickel-phosphorusNMR nuclear magnetic resonanceOC100 one cell, 100 kVOFHC oxygen-free high-conductivityORNL Oak Ridge National LaboratoryP parityP2 quadratic polynomial functionP4 quartic polynomial functionPDI polydispersity indexPEEK polyether ether ketonePLA polylactic acidxxList of AcronymsPLC programmable logic controllerPMT photo-multiplier tubePNPI Petersburg Nuclear Physics InstitutePSI Paul Scherrer InstitutePTB Physikalisch-Technische BundesanstaltPTFE polytetrafluoroethyleneQCD quantum chromodynamicsQED quantum electrodynamicsQFT quantum field theoryR radical functionRC resistor-capacitorRCNP Research Center for Nuclear PhysicsSCM superconducting magnetSD2 solid deuteriumSGN skewed generalized normal functionSI Syste´me international d’unite´s, or Interna-tional System of UnitsSLHS small liquid hydrogen sourceSN skewed normal functionSQUID superconducting quantum interference deviceSS stainless steelSSA simultaneous spin analyzerSUSY supersymmetry, or supersymmetricT time-reversalTOF time-of-flightTUCAN TRIUMF UltraCold Advanced NeutronUCN ultracold neutronUCNS universal cold neutron sourceUV ultravioletVCN very cold neutronW Weibull functionxxiSymbols~A magnetic vector potential.A asymmetry calculated from counts of UCNsin each of the two spin states in the Ramseymeasurement.a effective precession frequency seen in the ro-tating frame of a spin in the Ramsey sequence.AHe coefficient of single-phonon absorption forUCN losses in He-II.aˆ†k creation operator for a phonon of wave vector~k.aˆk annihilation operator for a phonon of wavevector ~k.AP coefficient of Paschen’s law.as scattering length.A0 asymmetry offset.Bˆ unit vector for the direction of a magneticfield.~B general magnetic field vector.b Redfield theory derivation: (ωx + iωy)/2.BAC magnetic field from an alternating current.BHe coefficient of two-phonon scattering for UCNlosses in He-II.bHg199 incoherent scattering length for neutrons with199Hg.Bi magnetic field Cartesian component with i =x, y, z.bi time-varying fluctuations from the averagemagnetic field, where i = x, y, z.Binc pseudomagnetic field describing the effect ofthe spin-dependent interaction of ultracoldneutrons (UCNs) with Hg atoms.BP coefficient of Paschen’s law.~B~v× ~E motional magnetic fieldB0,− static applied magnetic field magnitude, mea-sured when the electric field is anti-alignedwith the magnetic field.xxiiSymbolsB0,+ static applied magnetic field magnitude, mea-sured when the electric field is aligned withthe magnetic field.~B0 static applied magnetic field vector.B0 static applied magnetic field magnitude.~B1 oscillating applied magnetic field vector.B1 magnitude of ~B1.C interaction strength in QFT.C a geometrical coefficient in the calculation ofa frequency shift.c speed of light.Ccell capacitance of a EDM cell.C↓ coefficient of ψ↓ in superposition of states.CHe coefficient of phonon-roton scattering forUCN losses in He-II.cm specific heat capacity.cs speed of sound.C↑ coefficient of ψ↑ in superposition of states.cz Redfield theory derivation: ωz/2.D a geometrical coefficient related to the sam-pling of a specific magnetic field configura-tion. In the single cell analysis, this is usedin calculating the field sampled by UCNs dueto gravitational weighting. In the double cellit is used in calculating the component of thegradient seen in the magnetic field differencebetween two cells.d electric dipole moment.d˜ quark CEDM.de electron electric dipole moment.∆H difference in height between the centres of theUCN distributions in the two cells.∆HHg difference in height between the centres of theHg co-magnetometers in the two cells.dfalse false neutron electric dipole moment.dHg199Hg electric dipole moment.dn,bot measurement of the neutron electric dipolemoment obtained using the bottom cell only.dn,HV− measurement of the neutron electric dipolemoment obtained between the top and bot-tom cells at negative HV polarity.dn,HV+ measurement of the neutron electric dipolemoment obtained between the top and bot-tom cells at positive HV polarity.dn,top measurement of the neutron electric dipolemoment obtained using the top cell only.xxiiiSymbolsdn neutron electric dipole moment.dn,c measurement of the nEDM corrected for allsystematic corrections except those related togravitational shifts; used in the crossing-pointanalysis.dfn false EDM signal.E eigenvalue of energy in Schro¨dinger’s equa-tion.Ei components of electric field in Cartesian co-ordinates, with i = x, y, z.Ei components of electric field in cylindrical co-ordinates, with i = r, φ, z.E− magnitude of the part of the electric field par-allel to the precession axis of the ultracoldneutrons with negative HV polarity.E+ magnitude of the part of the electric field par-allel to the precession axis of the ultracoldneutrons with positive HV polarity.~E electric field vector.E magnitude of the part of the electric field par-allel to the precession axis of the ultracoldneutrons. Often equivalent to Ez.EB electric breakdown strength.Ek kinetic energy.E⊥ kinetic energy perpendicular to a surface.ER roton energy minimum.ES electromotive field due to Seebeck effect.E∗ energy at the intersection of the neutron andHe-II dispersion curves.~F force vector.F suppression factor of net rotational motiondue to filling.fabc three-gluon structure factor.fD fraction of a correctly detected spin states.fg frequency of a UCN bouncing due to gravity.Fµν electromagnetic field tensor.fP fraction of spin states that maintain polariza-tion.fs fraction of time spent inside the source vol-ume.fHe−II fraction of time UCNs spend in He-II.fw frequency of wall bounces.G gluon field.g gravitational acceleration at Earth’s surface.xxivSymbolsGg gravitational gradient associated with the dif-ference in the magnetic field sampled by UCNand Hg due the gravitational offset of UCN.GHg co-magnetometer gradient measured by thedifference between the top and bottom cells.Gl,m coefficients describing a magnetic field interms of the free-space harmonic polynomialbasis.G0,0 coefficient of l = 0, m = 0 basis polynomial,equivalent to a uniform field Bz.G1,0 coefficient of l = 1, m = 0 basis polynomial,corresponding to a linear gradient in Bz ver-sus z.G1,1 coefficient of l = 1, m = 1 basis polynomial,corresponding to a linear gradient in Bz ver-sus x.G1,−2 coefficient of l = 1, m = −2 basis polynomial,corresponding to a quadrupole field in the xyplane.G1,2 coefficient of l = 1, m = 2 basis polynomial,corresponding to a quadrupole field in the xyplane.G2,0 coefficient of l = 2, m = 0 basis polynomial,corresponding to a quadratic gradient in Bzversus z.G2,−1 coefficient of l = 2, m = −1 basis polynomial.G2,1 coefficient of l = 2, m = 1 basis polynomial.G2,2 coefficient of l = 2, m = 2 basis polynomial.G3,0 coefficient of l = 3, m = 0 basis polynomial,corresponding to a cubic gradient in Bz versusz.G3,−2 coefficient of l = 3, m = −2 basis polynomial.G4,0 coefficient of l = 4, m = 0 basis polynomial,corresponding to a quartic gradient in Bz ver-sus z.G5,0 coefficient of l = 5, m = 0 basis polynomial,corresponding to a fifth order gradient in Bzversus z.G7,0 coefficient of l = 7, m = 0 basis polynomial,corresponding to a seventh order gradient inBz versus z.gpi nucleon-pion couplings.gs strong coupling constant.gw weak coupling constant.Hˆ a Hamiltonian.~H magnetic field strength.xxvSymbolsH separation distance of the electrodes in thecentral region.H total energy.h Planck’s constant.~ reduced Planck’s constant.I current.~J volume current density vector.J volume current density.JµN conserved Noether current.~K surface current density vector.~k wave vector.k wavenumber.k1 slope used in the crossing-point analysis of asingle cell experiment.k2 slope used in the crossing-point analysis of adouble cell experiment.kB Boltzmann constant.L a Lagrangian.l an integer, often used as a polynomial orderin the harmonic basis ~Πl,m.~M magnetization.~m magnetic dipole moment.m an integer, often used as a polynomial orderin the harmonic basis ~Πl,m.mHe mass of a helium-4 atom.mn mass of a neutron.mr roton effective mass.N number of counts.n an integer.N0 maximum number of counts, typically a fitparameter in a storage lifetime measurement.n the number density of some species.nb number density of baryons in the Universe.nb¯ number density of anti-baryons in the Uni-verse.ndof degrees of freedom in a fit.N↓ number of UCNs detected in the spin downstate at the end of a Ramsey sequence.nγ number density of photons in the universe.n(H) total energy spectrum of UCNs.nˆ surface normal unit vector.xxviSymbolsni components of surface normal vector, with i =x, y, z.nHg199 number density of Hg atoms.Ntot total number of UCNs detected across the en-tirety of data taking.N↑ number of UCNs detected in the spin up stateat the end of a Ramsey sequence.N↑↓ number of UCNs detected in either spin stateat the end of a Ramsey sequence.P probability.p a generic parameter.p momentum.Pc pressure inside the prototype cell during astorage measurement.Pcv pressure measured at the cell valve during astorage measurement.Pf pressure in the prototype cell during filling fora storage measurement.PL probability of a UCN undergoing a diffuse re-flection when reflecting from a surface.PL,dPS probability of a UCN undergoing a diffuse re-flection when reflecting from a dPS surface.PL,NiP probability of a UCN undergoing a diffuse re-flection when reflecting from a NiP surface.PL,SS probability of a UCN undergoing a diffuse re-flection when reflecting from a SS surface.Pn Legendre polynomial of the first kind of ordern.P pressure.P↑↓ probability that a spin initially in the ψ↑ statewill be measured in the ψ↓ state following theRamsey sequence.P/V power per unit volume.~p momentum vector.Pz polarization in the z direction.Q electrical charge.~Q wave-vector transfer.R radius of the precession cells.r0 distance to the centre of some volume of in-terest.r distance, typically in a spherical or cylindricalcoordinate system.rˆ position unit vector.r′ distance, typically in a spherical or cylindricalcoordinate system.xxviiSymbols~r position vector.Rdet rate at which UCNs are detected.Re electrode radius.Rprod rate at which UCNs are produced.R ratio of neutron precession frequency to Hgprecession frequency.R0 ratio of neutron to Hg precession frequencywith all shifts corrected, ostensibly equal tothe ratio of gyromagnetic ratios.Rbot,HV− ratio of neutron to Hg precession frequencymeasured in the bottom cell with negative HVpolarity.Rbot,HV+ ratio of neutron to Hg precession frequencymeasured in the bottom cell with positive HVpolarity.Rc ratio of neutron to Hg precession frequencycorrected for all frequency shifts except thoserelated to gravitationally-dependent samplingby UCNs.R amplitude of reflected wave in scattering.R− ratio of neutron precession frequency to Hgprecession frequency measured with the elec-tric field anti-aligned with the magnetic field.R+ ratio of neutron precession frequency to Hgprecession frequency measured with the elec-tric field aligned with the magnetic field.Rtop,HV− ratio of neutron to Hg precession frequencymeasured in the top cell with negative HVpolarity.Rtop,HV+ ratio of neutron to Hg precession frequencymeasured in the top cell with positive HV po-larity.Si components of spin in Cartesian coordinates,with i = x, y, z.~S spin vector.S spin magnitude.s separation parameter of the working points.SE Seebeck coefficient.Sˆ vector of spin 1/2 operators.Sˆi components of spin-1/2 operator in Cartesiancoordinates, with i = x, y, z.T temperature.T period of some process.t time.T2 depolarization lifetime of ultracold neutrons.xxviiiSymbolsT2,m depolarization lifetime of ultracold neutronsdue to magnetic field fluctuations.T2,w depolarization lifetime of ultracold neutronsdue to wall bounces.T a generators of the colour group.td delay time.tf time spent by a UCN inside a cell during fill-ing, prior to the end of the filling period.tfill filling time of UCNs into a volume.T (k) spherical tensor operator of order k.t′ time, generally used as an integration variableto distinguish from t.TR free precession time in the Ramsey method.T amplitude of transmitted wave in scattering.U potential energy.V real part of the Fermi potential.vi component of velocity in cylindrical coordi-nates, with i = x, y, zvi component of velocity in cylindrical coordi-nates, with i = r, φ, zv speed, or magnitude of velocity.V volume.VAC AC voltage.VB breakdown voltage.vˆ unit velocity vector.V pq CKM matrix elements.vxy velocity in the xy plane.VS voltage produced by Seebeck effect.vsh UCN velocity due to small heating effects.~v velocity vector.W imaginary component of the Fermi potential./W+charged W-boson field.WdPS imaginary component of the Fermi potentialof a dPS-coated surface.Wg imaginary component of the Fermi potentialof the guides leading from IV1 to the detec-tors.WHe,v imaginary component of the Fermi potentialof helium-4 vapour.Wµ four-dimensional function.WNiP imaginary component of the Fermi potentialof a NiP-coated surface.Ws imaginary component of the Fermi potentialinside the vertical source.xxixSymbolsWSS imaginary component of the Fermi potentialof a SS-coated surface.x′ Cartesian coordinate in a secondary coordi-nate system.y′ Cartesian coordinate in a secondary coordi-nate system.Z electrical impedance.〈z〉 average z value of a population.zˆ unit vector in the z coordinate direction.α visibility of the central fringe.αe electric polarizability.αTE first Townsend coefficient.β depolarization probability per wall bounce.Γ relaxation tensor.Γ1,m longitudinal relaxation rate due to magneticfields.Γ2,m,B2 transverse relaxation term depending on thetransverse magnetic field squared.Γ2,m,BE transverse relaxation term depending on thelinear cross-term between the electric andmagnetic fields.Γ2,m,B2z transverse relaxation term depending on thesquare of the magnetic field in the z direction.Γ2,m,E2 transverse relaxation term depending on theelectric field squared.Γ2,m transverse relaxation rate due to magneticfields.γ Dirac matrix.γ gyromagnetic ratio.γHg gyromagnetic ratio of mercury.γn gyromagnetic ratio of the neutron.ΓP pressure-dependent UCN loss rate coefficient.γ Lorentz factor.γse secondary electron emission coefficient.δB0,i fluctuation of the main magnetic field duringa fill.∆Bz difference in Bz.∆δ a shift in R with sign dependent on the ori-entation of B0.∆E difference in energy levels.δ⊕ shift of R associated with Earth’s rotation.δg shift of R associated with the difference infield between UCNs and Hg due to gravita-tional weighting.xxxSymbolsδGHg a shift in GHg due to a difference between thetwo co-magnetometers.δi a positional offset in Cartesian coordinates,with i = x, y, z.δ± a shift in R which takes different values fordifferent orientations of B0.∆z difference in coordinate z.∆ω width of the central fringe of the Ramsey se-quence.δω frequency shift of a precession frequency.δωAC frequency shift depending on an alternatingcurrent field.δωB2 frequency shift term depending on the mag-netic field squared.δωBE frequency shift term depending on the cross-term between electric field and magnetic field.δωE2 frequency shift term depending on the electricfield squared.δω⊕ frequency shift due to Earth’s rotation.δωHg generic frequency shift of the Hg precessionfrequency.δωn generic frequency shift of the neutron preces-sion frequency.∆ωTB difference in neutron precession frequency be-tween the top and bottom cells.0 vacuum permittivity.µναβ Levi-Cevita symbol.r dielectric constant.η ratio of the difference between baryon and an-tibaryon densities to the density of photons inthe universe.θ an angle.θ¯ CP-violating phase of QCD.Λ BSM energy scale.λ difference between frequency of ~B1 and ~B0fields.λdB de Broglie wavelength.µ loss rate for UCNs.µ0 magnetic permeability of free space.µn neutron magnetic dipole moment.Πx,l,m x component of the polynomial basis functionof order l and m.Πy,l,m y component of the polynomial basis functionof order l and m.Πz,l,m z component of the polynomial basis functionof order l and m.~Πl,m vector of basis functions of order l and m inthe free-space harmonic polynomial basis.xxxiSymbols~Π0,1 vector of basis functions of order l = 0 andm = 1, corresponding to a constant field in xdirection.~Π0,−1 vector of basis functions of order l = 0 andm = −1, corresponding to a constant field iny direction.~Π1,−2 vector of basis functions of order l and m,corresponding to a quadrupole field in the xyplane.Πz,2,0 z component of the polynomial basis functionof order l = 2 and m = 0, corresponding to aquadratic gradient in Bz.ρ density.ρ1,0 density operator coefficient of σz.ρ1,1 density operator coefficient of σ+.ρ1,−1 density operator coefficient of σ−.ρˆ density operator.σ standard deviation.σO uncertainty in the origin of a coordinate sys-tem.σdn,stat statistical uncertainty of the nEDM measure-ment.σE electrical conductivity.σk1 uncertainty in the slope used in the crossing-point analysis of a single cell.σk2 uncertainty in the slope used in the crossing-point analysis of a double cell.σl loss cross section.σ− spherical Pauli matrix σx − iσy.σµν commutator of Dirac matrices γµ and γν .σN↓ uncertainty of spin down counts.σN↑ uncertainty of spin up counts.σ+ spherical Pauli matrix σx + iσy.σR standard deviation of the frequency ratio.σs scattering cross section.σx Pauli matrix of coordinate x.σy Pauli matrix of coordinate y.σz Pauli matrix of coordinate z.σωn standard deviation of neutron precession fre-quency measured by Ramsey’s method.σωHg standard deviation of Hg precession fre-quency.τ duration of the ~B1 pulse applied during theRamsey method.τ1−ph storage lifetime of UCNs in He-II due to singlephonon absorption.τ2−ph storage lifetime of UCNs in He-II due to twophonon scattering.xxxiiSymbolsτβ beta decay lifetime of neutrons.τc correlation time.τh storage lifetime of UCNs due to a hole intowhich the UCNs are lost.τr characteristic time constant of the decay ofrotational motion.τr-ph storage lifetime of UCNs in He-II due tophonon roton scattering.τs storage lifetime of UCNs.τHe−II storage lifetime of UCNs due to all upscatter-ing processes in He-II.τvap storage lifetime of UCNs due to scattering onhelium vapour.τw storage lifetime of UCNs due to wall losses.φ azimuthal angle or phase.φB magnetic scalar potential.ϕ generic field in QFT.φˆ unit vector in azimuthal direction.Φn neutron flux.φTRIUMF latitude of TRIUMF.χ2 standard goodness-of-fit parameter.χm volume magnetic susceptibility.Ψ time-dependent quantum mechanical wave-function.ψ quantum mechanical wavefunction.ψ↓ wavefunction of the spin eigenstate anti-aligned with a static magnetic field.ψ↑ wavefunction of the spin eigenstate alignedwith a static magnetic field.Ω precession frequency relative to the magni-tude.ω angular frequency.ω0 precession frequency due to a magnetic fieldB0.ωn,bot neutron precession frequency measured in thebottom cell.ωn,− precession frequency shifted by an electricfield anti-aligned with the magnetic field.ωn,+ precession frequency shifted by an electricfield aligned with the magnetic field.ωn,± precession frequency including a shift fromelectric field of differing orientations.ωn,top neutron precession frequency measured in thetop cell.~ω0 precession frequency vector.ω1 frequency of the applied oscillating or rotatingmagnetic field in the Ramsey sequence.σω1 uncertainty of the B1 pulse frequency ω1.xxxiiiSymbolsσA uncertainty of the asymmetry.ωAC alternating current frequency.ω⊕ rotational frequency of Earth.ωHg mercury precession frequency.〈ωHg〉 precession frequency of Hg averaged over aRamsey sequence.〈ωHg〉− precession frequency of Hg averaged over aRamsey sequence, measured with the electricfield anti-aligned with the electric field.〈ωHg〉bot,HV− precession frequency of Hg averaged over aRamsey sequence, measured in the bottomcell with negative HV polarity.〈ωHg〉top,HV− precession frequency of Hg averaged over aRamsey sequence, measured in the top cellwith negative HV polarity.〈ωHg〉+ precession frequency of Hg averaged over aRamsey sequence, measured with the electricfield aligned with the magnetic field.〈ωHg〉bot,HV+ precession frequency of Hg averaged over aRamsey sequence, measured in the bottomcell with positive HV polarity.〈ωHg〉top,HV+ precession frequency of Hg averaged over aRamsey sequence, measured in the top cellwith positive HV polarity.ωHg,i precession frequency of Hg during a fill.ωi working point frequency.σωi standard deviation of applied frequency at aworking point i.ωn neutron precession frequency.ωn neutron precession frequency determinedfrom a single Ramsey sequence.ωφ angular velocity about the z axis.ωx precession frequency due to a fluctuatingmagnetic field bx.ωy precession frequency due to a fluctuatingmagnetic field by.ωz precession frequency due to a fluctuatingmagnetic field bz.xxxivChapter 1Motivation1.1 Physical Symmetries and ViolationsThe concept of symmetry has been a cornerstone of particle physics since the early 20th century.A physical system is said to possess a symmetry if the properties of that system are unchangedunder an associated transformation. Such transformations can be either continuous, such as atranslation in space, or discrete, such as a mirror reflection. These two categories of symmetryare equally important in particle physics, but with different implications.Continuous symmetries are linked with the conservation laws that are fundamental to allareas of physics. In 1915, Emmy Noether proved that any continuous symmetry in a physicalsystem must have a corresponding conservation law [9]. More formally, consider a LagrangianL describing a field ϕ, and a continuous transformation ϕ→ ϕ+ Dˆϕδξ, where ξ parametrizesthe transformation, δξ is an infinitesimal change in ξ, and Dˆ = ∂∂ξ |ξ=0. If this transformationchanges the Lagrangian only by the addition of a four-divergence, such that DˆL = ∂µWµ, whereWµ is some arbitrary function, then there exists a currentJµN =∂L∂(∂µϕ)Dˆϕ−Wµ, (1.1)and, for ϕ satisfying the equations of motion, ∂µJµN = 0.The application of Noether’s theorem to spacetime transformations yields some of the mostfoundational laws of physics. Translational symmetry in space and time gives rise to conser-vation of momentum and energy respectively, while rotational symmetry yields conservation ofangular momentum.Noether’s theorem also applies to the internal gauge symmetries of fields in quantum fieldtheory (QFT). The theory of quantum electrodynamics (QED) is symmetric under the multi-plication of the field by a phase factor eiα(x), where α(x) is an arbitrary function of spacetime.This symmetry, known as U(1) symmetry, results in the conservation of electromagnetic chargein QED. The other field theories of the Standard Model have similar symmetries (SU(2) for theweak interaction and SU(3) for the strong interaction), which also result in the conservationof the underlying charges and particle numbers. Many of the most fundamental properties ofphysical laws are therefore connected to continuous symmetries.Discrete symmetries do not correspond to conservation laws, but have proven to be justas important, if more subtle, in the development of the understanding of fundamental physics.There are three discrete transformations of particular importance:1. C, charge-conjugation: q → −q, which exchanges the signs of all charges;2. P, parity: (x, y, z)→ (−x,−y,−z), which exchanges the signs of spatial coordinates;3. T, time-reversal: t→ −t, which reverses the direction of time.11.1. Physical Symmetries and ViolationsWhen these three transformations are applied simultaneously, the collective symmetry, knownas CPT symmetry, is universal. This result, known as the CPT theorem, was independentlyarrived at by a number of physicists [10, 11], and formally applies to any local quantum fieldtheory with Lorentz invariance [12, 13]. A large number of experiments have tested Lorentzinvariance, and by extension the CPT theorem, and have turned up no violation [14]. Physiciststherefore operate under the continued assumption that the combined discrete symmetry CPTis a symmetry of the Universe.Although CPT symmetry is universal, the individual symmetries C, P, and T, are not. Thesesymmetries do apply to many physical systems, but for each of them it has proven possible tofind violations. The study of these symmetry violations has been a critical research area inparticle physics beginning with Wu’s measurement of parity violation in weak interactions inthe 1950s [15]. This violation of P symmetry also violated C symmetry, such that the processwas symmetric under the combination of the two transformations. Following this result, it wassuspected that the Universe was symmetric under this combination, known as CP symmetry [16].However, within a decade, CP symmetry was found to also be violated in kaon decay [17]. Sincethe 1990s, further CP-violating processes have been measured, in neutral kaons by the KTeV [18]and NA48 [19] results, then in B mesons in BaBar [20] and Belle [21], and most recently, inthe D0 meson at LHCb [22]. The discovery of violation of CP symmetry has played a majorrole in the development of the Standard Model, and all of the CP-violating processes measuredso far are captured by the Cabibbo-Kobayashi-Maskawa (CKM) matrix, which describes themixing of quarks under the weak interaction [23, 24]. The CP-violating contribution of theCKM matrix can be written as a contribution to the LagrangianLCKM = igw√2V pqψ¯uL,p /W+ψdL,q + h.c., (1.2)where gw is the weak coupling constant, ψ¯uL,p are the left-handed up-type anti-quark wavefunc-tions of generation-p, ψdL,q the left-handed down-type quark wavefunctions of generation-q, Vpqare the elements of the CKM matrix, and /W+is the charged W-boson field.A significant implication of the CPT theorem is that there is a correspondence betweendifferent symmetry violations. In all CP-violating systems, in order for the CPT theorem tohold, it must be the case that the system also violates T symmetry. This T violation is requiredto cancel the CP violation when the triple transformation is applied. Violation of time-reversalsymmetry has proven more challenging to measure, but has been identified in BaBar alongsidethe measured CP violation in B mesons [25].The violation of CP symmetry remains an area of great interest in both experimental andtheoretical research. Current efforts are driven primarily by an interest in extending the Stan-dard Model. This search for beyond-the-standard-model (BSM) physics has arisen due to anumber of perceived deficiencies in the Standard Model.In physical cosmology, one unanswered question is that of baryonic asymmetry: the pres-ence of more matter than antimatter in our Universe. Theories of baryogenesis attempt toexplain this imbalance through hypothetical processes which would asymmetrically producemore baryons than anti-baryons. In 1967, Sakharov proposed a set of conditions that must besatisfied in order for the process of baryogenesis to be achieved [26]:1. baryon number violation;2. C and CP symmetry violation;3. the occurrence of the process out of thermal equilibrium.21.1. Physical Symmetries and ViolationsThe reasoning behind these conditions is straightforward. The necessity of the first conditionis immediately clear: there must be a process which creates baryons without creating anti-baryons. The second and third conditions are then required to prevent this process from beingbalanced by opposing processes. Without C symmetry violation, there would be a correspondingprocess that creates anti-baryons without baryons, which would proceed at the same rate. Thebaryons and anti-baryons created by the processes would annihilate, cancelling out the baryonnumber violation. However, C symmetry violation alone is not sufficient. If the process hasCP symmetry, then the creation of right-handed anti-baryons would occur at the same rate asthe creation of left-handed baryons (and similarly for other combinations of handedness andcharge). This would also result in the cancellation of the baryon number violation, and so CPsymmetry must also be violated. Finally, the process occurring out of thermal equilibrium isrequired to prevent the reverse process, which destroys baryons, from occurring at the samerate as the forward process.The baryonic asymmetry is characterized by the parameterη =nb − nb¯nγ, (1.3)where nb is the number density of baryons, nb¯ that of antibaryons, and nγ that of photons. Usingdata from the Planck collaboration it can be concluded that η ∼ 10−10 [27, 28]. Estimates ofthis parameter in the Standard Model vary, but it is generally agreed that η . 10−18 [29, 30].As it currently stands, the Standard Model therefore does not contain sufficient CP violationto satisfy Sakharov’s second condition and explain the amount of baryonic asymmetry presentin the Universe.This problem could be solved by the discovery of new sources of CP-violating physics beyondthe Standard Model. Experimental searches hope to identify a new source of CP violation,capable of satisfying Sakharov’s conditions and explaining the matter-antimatter asymmetry ofthe Universe.One possible solution to the deficiency of CP violation could come from supersymmetric(SUSY) extensions of the Standard Model. SUSY theories introduce a new symmetry under theexchange of bosons and fermions, and were first developed in response to the hierarchy problem— the observed discrepancy between the strengths of the weak and gravitational forces [31]. Thisnew exchange symmetry requires the introduction of a supersymmetric partner for each existingparticle, and also introduces many new parameters which could violate CP symmetry [32]. Atthe time of writing, no evidence of supersymmetric partners has appeared, but such theorieshave not been ruled out [33, 34]. Further direct searches require the construction of ever morepowerful colliders. As an alternative to direct searches, SUSY theories can also be constrainedindirectly by searches for CP violation, which limit the allowed parameter values.Another unanswered question in the Standard Model is the strong CP problem. As describedby the theory of quantum chromodynamics (QCD), the strong interaction permits CP violationin the form of a CP-violating phase θ¯, which appears in the QCD Lagrangian asLθ¯ = θ¯gs232pi2GaµνG˜aµν, (1.4)where gs is the strong coupling constant and Gaµν is the gluon field. Measurements of theneutron electric dipole moment, discussed further in Section 1.2, show that this parameter isconstrained such that θ¯ < 10−9 [35]. The strong CP problem asks why this free parameter takeson such a small value.31.2. Electric Dipole MomentsOne popular resolution of the strong CP problem is the addition of a new symmetry and anew scalar field, typically referred to as the axion field. This theory was originally developedby Peccei and Quinn [36], and variations of the concept have since appeared as experimentshave failed to detect the original axion [37–41]. Axion-like-particles have subsequently become aleading candidate in dark matter searches, drawing a possible connection between CP violationand yet another cosmological question [42].Past experimental measurements of CP violation have contributed significantly to the de-velopment of the Standard Model. Future searches have great potential in helping to illuminateseveral unanswered problems in physics. The following section will elaborate on one type ofexperiment connected to CP violation: searches for electric dipole moments.1.2 Electric Dipole MomentsA classical electric dipole moment (EDM) d exists when there is a separation of two charges±Q by a distance z, such thatd = Qz. (1.5)In the presence of an electric field E, an EDM contributes to the Hamiltonian of the system asHˆ = d · E. (1.6)Such an EDM does not violate CP symmetry. Under charge-conjugation, both d(∝ Q) and Ewill reverse sign, and under parity, both d(∝ z) and E will reverse sign, so all reversals canceland the classical Hamiltonian of an EDM in an electric field has CP symmetry.The situation changes when moving to a quantum-mechanical framework. Consider anEDM in the ground state of a spin-1/2 particle. The Wigner-Eckart theorem states that, foran eigenstate of total angular momentum 〈j,m|, and a spherical tensor operator T (k) of orderk, with elements labelled T (k)q (with q = −k...k), the elements of the transition matrix mustsatisfy〈j,m|T (k)q|j′,m′〉 = 〈j′,m′, k, q|j,m〉 〈j||T (k)||j′〉 . (1.7)This means that the elements of such an operator can be represented in terms of a Clebsch-Gordon coefficient 〈j′,m′, k, q|j,m〉, and a constant reduced matrix element 〈j||T (k)||j′〉 whichdoes not depend on m, m′, or q. Both the spin and EDM are vector operators of the form(vx, vy, vz), and can thus be written as spherical operators of order k = 1 such thatT−1 =1√2(vx − ivy),T1 =1√2(vx + ivy),T0 = vz.(1.8)Applying the Wigner-Eckart theorem to the spin S of a spin-1/2 particle gives〈12,m|Sq|12,m′〉 = 〈12,m′, 1, q|12,m〉 〈j||S||j′〉 . (1.9)Similarly, for the EDM,〈12,m|dq|12,m′〉 = 〈12,m′, 1, q|12,m〉 〈j||d||j′〉 . (1.10)41.2. Electric Dipole MomentsEquations 1.9 and 1.10 both contain the same constant, and so it is possible to combine themto write〈12,m|dq|12,m′〉 = 〈12,m|Sq|12,m′〉 〈j||d||j′〉〈j||S||j′〉 . (1.11)Thus, for a spin-1/2 particle, a non-zero EDM in the ground state must be parallel to the spin.The Hamiltonian for the EDM can then be written asHˆ = d~S · ~E. (1.12)The symmetry argument for this Hamiltonian is most easily approached by considering T ratherthan CP directly. Under time-reversal, the electric field and the constant d remain unchanged,but the spin is reversed, meaning that Hˆ does not possess T symmetry. By the CPT theorem,it follows that CP symmetry must also be violated, so as to balance the T violation in the triplesymmetry and obey the CPT theorem. So a permanent EDM of a spin-1/2 particle violatesCP symmetry.This result can be extended to any system for which well-defined angular momentum eigen-states exist, forbidding the presence of permanent EDMs in such systems without CP violation.This includes molecules, many of which are commonly said to have large permanent EDMs.The resolution of this seeming contradiction lies in the physical mechanisms by which theseEDMs are produced.In a classical description, maintaining a separation of charges will require the applicationof an additional force. What is relevant to this discussion is the application of an electric field,which polarizes a charge distribution, giving an induced EDM. In this case the EDM can bewritten as d = αeE, where αe is the electric polarizability of the system. This leads to Hˆ ∝ E2,which has CP symmetry.The so-called permanent EDMs in molecules are in fact induced EDMs. They appear to bepermanent only due to the presence of quasi-degenerate states in molecular systems [43, 44].Mixing of these states in the presence of an electric field results in behaviour that appears linearin E, but at very low fields would be observed to be quadratic [45]. These EDMs therefore onlyappear to be permanent in classical contexts.So, in this quantum-mechanical framework, true permanent EDMs can only exist due tothe presence of mechanisms that violate CP symmetry. Such mechanisms allow a system tohave spin-dependent (linear) interactions with electric fields. An alternative way of defining apermanent EDM is therefore as a measure of how strongly a system’s spin couples directly toan external electric field.Given that permanent EDMs violate charge parity (CP) symmetry, attempting to measurethem has great potential for probing CP-violating physics. The focus of experimental workin this direction is on spin-1/2 systems, which are the simplest to study. Different spin-1/2systems can be shown to have different theoretical CP-violating contributions [35]. Figure 1.1,adapted from Ref. [8], shows the contributions to the EDMs of different systems starting atthe level of fundamental theory. These contributions will be described in the remainder of thissection.At the fundamental level, the Standard Model contribution is present in the CKM ma-trix contribution and the QCD phase, and various BSM contributions can be considered fromSUSY [35] and other theories [46, 47].All fundamental contributions can be described as a model-independent effective field theory.Following the description by Engel et. al [48], the different contributions to a CP-violating51.2. Electric Dipole MomentsFundamentaltheory Effective fieldtheoryLow energyscale Nucleus levelAtomic/molecular levelFigure 1.1: Illustration of the connections between different theoretical sources of CP violationand measurable systems, based on the description of Ref. [8]. Boxes with dashed borders arethe parameters of theories at different levels, which must be extracted from the measurablesystems indicated by boxes with solid borders. The different terms at each level are describedin detail in the text.61.2. Electric Dipole MomentsLagrangian can be considered. First, for any fermion there is an EDM termLEDM = − i2dpψ¯pσµνγ5ψpFµν , (1.13)where ψp is the wavefunction of fermion p, σµν =14 [γµ, γν ] is 1/4 of the commutator of theDirac matrices γi. This term couples the spin of the fermion to the electromagnetic field tensorFµν . The strength of this interaction di is the fermion EDM. The quarks can acquire a separatechromoelectric dipole moment (CEDM), which is analogous to an EDM, in the formLCEDM = − igs2d˜pψ¯q,pσµνγ5Taψq,pGaµν , (1.14)where ψq,p is the wavefunction of quark p, ψ¯q,p the wavefunction of anti-quark p, and Ta arethe generators of the colour group. This term couples the spin of the quark to the gluon fieldtensor Gaµν . The strength of this interaction d˜p is the quark CEDM. It might naively seemthat a term that describes coupling to the strong interaction should be irrelevant to discussionsof EDMs. However, it can be shown that CEDMs can contribute significantly to measurableEDMs [35].The next term, introduced by Weinberg [49], is the 3-gluon interactionLG = gsΛ2CGfabcGaµρGρbνGcαβµναβ , (1.15)where fabc is the three gluon structure factor, and µναβ is the Levi-Civita symbol. Thiscontribution has interaction strength CG and Λ the BSM energy-scale.There are a total of five CP-violating four-fermion terms. The three semileptonic terms areLeq =iC ledq2Λ2(ψ¯eγ5ψeψ¯dψd − ψ¯eψeψ¯dγ5ψd)− iC(1)lequ2Λ2(ψ¯eγ5ψeψ¯uψu − ψ¯eψeψ¯uγ5ψu)− iC(3)lequ2Λ2µναβψ¯eσαβψeψ¯uσµνψu,(1.16)where ψe are lepton wavefunctions, ψu are up-type quark wavefunctions, and ψd are down-typequark wavefunctions, with ψ¯ denoting the anti-particle wavefunctions. These terms capturecouplings between leptons and quarks with strengths determined by the coefficients C ledq, C(1)lequ,and C(3)lequ, and violate CP symmetry when there is a non-zero imaginary component of thecoefficients.The two hadronic four-fermion terms areLqq =igs2C(1)quqd2Λ2(ψ¯uγ5ψuψ¯dψd + ψ¯uψuψ¯dγ5ψd− ψ¯dγ5ψuψ¯uψd − ψ¯dψuψ¯uγ5ψd)+ igs2C(8)quqd2Λ2(ψ¯uγ5Taψuψ¯dTaψd + ψ¯uTaψuψ¯dγ5Taψd− ψ¯dγ5T aψuψ¯uT aψd − ψ¯dT aψuψ¯uγ5T aψd).(1.17)These terms capture couplings between quarks with strengths determined by the coefficientsC(1)quqd and C(3)quqd, and as with the semileptonic terms violate CP symmetry when there is animaginary component of the coefficients.71.2. Electric Dipole MomentsThe final term arises from a quark-Higgs operator, with the resulting CP-violating termLqH =i Im(Cφud)3Λ2(ψ¯uψuψ¯dγ5ψd − ψ¯uγ5ψuψ¯dψd+ 3(ψ¯uTaψuψ¯dγ5Taψd − ψ¯uγ5T aψuψ¯dT aψd)),(1.18)where Cφud is the interaction strength, with the dependence on the imaginary component madeexplicit.Combining all of these terms together with the CKM term in Equation 1.2 and the QCDphase term in Equation 1.4 gives the leading-order CP-violating Lagrangian. All of these differ-ent interactions allow the construction of Feynman diagrams containing CP-violating vertices,which allow a system to interact in a spin-dependent fashion with external photons, contribut-ing to the EDM. For brevity, the description of these diagrams is not included here, but can befound in Refs. [35] and [8].With most of the CP-violating terms involving interactions between quarks, which are exper-imentally inaccessible, it is appropriate to further reduce the description to the nuclear-energyscale, as shown in figure 1.1, with interactions involving leptons, photons, pions, and nucleons.The lepton EDMs remain at this energy level without any further contribution. The semilep-tonic terms become lepton-nucleon couplings Ci. Nucleon EDMs dn and dp (as well as otherhadron EDMs dh) receive contributions from the quark EDM/CEDM terms and from the quarkand gluon coupling terms. Nucleon-pion couplings gpii also appear at this scale, receiving con-tributions from the quark EDM/CEDM terms, quark/gluon couplings, and directly from θ¯. Forthe sake of brevity, the details of this nuclear energy-scale formulation are not described here,but detailed descriptions of these contributions are well-summarized by Engel et. al. [48] andChupp et. al. [8].Within this effective framework describing the CP-violating contributions, it is possible toconsider how different physical systems can manifest measurable EDMs, and thus understandthe landscape of EDM searches. The contribution of the different terms that have been describedto different EDMs is again shown in Figure 1.1, and will be described in the subsequent sections.Section 1.2.1 gives an overview of EDM searches excluding the neutron, while Section 1.2.2evaluates the history and status of searches for the neutron electric dipole moment (nEDM).1.2.1 Non-neutron EDM SearchesThe landscape of EDM searches can be broadly divided into the hadronic and leptonic sectors.Hadronic sector EDM searches are largely focused on diamagnetic atoms, and also include theneutron, which will be discussed in Section 1.2.2. Other hadronic systems of interest includelight nuclei and the proton. Leptonic sector EDM searches have primarily focused on measuringthe electron EDM in paramagnetic atoms or solid state systems, with some work having beendone on the muon and tauon. These different systems are sensitive to different terms in theeffective field theory, forming a complementary landscape of searches.Tables 1.2 and 1.1 attempt to summarize the current state of hadronic and leptonic EDMsearches, respectively, not including searches for the neutron EDM. The results for EDMsearches are typically given in units of electron charge centimetres (ecm) — the magnitudeof an EDM produced by a 1 cm separation of charges ±e. With the ever-changing landscapeof experimental physics, this table is likely incomplete and already out-dated, but it shouldprovide a representative sample of the state of the field.81.2. Electric Dipole MomentsDiamagnetic systems are most sensitive to the EDMs of their individual nucleons dn anddp, as well as to pion-nucleon interactions, and are also sensitive to lepton-nucleon interactionsdue to presence of atomic electrons. Diamagnetic systems are also subject to Schiff’s shieldingtheorem, which states that the nuclear electric dipole moment is shielded by the rearrangementof the atomic electrons in the presence of an external electric field [50]. Schiff’s theorem doesnot apply perfectly to a finite-sized nucleus, and so residual symmetry-violating moments (oftenreferred to as the nuclear Schiff moment) in diamagnetic systems can still be measured, butthey are greatly suppressed [51]. Diamagnetic systems are therefore complicated to interpret,as they are broadly sensitive to different contributions.Ionized light nuclei, such as deuterons, tritons, protons, and 3He++, are also sensitive tothe nucleon EDMs and pion-nucleon couplings. These systems would avoid Schiff shieldingand lepton-nucleon interactions by virtue of having no electrons. No measurements have beenachieved of these systems yet, but there is now significant interest in studying them usingstorage rings [52]. A storage ring measurement has been developed for the deuteron [53, 54],but the final results are still awaiting publication at the time of writing [55].One final oddity among hadronic EDM searches is the Λ hyperon measurement performedat Fermilab [56]. This particle would have its own effective EDM dΛ, resulting from the quark(C)EDMs and quark interaction terms. However, with the exception of the proton and neutron,baryons are extremely short-lived and therefore challenging to study at high-precision. Com-pared to nuclear systems, the sensitivity of such a measurement is the worst by many orders ofmagnitude, and there is little motivation to spend effort in this area.Paramagnetic systems, possessing one unpaired electron, are sensitive to the electron EDMde, as well as to lepton-nucleon couplings Ci. Schiff’s theorem is avoided by the unpairedelectron in a paramagnetic system due to relativistic effects [57]. Most recent studies of para-magnetic systems focus on highly polar molecules, for which the large internal electric fieldincreases the experimental sensitivity [58].The electron EDM can also be studied in solid-state systems. In such systems, a large densityof unpaired electrons can be achieved, enhancing the signal obtained from an EDM. Past solid-state experiments have been developed that measured induced magnetization in an electricfield [59, 60] and induced electric polarization in a magnetic field [61]. However, measurementsof these systems have not approached the sensitivity of atomic systems.The greatest focus of EDM searches in leptonic systems has been on the electron. This istheoretically motivated by the concept of lepton flavour universality (LFU), which asserts thatelectroweak interactions have the same strength for each of the three lepton generations [62, 63].Under this assumption, the electron EDM can be scaled by the mass ratio to constrain the EDMsof the other leptons. Regardless, some measurements have been obtained for both the muonand tau lepton. A measurement of the muon EDM exists from a secondary analysis of the BNLmuon g − 2 storage ring experiment [64], while the tau EDM has been measured through ananalysis of the reaction e+e− → τ+τ− by the Belle collaboration [65]. Dedicated muon storageexperiments are under development to improve the muon result [66, 67]. This new developmenthas been motivated by the possibility that LFU may be violated [68, 69].91.2.ElectricDipoleMomentsSystem Group Technique Status Ref.Paramagnetic AtomsXem — Warm beam de < 3.1× 10−22 ecm (95% UL) [70]Cs Amherst Vapour cell de < 1.2× 10−25 ecm (95% UL) [71]Cs Penn. State Optical trap Under development [72]Tl Berkeley Warm beam de < 1.9× 10−27 ecm (95% UL) [73]YbF Imperial College Warm beam de < 1.2× 10−27 ecm (95% UL) [74]ThO ACME Cold beam de < 1.1× 10−29 ecm (90% UL) [58]HfF+ JILA/NIST/UC Ion trap de < 2.1× 10−29 ecm (90% UL) [75]BaF NL-eEDM Cold beam Under development [76]BaF EDM3 4K Ar Matrix Under development [77]Fr CYRIC/RIKEN Atom trap Under development [78, 79]Fr TRIUMF Atomic fountain Under development [80]Solid State SystemsGd3Ga5O12 — Induced magnetization de = (−5.57± 7.98± 0.12)× 10−25 [60]Eu0.5Ba0.5TiO3 Yale Induced magnetization de < 6.05× 10−25 (90% UL) [59]Gd3Fe5O12 Amherst Induced polarization de < 5× 10−24 (90% UL) [61]Othersµ BNL g − 2 Storage ring dµ < 1.8× 10−19 ecm (95% UL) [64]µ PSI Storage solenoid Under development [66]µ J-PARC Storage solenoid Under development [67]τ Belle e+e− → τ+τ− dτ = (1.15± 1.70)× 10−17 [65]Table 1.1: Summary of current measured EDM limits and experiments under development in the leptonic sector.101.2.ElectricDipoleMomentsSystem Group Technique Status RefDiamagnetic Atoms199Hg Washington Vapour Cell dHg < 7.4× 10−30 ecm (95% UL) [81]199Hg quMercury Optical trap Under development [82]129Xe Mainz Vapour cell dXe < 1.5× 10−27 ecm (95% UL) [83]129Xe PTB/UM/TMU/MSU Vapour cell dXe < 1.4× 10−27 ecm (95% UL) [84]129Xe RIKEN Vapour cell Under development [85]225Ra ANL Optical trap dRa < 1.4× 10−23 ecm (95% UL) [86]205TlF Yale Cold beam dTl < 6.5× 10−23 ecm (95% UL) [87]205TlF CeNTREX Cold beam Under development [88]171Yb USTC/MSU Optical trap dYb < 1.5× 10−26 ecm (95% UL) [89]Othersd JEDI Storage ring Analyzing [54, 55]p CPEDM Storage ring Under development [52]Λ Fermilab Beam dΛ = (−3.0± 7.4)× 10−17 [56]Table 1.2: Summary of the current measured EDM limits and experiments under development for hadronic systems, not includingneutrons.111.2. Electric Dipole Moments1.2.2 Searches for the Neutron EDMAmong those systems for which EDM searches have been performed, the neutron possessesunique properties that set it apart from other hadronic systems. At the nuclear energy scale ameasurement of the nEDM is sensitive only to the nEDM itself, with some contribution fromnucleon-pion couplings. As a bare nucleon it is not subject to Schiff shielding nor any lepton-nucleon coupling contributions. Unlike proposed experiments for the proton or charged lightnuclei, the neutron’s neutrality has made it substantially easier to study, as, sans-EDM, it hasonly a minimal response to an electric field (in the form of relativistic effects). These propertiesmake the neutron a unique system in EDM searches, and there has been a great deal of effortdevoted to searches for the nEDM.The history of nEDM measurements is presented in Table 1.3. The first experimental limiton the nEDM was set by an interpretation of a scattering experiment [90, 91]. The purposeof this scattering experiment had been to set a limit on a neutron-electron interaction, andthis limit was independently used by Ramsey and Purcell to calculate an upper limit on thenEDM. In that same publication, Ramsey and Purcell announced that they were undertakingthe first experiments to directly measure the nEDM. The details of that first measurementwere never officially published, but the result was referenced in other publications as beingdn < 5× 10−20 ecm [92].The official first direct measurement was published in 1957 by Ramsey, Purcell, and Smith,and reported the limit of dn < 4×10−20 ecm [93]. This measurement had two defining features:the use of magnetic resonance techniques and the use of a neutron beam.Ramsey’s method of separated oscillating fields is a magnetic resonance technique used tomeasure the Larmor frequency of a system [124], and has become the dominant technique foralmost all nEDM measurements. A detailed description of this method is given in the nextsection, but some of the key results are useful to refer to here to compare nEDM measure-ments. The principle of Ramsey’s method is that the neutron spins undergo a long period offree precession, sandwiched between two short interaction periods. This free-precession periodimproves the sharpness of the resonance, with a statistical uncertaintyσ ∝ 1αETR√Ntot, (1.19)where α is the neutron population’s spin polarization, E is the electric field strength, TR is thefree-precession time, and Ntot is the total number of neutrons detected.The 1957 measurement was built around the use of a neutron beam. In the beam-typeexperiment, a beam of neutrons travels between two flat plates providing an electric field [93].The beam method was dominant in nEDM measurements until the 1980s. Over a 20 year period,beam measurements improved the upper limit on the nEDM by four orders of magnitude. Thefinal beam measurement at Institut Laue-Langevin (ILL), published in 1977, improved the limitto dn < 3× 10−24 ecm [100].Beam measurements were able to achieve large neutron counts, but had limited free-precessiontimes, ranging from 1 ms in the first measurement to the order of 10 ms in later measurements.Further, beam measurements struggled with systematic uncertainty arising from relativistic,velocity-dependent effects. This motivated the shift away from beam measurements to mea-surements involving the storage of UCNs.The unique behaviour of UCNs, described in detail in Section 1.3, make it possible forthem to be confined in material cells for times up to ∼103 s (ultimately limited by the neutronlifetime). Given Equation 1.19, this feature was seen to have great potential for improving121.2. Electric Dipole MomentsGroup Year Technique Result RefANL 1950 Scattering < 3× 10−18 ecm [90, 91]ORNL 1951? Beam < 5× 10−20 ecm [92]ORNL 1957 Beam < 4× 10−20 ecm [93]ORNL 1967 Beam < 7× 10−22 ecm [94]MIT 1967 Bragg Ref. < 8× 10−22 ecm [95]ORNL 1968 Beam < 3× 10−22 ecm [96]BNL 1969 Beam < 1× 10−21 ecm [97]ORNL 1969 Beam < 5× 10−23 ecm [98]ORNL 1973 Beam < 1× 10−23 ecm [99]ILL 1977 Beam < 3× 10−24 ecm [100]PNPI 1980 UCN < 1.6× 10−24 ecm [101]PNPI 1981 UCN < 6× 10−25 ecm [102]ILL 1984 UCN < 8× 10−25 ecm [103]PNPI 1986 UCN < 2.6× 10−25 ecm [104]ILL 1990 UCN < 12× 10−26 ecm [105]PNPI 1992 UCN < 9.7× 10−26 ecm [106]ILL 1999 UCN < 6.3× 10−26 ecm [107]ILL 2006 UCN < 2.9× 10−26 ecm [108]CryoEDM — UCN Cryo Cancelled [109, 110]PNPI/ILL 2014 UCN < 5.5× 10−26 ecm [111, 112]ILL 2016 Reanalysis < 3.0× 10−26 ecm [113]PSI nEDM 2020 UCN < 1.8× 10−26 ecm [114]PSI n2EDM — UCN 2C Development [115]TUCAN — UCN 2C Development [116]LANL — UCN 2C Development [117]PNPI/ILL — UCN 2C Development [112, 118]PanEDM — UCN 2C Development [119]nEDMSF — UCN Cryo Development [120](nEDM@SNS)ESS — Beam Development [121]J-PARC — Crystal Development [122, 123]Table 1.3: Summary of the history of nEDM measurements and experiments under development.In UCN experiments, 2C denotes a “two cell” experiment, contrasting with earlier experiments,which had a single storage cell.131.2. Electric Dipole MomentsnEDM measurements. The first measurement using UCNs at the Petersburg Nuclear PhysicsInstitute (PNPI) in 1980 improved the limit to dn < 1.6 × 10−24 ecm [101]. In the 40 years ofmeasurements using UCNs, the nEDM limit has improved by another two orders of magnitude,with the most recent measurement in 2020 by the Paul Scherrer Institute (PSI) reportingdn < 1.8× 10−26 ecm [114].At the time of writing, there are six nEDM measurements under development using UCN.Five of those are room-temperature experiments, in which a source of UCNs is connected to aroom-temperature experiment [112, 115–117, 119]. These five experiments all aim to reach ameasurement sensitivity on the order of 10−27 ecm. The sixth measurement, formerly knownas nEDM@SNS, now known as nEDMSF, is a fully cryogenic experiment, in which UCNs areproduced inside the measurement apparatus [120]. This experiment aims for a sensitivity onthe order of 10−28 ecm.Although the free-precession times were greatly improved by using UCN, there are also anumber of disadvantages that have limited progress of nEDM measurements. The most criticalis the difficulty of producing UCNs in large densities. The efforts of those groups working onnew measurements has been largely focused on improving UCN densities by constructing newUCN sources capable of delivering greater numbers of UCNs to experiments. The details ofUCN sources are discussed in Section 1.3.2.In addition to the substantial effort on new UCN measurements, there is also a renewedeffort to perform beam measurements. With the very high neutron fluxes available at modernneutron spallation sources, it is now possible to offset the lower precession time and make such ameasurement competitive with UCN measurements. A new cold neutron beam measurement isunder development at ILL and planned to be installed at the European Spallation Source [121].This measurement will also use a pulsed neutron beam, rather than a continuous neutronbeam, making it possible to use time-of-flight methods to address velocity-dependent systematicuncertainties in the beam experiment.The story of nEDM measurements can be rounded out with a mention of crystal-scatteringmeasurements. Among the beam measurements of the 1960s, a single experiment was performedat MIT using a technique based on Bragg reflection [95]. This measurement reported a limitof dn < 8 × 10−22 ecm, only slightly worse than the beam measurement in the same year thatreported dn < 7×10−22 ecm [94]. Neutrons in scattering measurements are exposed to extremelylarge electric fields inside the crystal compared to other methods. Despite this advantage, thistechnique was not developed any further at that time. However, a newly proposed experimentat J-PARC will look at Pendello¨sung interferometry in crystals [123].1.2.3 Measurement Technique for the nEDMThe majority of measurements of the nEDM are based on making a determination of theprecession frequency of the UCNs in the presence of an electric field. The interaction of anEDM with the electric field will cause a frequency shift. The precession frequency of a neutronin the presence of a magnetic field ~B0 and an electric field ~E given by~ωn = 2µn~S · ~B0 + 2dn~S · ~E, (1.20)where µn is the magnetic dipole moment, dn is the electric dipole moment, and ~S is the spin.The frequency shift, and thus dn, can be determined by measuring the precession frequency forat least two values of ~E. The greatest sensitivity is achieved by measuring the frequency withopposite orientations of the electric field: aligned and anti-aligned with the magnetic field. The141.2. Electric Dipole MomentsFigure 1.2: An illustration of Ramsey’s method of separated oscillating magnetic fields. Apopulation of spins, ~S, is prepared in a polarized state relative to a magnetic field ~B0. Anoscillating magnetic field pulse ~B1 rotates the spin states into the transverse plane. The spinsare then allowed to precess for a long time before the application of a second pulse, which rotatesthe spin states out of the transverse plane. The illustration shows a sequence with the spinsending in the fully-reversed polarization state relative to their initial state, which is commonlyreferred to as an optimized sequence with ω1 = ωn. In the nEDM measurement, an electricfield ~E is also applied, and the sequence is repeated with ~E both aligned and anti-aligned tothe magnetic field.frequencies ωn,± measured for the two electric field polarities in this case are~ωn,± = 2µn| ~B0| ± 2dn| ~E|. (1.21)The nEDM can then be determined bydn =~ωn,+ − ~ωn,−4| ~E| . (1.22)The measurement of the precession frequency of neutrons for nEDM measurements has gener-ally been achieved using Ramsey’s method of separated oscillating magnetic fields [124], whichis illustrated in Figure 1.2. This method allows the precession frequency of a spin populationto be precisely determined. Ramsey’s method begins with a population of spins polarized inthe eigenstate ψ↑ along ~B0 = | ~B0|zˆ. The spins are subjected to a pulsed magnetic field ofmagnitude | ~B1| of duration τ , which is oriented in the xy-plane and oscillates with a frequencyω1. In the typical derivation of this method, ~B1 rotates in the xy-plane, but the same effectcan be achieved by a linearly oscillating field. The effect of this pulse is to rotate the spins intoa superposition ψ = C↑ψ↑ + C↓ψ↓. The spins are then allowed to precess freely about ~B0 fora long time TR. After the free precession period, a second pulse is applied, in-phase with theoriginal, and with the same magnitude and duration.The probability P↑↓ that a spin that was initially in the ψ↑ state is measured to be in thestate ψ↓ following the Ramsey sequence is given by the equationP↑↓ =(Ωa)2sin2(aτ)(λ2asin(λTR/2) sin(aτ)− cos(λTR/2) cos(aτ))2, (1.23)151.2. Electric Dipole Moments27.5 28 28.5 29 29.5 30 30.5 31Applied frequency f = /2 (Hz)00.20.40.60.81P = /4 = /8Figure 1.3: Plot of the calculated probability that a neutron spin in the state ψ↑ will bemeasured to be in the ψ↓ state following the sequence of Ramsey’s method. A holding field of| ~B0| = 1µT and precession time TR = 10 s are assumed. The short precession time is chosen tobetter illustrate the features of the curve. Two curves are shown, with Ωτ = pi/4 and Ωτ = pi/8,illustrating the effect of tuning the ~B1 magnitude and duration.where Ω = γn| ~B1| is the precession frequency relative to ~B1, λ = ω1 − ωn is the differencebetween the neutron precession frequency ωn and the frequency ω1 of the pulsed magnetic field,and a =√λ2+Ω22 . A spin that has undergone the transition from one state to another is referredto as having undergone a spin flip. A derivation of this result is included in Appendix A.1.The probability P↑↓ is plotted in Figure 1.3. The greatest probability of a spin-flip occurringis at the central fringe, where |ωn − ω1| Ω. At the central fringe, the equation simplifies toa simple cosine with the formP↑↓ =12(1 + cos(ω1 − ωn∆ω)), (1.24)where ∆ω gives the width of the central fringe. The width has the approximate form∆ω =pi(TR +4τpi) . (1.25)These equations are derived in Appendix A.2.161.3. Ultracold NeutronsMeasurements that depend on the absolute number of neutrons can have challenges dueto variations in counts, which could introduce large uncertainty in the result if not properlycorrected. In the case of the nEDM measurement, the dependence on absolute counts can beremoved by counting the number of neutrons in both spin states (N↑ and N↓) and calculatingthe asymmetryA =N↑ −N↓N↑ +N↓= A0 − α cos(pi(ωn − ωi)∆ω), (1.26)which is derived in Appendix A.2.2. In this equation, A0 is a constant offset, which could arisedue to a spin-dependent efficiency in the detection, and α, known as the visibility, accountsfor the overall polarization of the neutron population, as well as the efficiency of distinguishingspins.The four parameters A0, α, ∆ω, and ωn are unknown. To determine the frequency, a Ramseysequence is performed at four different applied frequencies, known as the working points, whichare given byωi = ωn ± ∆ω2(1± s), (1.27)where s is the separation of the working points.1 The asymmetry is measured at each point, andfit to Equation 1.26. From this fit, the precession frequency ωn is obtained. If this procedure isperformed in presence of an electric field ~E, then the measured precession frequency becomesωn,± depending on the orientation of ~E. These frequencies can then be used to determine dn.The theory of the Ramsey method applied to measuring the nEDM is relatively straight-forward. In practice, the desired high precision of the measurement makes it significantly morecomplicated to implement. This topic will be returned to in Chapter 4, where the componentsof the TRIUMF UltraCold Advanced Neutron (TUCAN) EDM spectrometer are described, andthe contributions to the overall precision of the experiment are elaborated upon.1.3 Ultracold Neutrons1.3.1 Properties of Ultracold NeutronsThe behaviour of free neutrons depends on their kinetic energy, and it is common to distinguishthe different energy ranges with unique names. Table 1.4 lists several notable energy rangesand their properties: velocity, calculated from v =√2Ek/mn, wavelength, calculated fromλdB = h/mnv, and equivalent temperature, calculated from T = Ek/kB.High energy neutrons are significantly relativistic, with v > 0.2c. They occur naturally incosmic rays and inside neutron stars. Below this level are fast neutrons, which are producedin nuclear reactions. The production of neutrons for experiments generally involves nuclearreactions, and so all neutrons used in EDM experiments begin as fast neutrons.Lower energy neutrons can be produced through moderation of fast neutrons, in which theneutrons lose their energy through successive collisions in a medium, bringing them into thermalequilibrium. In the absence of cooling, a moderator will be at room temperature, and so thename “thermal neutrons” is given to the specific energy that is achieved when neutrons aremoderated to a temperature of approximately 20◦C. The cold and very cold energy ranges areachieved in thermal equilibrium with cryogenic systems.1Since ωn and ∆ω are not known exactly, some initial studies are required to determine the best choice ofexperiment parameters. This equation is a good theoretical representation of the choice of working points in areal experiment.171.3. Ultracold NeutronsRange Energy (eV) v (m/s) λdB (nm) T (K)High energy > 20× 106 > 62× 106 < 6.4× 10−6 > 2.3× 1011Fast > 1× 106 > 14× 106 < 2.9× 10−5 > 1.16× 1010Thermal 0.025 2188 0.18 290Cold < 0.025 < 2188 > 0.18 < 290Very cold < 7× 10−4 < 366 > 1.0 < 8Ultracold < 3.5× 10−7 < 8.2 > 48 < 0.004Table 1.4: Commonly referenced neutron energy ranges and their properties.At the very bottom of the neutron energy scale are UCNs, with energies below ∼350 neV.This classification arises due to their unique behaviour compared to neutrons of other energies.Each of the four fundamental forces play a role in this behaviour, and will be considered inturn.Strong InteractionThe interaction of UCNs with nuclei through the strong interaction is the most important, asit defines the UCN energy scale.A neutron experiences the strong interaction when scattering from a single nucleus. Fermishowed that, for the purposes of calculating interaction matrix elements in the first Born ap-proximation, the nucleus can be described by a pseudopotential [125]V = 2pi~2mnasδ(3)(r), (1.28)where as is the scattering length. If a particle has a wavelength long enough for it to interactwith a collection of nuclei, the total potential becomesV = 2pi~2mn∑ias,iδ(3)(r − ri), (1.29)with as,i the scattering length and ri the location of nucleus i.UCNs have typical wavelengths of ∼ 50 nm, while typical interatomic spacings in solidmaterials are on the order of 0.1 nm. The result is that UCNs interact with thousands of nucleiwhen approaching a surface. It can be shown that in the case of an interaction with a densecollection of potentials, the potential seen by a UCN is a constant [126],V =2pi~2mn∑inias,i, (1.30)where i now indexes the type of nucleus and ni is the number density.UCNs therefore reflect from a potentialU ={0, x < 0,V − iW , x > 0, (1.31)which includes an additional imaginary component W that describes absorption or other lossmechanisms. This loss component can be calculated asW =~2∑iniσl,iv, (1.32)181.3. Ultracold Neutronswhere σl,i is the loss cross-section and v is the neutron velocity. Schro¨dinger’s equation takesthe form− ~22mn∇2ψ + Uψ = Eψ. (1.33)Assuming an incident wave of the form eikx the solution isψ ={eikx +Re−ikx, x < 0,T eik′x, x > 0,(1.34)where R is the amplitude of the reflected wave with wavenumber k =√2mE⊥/~, and Tis the amplitude of the transmitted wave with wavenumber k′ =√2mn(E⊥ − U)/~. Here,E⊥ = E cos2(θ) denotes the surface-normal component of the energy of the UCN at angle ofincidence θ.At the boundary of the potential, it is required that(1ψdψdx)x<0=(1ψdψdx)x>0=ik(1−R)1 +R= ik′, (1.35)which can be solved to giveR =√E⊥ −√E⊥ − U√E⊥ +√E⊥ − U. (1.36)If W = 0, so that U = V is real, the behaviour of neutrons of different energies can beeasily seen. If E⊥ > V , then k′ is real, and the reflection probability |R|2 < 1, so that some ofneutrons are transmitted through the surface. If E⊥ < V , thenR =√E⊥ − i√U − E⊥√E⊥ + i√U − E⊥, (1.37)and then |R|2 = 1, and k′ is imaginary, so that the only penetration of the wavefunction pastthe potential is an exponentially decaying term.This result leads to the definition of UCNs. A neutron is referred to as ultracold if it satisfiesthe condition E < U for some material, where E⊥ has been replaced by the total energy. Inthis case, for any angle of incidence, a UCN is reflected, meaning that it can be confined insidea vessel made of that material. Table 1.5 shows the value of this potential for a number ofmaterials. The largest known potential is for 58Ni, which defines the upper energy limit forUCNs at 347 neV. It is common to see UCN defined as having lower energies than this, sincemore commonly used materials do not exceed 200–250 neV.Returning to Equation 1.36, the effect of the imaginary componentW can now be considered.From Table 1.5, common materials generally have W V , so by applying a Taylor expansionand keeping only the leading order correction, it can be shown that|R|2 = 1− 2WV√E⊥V − E⊥ = 1− µ(E, θ), (1.38)where µ(E, θ) is called the loss rate. In literature, it is common to report the value W/V tocharacterize the losses for different materials. These wall losses limit the lifetime of UCNs whenstored in material vessels. The loss probability for a UCN stored in a material vessel can bemodelled as an exponential distributionP loss = e−t/τs , (1.39)191.3. Ultracold NeutronsMaterial V (neV) W (neV) W/V58Ni 347 2.06× 10−1 5.94× 10−4SS316 183 8.52× 10−2 4.66× 10−4NiP 213 1.21× 10−1 5.68× 10−4DLC 243 5.65× 10−2 2.33× 10−4C8D8 (dPS) 171 4.70× 10−2 2.75× 10−4C2D4 (dPE) 209 6.07× 10−2 2.90× 10−4SiO2 91 1.73× 10−2 1.90× 10−4Table 1.5: Fermi potentials of some commonly-used materials in UCN experiments, calculatedusing data from Ref. [7]. Values of W are calculated including absorption and incoherentscattering losses.where τs is called the “storage lifetime”. The storage lifetime is a combination of all differentloss channels and can be decomposed asτs =(∑iτ−1i)−1, (1.40)where τ−1i is the loss rate for some mechanism and the sum is over all such mechanisms. Theτi will typically include the wall losses τw, as well as the free neutron lifetime from beta decayτβ, but can also include other mechanisms such as losses to holes or to fluid media. Subsequentchapters will describe different contributions to the storage lifetime in various situations.GravityAt the Earth’s surface, the gravitational potential for a neutron of mass mn = 1.674× 10−27 kgisUg = mngz = (102.5 eV/m) z. (1.41)This is comparable to the energy scale for UCNs, and so gravity plays a significant role in UCNdynamics. A UCN of 100 neV energy will not be able to climb over a wall of 1 m height — animportant consideration when extracting UCNs from sources. This dependence on gravity willalso affect how UCNs fill a space with significant vertical dimension.In a quantum mechanical treatment, the vertical probability distribution is obtained fromSchro¨dinger’s equation,− ~22mn∇2ψ +mngzψ = Eψ, (1.42)for which the solutions are the Airy functions of the first kindψ ∝ Ai (z′ − E′) , (1.43)where z′ =(2mn2g/~2)1/3and E′ =(2/~2mng2)1/3are the rescaled height and energy. Theboundary condition ψ(0) = 0 quantizes the energies to beEi = −xAi,i(~2mng22)1/3= −xAi,i(6.02× 10−4) neV, (1.44)201.3. Ultracold Neutronswhere xAi,i is the ith zero. The first energy level is E1 = 1.41 peV, and so the energies ofquantized gravitational levels are small compared to typical UCN energies. With increasingenergy, these levels grow closer together, and in fact, E14377 = 1.00 neV is the first energy levelto exceed a single neV. This means that for almost all UCNs, the quantization with respect togravitational potential energy is not relevant.In a classical mechanical treatment, the probability of finding a UCN at a height z isP(z) = 1T√2mnH− Ug(z) =√g8h√2mnmngh−mngz =12√h1√h− z , (1.45)where H is the total energy and h = H/mng is the maximum height. From this equation, aUCN is most likely to be found at the peak of its trajectory.This result would be observed for a single UCN bouncing repeatedly on a surface, but is notgenerally seen in experiments with large populations of UCNs. In that case, the UCN populationis more accurately modelled as a gas in mechanical equilibrium (the validity of this assumptionis examined in Ref. [126]). For the UCN gas, the distribution is obtained from considerations ofphase space density. Following the derivation in Ref. [126], under the assumption of mechanicalequilibrium, the phase space density for a given energy is a constant ρ(H). Then the real spacedensity at z is n(H, z) = 4piρ(H)p(z)2dp(z), where p(z) is the momentum. From conservationof energy, p(0)dp(0) = p(z)dp(z), from which it follows that4piρ(H)p(z)2dp(z) = 4piρ(H)p(z)p(0)dp(0) = 4piρ(H)p(z)p(0)p(0)2dp(0), (1.46)and thusP (z) ∝ n(H, z) ∝ p(z)p(0)∝√H−mngzH . (1.47)In mechanical equilibrium, the smaller momentum at greater heights results in a smaller avail-able phase space, and thus fewer UCNs at greater height. This condition is achieved in manyUCN experiments due to the randomization of UCN trajectories by wall reflections.ElectromagnetismWith net zero charge, UCNs do not respond directly to electric fields. However, they dopossess a magnetic dipole moment µn = 0.966× 10−26 J/T, through which they can experienceelectromagnetic interactions. The magnetic potential is given byUm = −µn~S · ~B, (1.48)where µn is the magnetic dipole moment and ~B the magnetic field. If the spin is parallel to themagnetic field, then this becomesUm = ±(60.3 neV/T)| ~B|, (1.49)where the sign is determined by the orientation of the spin relative to the field. This potentialis also of similar magnitude to UCN energies for fields of a few tesla. A 60 neV UCN will beunable to pass through a 1-T field, assuming that its spin is aligned with the magnetic field.Alternatively, if the spin is anti-aligned with the magnetic field, then a 1-T field will increasethe UCN energy to 120 neV, accelerating it. This makes it possible to polarize a population ofUCNs by passing them through a large field that will reject one spin state (known as low-fieldseekers) and accept the other (known as high-field seekers).211.3. Ultracold NeutronsWeak InteractionWeak interactions drive the free neutron beta decayn→ p+ e− + ν¯e, (1.50)with a lifetime of τβ = (878.4±0.5) s [127]. This free neutron lifetime contributes to the storagelifetime τs and sets the upper limit for the duration of experiments using UCNs, with only 5%of UCNs not having undergone beta decay after 3τβ.Summary of UCN PropertiesUltracold neutrons can be confined by materials, by gravity (in the vertical direction), and bymagnetic fields (if the UCNs are polarized). Their interaction with magnetic fields makes itpossible to produce a fully polarized population by applying a sufficiently strong magnetic field,enabling spin-dependent measurements. Beta decay limits the possible duration for which UCNcan be confined, with times on the order of 103 s still achievable, although this is further limitedby wall losses. The combination of properties possessed by UCNs makes them very powerful forEDM searches, since a fully polarized population of UCNs can be stored for a relatively longtime.1.3.2 Production of Ultracold NeutronsThe greatest challenge to using UCN is their production. Large fluxes of high energy neu-trons can be produced in reactors and accelerator-based spallation sources. These are readilyconverted into thermal neutrons via moderators, with the best reactors producing thermalneutron fluxes of Φn = 1015 n cm−2s−1, and spallation sources producing fluxes of Φn =1017 n cm−2s−1 [128]. Thermal neutrons can be further moderated to cold temperatures incryogenic materials, but moderation to the < 4 mK equivalent temperatures of UCNs is notpractical. Such temperatures can be achieved in dilution refrigerators [129], but the availablecooling power is too small to use such devices for cooling large neutron fluxes.UCN sources typically start with some form of moderation to reduce the neutron energy, anddiffer in the final step of converting moderated neutrons to UCNs. Sources can be categorizedby whether they employ conservative or non-conservative forces for this final step. Conservativesources must obey Liouville’s theorem, which requires that the density in phase space remainsconstant. These sources can shift some region of phase space into the UCN energy range, butcannot compress more UCNs into that phase space — they are limited by the original phasespace density from the moderator. Non-conservative sources are able to overcome this limitationto produce higher densities.Conservative UCN SourcesConservative sources rely on the fact that UCNs are still present in the low energy tail of anythermal/cold neutron distribution. At thermal equilibrium with a moderator of temperatureT , the density of neutrons is given by [126]ρ(v)dv =2Φnβv2β2e−v2/β2 dvβ, (1.51)221.3. Ultracold Neutronswhere β =√2kBT/mn and Φn is the thermal neutron flux. Integrating this density up to someUCN cut-off energy V givesρUCN =23Φnβ(VkBT)3/2. (1.52)Assuming a reactor flux of Φn = 1015 n cm−2s−1, in a 20-K deuterium moderator, the maximumachievable density of UCNs would be 2× 104 cm−3, so a 10 l vessel of deuterium would contain20 × 107 UCNs. This is not insignificant, but such a density cannot be practically extractedfrom a moderator into experiments. This is due to a combination of losses from interactionswith the moderator and with the vessel walls, reduction of the density due to dilution out ofthe moderator volume, and losses associated with extracting UCNs away from the moderator.The first conservative UCN source was constructed at PNPI, and was later followed by twoimproved sources [130]. All of these were based on the above principle of extracting the low-energy tail of a moderated neutron spectrum. The cold beryllium source (CBS) used a berylliummoderator, the small liquid hydrogen source (SLHS) used hydrogen, and the universal coldneutron source (UCNS) used a mixture of hydrogen and deuterium. These moderators wereplaced near the centre of the WWR-M reactor at PNPI, and neutrons were extracted usinga vertical guide. The loss of energy to overcome the gravitational potential when climbingto the top of the guide serves to shift the neutrons to the UCN energy level. The extracteddensity of UCNs for the UCNS was 16 cm−3 — substantially less than the expected density ina moderator. A similar source was constructed at the SM-2 reactor at RRI, using horizontalextraction instead of vertical, which reported an extracted density of only 0.015 cm−3 [131].Other conservative sources have employed “turbines”. These turbines are moving surfacesthat reflect cold neutrons and reduce their velocities to the ultracold regime. Consider a neutronwith velocity v, and a surface moving at velocity vt. With respect to the surface, the neutronmoves at velocity v − vt. If the neutron reflects elastically from the surface, then its velocityis reversed and becomes vt − v. Then, in the lab frame, the velocity is 2vt − v. Neutronsthat satisfy 2vt − v <√2V /mn will become UCNs. The advantage of such a technique isthat the UCNs are only produced at the turbine, by shifting higher-energy neutrons towardsUCN energies. As a result, turbine sources have reduced losses associated with production andextraction, and can output higher densities.The turbine principle has been successfully implemented in a number of sources [4, 132–135].The most successful of these has been the ILL PF2 reactor source, which was the driving sourcebehind the ILL nEDM effort [113]. This source continues to operate and is capable of achievingdensities of 36 cm−3 in experiments [134].Non-Conservative UCN SourcesNon-conservative sources, also called superthermal sources, are based on specific systems thatallow neutrons to be cooled to energies below the temperature of the system at a high rate.The principle can be demonstrated by considering neutrons scattering in a system with twoenergy levels, EUCN and EUCN + ∆E. The principle of detailed balance requires that the crosssections of the scattering process obeyEUCNe−EUCN/kBTσs(EUCN → EUCN + ∆E)= (EUCN + ∆E)e−(EUCN+∆E)/kBTσs(EUCN + ∆E → EUCN),(1.53)231.3. Ultracold Neutronswhich can be rearranged to giveσs(EUCN → EUCN + ∆E)=(EUCN + ∆E)EUCNe−∆E/kBTσs(EUCN + ∆E → EUCN).(1.54)Assuming that the cross section of the down-scattering process σs(EUCN + ∆E → EUCN) doesnot have a significant temperature dependence, and that ∆E kBT EUCN, then the up-scattering cross section σs(EUCN → EUCN + ∆E) can be greatly suppressed. It follows that,if there is an incoming flux of neutrons with temperature Tn ≥ ∆E/kB, then there will be asignificant number of down-scattering events, but the rate of the reverse up-scattering processwill be small. A population of UCNs will therefore build up, withnUCN ∝ EUCN∆Ee∆E/kBT . (1.55)The premise of a non-conservative source is to find a material that exhibits such behaviourin interactions with neutrons. This can be achieved when the dispersion curve of the materialand the dispersion curve of a free neutron intersects. Then, the down-scattering process occursthrough the emission of an excitation (e.g. a phonon, roton, etc.) into the material.Sources based on the superthermal conversion process have been built using superfluidhelium-4 (He-II) and solid deuterium (SD2). Table 1.6 summarizes the properties of the sourcesthat have been operated or are under development.He-II was the first material considered as a possible superthermal UCN converter, in the1970s [136]. This consideration was driven largely by the fact that 4He does not absorb neutrons,making it possible to achieve a large storage lifetime for UCNs in an isotopically pure productionvolume. The first demonstration of UCN production in He-II was achieved at ILL in 1983 [137],although this source was not used for experiments. This was followed at ILL by the SUN I andSUN II sources starting in 2010 [138, 139], and the SuperSUN source in 2023 [140]. The cancelledCryoEDM experiment also included an in-experiment He-II source using a cold neutron beam,which had achieved UCN production by 2003 [110, 141]. Another He-II source was built atthe Research Center for Nuclear Physics (RCNP) in the early 2000s [142], and parts of thissource were moved to and operated at TRIUMF2 from 2017 to 2019 [3]. Several new He-IIsources are under development: an un-named source at PNPI [130, 143]; an un-named sourceat Los Alamos National Laboratory (LANL) [144]; and the TUCAN source at TRIUMF [145].The nEDMSF, formerly nEDMSNS, experiment also includes an in-experiment source usingHe-II [120].The first demonstration of UCN production in SD2 was accomplished at PNPI [130], whichlater developed into the SDS in 1997 [146]. Several sources world-wide are currently in operationusing SD2 [147–150], with one more in development [151]. In SD2, neutron losses are significant,with the achievable storage lifetime limited to 70 ms, but this is offset by a large productioncross-section [152]. The performance of these sources also depends on the quality of the SD2crystal, as inhomogeneities limit the ability to extract UCNs, and so significant effort is requiredto produce a good UCN converter using SD2 [153, 154].Both types of sources benefit from the presence of a cold moderator, as the neutron energiesat which the conversion is most likely are below thermal neutron energies. A 20-K moderatoris ideal for He-II sources [136], while a 40-K moderator is ideal for SD2 [152]. In several SD22Note that, despite the capitalization, the name “TRIUMF” is not an acronym. It originally stood for “TriUniversity Meson Facility”, but that name is officially no longer used, and “TRIUMF” is instead a stylized name.241.3. Ultracold Neutronssources, the SD2 has doubled as a cold moderator, although the PULSTAR and LANL sourcesuse additional solid CH4 and polyethylene moderators, respectively [147, 150]. He-II sourceshave used liquid D2 [139] and solid D2O [3, 142] cold moderators.Despite their promise, superthermal sources have struggled to achieve large density improve-ments over turbines. A “standard bottle” experiment, comparing European sources, found thatthe PSI UCN source had the best performance, with an extracted density of 22 cm−3, comparedto the ILL PF2 turbine with a density of 20 cm−3 [155]. The SUN II source, the only He-IIsource in that comparison, reported a maximum density of only 7.2 cm−3. The LANL sourcehas reported a density in-experiment of 39 cm−3 [150]. Achieving a truly significant improve-ment in UCN densities, and thus nEDM measurement precision, is the goal of those sourcesunder development, each of which aims for experiment densities on the order of 100 cm−3.1.3.3 Simulating Ultracold NeutronsMuch of the work that will be presented in this dissertation revolves around the behaviourof UCNs in a variety of situations. This behaviour has been modelled using PENTrack [156].PENTrack is a specialized Monte Carlo simulation package, built specifically for UCNs. Itlargely models the dynamics of UCNs using the properties that have been outlined in 1.3, withUCN trajectories determined by the equation of motion~¨x =1γmn(~F − 1c2(~˙x · ~F)~˙x), (1.56)where ~¨x is the acceleration, ~˙x is the velocity, ~F is the force, and γ is the Lorentz factor (not tobe confused with the gyromagnetic ratio). The dynamics of UCN spins will be a major subjectof later chapters. In PENTrack, a neutron is assigned a spin vector ~S, which evolves accordingto the Bargmann-Michel-Telegdi (BMT) equationd~Sdt=(−2µnγ~~B′ + ~ωT)× ~S, (1.57)where γ is the Lorentz factor and µn is the magnetic moment. Here ~B′ is the magnetic field inthe rest frame of the particle~B′ = γ ~B + (1− γ)( ~B · ~˙x) ~˙x~˙x2− γc2~˙x× ~E, (1.58)where ~ωT is the Thomas precession axis~ωT =γ2c2(γ + 1)~¨x× ~˙x, (1.59)where ~¨x is the acceleration. Both the equation of motion and the BMT equation are numericallyintegrated to reproduce the behaviour of a UCN as it travels through an experiment. There area few other general features of PENTrack that will be referred to throughout this dissertationand that are worth summarizing here.PENTrack allows the full geometry of the experiment to be imported from CAD softwareas an STL file [157]. This makes the implementation of geometries convenient, as experimentalequipment for UCNs typically requires a CAD model to be made before the equipment can bebuilt. However, since STL files describe geometries as meshes of triangles, this does introduce251.3.UltracoldNeutronsSource First Operation Converter Cold Moderator Neutron Source Ref.ILL1 1983 He-II Liquid D2 Reactor [137]Mark-3000 1992 He-II Liquid H2 Reactor [158]PNPI SDS 1997 SD2 None Reactor [143, 146]RCNP 2002 He-II Solid D2O Spallation [142]CryoEDM2 2003 He-II Liquid D2 Reactor [141]LANL 2008 SD2 Polyethylene Spallation [150]SUN I + II 2010 He-II Liquid D2 Reactor [138, 159]PSI UCN 2011 SD2 None Spallation [148]TRIGA Mainz 2014 SD2 None Reactor [149]TRIUMF 2017 He-II Solid D2O Spallation [3]SuperSUN 2023 He-II Liquid D2 Reactor [140]TUCAN Source Planned 2025 He-II Liquid D2 Spallation [145]nEDMSF3 Development He-II Unknown Spallation [120]PNPI Development He-II Liquid D2 Reactor [130, 143]PULSTAR Development SD2 Solid CH4 Reactor [147]FRM-II Development SD2 SH2 Reactor [151]LANL4 Concept He-II Liquid D2 Spallation [144]Table 1.6: Summary of superthermal UCN sources and their main features. The source (1) at ILL was never used for experiments,but was the first demonstration of the use of He-II for UCN production. The source (2) was the first prototype for the in-experimentsource for the CryoEDM experiment, which was subsequently cancelled. The source (3) is also an in-experiment source for the formernEDM@SNS experiment, now renamed the nEDMSF experiment; this source was intended to use the SNS spallation source but, at thetime of writing, the destination of this experiment is unknown. The source (4) is at a conceptual stage described in Ref. [144].261.4. Outline of Dissertationsome limitations compared to models formed from mathematical primitives. The resolutionthat can be achieved by STL files is limited by the floating point precision of the coordinatesystem. This makes it difficult to implement very thin geometries. Additionally, the use oftriangular meshes means that surfaces will never be completely smooth in PENTrack. Theselimitations are discussed further in sections where they became relevant.The previous description of UCNs reflecting from a surface is a strictly correct description.However, it will not be correct when applied to the surfaces in CAD models. This is because realsurfaces have roughness: they are not perfectly smooth, but instead have many small featuresthat are not visible to the eye. When a UCNs reflects from a surface, it is reflecting fromthe microscopic surface, and this must be captured in any modelling of UCNs dynamics. Thisis typically done by treating the reflection angles from a surface as obeying some probabilitydistribution, and there are several different models for this [160–162]. The model that will beused in the PENTrack simulations in this dissertation is Lambert’s diffuse reflection model.With this model, a probability PL is assigned to each material surface. When a UCN reflectsfrom a surface, it undergoes specular reflection with probability (1−PL), and diffuse reflectionwith probability PL. A diffusely reflected UCN is scattered into a direction according to a cosinedistribution of angles about the surface normal.1.4 Outline of DissertationThe topic of this dissertation is the TUCAN source and EDM experiment at TRIUMF. PartI, covering Chapters 2 and 3, is focused on the source, while Part II, covering Chapters 4–7, isfocused on the experiment.In Part I, the outcome of some UCN source experiments performed at TRIUMF will bediscussed. Chapter 2 describes the TUCAN source design in detail, as well as the prototypesource operated at TRIUMF from 2017–2019. Chapter 3 reports on experiments with theprototype source related to the storage lifetime of UCNs in He-II. These studies are critical forunderstanding the performance of He-II sources.In part II, a collection of studies for the nEDM experiment design will be presented, largelyfocused on the central region of the experiment in which the UCNs are stored during themeasurement. Chapter 4 describes details of the nEDM measurement technique, general con-tributions to systematic uncertainties, and the overall design of the TUCAN EDM experiment.Chapter 5 describes the construction of a prototype storage cell for UCNs and measurements ofthe storage lifetime in said cell. Chapter 6 describes simulation studies of the electric field of thecentral region, used for high voltage (HV) design and calculation of systematic uncertainties.Chapter 7 describes simulation studies of magnetic properties and interactions in the centralregion, also relating to systematic uncertainties. The outcome of this work is a design of thecentral region and recommendations on how to achieve a 10−27 ecm systematic uncertainty.27Part IUltracold Neutron Dynamics28Chapter 2Ultracold Neutrons at TRIUMF2.1 IntroductionIn order to meet the UCN density requirements for an improved measurement of the nEDM,the TUCAN collaboration is building a new UCN source. This source has been designed withthe goal of achieving a statistical uncertainty of σstat = 10−27 ecm on the nEDM within 400days of data taking, corresponding to two actual years of operation. It is based on the principlesof superthermal UCN production described in Section 1.3.2 using He-II as a converter.Prior to beginning construction of the TUCAN source, a prototype source was installed andoperated at TRIUMF, with operation starting in 2017 and concluding in 2019. This prototypeused a new proton beamline and spallation source at TRIUMF, with the moderators and He-IIcryostat from the RCNP source. The goal of the 2017–2019 operation period was to developoperational experience with a UCN source at TRIUMF, and to perform a series of measurementsto better understand UCN production, transport, and storage for design of the TUCAN source.In the following chapters, the outcomes of a subset of these measurements will be described.This chapter outlines the details of the TUCAN source and the prototype source, as well asgeneral data acquisition during the experimental runs. Chapter 3 describes experiments relatingto the storage lifetime inside the source.2.2 The TUCAN Source2.2.1 Beamline 1UThe fast neutron flux required for the TUCAN source is produced through spallation of atungsten target. In this spallation process, high-energy protons incident on the target interactwith individual nucleons due to their short de Broglie wavelength. These nucleons receive a largeamount of kinetic energy. Some high-energy nucleons escape the nucleus and induce secondaryspallation reactions in other nuclei, while others deposit their energy in the original nucleus,leaving it in a highly excited state. The excited nucleus returns to its ground state primarily bythe evaporation of neutrons. The resulting cascade from a single incident proton can produceupwards of ten neutrons per proton [163].Spallation neutron sources have several advantages for maximizing UCN production for annEDM experiment. They are well-suited for pulsed production, as the proton beam can beswitched on and off easily. This allows for more efficient operation, with lower heat loads whenUCNs are not being filled into an experiment. Spallation targets can also be purpose-built fora UCN source, allowing the production volume to be placed very close to the neutron source,which maximizes the cold neutron flux. The combination of these properties make it possibleto achieve greater peak UCN production rates, which is ideal for an experiment where UCNswill be stored for a long period of time.The protons that are incident on the TUCAN target are produced by the TRIUMF maincyclotron. The TRIUMF cyclotron accelerates H− ions produced using a DC cusp source [164]292.2. The TUCAN Source(a) The primary section of Beamline 1 (BL1), labelled Beamline 1V due to its location in the cyclotron vault,which branches into BL1B, BL1A, and BL1U.(b) Beamline 1U, which branches off of BL1 after the septum magnet 1USEPT.Figure 2.1: Schematics of the BL1 facility at TRIUMF, focused on BL1U. These diagrams werepreviously published in Ref. [1], and their use here is permitted by the TUCAN collaboration.302.2. The TUCAN Sourceand pre-accelerated in a 300 keV electrostatic injection beamline [165]. The use of H− ionsmakes it possible to extract protons into multiple beamlines at once. This is achieved by theuse of stripper foils, made of pyrolitic graphite, which strip the electrons from the ions [166].The resulting protons follow paths out of the cyclotron into the four primary beamlines: BL1,BL2A, BL2C, and BL4 [167].Construction of the TUCAN source required the construction of a new proton beamline atTRIUMF to bring the protons to the source [1]. This beamline is part of the BL1 facility, shownin Figure 2.1. BL1 is nominally capable of providing a total beam current up to 170µA at anenergy of 500 MeV [167], but typical operation is at 100µA and 483 MeV. BL1 splits into twomutually-exclusive beamlines 1A and 1B, controlled by the use of the static bender magnet 1BVB2. BL1B is used for proton irradiation, while BL1A is used for muon production, neutronirradiation, and UCN production.The UCN beamline 1U (BL1U) operates parasitically on BL1A. This is achieved by theuse of a kicker magnet, labelled 1VK6 in Figure 2.1 which kicks a fraction of beam pulsesupwards into a secondary beam tube [168]. Protons in this tube pass through the field of aseptum magnet, providing an angular deflection into BL1U. The remainder of BL1U consistsof additional bending magnets which further deflect the beam, and quadrupoles for steeringand focusing. A variety of diagnostic devices are also present for measuring the beam positionand size. More details of the construction and operation of the beamline can be found in Ref.[1]. The nominal operational current for BL1U is 40µA, requiring a kick fraction of 2/5 at thetypical 100µA current in BL1A, or 1/3 at 120µA.The spallation target at the end of beamline 1U is shown in Figure 2.2. It is a set of fivetungsten blocks, with cross-sectional area 78×56 mm2, contained in a stainless steel flask with adedicated water cooling loop [2]. The separation of the target into five blocks, two 30 mm thick,and the others 20 mm thick, improves the cooling of the blocks by allowing the cooling waterto flow between them. A 0.86-mm-thick tantalum cladding, applied by hot isostatic pressing tothe outside of the blocks, prevents corrosion in the cooling water. The target flask is isolatedin a helium-filled crypt connected to the beamline by an aluminium window of 1 mm thickness.The crypt is encased in lead shielding, which is relatively transparent to the spallation neutronsbut absorbs gamma radiation that would otherwise heat the UCN source.2.2.2 ModeratorsFast neutrons produced in the target need to be moderated to cold energies in order to maximizeUCN production. The moderator geometry and overall composition for the TUCAN source wasanalyzed extensively to achieve optimal UCN production [5]. The final geometry of the sourceis shown in Figure 2.3.Room temperature moderation removes a large amount of energy from the fast neutrons,reducing the thermal load on the cold moderators. The primary room-temperature moderatoris heavy water (D2O), contained in an upper and lower tank which together surround the coldmoderator volume. The D2O is circulated out of the tanks to remove heat. The D2O tanksare surrounded by a layer of graphite bricks. The graphite serves as a reflector, redirectingneutrons back into the production volume, while also acting as an additional room temperaturemoderator.The cold moderator for the TUCAN source is liquid deuterium (LD2) at 20 K. The useof LD2 over LH2 has been shown to be preferable due to the lower neutron absorption, whileother moderators considered failed to reduce the neutron temperature sufficiently [5], makingliquid LD2 the optimal cold moderator for a He-II source. Cooling of the LD2 is maintained312.2. The TUCAN SourceCooling water inletBeam directionCooling water outletTantalum-clad tungsten blocksFigure 2.2: Photograph of the spallation target used for the TUCAN source. The five tantalum-clad tungsten blocks are contained in a stainless steel flask, through which cooling water cir-culates. The beam enters the flask through a 1.3-mm-thick window. This photograph waspreviously published in Ref. [2], and its reuse here is permitted by the TUCAN collaboration.322.2. The TUCAN SourceFigure 2.3: Geometry of the TUCAN source cut through the central plane. The proton beamis directed into the page and hits the target, indicated in the lower right corner.by a gravitationally-driven thermosyphon. As the LD2 is heated, the change in density withtemperature causes the LD2 to rise in the moderator volume and flow through a section of pipetowards a heat exchanger, which is cooled by a cold head. The cooled LD2 sinks downwardsand flows back through another pipe to replace the heated LD2.2.2.3 UCN Production VolumeIn the TUCAN source, cold neutrons are converted to UCNs in a 28 l quasi-spherical bottleof isotopically-pure 4He cooled below 2.17 K to the He-II state. The bottle is located in thecentre of the moderator volumes, and is connected to a primary heat exchanger, referred to asHEX1, by a 2.35-m-long 15-cm-diameter conduction channel, which is also filled with He-II.This channel doubles as the extraction guide through which UCNs diffuse out of the sourcetowards experiments. The temperature of HEX1 is maintained by a 3He cryostat. These majorfeatures of the source are shown in Figure 7 2.3.In the cryostat, 3He is cooled through a series of heat exchangers before entering HEX1.Initial cooling to ∼10 K is achieved in a heat exchanger with the returning gas from the 3He andnatural helium pumping systems. The 3He then flows through another heat exchanger in a bathof natural helium, provided by a liquefier, which cools it to ∼4 K. The next step of 3He coolingis by yet another heat exchanger in a 1 K natural helium bath, which is itself cooled by pumpingto reduce the vapour pressure and thus the temperature through evaporative cooling. This stepreduces the 3He temperature close to 1 K. Finally, the 3He is filled into HEX1. At this pointthe 3He is a liquid, and additional pumping of the vapour pressure reduces the temperaturebelow 1 K. The expected performance of the cryostat is calculated in Ref. [169], with the final332.3. The RCNP Source at TRIUMFtemperature of the He-II at HEX1 expected to be between 1.0–1.1 K when accounting for theKapitza conductance of the heat exchanger. These calculations assume a heat load of ∼10 Wto the He-II determined from the moderator simulations of Ref. [5].The long conduction channel between the production volume and the cryostat was includedin the design in order to remove the cryostat from the high-radiation environment above thetarget. This has multiple benefits for operation of the source, largely related to activation.The primary motivation was the activation of the copper of HEX1 and the 3He [170]. Earlysimulations suggest that, at a distance of 60 cm, an additional 8 W of heat load would be presentin HEX1, which was reduced to ∼0.1 W in the final design. The same simulations indicated ahigh activation of the 3He due to production of 3H — over 1000 mBq/day at 60 cm, reducedto 10 mBq/day in the final design. Additionally, in the final geometry, the total activation ofmaterials in most areas around the cryostat is expected to be below 105 Gy over the lifetimeof the source, reducing the risk that components of the source will fail due to irradiation.The reduced activation also makes it possible to access the cryostat for maintenance on shorttimescales, with the expected dose rate around the cryostat being ≤100µSv/hr 24 hours afterstopping beam, while the dose around the target is expected to exceed 1 mSv/hr 4 weeks afterstopping beam. The caveat of the long conduction channel is that heat conduction in He-IIlimits the temperature achieved in the production volume itself, with calculations predicting atemperature between 1.1–1.2 K [169].Additional UCN guide elements lead from HEX1 to the exit of the source. On the down-stream side of HEX1, there is a vertical guide section of ∼40 cm height, consisting of two 90◦elbows, referred to as the S-bend, which serves to contain the He-II gravitationally. Above theS-bend, further guide tubes lead out to a UCN-compatible valve, which can be closed to containUCNs in the source during production. The guides between the S-bend and the valve providespace to transition from the ∼1 K temperature of the He-II to room temperature at the valve.Beyond the valve is an aluminium foil which contains the 4He vapour in the source. UCNsproduced in the source exit the He-II volume at the S-bend, then travel through the foil. Thefinal geometry of the foil, which was studied in simulations in Ref. [6], is still being determinedat the time of writing, but the thickness will be no more than 100µm. On the other side of thefoil, additional guides lead out to the experimental area.Because the UCNs potentially spend tens of seconds in the He-II in the TUCAN source,during both production and extraction, the performance of the TUCAN source will dependheavily on the behaviour of UCNs in He-II. Furthermore, the long guide tubes, as well as partssuch as the separating foil and elbows, represent significant extraction losses that limit thepotential performance of the source. Prior to the construction of the source, these elementswere studied using a lower-density prototype source.2.3 The RCNP Source at TRIUMFThe prototype He-II UCN source at TRIUMF was originally built and operated at RCNP,Osaka [142], and was moved (with the exception of the target) to TRIUMF in 2017. Like theTUCAN source, it uses a 3He cryostat to cool He-II through a heat exchanger, but it wasdesigned with a substantially lower cooling power, as well as a few other key differences. Thegeometry of the the prototype source is shown in Figure 2.4. Due to the long vertical guideleading out of the cryostat, this source is commonly referred to as the vertical source.For the vertical source, BL1U and the spallation target were configured as planned for theTUCAN source, but with differences in operation imposed by the design of the cryostat. At342.3. The RCNP Source at TRIUMFFigure 2.4: Geometry of the prototype UCN source at TRIUMF. This figure was previouslypublished in Ref. [3], and its reuse here is permitted by the TUCAN collaboration.RCNP, the source had been intended to operate with 1µA of 392 MeV proton beam [142],much lower than the intended 40µA of 483 MeV beam at TRIUMF. Heating tests at TRIUMFalso showed that the He-II temperature increased above ∼75 mW of applied heating (with anestimated ∼230 mW of static heat load), suggesting that this was the limit of the cryostat’scooling power. The vertical source was therefore limited to only 1/40th of the intended powerof the TUCAN source. Studies at higher beam current showed increased temperature of theHe-II above 1µA [3].The vertical source was also able to operate at a lower He-II temperature, between 0.9–1.0 K [3], compared to the 1.1–1.2 K expected in the TUCAN source. This lower temperaturewas possible due to several factors. Because the production volume in the vertical sourcewas inside the cryostat, there was no significant temperature gradient between the productionvolume and the primary heat exchanger. The lower overall heat load also plays a role inreducing the He-II temperature. In He-II, in the turbulent flow regime where the TUCANsource is expected to operate, the temperature gradient obeys the Gorter-Mellink equationdTdx= −f(T )qm (2.1)where q is the heat flux, m ≈ 3 and f(T )−1 is analogous to thermal conductivity [171]. Different352.4. Experimental Campaignsmodels for f(T )−1 exist, but all models and measurements show maximum thermal conductionclose to 2.17 K, at the transition between normal and superfluid He-II. Another, similar effectcomes from the temperature difference in the heat exchanger between the He-II and 3He. Thecombination of Kapitza resistance at the surface and thermal conductivity in the bulk of theheat exchanger requires a temperature difference between the two fluids, which scales with theheat flux. These factors lead to a higher design temperature in the TUCAN source to be ableto handle the higher heat load and maximize UCN production [170].The vertical source used D2O and graphite as room temperature moderators, but differedfrom the TUCAN source by the choice of cold moderator. In the vertical source, cold moderationwas accomplished in a vessel of solid D2O, cooled by a cold head. The resulting cold neutronspectrum in solid D2O was found to be sub-optimal for UCN production. This is believed to becaused by the effective temperature of the moderator being limited to ∼80 K [172], which leadsto a lower cold neutron flux at those energies where conversion to UCNs can occur in He-II.The under-performance of this moderator drove the use of LD2 in the TUCAN source.The overall geometry of the production volume and moderator layers in the vertical sourcealso differs from the TUCAN source design. Qualitative differences, such as the more completeenclosure of the production volume by the moderators, can be seen by comparing Figures 2.3and 2.4. The most notable quantitative difference is the volume of the He-II in the productionvolume, which is 20 l in the TUCAN source and 8 l in the vertical source. A larger volumeallows more cold neutrons to be captured and converted to UCNs.Unlike the TUCAN source, UCNs were extracted from the vertical source using a tall verticalUCN guide connected to the production volume, visible in Figure 2.4. Due to the effect of thegravitational potential on UCNs, this extraction method imposes an energy cutoff for extractedUCNs. Measured from the bottom of the production volume to the bottom of the extractionguide, this potential is 129 neV, and measured from the top of the production volume to thebottom of the extraction guide, this potential is 61 neV. In contrast, the corresponding potentialsfor the TUCAN source are 61 neV and 24 neV. This should allow a greater number of lower-energy UCNs to escape from the TUCAN source compared to the vertical source.The differences between the vertical source and the TUCAN source are summarized in Table2.1. The differences in terms of cooling power, moderator choice, and extraction limited theoverall performance of the vertical source. Studies of the vertical source concluded a produc-tion rate of (2.06 ± 0.02) × 104 UCNs/s at 1µA [3], while the TUCAN source is expected toachieve (1.5± 0.1)× 107 UCNs/s at 40µA [5]. Regardless, the vertical source was still able toproduce sufficient UCNs for measurements characterizing the performance of components forUCN production, storage, and transport.2.4 Experimental Campaigns2.4.1 OverviewA total of three experimental campaigns were undertaken with the vertical source, in eachof 2017, 2018, and 2019. The 2017 campaign was focused largely on first measurements ofUCN production in the source. The results of the 2017 measurements are detailed in thedissertation of T. Andalib [173] and in the associated publication [3]. During 2018 and 2019, themajority of measurements focused on the testing of guide components for the source, but furthermeasurements of source performance were also included. Measurements of polarized UCNsduring 2018 are detailed in the dissertation of S. Hansen-Romu [174]. A series of measurementsof UCN transmission properties, in particular the probability of diffuse reflection, was attempted362.4.ExperimentalCampaignsComponent Vertical source TUCAN sourceBeam current 1µA 483 MeV 40µA 483 MeVUCN converter He-II He-IIConverter heat load ≤ 0.3 W ∼10 WConverter temperature 0.9–1.0 K 1.1–1.2 KConverter volume 8 l 20 lWarm moderator D2O and graphite D2O and graphiteCold moderator Solid D2O Liquid D2Potential from bottom of production volume 129 neV 61 neVPotential from top of production volume 64 neV 24 neVEstimated production rate (2.06± 0.02)× 104 UCNs/s (1.4± 0.1)× 107 UCNs/sTable 2.1: Summary of the differences between the vertical source and the planned TUCAN source at TRIUMF.372.4. Experimental Campaignsduring 2019 and is detailed in the dissertation of S. Sidhu [6].Across the 2018 and 2019 runs, a large number of different measurements were performed.These different measurements correspond to both different experimental geometries (i.e. ex-changing of components) and different measurement types (i.e. different experiment sequences).To distinguish between experiments, each was assigned a unique name. In 2018, each measure-ment was assigned a name in the format TCN18-XXX, where XXX is a three-digit number. In2019, this naming was changed to TCN19-XXXYYYYY, where XXX remained a three digitnumber which corresponded to the particular experiment geometry, and YYYYY was an al-phanumeric code between 2–5 characters describing the particular experimental sequence. Thisnaming scheme has also been reused for other TUCAN experiments with UCNs, including thosein Chapter 5. These names will be introduced and used throughout the subsequent chapters.2.4.2 DetectorsThe primary form of data collected during the experimental campaigns was UCN counts indetectors versus time. Two detectors were used: a 6Li scintillating detector and a 3He gasproportional detector.The 6Li scintillating detector, developed and built at the University of Winnipeg [175], isbased on the reaction6Li + n→ α(2.05 MeV) + t(2.73 MeV), (2.2)where t is a 3H nucleus, commonly called a triton. The neutron-capturing elements of thedetector are 160-µm-thick pieces of GS20, a cerium-activated lithium aluminosilicate glass en-riched to an isotopic concentration of 95% 6Li [176]. The GS20 also acts as a scintillator andemits light proportional to the energy of the alpha and triton particles released in the captureinteraction. The light is captured by a photo-multiplier tube (PMT) attached to the glass,producing a detection signal. To allow this detector to handle high detection rates, the detectoris split into nine segments with individual PMTs.In this detection scheme it is possible for a reaction at the surface of the GS20 to emit oneof its products away from the glass, which results in the detector not capturing the full energyof the reaction. To mitigate these losses, a second 60µm 6Li-depleted layer of GS30, with aconcentration of 99.99% 7Li [176], is attached to the neutron-facing side of the GS20. Thisadditional layer is largely transparent to UCNs, but does act as a scintillator, preventing thereaction products from escaping.The material properties of GS20 and GS30 relevant for UCN interactions have been cal-culated based on the nominal compositions of the glasses [177]. The real Fermi potentials are103.4 neV and 83.1 neV respectively. The detector will therefore reflect UCN with energies below103.4 neV. The imaginary Fermi potentials are 1.237 neV and 1.23× 10−4 neV respectively.The 3He detector is a DUNia-10 proportional gas detector [178], based on the reaction3He + n→ p(573 keV) + t(191 keV). (2.3)This detector contains a gas mixture of roughly ∼12 mbar of 3He, ∼26 mbar of CO2, and Arup to a total pressure of 1.1 bar. The gas volume is connected to the UCN guides through a100-µm-thick aluminium foil window, which allows UCN to pass into the gas volume wherethey interact with 3He. The proton and triton ionize the Ar, and the ionization products arecaptured on an anode and cathode to produce a signal. Although the mechanism of the detectoris that of a “proportional” gas detector, the very low energy of the UCNs makes it impossibleto distinguish their energies using this detector.382.4. Experimental CampaignsThe material properties of the foil window are important for modelling and understandingthe behaviour of the detector. The foil is a high-purity aluminium foil, with a calculatedreal Fermi potential of 54.1 neV using NIST data [7]. The imaginary Fermi potential has beendetermined from measurements of the interaction cross-section of 6.2 m/s neutrons, with a valueof 2.81× 10−3 neV [179].2.4.3 Data AcquisitionIn addition to the detector data, a variety of other data was collected during the experimentalcampaigns. Control and data acquisition for the vertical source runs was spread across severalsystems:• Source programmable logic controller (PLC): slow control of the cryostat and con-nected guide systems was handled through a Schneider Electric PLC. This included alldata channels for pressure, flow, and temperature sensing, as well as control of pumps,heaters, and valves.• Beamline PLC: a separate PLC was built to handle control and readout of the BL1Uelements and diagnostics.• Transfer line Labview: transfer of liquid helium into the cryostat was handled by anindependent control system written in Labview.• Keithley picoammeter: The signal from a thermal neutron detector placed close tothe source was read out using a Keithley picoammeter.• CAEN V1720: The signals from the 6Li scintillating detector were read out from aV1720 waveform digitizer.• CAEN V785: The signals from the 3He proportional gas detector were read out from aV785 peak-sensing analog-to-digital converter (ADC).The data from these systems, with the exception of the Labview system, was combined by acentral maximum integrated data acquisition system (MIDAS) control system [180] into singledata files. The MIDAS system also implemented an experiment sequencer that allowed pro-grammable control of the elements required for each measurement. The data collected fromMIDAS is organized into runs, with a new run created each time the sequencer was started.Within a run, the data is further broken down into individual cycles. A cycle consists of asequence of states, such as irradiation of the target, detection of UCNs, etc., which depend onthe details of the measurement being performed.2.4.4 Raw Data ProcessingAll raw data collected during the experimental campaigns underwent the same kind of processingand cleaning prior to analysis. The purpose of this treatment is to reduce background countsand discard bad data.The 6Li detector uses pulse-shape discrimination to determine whether an event is a neutron.The accumulated charge after the initial event trigger is calculated for two windows: QL is thecharge in a long window of 200 ns and QS is the charge in a short window of 40 ns, both given392.4. Experimental Campaignsin ADC units. An event is accepted as a neutron if it satisfies the conditionsQL −QSQL> 0.3,QL > 2000.(2.4)Pulse shape discrimination is necessary for the 6Li detector because it has a high sensitivity togamma background. In contrast, the 3He detector sees a low background, and neutrons can bedistinguished only by the requirement that a sufficient amount of charge is deposited, such thatQ > 300. (2.5)Many cycles were excluded from analysis due to an assortment of issues, including:• The beamline went offline, indicated by the measured beam current dropping below 0.1µA• The beamline displayed instability, indicated by fluctuations in the beam current of morethan 0.02µA• The run was aborted during a cycle, indicated by low counts in the 6Li detector• Failure of a UCN valve to be actuated according to the sequencer, indicated by limitswitches• Spurious counts due to light shining into the 6Li detector, which could be indicated by theion gauge having a reading or by the distribution of counts in the different 6Li segmentsshowing a deviation from the normal behaviour• UCNs remaining in the guides after a cycle, indicated by a high background rate at thebeginning of a cycle• Unsuitably low counts in either detector during a given part of the cycle, for which thecondition varied depending on the type of experimentIn some cases, specific detector events were also excluded. This occurred when a spike in eventswas observed correlated with valve actuation, indicating a signal due to noise. This was onlyobserved in experiments where pressure in the guides was high.Both detectors saw some amount of background not corresponding to UCNs, and so allexperiments had background detector rates subtracted. Those rates were determined by exam-ining the counts during periods which had both the beam off target and the detectors isolatedfrom UCNs. The 3He detector saw a typical background rate of 0.0345±0.0015 s−1 in 2018 and0.0345 ± 0.0022 s−1 in 2019. The 6Li detector saw a typical rate of 2.140 ± 0.013 s−1 in 2018and 1.543± 0.012 s−1 in 2019.40Chapter 3Lifetime of Ultracold Neutrons inHe-II3.1 UCN Production and Storage Lifetime in He-IIThe subject of this chapter is the behaviour of UCNs inside a volume of He-II. The focus willbe on the storage lifetime of UCNs in He-II, which has great relevance for the performance ofHe-II UCN sources such as the TUCAN source. In these sources, since UCNs are created insideHe-II, they will necessarily spend some time inside the volume, and the storage lifetime willdetermine how many UCNs can be extracted.In Chapter 1.3.2, the general theory of how UCNs are produced in He-II sources was de-scribed. This process, referred to as downscattering, occurs through the emission of a quasi-particle, which in the case of He-II is a referred to as a phonon.3 Downscattering is madepossible by the intersection of the He-II dispersion curve and the neutron dispersion curve,shown in Figure 3.1, which allows the conservation of momentum and energy in this process.In the reverse process, referred to as upscattering, a UCN absorbs a phonon from the He-II. The energy and momentum absorbed by the UCN convert it back into the cold neutronregime. These cold neutrons can be expected to escape the source and be lost in some materialinteraction elsewhere, and so upscattering introduces loss channels for UCNs in a He-II volume.These losses contribute to the storage lifetime of UCNs.The theoretical calculation of storage lifetimes due to uspcattering processes was first pub-lished by R. Golub [182], and a somewhat more detailed derivation was later published by C.Jewell [183]. In the remainder of this section the arguments and results of these calculationswill be repeated, and the subsequent attempts to measure the upscattering contributions willbe described.The calculation of upscattering losses is based on the description of the density and velocityof the He-II using second quantization, following the approach of Landau and Khalatnikov [184].This yields an operator for the fluctuating part of the He-II densityn′(r) =1mHe√ρ0~2csV∑~k√|~k|(aˆkei~k·~r + aˆ†ke−i~k·~r), (3.1)where ρ0 is the equilibrium mass density free of any excitations, mHe is the mass of a helium-4atom, cs is the speed of sound in He-II, V is the normalization volume, and aˆ†k and aˆk are thecreation and annihilation operators for a phonon with wave vector ~k. A similar expression can3While the term phonon is most commonly used to describe a quantized acoustic vibration in a solid, it isalso used in the description of similar collective excitations in He-II. These excitations have an approximatelylinear dispersion relation at low momentum, analogous to acoustic vibrations in solids.413.1. UCN Production and Storage Lifetime in He-II-1)Energy/kB (K)Liquid Helium Phonon-Roton DispersionNeutron DispersionFigure 3.1: Plot of the dispersion curves of free neutrons and He-II. The He-II curve was plottedusing data published in Ref. [181], and the small fluctuations in the curve are an artifact ofthat data. Excitations in the low-momentum region below the peak of the He-II dispersioncurve are phonons. Excitations in the high-momentum region around the minimum are rotons.Downscattering of neutrons to UCNs occurs at the intersection of the He-II curve with the freeneutron curve.423.1. UCN Production and Storage Lifetime in He-IIFigure 3.2: Feynman diagrams of the three upscattering processes in He-II relevant for UCNlosses: (a) single phonon absorption, (b) two-phonon scattering, (c) phonon-roton scattering.be written for the velocity, giving~v(r) =√cs~2ρ0V∑~k~k√|~k|(aˆkei~k·~r + aˆ†ke−i~k·~r). (3.2)There are three upscattering processes of interest for UCN losses, which are shown in Figure3.2. Each of these will be described, and their loss rate derived, in turn.The simplest contributing process is the absorption of a single phonon by a UCN. Thepotential for such an interaction isU1−ph =2pi~2asmnn′(r)=2pi~2amnmHe√ρ0~2csV∑k√|~k|(aˆkei~k·~r + aˆ†ke−i~k·~r),(3.3)where as is the scattering length of neutrons from helium, and mn is the neutron mass [184].The single phonon absorption is described by the transition from a state with neutronmomentum ~pi and phonon occupation number n~k1 to a state with neutron momentum ~pf andphonon occupation number n~k1 − 1. Assuming plane wave states for the neutron given by〈pi| = eipi·~r/~/√V and 〈~pf | = ei~pf ·~r/~/√V, the transition matrix element for this scattering is〈f, n~k1 − 1|U1−ph|i, n~k1〉=(2pi)4a~2mnmHeV3/2√ρ0~2c√~k1√n~k1δ(~Q− ~k1) (3.4)where ~Q = (~pf − ~pi)/~ is the neutron wave-vector transfer. The total transition rate, which isthe inverse of the UCN storage lifetime due to this process, can then be calculated by Fermi’sgolden rule, resulting in1τ1−ph=2pi~∫ ∫|〈f, n~k1 − 1|U1−ph|i, n~k1〉|2δ[~cs(|~k1| − | ~Q|2/k∗)]× V d3~k1(2pi)3V d3Q(2pi)3,(3.5)433.1. UCN Production and Storage Lifetime in He-IIwhere k∗ = 2mncs/~ is the intersection of the neutron dispersion curve with the linearizedphonon dispersion curve [185], and the delta function captures the required energy conservationin a single phonon transition. Evaluation of this integral [183] gives1τ1−ph= n0σs~k∗mE∗mHecs2e−E∗/kBT = AHee−E∗/kBT , (3.6)where σs = 4pias2 is the scattering cross-section, and E∗ = ~csk∗ is the phonon energy at thedispersion curve intersection. The exponential temperature dependence arises from the assumeddistribution of the phonon occupation number n~k1 . The phonons should obey Bose-Einsteinstatistics, but if the temperature is small compared to the intersection energy, then this can beapproximated using Boltzmann statistics. The typical operating temperature of a He-II UCNsource is close to 1 K, and the intersection energy has an equivalent temperature around 12 K,so this is not an unreasonable assumption.At the same time as making this derivation, Golub noted that it had some limitations [182].An earlier calculation by Golub and Pendlebury [186] had yielded the result1τ1−ph= n0σs~k∗m2S(k∗)e−E∗/kBT = AHee−E∗/kBT , (3.7)where S(k∗) is the structure factor of the scattering. A comparison of these two results showedthat the single-phonon scattering in measurements of S(k∗) was roughly half of that neededfor the calculations to match. This suggests that the single-phonon calculation is not entirelyaccurate, with these interactions occurring at perhaps no more than half the rate suggested.While single-phonon scattering requires that the neutron wave-vector transfer and thephonon wave-vector are matched, it is also possible to conserve momentum and energy byinvolving multiple interactions. It is not possible for upscattering to occur by a UCN simplyabsorbing one phonon and then emitting another, as the two phonons must fall on the disper-sion curve, and such a process would leave the neutron as a UCN. However, multiple scatteringprocesses involving phonon-phonon interactions can cause upscattering.The phenomenological hydrodynamic potential of He-II makes it possible for phonons to in-teract and produce virtual phonons with energy and momentum off of the dispersion curve [184].These virtual phonons can then be absorbed by a UCN scattering from the U1−ph potential.The dominant multiple phonon process involves the potential for a three-phonon interactionvertexU3mHe=~v · (n′~v)2+16∂∂n(cs2n)n′3, (3.8)where ~v is the velocity of the He-II, and n is the total density including both the equilibriumand fluctuating components. Two relevant types of interaction are possible through this vertex.In the first, one phonon absorbs another, and is then absorbed in turn by the UCN. Since theneutron must still fall on its own dispersion curve, this process can only upscatter neutronsto energies above the peak of the He-II dispersion curve. This process therefore requires high-energy phonons, and so is expected to be highly suppressed at low temperatures. In the secondinteraction, one phonon emits a second, virtual phonon, and the off-curve virtual phonon isthen absorbed by the UCN. This process can occur with low energy phonons, and can thereforecontribute significantly to upscattering. Although this process involves a three phonon interac-tion, it is given the name “two phonon scattering”, as the Feynman diagram, shown in Figure3.2b, has two external phonon edges.443.1. UCN Production and Storage Lifetime in He-IIThe scattering rate calculation proceeds in the same manner as for the single-phonon scat-tering. The transition matrix element to be calculated is〈f, n~k1 − 1, n~k2 + 1, n~k3 |U1−ph|i, n~k1 − 1, n~k2 + 1, n~k3 + 1〉~c(~k1 −Q− ~k2)×〈i, n~k1 − 1, n~k2 + 1, n~k3 + 1|U3|i, n~k1 , n~k2 , n~k3〉,(3.9)which has gained an energy factor in the denominator as it is now a second order scattering.Carrying out the same integration as for the case of single-phonon absorption proceeds in thesame fashion, with the energy conservation condition changed to accommodate the additionalphonon, and an additional integration over the second phonon wave-vector introduced. Thefinal result is1τ2−ph=14pi(Tas~cs)2( Tmncs2)2( TmHecs2)2 T~I = BHeT7, (3.10)where I is a dimensionless integral [182]. The form of this numerical integral is not particularlyenlightening, but it is worthwhile to note that it is not independent of temperature. Jewellreported values of I = 1.07 at T = 1.0 K, and I = 1.01 at T = 1.2 K [183]. Fortunately,this is not a large effect compared to the T 7 dependence, so it is not unreasonable in a smalltemperature range to assume a constant value of BHe.A similar two-interaction process, shown in Figure 3.2c can also occur involving the highermomentum roton excitations. In this process, a roton scatters from a UCN by the exchange ofan off-curve phonon. The phonon-roton interaction potential is given byU r−ph = −12(~pr · ~v + ~v · ~pr), (3.11)where ~pr is the momentum operator for the rotons. A similar calculation as for two-phononscattering yields [183]1τr-ph=32√pimr2mmHe(|~k0|as2)~|~k0|mncsTmHecs2√2mrcs2T~e−ER/kBT=CHeT3/2e−ER/kBT ,(3.12)where mr is the effective roton mass, |~k0| is the wave-vector magnitude at the roton minimumof the dispersion curve, and ER is the energy at the roton minimum.These three processes combine to give the storage lifetime of UCN in He-II as1τHe−II=1τ1−ph+1τ2−ph+1τr-ph= Ae−E∗/kBT +BT 7 + CT 3/2e−ER/kBT . (3.13)We have that E∗/kB = 12 K and ER/kB = 8.6 K. Golub calculated, at 1 K, A = 130 s−1[187], B = 7.6 × 10−3 s−1K−7, and C = 18 s−1K−3/2 [182]. Therefore, at 1 K, τ1−ph = 1252 s,τ2−ph = 132 s, and τr-ph = 301 s. This tells us that the upscattering in He-II at typical UCNsource temperatures should mostly be dominated by the two phonon interaction, although thesingle phonon and roton-phonon contributions are not insignificant.There have since been several attempts to measure τup and extract the temperature depen-dence. The results of those attempts are summarized in Table 3.1. Contrary to the theoretical453.2. Experiment GeometryYear Temp. Range (K) A s−1 B s−1K−7 C s−1K−3/2 Ref.1983 0.75–1.15 0 0.0076 0 [137]1992 0.45–1.40 0 0.008 0 [158]2014 0.7–1.5 0 0.0085–0.0096 0 [188]2016 1.1–1.95 < 61 0.004–0.016 < 5.3 [189]2017 0.85–1.5 0 0.008–0.016 0 [3]Table 3.1: Summary of past experimental measurements of He-II upscattering coefficients.prediction, these measurements have found that the storage lifetime can be described well withonly the two phonon scattering term. Furthermore, as these measurements have evolved, theyhave increasingly indicated that the value of BHe could be higher than predicted.The 2017 measurement shown in Table 3.1 was performed at TRIUMF using the verticalsource. This measurement was limited by a large uncertainty on the temperature of the He-II. At this time, high-precision Cernox temperature sensors were installed in the cryostat incontact with the source volume and heat exchanger components. However, these sensors weredetermined to be inaccurate, as they were not directly in the source bottle and were thereforesubject to uncharacterized offsets due to thermal conductivity. It was decided that the temper-ature could be better determined using a vapour pressure measurement, but during the 2017run only a Baratron with a range of 0–1000 torr was installed to measure the pressure inside thesource volume. Since the vapour pressure of 4He is 0.12 torr at 1 K, the resolution provided bythis sensor (nominally 0.1 torr) was not sufficient for a precise measurement at low temperature.Prior to the 2018 experimental run, a new Edwards Barocel 600AB with a 0–2 torr range wasinstalled, with the goal of enabling an improved measurement of the temperature of the He-II.The remainder of this chapter discusses measurements of the vertical source storage lifetimeand steady-state production rate, with a focus on the temperature dependence of these results.3.2 Experiment GeometryThe configuration of the vertical source and experimental area during these experiments isshown in Figure 3.3. This schematic shows only the internal components of the geometry thatare visible to UCNs. The materials used in the different parts of the geometry are described inSection 3.3.3.UCNs are created in the source bottle, a nickel-coated vessel which is filled with He-II toa nominal level of 618 mm. The UCNs can exit the He-II and source by a vertical stainlesssteel guide tube. The exit from the bottle is through a narrow neck of 49 mm diameter, whilethe rest of the guide tubes are 85 mm diameter. From the fluid level, the UCNs must travel658 mm vertically to reach the a horizontal guide tube section. This horizontal section is2945 mm long and terminates at a UCN isolation valve referred to as isolation valve 1 (IV1),which is a commercial valve manufactured by VAT (17240-CE44 with expanding protectionring). This volume, from the source bottle to IV1, is referred to as the source volume, orsimply the source. During all experiments, the source volume is filled with helium vapour,with pressure ranging from 0.16 to 16.39 mbar depending on the temperature of the source.Temperature measurements along the length of the guides indicate that this vapour has aroughly linear temperature gradient, from the temperature of the He-II at the fluid level up toroom temperature at the valve.The Edwards Barocel 600AB, along with other pressure gauges, was connected to the source463.2.ExperimentGeometry1270HH2945 97 3279 34457322821026181117658 He-II level3He detector6Li detectorSource BottleUCN Valve IV1UCN Valve IV2Burst disk and pressure gauge connectionHeat exchanger connectionFigure 3.3: Schematic of the geometry of the vertical source and experimental configuration for experiments relating to the behaviour ofUCNs in He-II. The top view shows the layout from above, and the bottom view shows the layout from the side. All dimensions given inmm.473.3. Storage Lifetime Measurementvolume through a small tube located at a burst disk on the guides just upstream of IV1. At thislocation, the gauges should operate at room temperature. The Cernox temperature sensors wereinstalled in the connection between the heat exchanger and the source bottle, located belowthe bottle neck. To reduce UCN losses into the heat exchanger, the flow of He-II between thebottle and the heat exchanger is made through narrow slits. This is believed to have been amajor contribution to the poor performance of the sensors.After IV1, UCNs must be extracted through several meters of concrete and steel radiationshielding to the experimental area. The experimental area is offset from the source, and sothe extraction guides are angled as shown in Figure 3.3. Two 45◦ angled guides, separated by1270 mm of straight guide tube, account for this offset. After the second angled guide, there is3279 mm of straight guide leading up to a second UCN isolation valve, named isolation valve2 (IV2). The radiation shielding terminates approximately 750 mm upstream of IV2. Theguide volume downstream of IV1 is also filled with helium vapour during experiments. Dueto pumping on the guide volumes downstream of IV1, the pressure of this vapour fluctuatesduring measurements and is not precisely known. However, these guide sections are known tohave been at room temperature and so the vapour density is not a major consideration in themeasurements.Located between the end of the radiation shielding and IV2 is a T-shaped guide section.The vertical section of this guide is connected to the horizontal section by a “pinhole” of 15 mmdiameter. This allows a fraction of UCNs to leak out of the horizontal guide section. Thevertical guide section of 573 mm connects to the 3He detector, which is used as a secondarymonitor detector.After IV2, there is a 90◦ bend that leads down to the main 6Li detector. From the centreof the guide at the horizontal level to the detector surface there is a 1117 mm vertical drop.In total, a UCN created at the bottom of the source bottle needs to travel roughly 10.5 mto reach the 6Li detector. In practice, path lengths may be much longer due to reflections.3.3 Storage Lifetime Measurement3.3.1 Experiment DescriptionThis section describes a direct measurement of the storage lifetime of UCNs in the He-II in-side the source volume. By varying the temperature of the He-II between 1.0 and 1.8 K atemperature-dependent measurement of the storage lifetime was obtained. Analysis of thisdata allows the temperature-dependent component of the storage lifetime to be extracted andcompared to the theoretical model and past experiments.The measurement of the lifetime was a three stage process. In the first stage, the targetwas irradiated at a nominal beam current of 1 µA over 60 s to accumulate UCNs in the source,during which time IV1 remained closed. After this production stage, the beam was turned offand IV1 remained closed, so that the UCNs were kept in the source for a variable storage timet. After the storage time had elapsed, IV1 was opened and the UCNs were emptied into thedetectors for counting over 100 s. This process was repeated for eight different storage times inthe sequence (0, 60, 2, 40, 4, 25, 8, 15) s to produce a curve of detected UCN vs storage time.The temperature was controlled by adjusting the cooling power of the heat exchanger tothe He-II. This was done by changing the temperature of the 3He, which was achieved bythrottling the pumping of the vapour using a proportional flow valve in the pumping system.Before proceeding with each storage measurement, the temperatures of the He-II volume weremonitored until they appeared stable. This stabilization time varied, and was likely not perfect,483.3. Storage Lifetime Measurementbut additional changes in the temperature of the He-II during the measurements are capturedas uncertainty in the analysis.3.3.2 Analysis MethodsThe UCN count data was processed to account for background and beam fluctuations as de-scribed in Section 2.4.4. The pre-processed counts Npp(t) (for a storage time t) were thennormalized to the average beam current I¯beam during the irradiation to account for variationsin the beam between different storage times,N(t) =Npp(t)I¯beam(t). (3.14)The beam current was determined based on the extracted current into BL1 and the kickerfraction into BL1U. The extracted current is determined directly from the electron capture onthe stripper foils in the main cyclotron [166], while the kicker fraction is a setting of the kickermagnet. This method is considered to provide a very accurate and precise determination ofthe current, and no additional uncertainties have been considered in this analysis other thanthe fluctuations of the beam. The validity of the normalization depends on the linearity of therelationship between counts and this current, and this was established in the UCN measurementsdescribed in Ref. [3]. The entire beam is assumed to be incident on the target, as this is how thebeamline is designed and tuned, but deviations from this would only be important if differentbeam currents resulted in different fractions of the beam on the target. The standard deviationof the beam current was combined with that of the counts to obtain the uncertainty on thenormalized countsσ(N(t)) = N(t)√(σ(Npp(t))Npp(t))2+(σ(Ibeam(t))I¯beam(t))2. (3.15)To extract the storage lifetime of UCN in the source, the normalized detected counts N(t)were fit to a single exponential function to obtain a set of temperature-dependent storagelifetimes τs(T ). A single exponential is an approximation in describing the storage lifetimeof a UCN population. Since loss rates are typically energy-dependent, a single exponential isan approximation from the more accurate sum of exponential functions (one for each energypopulation). In practice, the long vertical extraction of the vertical source results in a relativelynarrow energy spectrum for detected UCNs, and it was observed that the storage-lifetimedata tended to fit well to a single exponential. Additionally, the simulation and experimentalresults undergo the same fitting, which should help to mitigate any deviation from the singleexponential. The experimental results and fitted curves for the eight temperatures for whichthe storage lifetime were measured are shown in Table 3.2.The temperature is determined from the vapour pressure measurement using an empir-ical equation for the vapour pressure of liquid helium established by Clement, Logan, andGaffney [190], which has a reported accuracy of σCLG(T ) = 2 mK. An uncertainty must beapplied to the temperatures determined in this fashion, accounting not only for the accuracy ofthe equation, but also for the accuracy of the pressure measurement, and the variation of thetemperature during a measurement of the storage lifetime.The Edwards Barocel 600AB has a nominal accuracy of 0.15% in its pressure determination.The corresponding uncertainty on the temperature measurement is determined with a MonteCarlo approach. For a given measured pressure P , the accuracy 0.15P/100 is taken as the493.3. Storage Lifetime MeasurementT (K) σ(T ) (K) τs,6Li (s) σ(τs,6Li) (s) τs,3He (s) σ(τs,3He) (s)1.805 0.005 4.88 0.22 6.46 2.341.786 0.011 5.40 0.20 3.36 0.961.661 0.004 7.42 0.14 7.29 1.051.652 0.012 7.34 0.16 6.37 0.991.377 0.038 14.40 0.12 13.91 0.831.231 0.003 21.57 0.14 20.44 0.971.217 0.006 21.83 0.14 21.44 0.981.037 0.003 30.01 0.13 31.27 1.09Table 3.2: Measured storage lifetimes and uncertainties for the temperature-dependent storagelifetime experiment. Fitted lifetimes are given for both the 3He and 6Li detectors. The 3Heresults have significantly higher uncertainty due to lower counts, which is a result of the detectorbeing connected to the guide volume only through a small hole.standard deviation of a Gaussian distribution with mean P . A sample population of 100 samplesis generated from this distribution and the temperature is determined for each sample. Thestandard deviation of the resulting temperature distribution can be taken as the uncertaintyσP (T ) due to the pressure measurement.During a measurement of the storage lifetime there was some variation in the pressure. Tomitigate the potential for such a drift to contribute a systematic effect, the order of the storagetimes was shuffled to avoid measuring the shortest storage time first and the longest storagetime last (or vice versa). To simplify the analysis, each storage lifetime measurement is treatedas occurring at the average temperature over the range of storage times. The temperaturevariation is accounted for by taking half the range of the temperatures as an uncertainty, suchthatσVar.(T ) =Tmax − Tmin2. (3.16)The Edwards Barocel 600AB has a response time constant of 8 ms. Temperature fluctuationsfaster than this may therefore not be captured by this uncertainty. Since this time constant ismuch shorter than the measurement times, such fluctuations would contribute to the averagetemperature at significantly less than the 1% level.The three uncertainties described can finally be combined to obtain the total uncertaintyof the temperatureσ(T ) =√σP (T )2 + σVar.(T )2 + σCLG(T )2. (3.17)The analysis of the τs vs t data is based around a Monte Carlo simulation of the experiment,developed using the PENTrack software package for simulations of UCNs [156]. PENTrackallows the full geometry of the experiment to be imported from CAD models (built usingSolidworks), so as to capture all geometric effects. The guide geometry shown in Figure 3.3is in fact a drawing of the model used for this simulation. PENTrack implements a completemodel of the UCN trajectories through this geometry, accounting for all loss channels throughthe definition of material parameters. The details of this model will be described in the Section3.3.3.To analyze the temperature dependence, certain model parameters will be taken as freeparameters. These parameters are adjusted in the simulation, with the best replication of themeasured lifetime results over the temperature range indicating the best fit of the temperature-dependence. In this analysis, the goodness-of-fit is determined by a modified χ2, combining both503.3. Storage Lifetime Measurementthe experimental and simulation uncertainty and summing over the temperatures to obtainχ2 =∑T(τs,exp − τs,sim)2σ(τs,exp)2 + σ(τs,sim)2, (3.18)where the subscript “exp” denotes the results of the experimental measurements, and “sim”denotes the results of the simulations.The experimental uncertainty must include the uncertainty in the temperature measure-ment. This is combined with the fit uncertainty of the storage lifetime σ(τs,exp,fit) througherror propagation to obtain the total experimental uncertainty. This calculation is slightlycomplicated by the fact that the lifetimes add as τs−1. In general, for x = 1/y, the propagateduncertainty is σ(x) = σ(y)/y2, so thatσ(τs) =1(1/τs)2σ1/τs = τs2σ1/τs . (3.19)The uncertainty in τs−1 isστs−1 =√σfit(τs−1)2 + σT (τs−1)2=√σfit(τs,exp)2τs4,exp+(∂(τHe−II−1)∂T)2σ(T )2=(σfit(τs,exp)2τs4,exp+(AHeE∗e−E∗/kBTkBT 2+ 7BHeT6+ CHee−ER/kBT(32T 1/2 +ERkBT−1/2))2σ(T )2)1/2.(3.20)Combining Equations 3.19 and 3.20 gives the definition of σ(τ). This approach to treating thetemperature uncertainty is model-dependent, so that the temperature uncertainty contributesmore or less depending on the value of the parameters AHe, BHe, and CHe. There is some logicto this: if the temperature dependence were very weak, then changes in temperature should notcontribute significantly to the uncertainty in the lifetime. A model-independent approach couldbe achieved by numerically differentiating the experimental data with respect to temperatureto obtain(∂(τHe−II−1)∂T), but, given the strong temperature-dependence, this would require high-resolution data to be accurate, so such an approach has not been used here.3.3.3 Simulation ModelThe accuracy of the analysis described in the previous section will depend on whether thesimulation model can capture all effects that impact the measured storage lifetime. This includesboth the contributions to the storage lifetime in the source, and extraneous effects that canmodify the measured lifetime away from the true lifetime.The storage lifetime τs is a combination of several loss rates: the true lifetime in He-II,τHe−II−1(T ), scattering on helium vapor above the liquid surface, τvap−1(T ), absorption orupscattering at the walls of the source volume, τw−1, and beta decay τβ−1:τs−1 = fHe−IIτHe−II−1 + (1− fHe−II)τvap−1 + τw−1 + τβ−1. (3.21)Since only a small part of the volume of the source is filled with He-II, UCNs only spenda fraction of their storage time in the He-II, with the rest of it being spent in the helium513.3. Storage Lifetime MeasurementMaterial V (neV) W (neV)Source Bottle (NiP) 213 0.1–0.14Source Guides (SS) 183 0.1–0.14He-II 18.8 BT 7, B = 0.007–0.016He Vapour 0 ∼ nHe,v(T, P )√TExtraction Guides (SS) 183 0.10–0.206Li Depleted Glass 83.1 1.237× 10−46Li Enriched Glass 103.4 1.237Aluminium Foil 54.1 2.81× 10−3Table 3.3: Summary of material properties used in the PENTrack simulation. The values andranges are explained in the main text.vapour. To account for this, the contributions of the He-II and helium vapor are modifiedwith a factor fHe−II, corresponding to the fraction of time the UCNs spend in the superfluidhelium while confined in the source. This factor introduces significant difficulty in analyzingthis experiment, as fHe−II cannot be readily determined experimentally. It is also not a simpleconstant, as fHe−II evolves with storage time as the distribution approaches an equilibriumcondition, shown in Figure 3.4. Furthermore, fHe−II can differ for different source conditions,especially temperature, as higher temperatures greatly increase the probability of upscatteringin the He-II, such that UCNs that are detected at the highest temperatures must spend relativelylittle time inside the He-II in order to survive. This is illustrated in Figure 3.5.While the combination of these four loss channels together describes the storage lifetimeof UCNs in the source, this is not necessarily the same as the measured storage lifetime. Asmentioned previously, the single storage lifetime is actually an average over the different storagelifetimes of different energy populations. This means that any energy-dependent effect couldchange the weighting of this average and modify the measured storage lifetime. This couldoccur due to the extraction of the UCNs out of the source, or in the detectors themselves.All of these effects must be considered and modelled in the analysis. This is most easilydone in a Monte Carlo simulation rather than attempting to construct an analytical model,which justified the use of PENTrack in analyzing these measurements. We will describe howeach effect is treated in turn, and summarize the material properties used in this modelling inTable 3.3.Source Wall LossesThe interaction of UCNs with materials on reflection was described in Section 1.3. Theseinteractions depend on the real part of the Fermi potential V and the imaginary part W . Thereal part of the Fermi potential, V , has been measured for many materials and is generallyconsistent with calculations using Equation 1.30, with minimal variation between materials ofthe same expected composition. The imaginary part, W , tends to be less well-determined.This can be due to slight variations in material composition, such as impurities with highneutron absorption cross sections; the cleanliness of a surface, as organic contaminants containlarge amounts of neutron-absorbing hydrogen; or even just the amount of water adsorbed ontoa surface during a particular experiment. The result is that loss rates from wall collisionscan vary significantly even when using materials that have been manufactured to have similarcompositions. A practical method for treating wall losses is to consider W as a free parameter,which must also be fit to experimental data for any material used.523.3. Storage Lifetime Measurement0 0.2 0.4 0.6 0.8Fraction of time spent in He-II00.010.020.030.040.050.060.07Probabilityt = 2 st = 15 st = 60 st = 120 sFigure 3.4: Histogram showing the distribution of fHe−II, the fraction of time spent by UCNsin the He-II volume, for different storage times in the source. These results are taken fromsimulations with AHe = CHe = 0, BHe = 0.007 s−1K−7, Ws = 0.10 neV, and only UCNs that aredetected in the experiment are included. The evolution of fHe−II over time is clearly visible.0 0.2 0.4 0.6 0.8Fraction of time spent in He-II00.050.10.150.2ProbabilityB = 0.007, Ws = 0.010, T = 1.03 KB = 0.007, Ws = 0.014, T = 1.03 KB = 0.016, Ws = 0.010, T = 1.03 KB = 0.007, Ws = 0.010, T = 1.80 KFigure 3.5: Histogram showing the distribution of fHe−II, the fraction of time spent by UCNsin the He-II volume, for different conditions during storage in the source. These results are fora storage time of 15 s, and only UCNs that are detected in the experiment are shown. Smallshifts are seen associated with BHe and Ws, while a large difference is visible between low andhigh temperatures.Wall losses inside the source volume are a primary contribution to the measured storage life-time. This volume is made of two materials. The source bottle is coated with electroless nickel-phosphorus (NiP) (applied by an electroless plating process), while the guides are uncoatedstainless steel (SS). These two materials have different Fermi potentials, with V NiP = 213 neVand V SS = 183 neV. The most correct treatment would be to assign an imaginary Fermi poten-533.3. Storage Lifetime Measurementtial to each of these materials. However, it is not clear whether the experiment has sufficientsensitivity to distinguish the contributions of the two materials. Furthermore, having separatevalues increases the dimension of the parameter space to be studied, which greatly increasesthe computation time required to run such simulation studies. Therefore, a single averageimaginary Fermi potential value Ws is used inside the source, applied to both materials.This treatment could create some bias, as the loss rate on reflection is proportional to W/V ,and so this gives the NiP surface a 1 − V SS/V NiP = 14% lower loss probability. However, itwould be equally biased to treat the two materials as having the same loss rate, as there is noguarantee that this is true either. There is no natural assumption that can be made on thedistribution of W values in the source. Therefore this assumption is as good as any, but thislimitation in the treatment of the surface losses must be acknowledged.All of the surfaces inside the source are polished by a combination of abrasive polishing andelectropolishing to reduce surface roughness [191], and a value of PL = 3% is assumed for theLambert diffuse reflection. This is based on previous measurements [3], as well as additionaltransmission measurements that were performed at TRIUMF during these same experimentalcampaigns. When the source was studied at RCNP, a higher value of 6% was used [191]. Thisdiscrepancy could be due to differences in simulation modelling between these studies. Theeffect of this parameter is not explored here, as it has generally been observed that it does nothave a significant effect for storage lifetime experiments.He-II UpscatteringIn the simulation, the He-II is modelled as a material volume occupying the same space as inthe experiment, up to the nominal level shown in Figure 3.3. This level may have differed inthe actual experiment, and will be discussed further in the results. This material is assigned acomplex Fermi potential, with the real part given by the expected theoretical potential V He =18.8 neV, and the imaginary part given byWHe−II(T ) =12~τHe−II−1(T ). (3.22)The analysis will largely focus on the two-phonon scattering term, following on the results ofprevious measurements which indicated that this was sufficient to model the He-II upscattering.Ignoring one-phonon and phonon-roton contributions, the losses in He-II are then describedentirely by τHe−II−1 = BHeT 7, where BHe will be varied to determine the best fit. The valuesof WHe−II(T ) are calculated using this definition. The entire He-II volume is assumed to be atthe same temperature during a given measurement (as given by the vapour pressure).Vapour LossesThe losses in vapour can be described in a similar way to those in He-II, using a material volumewith appropriate V and W [192]. In this case the imaginary Fermi potential isWHe,v = ~(nHe,v(T , P )σs,He,f√2kBTpimHe), (3.23)where nHe,v(T , P ) is the number density of the helium vapour at temperature T and vapourpressure P , kB is the Boltzmann constant, mHe is the helium atomic mass, and σs,He,f is thefree scattering cross section, which is related to the bound scattering cross section byσs,He,f =σs,He,b(1 +mn/mHe)2. (3.24)543.3. Storage Lifetime MeasurementHe-II fill surface(1-2 K)He vapour volumes10 cm15 K20 K30 K45 K60 K75 K90 KIsolation valve 1Room temperature vapour fill volume (290 K)He-II fill volume105 K120 K130 K135 K140 K145 K150 K155 K160 K165 K170 K175 K180 K185 K190 K195 K200K205 K210 K215 K220 K230 K240 K250 K260 K270 K290 KFigure 3.6: Picture of the Solidworks model of the vertical source showing the helium vapourfilled up to IV1. Temperatures are indicated on the faces of each of the vapour volumes, with thetemperature at the He-II fill level being the variable temperature of the He-II. While the He-IItemperature was determined using a vapour pressure measurement, the vapour temperaturesare based on temperature sensor readings on the outside of the guides and an assumed lineargradient between those measured values.With σs,He,b = 1.34 b [193], the free scattering cross section is σs,He,f = 0.855 b. The vapourlosses depend on both the temperature of the He-II, which determines the vapour pressure,and the temperature of the vapour throughout the length of the guides. The vapour in thesimulation is modelled as a series of volumes along the length of the guides, shown in Figure3.6. Each volume is 10 cm long, starting at the surface of the He-II and extending throughthe guides up to a point just before IV1, where the guides transition out of the cryogenicregion. After this region the remainder of the volume, out to the detectors, is treated as filledwith a vapour at a room temperature of 290 K. The temperatures between the He-II and theroom temperature volume are shown in Figure 3.6, and were estimated based on temperaturesensors along the guides. Each vapour volume is assigned a potential averaged between the twotemperatures of its faces, given by WHe,v = (WHe,v|T=Tmin +WHe,v|T=Tmax)/2.The values of WHe,v range from 2× 10−10–1× 10−7 neV, with the larger values at the largerHe-II temperatures and in the densest part of the vapour (nearest to the He-II). While smallrelative to the other loss channels, this does play a noticeable role in modelling the storagelifetime, with the greatest effect at the highest source temperatures.After this analysis was completed, it was noted that the value used for σs,He,b = 1.34 bwas not consistent with recent measurements. A measurement from 2019 reported σs,He,b =1.188±0.005 b [194], while a measurement from 2020 reported σs,He,b = 1.206±0.002 b [195]. Toaccount for this discrepancy an additional uncertainty is introduced to the measured lifetimes,given byστ,vap. = τexp −(1τexp− 1τ¯1.34+1τ¯1.19)−1, (3.25)where τ¯1.34 and τ¯1.19 are the average vapour lifetimes for σs,He,b = 1.34 b and σs,He,b = 1.19 brespectively. These averages were calculated over the temperature gradient in the vapour and are553.3. Storage Lifetime Measurementnot weighted by the time fraction that the UCNs spend in the vapour, so this uncertainty is likelylarger than needed. This should therefore appropriately capture any systematic temperature-dependent shifts that could be present between our simulations and the experimental resultsdue to the discrepancy in the scattering cross section.Beta DecayThe beta decay of neutrons is described as an exponential lifetime, which was taken to be880.1 s. This lifetime is slightly mismatched from the current literature average of 878.4 ±0.5 s [127], which has been updated since the neutron lifetime value was first programmedinto the PENTrack model. This can be corrected, given that the beta decay lifetime has notemperature or geometric dependence, but this correction will change the total storage lifetimeby 2 ms at most, so it is not significant on the level of our storage lifetime measurements.Transport LossesTo reach the detectors, the UCNs must diffuse down the 6 m-long stainless steel (SS) guidelength after IV1, during which time they will experience further losses on the guide walls. Theimaginary Fermi potential Wg for these extraction guides could be different than the value Wsinside the source, and so may have different energy-dependent loss rates. These transport lossescould therefore have some impact on the measured storage lifetime, if the energy spectrumis significantly different between the longest and shortest storage times. From other storagelifetime experiments performed on this guide volume, it is believed that Wg of the guidesbetween IV1 and the detectors is in the range of 0.10–0.20 neV, with the most likely value being0.15 neV. The relatively large range for this parameter is to account for a change in the guideconfiguration between the measurements of the guide storage lifetime and these measurementsof the storage lifetime in the source. As with the source, a value of PL = 3% is assumed basedon prior measurements [3].There is an additional source of losses during transport from the two valves IV1 and IV2.When purchased, it was believed that all of the components of these valves that were visibleto UCNs were made of SS, and could be treated as having the same properties as the guidesthemselves. Later, this was found to be incorrect. A small part of the valves was plated insilver, likely to prevent galling between moving SS parts. This area of silver plating, indicated inFigure 3.7, is visible to UCNs when the valves are open. This surface in both valves is modelledas pure silver, with theoretical parameter values of V Ag = 90.45 neV and WAg = 0.290 neV.Although the surface area of this part is small, the significantly lower V and large W comparedto SS could lead to a significant effect in the modelling.Detector LossesThe two detectors both have windows with significant Fermi potentials that the UCNs mustpass through before being detected. The 3He detector has a 30 µm aluminium foil, with V Al =54.1 neV and WAl = 2.81× 10−3 neV [179]. The 6Li detector has a 60 µm 6Li-depleted (GS30)layer, with V GS20 = 83.1 neV, and UCN must also penetrate into the6Li-enriched layer (GS20),with V GS30 = 103.4 neV. The glasses have WGS20 = 1.237 neV and WGS20 = 1.237× 10−4 neV.All of these values were taken from Ref. [177].The detectors are located at the bottom of vertical drops, shown in Figure 3.3, which increasethe energy of the UCNs such that they are expected to be above the energy of the potentialbarriers V . However, if these drops are not sufficient, then the windows could block UCNs below563.3. Storage Lifetime MeasurementValve opening gap(1.66 mm width)Stainless steel valve adapter guide (⌀ 85 mm)Stainless steel valve opening ring segment(19.02 mm width)Silver-plated valve opening ring segment(1.78 mm width)Stainless steel valve adapter guide (⌀ 85 mm)Figure 3.7: Cut-through picture of the Solidworks model of IV2, indicating the internal surfacesand geometry in the open position. Most of the exposed surface of the valve is SS, but there is asmall silver-coated section. Additionally, a small gap between the SS and silver-plated sectionsis visible.the energy threshold from being detected, which could change the measured storage lifetimeif the energy spectrum is different for the different storage times. This insufficiency couldoccur due to non-specular reflections on the guide walls during the vertical drop, which couldchange the component of the kinetic energy normal to the windows. Furthermore, as with theextraction guides, the losses from the imaginary Fermi potentials in the windows could changethe lifetime.3.3.4 ResultsThe goal of this analysis is to examine the temperature dependence of τHe−II. Thus, the firstquestion to ask is whether the measured τs vs T data fits to a model using the calculatedtheoretical values for AHe, BHe, and CHe. Past experiments have indicated that the answer tothis question is no, but this analysis is repeated here for completeness.Figure 3.8 plots τs vs T , comparing experiment to models with A = 130 s−1 , B = 7.6×10−3s−1K−7, and C = 18 s−1K−3/2, with variable values of Ws. The residuals between simulationand experiment show clearly that the theoretical model does not replicate the structure ofthe temperature-dependence seen in the experiments. Given this result, the remainder of theanalysis will follow the approach of previous publications and attempt to model the lifetimeusing only BHe.To find a best-fit model for the experimental data, the parameters (BHe,Ws) were varied ina grid search and the value of χ2 for each parameter set was calculated. The minimum χ2 valuegives the best fit, and the difference ∆χ2 gives confidence intervals. Since this study has twoparameters, the confidence interval for (68.3%, 90%, 95.4%, and 99%) are respectively given by∆χ2 = (2.30, 4.61, 6.17, 9.21) [196]. This study was repeated for three different values of Wg, at0.10, 0.15, and 0.20 neV, to check for a dependence on the losses in the extraction guides.573.3. Storage Lifetime Measurement1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.951015202530Storage lifetime s (s)ExperimentWs = 0.080 neVWs = 0.090 neVWs = 0.100 neVWs = 0.110 neVWs = 0.120 neV1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9Temperature (K)-4-20246Residual s (s)Figure 3.8: Plot of experimental storage lifetime results vs. temperature, alongside simulatedresults using the theoretical He-II upscattering model (with parameters A = 130 s−1 , B =7.6×10−3 s−1K−7, and C = 18 s−1K−3/2), with different values of the imaginary Fermi potentialWs in the source volume. The lower plot shows the residuals between experiment values andsimulation values, and it can be seen that the theoretical model does not reproduce the structureof the experimental data.583.3.StorageLifetimeMeasurement0.009 0.01 0.011 0.012 0.013 0.014 B (s-1K-7)0.080.090.10.110.12 W (neV)34567892(a) Wg = 0.10 neV0.009 0.01 0.011 0.012 0.013 0.014 B (s-1K-7)0.080.090.10.110.12 W (neV)34567892(b) Wg = 0.15 neV0.009 0.01 0.011 0.012 0.013 0.014 B (s-1K-7)0.080.090.10.110.12 W (neV)34567892(c) Wg = 0.20 neVFigure 3.9: Heatmap plots showing the value of ∆χ2 for different simulation values of (BHe,W ) compared to the experimental storagelifetime results. Each of the three plots is for a set of simulations with a different value of the imaginary Fermi potential Wg for the guidesoutside the source. The upper end of the colour scale corresponds to the 99% confidence level, while the lower end corresponds to the68% confidence level. White squares are parameter values that were not simulated as they appeared to be far outside of the parameterrange. No significant difference in the accepted range of BHe is visible between the three values of Wg. The resulting 95% confidenceinterval for BHe, taken across all three plots, is 0.010 ≤ BHe ≤ 0.014 s−1K−7.593.3. Storage Lifetime MeasurementThe results of the analysis, fitting only to lifetimes as measured by the 6Li detector, areshown in Figure 3.9. The results of the 3He detector will be discussed momentarily. All threeWg values give similar results, but with different confidence intervals. The tightest confidenceinterval is produced by Wg = 0.15 neV, for which only the value of BHe = 0.011 s−1K−7 isaccepted at the 68% level. This expands significantly for the other values of Wg. The 95%confidence interval will be considered for the final results. For Wg = 0.15 neV the range is0.01 ≤ BHe ≤ 0.013 s−1K−7, for Wg = 0.10 neV, the range is 0.01 ≤ BHe ≤ 0.0135 s−1K−7,and for Wg = 0.20 neV, the range is 0.105 ≤ BHe ≤ 0.0140 s−1K−7. There may be some smallsensitivity to the value of Wg, and these three ranges can be combined to obtain a final rangefor BHe.The lifetimes as measured by the 3He detector were excluded from the above results. Figure3.10 shows the ∆χ2 values for Wg = 0.15 neV when considering only the3He detector. Thereare two major observations from this analysis.First, the 3He detector produces a larger range of accepted values than the 6Li detector.This is to be expected, as the 3He detector was connected to the guides through a small openingand saw significantly fewer counts. This leads to a higher uncertainty in the lifetimes obtainedfrom this detector, and so these results do not constrain the parameter values as strongly asthe 6Li detector. Therefore, including the 3He results does not improve the analysis.However, the second observation is that the map of ∆χ2 contains a lot of structure for the3He detector, with poorly fitting parameter pairs interspersed among the best fitting values.This introduces a problem when combining the data from the two detectors, as the noise in the3He analysis tends to constrain the 6Li analysis to a narrower range of BHe than the6Li dataon its own. For this reason, the 3He data is excluded in the determination of the parameterrange.It is currently not clear why this noise appears. It is possible that it is related to the combineduncertainty of the simulations and experiments. Ideally, the simulation uncertainty would benegligible compared to the experimental uncertainty. However, this would have taken a lotmore simulation time to achieve, and instead sufficient simulations were run for the uncertaintyof the simulated 6Li results to be below the uncertainty of the corresponding experimentalresults, with most simulated lifetimes having uncertainty on the order of 30% of that of theexperimental results. In the case of the 6Li analysis this appears to produce reasonable results.However, for the simulated 3He results there are many cases where the uncertainties are larger— between 50% and 80% of the experimental uncertainties. These larger uncertainties may beleading to some sort of noise in the analysis, in which the simulation results fluctuate far awayfrom, or very close to, the experimental results. Investigating this would require significantlyincreasing the simulation statistics. Since the uncertainty scales as√N , and simulation timescales linearly in N , this would require a significant increase in total computing time (potentiallyhundreds of additional CPU-years when running on current DRAC cluster hardware). Due tothe required computation time, these studies have not yet been pursued, so this explanationis only speculative. However, when cutting the 6Li analysis to include only 1/10th of thesimulation statistics a similar behaviour was observed, which supports the idea that this isconnected to simulation statistics.From this analysis, the 95% confidence interval for BHe is 0.010 ≤ BHe ≤ 0.014 s−1K−7. Thisresult is consistent with the previous published result of Ref. [189], but is somewhat narrower.It is observed than in modelling this data, when assuming AHe = CHe = 0, a higher BHe valueis required compared to the theoretical BHe = 0.0076 s−1K−7.Table 3.4 lists, for each BHe value, the minimum value of χ2/ndof , alongside the correspond-ing Ws value. Here ndof = 6 is the number of degrees of freedom in the analysis of the6Li603.3. Storage Lifetime Measurement0.009 0.01 0.011 0.012 0.013 0.014 B (s-1K-7)0.080.090.10.110.12 W (neV)34567892Figure 3.10: Heatmap plot of ∆χ2 for different simulated values of (BHe,W ) for the3He detector.The imaginary Fermi potential for the guides outside the source Wg = 0.15 neV. The range ofparameter values that could fit this data does overlap with that for the 6Li data, but there issignificant noise in the analysis.613.4. Production Rate in He-IIWg=0.10 neV Wg=0.15 neV Wg=0.20 neVBHe (s−1K−7) W (neV) χ2/ndof W (neV) χ2/ndof W (neV) χ2/ndof0.0085 0.115 3.60 0.115 4.62 0.120 4.670.0090 0.115 3.41 0.115 2.80 0.115 2.600.0095 0.110 2.14 0.110 2.21 0.115 2.860.0100 0.110 1.00 0.110 1.10 0.110 1.940.0105 0.110 1.08 0.110 1.26 0.110 1.010.0110 0.105 0.74 0.105 0.58 0.105 0.710.0115 0.105 0.98 0.105 1.48 0.105 1.100.0120 0.100 0.84 0.100 1.36 0.105 1.370.0125 0.100 1.26 0.100 1.17 0.100 0.910.0130 0.100 1.82 0.095 1.05 0.100 1.570.0135 0.095 0.89 0.100 1.83 0.095 1.730.0140 0.95 2.04 0.095 2.28 0.095 1.830.0145 0.90 2.42 0.095 2.37 0.095 2.37Table 3.4: Best-fit values of Ws for each of the simulated values of BHe, with the obtained χ2value divided by the degrees of freedom ndof = 6.data (eight storage lifetimes minus two fitted parameters). The minimal χ2 value for all threeWg values was at BHe = 0.011 s−1K−7, with χ2/ndof < 1 in all three cases. It does appearthat there may be some overestimation of the uncertainty in this analysis. As it stands themodelling with BHe alone appears to fit the data well.In the design of the TUCAN source and subsequent nEDM experiment optimization, a valueof BHe = 0.016 s−1K−7 was assumed as a worst case. The analysis of this experiment supportsa slightly lower value. This serves as early validation of this assumption in the design of theTUCAN source.3.4 Production Rate in He-II3.4.1 Experiment ConfigurationImmediately following the storage lifetime experiment, we performed a second measurement ofsteady-state UCN production in the source, using the same geometry as described in Section3.2. The temperature of the He-II was varied continuously in this experiment from 1.1 to 1.8 K,yielding a curve of UCN count rate vs temperature. This experiment can be interpreted as anindirect measurement of the storage lifetime, and can be compared to the results of the storagelifetime experiment.For this experiment, the valve IV1 was kept open while the target was irradiated. Thevariation of the temperature was achieved in the same way as in the storage lifetime experiment,by adjusting the flow of the 3He to reduce the cooling power of the primary heat exchanger.At each temperature, counts were allowed to accumulate in the detectors for several minutesto allow the source to reach saturation conditions. Two sets of data were taken: the first whileincreasing the temperature, and a second while decreasing the temperature back to its originalvalue.623.4. Production Rate in He-II0.8 1 1.2 1.4 1.6 1.8 2Temperature (K)05001000150020002500)-1Aµ -1UCN count rate (sTemp. Sensor #11 warmupTemp. Sensor #16 warmupVapour Pressure warmupTemp. Sensor #11 cooldownTemp. Sensor #16 cooldownVapour Pressure cooldownFigure 3.11: Measured UCN count rate in the 6Li detector versus temperature as the sourcewas warmed up to 1.8 K and then back down to 1.1 K. The temperatures were measured in thesource using several temperature sensors and the vapour pressure. The temperature sensorsshow poor performance at low temperatures, as well as having a large hysteresis between coolingdown and warming up. The vapour pressure measurement provides a superior determinationof the source temperature. Some hysteresis is also visible in the counts in the vapour pressurecurve, but this is suspected to be due to changing conditions in source.3.4.2 Analysis MethodsThe analysis of this experiment is largely the same as the direct storage lifetime measurement.The treatment of the raw counts includes the same normalization to beam current and sub-traction of background rate. However, since this experiment was performed with beam oncontinuously, the background rate in the 6Li detector increased by 7.004± 0.015 s−1, while thatin the 3He detector increased by 0.0946± 0.0025 s−1. For this measurement, only counts in the6Li detector are used in the analysis, as the small pinhole connecting the 3He detector to theguides resulted in a very low count rate and large uncertainty.Figure 3.11 shows the results of these production experiments, with the production rateplotted against temperatures taken both from the vapour pressure measurement and from sev-eral temperature sensors attached to the production bottle. This data directly illustrates theunreliability of the temperature sensors. The sensors do not agree with each other, they exhibita strong hysteresis effect when cooling down versus warming up, and at low temperatures theyshow a very large spread in temperature vs. counts. In comparison, the vapour measurementappears much more self-consistent, although it does exhibit a small hysteresis. It is suspectedthat the hysteresis in the vapour measurement data is caused by some variation in the sourceconditions, such as a flow of contamination into the volume upstream of IV1 during this mea-surement, which could reduce the storage lifetime of the walls. This hypothesis is supported byobservations of the source lifetime over time, which are discussed in Section 3.5. With that inmind the warmup and cooldown data are analyzed separately.633.4. Production Rate in He-IIAnalysis of the data uses the same simulation model as described in 3.3.3, varying thematerial parameters BHe and Ws in the same way to extract the temperature dependence.Making a comparison to simulation requires the selection of a set number of points in the ratecurve. Counts were extracted for temperatures ranging from 1.0–1.8 K, in steps of 0.1 K. Foreach temperature, the count rate was taken to be the average value of the points seen in Figure3.11, with an uncertainty corresponding to the range.Unlike the storage lifetime experiment, this experiment depends on the absolute counts ofUCNs and requires some normalization between simulation and experiment. An initial attemptat this normalization took the experimental rate at 1.1 K and used that to scale the simulationdata. However, it was observed that this method caused the analysis to become poorly con-strained, as the ratios could be reproduced at a wide range of parameter values. This renderedit difficult to draw any conclusion from the analysis. Instead, in the analysis presented here,the simulations are scaled to an expected absolute production rate. The best estimate for theUCN production rate was 20600± 200 s−1µA−1. This was determined using simulations of thecold neutron flux from MCNP and described in the first publication on the operation of thevertical source at TRIUMF [3, 197].To analyze this production rate data the same MCNP-estimated production rate is used toscale the simulated counts. However, the uncertainty has been increased to 10%. This is due toquestions regarding the accuracy of the modelling of solid D2O, as well as the lack of a successfulmeasurement of the cold neutron spectrum at TRIUMF [198]. Furthermore, the conversion ofcold neutrons to UCNs in He-II has some observed temperature dependence [189, 199]. Basedon the available published data it is estimated that, in the measured temperature range andwith the assumed simulated cold neutron spectrum, the production rate could vary by up to5% due to this dependence. This is included as an additional uncertainty on the productionrate.The goodness-of-fit for this experiment was calculated asχ2 =∑Temp.(Rexp −Rsim)2σ2R,exp + σ2R,sim. (3.26)For this experiment, no temperature uncertainty is propagated into the uncertainty in therate. Since the measurement of the rate at any point in the data is nearly instantaneous,temperature variations for a given measured point is relatively small. The accuracy of thepressure measurement and the vapour equation are also small compared to the large uncertaintyon the absolute count rate, so these contributions can all be neglected for ease of analysis.3.4.3 Simulation ModelAt saturation, the counts per second (with background subtracted) give a steady-state rate thatshould be proportional to the number of UCN in the experiment volume. The number of UCNis proportional to the production rate and the storage lifetime in the experiment volume. Atsaturation density, the detection rate isRdet ∼ Rprodτs = P ((1− fHe−II)τHe−II−1 + fHe−IIτvap−1 + fsτw−1,s+(1− fs)τw−1,g + τβ−1 + τ−1det)−1,(3.27)where P is the production rate, τw,s is the lifetime due to wall collisions in the guides insidethe source (upstream of IV1), τw,g is the lifetime due to wall collisions in the guides outsidethe source (downstream of IV1) and τ−1det is the loss rate due to detections. The fractions of643.4. Production Rate in He-IItime spent in the different volumes are also now different. In this configuration there is a largervapour volume, and two different guide volumes. The fraction of time spent in the He-II islabelled f1. The fraction of time spent inside the source volume (during which time the UCNssee the loss rate τw−1,s ) as fs. Despite the increased complexity, this storage lifetime taken overthe full experiment volume contains the same temperature-dependence as the storage lifetimeof the source alone, and can thus also be used to probe the losses in the He-II.The contributing material properties that define different lifetimes were described in Section3.3.3. The only difference is that this analysis was performed prior to learning of the silverplating on the valve in Figure 3.7. This analysis was not repeated with the updated material asthe outcome of this analysis did not strongly constrain the parameter values, and so it was notconsidered a good use of computing time. This could introduce some systematic error, such asfavouring higher Ws and BHe in order to account for the losses to the silver. The simulationmodel is otherwise the same as for the lifetime experiment.Equation 3.27 makes it easier to understand why the production rate is needed to analyzethis experiment. The storage lifetime can be written as τs−1 = τs−1,0 + BHeT7, where τs,0captures all non-temperature-dependent effects, such as the wall losses from Ws. If attemptingto normalize the data using one of the measured data points at temperature T 0, then the ratioof rates isRdet(T )Rdet(T 0)=(Rdet(T 0)Rdet(T ))−1=τs−1,0 +BHeT7τs−1,0 +BHeT70=τs−1,0τs−1,0 +BHeT70+BHeT7τs−1,0 +BHeT70=11 + τs,0BHeT 70+τs,0BHeT71 + τs,0BHeT 70.(3.28)From this, it can clearly be seen that τs,0 and BHe are mathematically interchangable in theratio. This means that, while 3.27, which depends on the production rate, can be used todetermine Ws and BHe uniquely, the ratio formed by normalizing the data by itself does notconstrain the parameters sufficiently.3.4.4 ResultsSimulations of the experiment were performed for each of the selected temperatures, againvarying (BHe,Ws) to identify the best fit. Figure 3.12 shows maps of the values of ∆χ2 whileboth warming up and cooling down. In analyzing this experiment, a larger range of (B,W )than in the lifetime experiment is examined, with 0.004 ≤ B ≤ 0.022 s−1K−7 and 0.06 ≤ W ≤0.17 neV. This range was expanded due to the apparent greater range of parameters that couldfit the experimental data.The range of parameters that can fit the data is much wider than in the storage lifetimeexperiment, and this analysis yields only a lower limit at the 95% confidence level of B ≥0.0095 s−1K−7. An upper limit is not obtained as the allowable region extends beyond therange simulated, and an effort was not made to find the end of this range (if one exists), asthe possibility of a lower value is not of great interest. Table 3.5 gives the best-fit Ws for eachsimulated value of BHe.It is seen that χ2 1 in many cases. This is due to the inclusion of the large uncertaintyin the UCN production rate. It should be noted that the treatment of this uncertainty in thisanalysis is not completely rigorous, as the production rate uncertainties for each data pointare correlated, while this analysis has treated them as uncorrelated. It should be possible to653.4. Production Rate in He-II0.005 0.01 0.015 0.02 BHe (s-1K-7)0.060.080.10.120.140.16 Ws (neV)3456789(a) Warmup0.005 0.01 0.015 0.02 BHe (s-1K-7)0.060.080.10.120.140.16 Ws (neV)3456789(b) CooldownFigure 3.12: Heatmap of ∆χ2 for different simulated values of (BHe,W ), with Wg = 0.15 neV,for the production experiment. Due to the high uncertainty on the absolute production rate,this experiment only weakly constrains BHe ≥ 0.0095 s−1K−7. It is noted that the data takenwhile cooling down, which was the second set, appears to favour a slightly higher value of thesource imaginary Fermi potential Ws. This is suggestive of some contamination flowing into thesource during the measurement.663.4. Production Rate in He-IIWarmup CooldownBHe (s−1K−7) W (neV) χ2/ndof W (neV) χ2/ndof0.004 0.150 7.51 0.150 7.940.005 0.160 5.63 0.150 5.790.006 0.150 4.15 0.140 4.290.007 0.120 2.74 0.130 3.140.008 0.120 1.87 0.120 2.150.009 0.115 1.25 0.125 1.320.010 0.100 0.68 0.115 0.800.011 0.090 0.38 0.100 0.490.012 0.080 0.11 0.090 0.180.013 0.060 0.05 0.080 0.070.014 0.060 0.07 0.070 0.040.016 0.060 0.90 0.060 0.400.018 0.060 2.57 0.060 1.470.020 0.060 5.00 0.060 3.340.022 0.060 8.49 0.060 6.17Table 3.5: Best-fit values of Ws for each of the simulated values of BHe in the analysis of theproduction experiment, with the obtained χ2 value divided by the degrees of freedom ndof = 6.Results are given separately for the warmup and cooldown data. The χ2 values are notablysmall for many cases, which is due to the large production rate uncertainty, and is discussedfurther in the text.improve this analysis with better treatment of this uncertainty. However, some part of thisuncertainty reflects an apparent temperature-dependent variation of the production rate. Thiswas estimated from limited data in Refs. [199] and [189], and there is therefore uncertainty inexactly how the production rates at different temperatures are related. As this would makethe construction of a better treatment difficult, and as the results of this first analysis do notseem promising compared to the storage lifetime analysis, a refinement of the methodology hasnot been pursued. This unfortunately means that the χ2 values reflect the large systematicproduction rate uncertainty, which further limits the conclusions that can be drawn here.The parameter ranges obtained are consistent with those of the storage lifetime analysis.Greater confidence in the production rate would allow this experiment to better constrain theparameters, but as it stands it only serves to support the direct measurement of τs. Thebest-fit BHe are slightly higher than the value of 0.011 s−1K−7 obtained in the storage lifetimemeasurement, at 0.013 and 0.014 s−1K−7 for the warmup and cooldown data respectively. Thisstill falls below the worst-case value of 0.016 s−1K−7 used in the design analysis of the TUCANsource, and the difference in the best-fit values could be entirely due to an inaccuracy in theproduction rate.The results for the warmup favour lower Ws values compared to the cooldown. The warmupoccurred first, and so this is suggestive of a possible deterioration of the source conditionsduring the production experiment. This could be due to contamination introduced into thesource while running this experiment with IV1 open for roughly 7.5 hours. In the next sectionsome additional data on the changing source conditions over the course of the 2018 experimentalrun will be presented. A detailed analysis of this observation in the production rate has notbeen pursued due to the already low precision of this experiment, but it should be acknowledged673.5. Long-term Behaviour of Source Lifetimeas a possible systematic shift in the results.3.5 Long-term Behaviour of Source Lifetime3.5.1 Source Storage Lifetime vs. TimeThroughout the duration of the 2018 experimental campaign, source storage lifetime measure-ments were periodically repeated, generally once or twice per experiment configuration. Thegoal of these measurements was to track the performance of the source over time.These measurements were performed in largely the same way as described in Section 3.3.The only differences were that the temperature of the source was not controlled in any way,and that the geometry of the guides outside of the radiation shielding differed for the differentexperiment configurations. These geometry changes could impact the measured lifetime, sincethey could change the extraction of the UCN from the source in an energy-dependent way.Figure 3.13 shows the measured source storage lifetime over time. The lifetime did decrease,by roughly 20% over the course of a month. During the campaign, this was suspected tobe caused by a gradual reduction in the He-II level. This could occur due to the lack ofany barrier between the source and experiment guides, which allowed helium vapour into theguides after IV1. This vapour would then be lost to vacuum pumping, leading to a gradualdecrease in the total amount of helium inside the source. This slow decrease in the He-IIlevel could lead to increases in the temperature of the He-II, as the heat conductance to theheat exchanger would be reduced as the level fell below a certain level. Figure 3.14 shows thetemperature recorded during the source storage lifetime measurements over time (based on thevapour pressure measurement). In can be seen that there indeed was a gradual increase in thetemperature inside the source. This motivated refilling of the He-II, which is indicated in thefigures by two gray lines. Refilling the He-II seemed to impact the temperature, but it is notclear whether the storage lifetime was also restored.To understand the time evolution of the storage lifetime, four possible contributions mustbe considered:1. Changes of the downstream geometry,2. The changing He-II causing temperature changes,3. The changing He-II reducing the volume of He-II in the source thus changing the fractionof time (1− fHe−II) spent by the UCNs in He-II vs. in helium vapour,4. Changes of the imaginary Fermi potential Ws of the guides in the source.The results of the storage lifetime analysis in Figure 3.9 indicate that downstream geometriesdo not have a large impact on the measured storage lifetime, at least not in terms of Wg. Whileit is possible that some geometries may have significant effects, most of the geometries likelydo not, and for the moment changes in the experiment geometry can be neglected.Figure 3.15 shows results of a simulation of the storage lifetime for different He-II levels rel-ative to the nominal level, and at different temperatures. The connection to the heat exchangeris roughly 2.5 cm below the nominal He-II level in the simulation model, and has a diameterof roughly 2.5 cm. Given these dimensions, temperature increases should have been observedwith a level reduction of no more than 5 cm. The simulations suggest no significant changesin the measured storage lifetime with a level change of this magnitude. The 5 cm level changecorresponds to a 4% change in volume of He-II. Considering that τHe−II is only part of the total683.5. Long-term Behaviour of Source LifetimeNov 3 Nov 9 Nov 15 Nov 21 Nov 27 Dec 326283032343638Source storage Lifetime (s)6Li Detector3He DetectorFigure 3.13: Plot of the storage lifetime in the source measured over the duration of the 2018experimental run. A gradual reduction in the lifetime over the course of the month is clearlyvisible. The decreasing lifetime is attributed partly to a change in the temperature of the source,and partly to an apparent degradation of the source conditions. This degradation could be dueto the introduction of contamination onto the walls of the source or into the He-II volume.These forms of contamination are indistinguishable in the analysis and the degradation can bemodelled as an increase in the imaginary Fermi potential Ws of the walls of the source. Thesecauses are discussed in detail in the text. The two vertical gray lines indicate attempts atrefilling the He-II level to restore the lifetime.693.5. Long-term Behaviour of Source LifetimeNov 3 Nov 9 Nov 15 Nov 21 Nov 27 Dec 30.9511.051.11.151.2He-II Temperature (K)Figure 3.14: Plot of the temperature of the He-II in the source measured over the durationof the 2018 experimental run. The temperature was determined using the vapour pressure asdescribed for the temperature-dependent storage lifetime experiment. The two vertical linesindicate attempts at refilling the He-II level. The temperature was reduced by this refilling andapparently returned to its nominal level after the second refilling.703.5. Long-term Behaviour of Source Lifetime1 1.2 1.4 1.6 1.8Temperature (K)51015202530Storage Lifetime (s)Nominal level-10mm-25mm-50mmFigure 3.15: Simulated storage lifetime measurement results vs. temperature for differentreductions in the level of the He-II volume, varying from the nominal level to 5 cm lower.storage lifetime, that the lower level is replaced by He vapour (which still has some losses), andthat UCNs tend to spend more time at lower elevations (while this is the uppermost part ofthe He-II volume), this lack of a noticeable effect seems to be consistent.If the change in the storage lifetime was entirely due to changes in temperature, then itshould be observed that the lifetimes fall on a line defined by a constant BHe and Ws versusT . Figure 3.16 plots the storage lifetime vs. temperature for simulation models with differentvalues of Ws and a BHe = 0.010 s−1K−1 alongside the measured values. The slightly lowerBHe compared to the optimal value is used here as this analysis was completed before the finalresults of the storage lifetime analysis suggested BHe = 0.011 s−1K−1 as the best fit value. Thisshould not significantly influence the conclusions here, which are largely qualitative.Figure 3.16 shows that the storage lifetimes measured throughout 2018 do not fall on anyline. However, different groups of lifetimes do appear consistent with different curves. Two largegroups, labelled A and B, are respectively consistent with Ws = 0.10 neV and Ws = 0.11 neV.All measurements in group B occurred later than those in group A. A third group are isolated asgeometries that contained foils; one of these experiments specifically tested transmission throughan aluminum foil, and the other three had polarization foils. These foil measurements were notsequential in time, and do not appear to fall on any consistent curve, indicating a possiblegeometry effect due to the presence of the foils. A fourth group is identified as containing asingle measurement. This measurement was taken after the steady-state production experiment,and is consistent with Ws = 0.12 neV.There is therefore substantial evidence from these periodic lifetime measurements that thesource deteriorated over time, which can be modelled as a change in Ws from 0.10 neV to0.12 neV over the course of the 2018 experimental run. In the next section, one final experimentis presented. This experiment will attempt to provide one possible explanation for how this713.5. Long-term Behaviour of Source Lifetime0.95 1 1.05 1.1 1.15 1.2 1.25He-II Temperature (K)2025303540Source storage Lifetime (s)Group AGeometries with foilsGroup BPost steady-stateWs = 0.10 neVWs = 0.11 neVWs = 0.12 neVWs = 0.13 neVFigure 3.16: Measured “daily” storage lifetimes in the source vs. the measured temperatureof the He-II in the source. The black curves are simulation models with B = 0.010 anddifferent values of Ws. The experiment data is divided into four groups. The first group Ais generally consistent with the Ws = 0.10 neV line. The second group are those where theexperiment geometry contained foils, mostly in polarization experiments, which appears tohave a significant effect on the measured lifetime. The third group B is generally consistentwith the Ws = 0.11 neV line. The last group, a singular measurement, was taken after thesteady-state production measurement and is consistent with Ws = 0.12 neV.723.5. Long-term Behaviour of Source Lifetimedeterioration could have occurred.3.5.2 Source SpoilingIn a final experiment in 2018, the storage lifetime was measured while intentionally contami-nating the source volume, in a process referred to as “spoiling” the source. This was achievedby injecting a set amount of nitrogen gas into the source. The hypothesis was that this gaswould condense (below 77 K) and freeze (below 63 K) onto the cold sections of the guides. Theresulting layers of nitrogen on the guides would change the loss rate of the guides and thus thesource storage lifetime. It is also possible that the nitrogen could freeze inside the He-II, or onits surface.To inject the nitrogen, an additional pipe volume was connected to the pumping systemof the guides through a manual valve. This volume was connected to a pressure gauge and anitrogen cylinder. This section of pipe had a known volume of 103 cm3, and by filling it witha specific pressure of nitrogen, a known mass of nitrogen could be introduced into the sourcewhen the manual valve was opened.The results of this experiment are shown in Figure 3.17. The lifetimes are plotted againsttemperature, as an increasing temperature was observed as nitrogen was injected. It is unclearwhat mechanism drove this increase. Data on the sublimation pressure of nitrogen could notbe found in the 1 K temperature range, but is available at higher temperatures. At 20 K it isreportedly 1.47× 10−16 mbar [200], so if the nitrogen froze out close to the He-II, there shouldnot be any measurable pressure. However, at 60 K the sublimation pressure is 7.4 mbar [200], soif it froze out higher up in the guides there could be a significant pressure due to sublimation.Since the vapour pressure of 4He at 1 K is 0.16 mbar, this could certainly have altered thevapour pressure measurement. That said, if there was a significant amount of gaseous nitrogenthroughout the source, over time it would freeze out in the 1 K region where the sublimationpressure is lowest, ultimately reducing the measured pressure. Without further study it isdifficult to draw a conclusion about the exact distribution of nitrogen in the guides, and so noconclusion can be drawn either way as to the cause of this temperature effect.The increasing nitrogen mass in the source decreased the lifetime. In the first several points,this is not statistically significant when plotted against the temperature change. The third- andsecond-to-last points appear to fall onto a lower Ws curve, but their uncertainties are also toolarge to mark this as statistically significant. However, the final point shows a definite deviationfrom the curves that the other points are consistent with, indicating a higher Ws. This leadsus to conclude that the injection of nitrogen into the source did have the effect of increasingcontamination.The relevance of this result to the change of storage lifetime over time can be understoodby considering the imperfection of vacuum systems. While the source was always protectedfrom direct exposure to atmosphere by IV1, small quantities of gas can still enter a vacuumsystem through leaks, outgassing, diffusion through elastomer seals, or from trapped gas vol-umes. Each time an experimental guide configuration was changed, everything downstream ofa valve (either IV1 or one of the other valves used in experiments) would be exposed to atmo-sphere, potentially refreshing these gas sources. These sections of the geometry would alwaysbe pumped out before exposing the source to them, with a target pressure of 1 × 10−5 mbar.Still, it cannot be guaranteed that a sufficiently good vacuum was always achieved, or that eachexperiment assembly was sufficiently leak-free. Over the course of several weeks of experiments,it is therefore possible that the source deteriorated due to the introduction of gas.The volume of gas that produced a change in this experiment is not out of reach for a typical733.5. Long-term Behaviour of Source Lifetimevacuum system. An idea can be obtained by considering the typical gas loads in a vacuum sys-tem. Viton O-rings reportedly have permeation coefficients of 10−10 cm3/cm2/cm−1/s−1/Torrfor air [201].4 In the experiments at TRIUMF, G90 O-rings were used, which have a cross-sectionof 3.1 mm and a diameter of 89.4 mm. Such an O-ring would therefore have 3.4× 10−9 mbar-l/sof permeation into the experiment volume. Even with tens of such O-rings, it would take manyyears to account for the gas load required to change the storage lifetime significantly. However,O-ring permeation is a relatively small contribution to the gas load in a vacuum system, whichcan be up to 1× 10−5 mbar-l/s for a typical clean and well-sealed system [202]. This would stilltake many days to accumulate the quantities in Figure 3.17, but that would still be relevant forthe long-term performance of the TUCAN source.These typical gas loads would not account for the observed change in conditions after thesteady-state production measurement, which took 7.5 hours. However, there are a few reasonsto suspect that significant contamination could still have occurred. The first is that the spoilingexperiment only introduced nitrogen. Other residual gases in vacuum systems, like hydrogenor water vapour, could produce higher UCN loss rates and thus require significantly less gas toproduce a similar storage lifetime reduction. The other is that it is not clear that the system inthis experiment was such a good vacuum system. The 6Li detector, which has exposed acryliclight guides, has been noted as seeing a large amount of outgassing (primarily of water), andwhile the detector had its own direct pumping connection, gas from it could still diffuse intothe source. Therefore contamination of the source during this length measurement remains apossible explanation for the reduction in storage lifetime.It is possible to mitigate this issue through the use of a thin separation foil, through whichUCNs can pass but gas is blocked. Such a foil also serves to prevent the gradual loss of He-II, asobserved in these experiments. While the vertical source lacked such a foil, one will be presentin the TUCAN source. It is anticipated that this will reduce any possible contamination of thesource.4This unit is (volume of air)/(exposed area of O-ring)/(1/thickness)/second/(pressure difference)743.5. Long-term Behaviour of Source Lifetime0.95 1 1.05 1.1 1.15 1.2 1.25He-II Temperature (K)20253035Source storage Lifetime (s)050100150200250300350Accumulated Pressure Volume (mbar-l)Spoiling DataWs = 0.11 neVWs = 0.12 neVWs = 0.13 neVWs = 0.14 neVPressure volumeFigure 3.17: Plot of measured source storage lifetimes versus temperature while spoiling thesource by injecting nitrogen. The right axis and red curve show the calculated amount ofnitrogen injected into the system at each measured temperature. The black curves are simulatedstorage lifetimes for different values of Ws in the source. Most of the points are consistent witha change only due to temperature, but the final point shows a deviation from the simulationcurves that suggests a clear contamination of the source volume.75Part IIThe TUCAN EDM Central Region76Chapter 4The nEDM Measurement4.1 IntroductionIn Chapters 2 and 3, the efforts of the TUCAN collaboration to build a new high-density sourceof UCNs were discussed. The purpose of this source is to provide sufficient statistics to allowa measurement of the nEDM to a precision of ∼10−27 ecm. This improved precision will allowthe collaboration to place tighter constraints on theoretical sources of CP violation as discussedin Chapter 1.With the increased UCN density available from the TUCAN source, it will also be necessaryto build a more advanced nEDM spectrometer. This spectrometer must be designed to makeuse of the high UCN density, maximizing the number of UCNs that can be measured per unittime. It must also by designed with the goal of reducing the systematic uncertainty below the∼10−27 ecm threshold.The following chapters will describe some of the effort to design this new nEDM spectrome-ter. This chapter describes the overall design of the TUCAN EDM spectrometer, and presentsan overview of sensitivity contributions. The subsequent chapters describe the design of theso-called central region of the spectrometer. Chapter 5 describes the design and testing of aprototype storage cell to achieve a high UCN storage lifetime during the nEDM measurement.Chapter 6 describes the design of the electrodes and calculations of systematics related to theelectric field. Chapter 7 describes magnetic-field design considerations in the central region andincludes simulations and measurements related to the impact of material choices.4.2 The TUCAN EDM SpectrometerThis section will describe the TUCAN EDM spectrometer, which implements Ramsey’s method,which was described in Section 1.2.3, to search for an nEDM. The basic sequence of this mea-surement can be broken down into seven steps:1. Produce UCNs in a source.2. Polarize the UCNs and fill them into a storage volume.3. Apply magnetic fields ~B0 and ~B1 to the UCNs in the storage volume, as described inSection 1.2.3, with an electric field ~E aligned with ~B0.4. At the conclusion of the Ramsey measurement, empty the UCNs from the storage volumeinto spin-sensitive detectors to obtain N↑ and N↓ and calculate A.5. Repeat with different ~B1 frequencies to obtain A vs. ωi, and fit to obtain ωn,+.6. Repeat with ~E anti-aligned with ~B0 to obtain ωn,−.7. Compare ωn,+ and ωn,− to obtain a value for the nEDM.774.2.TheTUCANEDMSpectrometerGlm CoilsVacuum ChamberPrecession CellsUCN Guidesand SwitchesHV ElectrodeGround ElectrodesUCN ValvesSpin AnalyzersMagnetic Shieldingn×n CoilsCs MagnetometerArrayHg Co-magnetometerPrepolarizing CellAmbient Field CompensationB0 CoilB1 CoilsHg Co-magnetometer Readout LaserFigure 4.1: Illustration showing the layout of the major components of the TUCAN EDM spectrometer. This figure is not to scale: theinner vacuum chamber is ∼1 m diameter, while the exterior of the magnetic shielding is approximately 3.5× 3.5× 3.5 m.784.2. The TUCAN EDM SpectrometerThe different repetitions and combinations of measurements require some careful definition ofterminology to avoid confusion. The smallest unit of measurement is a single measurement ofasymmetry, and will commonly be referred to as a fill. Four fills, taken at different values of ωi,are combined into a single measurement of a precession frequency, and will be referred to as aRamsey sequence. Two (or more) Ramsey sequences, with oppositely oriented ~E, are combinedinto a single measurement of dn, and will be referred to as an EDM sequence.The measurement requires a UCN source, a method of polarization and spin transport, astorage volume, magnetic coils for ~B0 and ~B1, a HV system to apply ~E, and a spin-sensitivedetection system. In order to achieve a measurement at the level of ∼10−27 ecm, a numberof other components are required to reduce the statistical and systematic uncertainty. Theseinclude additional magnetic coils, magnetometry, and magnetic shielding.All of the components of the EDM spectrometer are shown in schematic form in Figure4.1. At the time of writing, the spectrometer has two versions, referred to as OC100 (one cell,100 KV) and 2C200 (two cells, 200 KV). These versions are part of a phased approach to theconstruction of the experiment, starting with one cell, 100 kV (OC100) and then upgrading totwo cells, 200 kV (2C200). Figure 4.1 shows the 2C200 spectrometer. The OC100 version isobtained by removing the upper cell and everything to which it is connected (except for the HVelectrode). The purpose of each of the components will be described in turn in the followingsection.4.2.1 Magnetic ShieldingThe creation of a uniform 1µT field requires a blank-slate magnetic environment. Such anexperiment is not possible in Earth’s magnetic field, which has a magnitude of ∼50µT. Toachieve a low field as a starting point for the nEDM measurement, the central region and coilswill be installed inside a magnetically shielded room (MSR).The TUCAN MSR, constructed by Magnetic Shields Ltd. [203], has five layers of high-permeability mu-metal, which redirects the magnetic flux of low-frequency fields around theinner volume, and one layer of copper, which suppresses high-frequency fields by eddy currents.The purpose the MSR is to achieve a shielding factor between 5 × 104 and 105, which wouldreduce Earth’s field to the nT level. Similar rooms have been constructed and validated for thePanEDM [204] and n2EDM [205] experiments, although these have aimed for higher shieldingfactors.4.2.2 Ambient Compensation CoilsThe TRIUMF main cyclotron has a substantial stray field, which has been measured to be atthe level of 400µT in the TUCAN EDM experimental area. To compensate this field and allowthe MSR to perform as desired, a pair of coils are positioned at the top and bottom of the MSR.These coils will generate a counteracting field such that the MSR only sees a field no greaterthan Earth’s field of ∼50µT. The cyclotron stray field is regarded as being highly stable, andso these coils would operate statically, without any need for feedback control.Additional compensation coils may be necessary to further compensate the ambient mag-netic field. This could be necessary to handle fields from other nearby experiments. Thesefields would exhibit more time variance and would therefore need to be dynamically controlled.Additionally, more coils could be added depending on the final performance of the MSR. Ifthe residual field is higher than expected, this could potentially be improved by using furthercompensation to reduce the field (e.g. by a factor of 10) at the location of the MSR. Determin-794.2. The TUCAN EDM Spectrometering the requirements for these coils is an ongoing process and is not discussed further in thisdissertation.4.2.3 Spin Polarization and TransportTo perform the nEDM measurement, UCNs produced in the TUCAN source must be polarizedand transported into the MSR. After the measurement, they must be transported out of theMSR into the spin analyzer.The initial polarization is achieved using a superconducting magnet (SCM). The SCM islocated directly at the exit of the TUCAN source, shown in Figure 2.3. UCNs pass throughthe bore of the magnet, inside which is a magnetic field with a nominal magnitude of 3.7 T.This field acts a spin-dependent potential of ±222 neV. As the UCN guides in the source areNiP-coated, with V = 183 neV, the magnitude of the magnetic potential exceeds the energyof all UCNs. This means that only one spin state can pass through the SCM, resulting in apopulation that is nominally polarized to 100%.To maintain the polarization, the UCNs must be transported in a field that varies graduallyenough that the spins behave adiabatically. The required condition is that the precessionfrequency is large compared to the change in the field direction seen by the UCNs, i.e. ~ω0 2pi dBˆdt . This can be achieved by winding the guides between the SCM and the MSR with coilsto apply a relatively large uniform field. Some early development of the spin-transport systemfor the TUCAN EDM experiment is described in the thesis of E. Klemets [206].4.2.4 The Central RegionA conceptual model of the central region is shown in Figure 4.2. It encompasses the precessioncells in which the UCNs are stored, and the surrounding vacuum chamber and high-voltageinterfaces. The concept model shows the design that will be used for OC100, alongside apossible implementation of 2C200, in which a duplicate top cell is added on top of the bottomcell. However, it is likely that 2C200 will include further modifications away from this design.The precession cells are the core of the central region. These cells contain the neutronsduring the Ramsey measurement. The top and bottom surfaces of the cylindrical cells are thetwo electrodes that apply the HV. In 2C200, the HV is applied to a single central electrode,which in OC100 acts as the top surface of the single cell. The electrodes are separated byplastic insulator rings, forming the walls of the cells. UCNs are filled into the cells throughentrances sealed by custom-built UCN valves. The cells are enclosed in a vacuum chamber toprovide an insulating vacuum for the HV, with the ground electrodes integrated into the topand bottom flanges of the chamber. In OC100, the cells will have an inner diameter of 0.5 mand an electrode separation of 0.1224 m. In 2C200, it is expected that the separation will beincreased to 0.16 m to make best use of the HV supply. For that reason many of the studiesthat will be discussed use a separation of 0.16 m. The design of the central region is the focusof the remaining Chapters 5, 6, and 7.4.2.5 Magnetic CoilsSeveral different magnetic coils are used to apply the magnetic fields necessary to perform annEDM measurement. In addition to the fields applied as part of the Ramsey measurement,there are specific coils intended to improve the overall field homogeneity and to apply gradientsas part of the analysis of systematics. These coils include804.2. The TUCAN EDM SpectrometerFigure 4.2: Concept model of the central region of the TUCAN EDM spectrometer. The topview shows the single-cell version (OC100), and the bottom view shows the double-cell version(2C200). UCNs are filled into the cells through UCN guides which connect to the cell valves onthe ground electrodes. The electrode separation is shown here as 122.40 mm, which is the OC100value, but this will likely be increased to 160 mm for 2C200, with a corresponding increase inthe vacuum chamber height.814.2. The TUCAN EDM Spectrometer• B0 Coil: provides the main holding field of 1µT, about which the neutrons precess. Thiscoil is a self-shielded cosine-theta-like coil.• B1 Coils: applies the pulsed low frequency (LF) field of ∼10 nT for the spin flip in theRamsey measurement. These coils are currently envisioned as saddle coils wrapped aroundthe vacuum chamber of the central region. In OC100, with the vacuum chamber beingmade of metal, eddy currents have some impact on the coil behaviour. This interface withthe central region is discussed in Chapter 7.• n× n Coils: an array of square coils surrounding the B0 coil, which are used to improvethe uniformity of the magnetic field by compensating gradients. The absolute uniformityof the magnetic field impacts the rate of depolarization of UCNs, and therefore contributesto the visibility α in the statistical uncertainty. The uniformity also has some bearing onsystematic uncertainties, but this is primarily handled using magnetometry. The goal forthe n×n coils is to reduce the variance of the field inside each cell such that σ(Bz) < 40 pT.• Glm Coils: auxiliary coils that can apply specific gradients for systematic studies orprecise tuning of the magnetic field. The specific need for these coils is elaborated inSection 4.3.3.4.2.6 MagnetometryDifferent types of magnetometry are used as part of the nEDM measurement in order to correctfield fluctuations and systematic contributions. Two types of online magnetometry are employedduring the measurement: internal, or co-habiting, magnetometry, in which a magnetometeroccupies the same volume as the neutrons, and external magnetometry, in which magnetometersare placed outside of the cells. Offline magnetometry is also used, which consists of field mapstaken using additional magnetometers outside of the measurement time.Co-Habiting MagnetometryThe co-habiting magnetometers, or co-magnetometers, provide in-situ measurements of themagnetic field inside the precession cells. This is used to correct fluctuations in the field seen bythe UCNs. Calculation of the nEDM requires the cancellation of the magnetic field contribution,which is only valid if the field is very stable. The frequency shift at a the level of dn ∼10−25 ecm (the typical sensitivity expected per pair of Ramsey measurements) in the TUCANEDM spectrometer would be 1.2µHz. A similarly sized frequency shift would be caused bya 40 fT magnetic field. Rather than attempting to stabilize a magnetic field to this level,the co-magnetometer is used to compensate magnetic field fluctuations. The need for a co-magnetometer for this purpose is motivated by possible contributions such as small magneticdipoles or leakage currents in the cells, which may not be detectable by external magnetometry.The TUCAN EDM experiment will implement co-magnetometers using Hg vapour filled intothe cells, building on the previous work of the ILL/RAL/Sussex group and the PSI group [207–209]. The vapour is polarized by an ultraviolet (UV) pump laser in pre-polarizing cells locatedabove and below the central region. After polarization, the vapour is filled into the precessioncells through small valves in the ground electrodes. The precession frequency of the Hg atomsis measured using a second UV probe laser that passes through optical windows in the insulatorrings of the cells. A notable difference from previous experiments is the use of co-magnetometersin two cells, which provides additional information about the gradient between the top and824.2. The TUCAN EDM Spectrometerbottom cells. The use of a co-magnetometer plays a major role in the sensitivity of the nEDMmeasurement, which will be discussed in depth in Section 4.3.6.External Magnetometer ArrayMany of the effects that contribute to the overall sensitivity of the nEDM measurement, dis-cussed in Section 4.3 are related to magnetic field gradients. To measure these gradients anarray of cesium magnetometers (CsMs) will be installed around the central region. The oper-ation of these CsMs is entirely optical, requiring no electronics near the measurement volume.Each magnetometer provides a scalar measurement of the magnetic field at its location, whichcan then be fit to obtain values of magnetic field gradients. These gradient values will be usedin the final data analysis to correct for systematic contributions, and will also be used in-situto control the currents of the n× n coils.Optically pumped CsMs were first used in the PNPI experiments, but only for the purposeof stabilizing magnetic field fluctuations [104]. The first use for explicitly extracting gradientsin an nEDM experiment was for the PSI nEDM experiment [210, 211]. The CsMs and thedesign of the array (optimized using a genetic algorithm) for the TUCAN EDM spectrometerhave been developed by W. Klassen [212–214].4.2.7 Spin AnalysisRamsey’s method requires spin-sensitive UCN detection to count the number of UCNs in agiven spin state. Such detection can be achieved by exploiting the interaction of UCNs withmagnetic fields, previously discussed in Section 1.3. A large magnetic potential will block onespin state while allowing the other to pass through. By itself, such a magnetic field allows onestate to be passed into a detector. The other state can be detected by flipping the spins of theUCNs before they reach the magnetic potential. Because the UCN population will arrive at thedetectors at different times, and with different velocities, this spin flip cannot be easily achievedin the same way as in the Ramsey method. Instead, an adiabatic spin flipper (ASF) is used.In an ASF, a uniform ~B1 is applied, but the holding field ~B0 is made inhomogeneous such thatit gradually reverses over the length of the ASF. With correctly tuned field magnitudes andgradients, the spins of the UCNs adiabatically follow the combined field and the population isinverted at the exit of the ASF [215–217].There are two different possible implementations of spin analysis for an nEDM experiment.The two types of analyzers are referred to as the alternating spin analyzer (ASA) and thesimultaneous spin analyzer (SSA). The ASA consists of only one of the spin-sensitive detectorsdescribed above, and was first used in the measurements at ILL [103]. With an ASA, thedifferent spin states are counted by periodically turning the ASF on or off. With an SSA,two analyzers joined by a forked UCN guide are used. By having one analyzer with an activeASF, and one without, the two states can be counted at the same time. This technique wasfirst developed as part of the PNPI experiments [101] and was also used in the PSI nEDMmeasurement [114]. The use of an SSA naturally results in higher UCN counts, as in the ASAone of the spin populations is decaying (due to wall losses and depolarization, as well as betadecay) while the other is being counted. Using an ASA also complicates calculation of theasymmetry, as the populations are being counted at different times, which could introducesome differences between the two populations.For the TUCAN EDM spectrometer, an ASA will be implemented for the OC100 phase,followed by an upgrade to an SSA. The spin analysis is achieved using a thin iron film magnetized834.3. Overview of Sensitivity Effectsby a dipole magnet. The field inside the foil is large enough to allow only one spin state topass through the foil. To measure the other state, an ASF is placed above the foil and usedto invert the spins of the UCNs. This analysis method is described in detail in the work of S.Hansen-Romu [174] and T. Higuchi [218]. For the SSA implementation, both arms include anASF. This makes it possible to periodically switch which state is detected by each detector fora given Ramsey cycle as a check on asymmetries in the detection.For the TUCAN EDM experiment, the spin analysis requires a detector capable of a highcount rate, expected to be on the order of 1 MHz. A variety of detectors have been developedcapable of achieving such a rate. The 6Li detector used for the experiments of Chapter 3 is onepossibility. Other technologies have been explored, including CASCADE gas electron multiplierdetectors for the PanEDM experiment [219], 10B ZnS:Ag scintillating detectors in the LANLEDM experiment [220], and CF4 scintillating gas detectors in the PSI n2EDM experiment [221].The TUCAN collaboration is also pursuing the CF4 technology as an alternative to the6Lidetectors.4.3 Overview of Sensitivity EffectsThis section presents an overview of effects that contribute to the overall sensitivity of thenEDM experiment. The focus is primarily on systematic effects, with the statistical sensitivityhaving been previously studied by S. Sidhu [6], but several small effects that increase statisticalsensitivity will be mentioned, and the requirements on depolarization will also be discussed.The theoretical statistical uncertainty on the nEDM measured using Ramsey’s method isσdn,stat ≈~2α| ~E|(TR + 4τpi )√Ntot(1 +pi2s28). (4.1)The derivation of this equation is given in Appendix A.3. The factor pi2s2/8 is the increase inuncertainty due to the choice of working points, recalling that s parametrizes the separation ofthe working points. A typical choice of s = 0.1 is therefore predicted to increase the uncertaintyby only 1.2%. The approximate nature of this equation is due to the assumption that theworking points are taken close to values ωn ± δω/2, for which the uncertainty is minimized.The statistical uncertainty depends linearly on the magnitude of the applied electric field| ~E|. This value is determined by the ability to sustain a high-voltage in the cells with the Hgvapour present. The electric field is discussed further in Chapter 6.The neutron counts Ntot and the timings of the Ramsey method τ and TR are interrelatedthrough the storage lifetime of the UCNs. This storage lifetime is impacted by geometric andmaterial factors. This topic is studied in detail in the dissertation of S. Sidhu [6]. From thesestudies it is assumed that the TUCAN EDM experiment will be performed with TR ≈ 150 s,and τ = 2 s. Measurements of the cell storage lifetime are presented in Chapter 5.The visibility α is determined by the depolarization lifetime of the UCNs, labelled T2, aswell as the analyzing power of the spin analyzer. The depolarization lifetime in the cell isdetermined by the probability of depolarization per wall bounce and the depolarization due tomagnetic field inhomogeneity. The contribution from wall bounces is simplyT2,w =1βfw, (4.2)where fw is the wall-bounce frequency, and β is the depolarization probability per wall bounce.Depolarization probabilities have been measured for a number of materials, with the values for844.3. Overview of Sensitivity Effectsmaterials used in precession cells typically not greater than 10−5. With a wall-bounce frequencyof fw ≤ 20 Hz, this gives a limiting polarization lifetime of T2,w ≈ 5000 s, or α = 0.97. Reachingthis lifetime requires the minimization of magnetic contributions, which is described in Section4.3.5.Before proceeding to discuss systematic contributions, it is useful to consider how the co-magnetometer measurements can be introduced into the measurement. This introduction hasan impact on both statistical and systematic uncertainties. To correct the magnetic field fluc-tuations, the ratio〈ωHg〉ωHg,iis introduced into the asymmetry equation, where 〈ωHg〉 is the averagefrequency measured over the four working points, and ωHg,i is the frequency measured at work-ing point i. The modified asymmetry equation isA = A0 − α cos(pi2∆ω(ωn − ωi)〈ωHg〉ωHg,i)= A0 − α cos(pi2∆ω(ωnωHg,i− ωiωHg,i)〈ωHg〉)= A0 − α cos(pi2∆ω(R− ωiωHg,i)〈ωHg〉).(4.3)The effect of using R in the modified equation can be seen by assuming a working-point-dependent variation δB0,i from the main field B0. At any working point, the ratio of frequenciesisRi = −γnB0 − γnδB0,i + 2~−1dnE +∑δωn−γHgB0 − γHgδB0,i + 2~−1dHgE +∑δωHg(4.4)=−γnB0−γHgB01 +δB0,iB0− 2 dnEγnB0~ − (γnB0)−1∑δωn1 +δB0,iB0− 2 dHgEγHgB0~ − (γHgB0)−1∑δωHg≈ γnγHg(1 +δB0,iB0− 2 dnEγnB0~− (γnB0)−1∑δωn)×(1− δB0,iB0+ 2dHgEγHgB0~+ (γHgB0)−1∑ δωHg) ,where this approximation is achieved by a Taylor expansion of (1−x)−1 ≈ 1+x, assuming thatall of the other terms are small shifts,≈ γnγHg(1 +δB0,iB0− 2 dnEγnB0~− (γnB0)−1∑δωn (4.5)− δB0,iB0+ 2dHgEγHgB0~+ (γHgB0)−1∑ δωHg),854.3. Overview of Sensitivity Effectswhere higher order terms involving small shifts multiplied by each other have been discarded,=γnγHg(1− 2 dnEγnB0~− (γnB0)−1∑δωn (4.6)+ 2dHgEγHgB0~+ (γHgB0)−1∑ δωHg) = R,where δωn and δωHg are any frequency shifts in the UCNs or Hg gas respectively. To leadingorder, the ratioR will be independent of δB0,i, so the fitting of the asymmetry can be performedfor R instead of ωn.If all frequency shifts not related to dn are removed, then the frequency ratio becomesR+ = γnγHg(1− 2 dnEγnB0~), (4.7)when ~E is aligned with ~B0, andR− = γnγHg(1 + 2dnEγnB0~)(4.8)when ~E is anti-aligned with ~B0. Then, dn can be determined asdn = ~γHgB0,−R− − γHgB0,+R+4E= ~〈ωHg〉−R− − 〈ωHg〉+R+4E. (4.9)The modification of the asymmetry equation in this way alters the treatment of statisticaluncertainty, which is derived based on the assumption of fitting the precession frequency, notR. Appendix A.3.3 includes a short derivation of the modified uncertaintyσRR ≈σωnωn√1 +σωi2σωn2+σωHg2σωn2R2. (4.10)The applied frequency ωi is assumed to be very precise and to have negligible σωi . Eachsupercycle is expected to have uncertainty on the order of ∼ 10−25 ecm, or 4µHz per Ram-sey measurement, while the Hg co-magnetometer aims for a 10 fT measurement per cycle, or0.24µHz. With R ≈ γn/γHg ≈ 3.8, this results in an approximately 3% increase in statisticaluncertainty due to the use of the co-magnetometer.There is some further contribution from the co-magnetometer from needing to multiply Rby 〈ωHg〉 in the calculation of dn. This term has a relative uncertainty σωHg2/4〈ωHg〉2, which isnegligible.The systematic uncertainty of the nEDM measurement is determined by any effects thatcould produce a signal that looks like an electric dipole moment. With the combination of UCNsand Hg, this can be caused by any frequency shift of either the neutrons or co-magnetometerwhich is correlated with the electric field polarity. Some analysis techniques additionally in-troduce effects from sources which are not correlated with the electric field. The study ofsystematic effects in the nEDM experiment is therefore the characterization and removal ofthose frequency shifts not related to dn, which includes all shifts for both the UCNs and Hggas of the co-magnetometer. To achieve σsys ≤ 10−27 ecm systematic uncertainty, a reasonablegoal is to limit the largest systematic to 5 × 10−28 ecm. Then, up to eight other systematiccontributions of 3× 10−28 ecm are permitted, or up to eighteen contributions of 2× 10−28 ecm.These levels will be used in the following discussions and subsequent chapters.864.3. Overview of Sensitivity Effects500160BCR4.750.515DETAIL BInsulator trench8520R6.35R11DETAIL CPlugged entrance cavityElectrodesWallFigure 4.3: Schematic showing the geometry of the cells assumed for systematic calculationsand PENTrack simulations. Dimensions are given in mm.874.3. Overview of Sensitivity Effects0 50 100 150UCN energy (neV)050010001500200025003000UCN counts (per 3 neV)Top kinetic energyBottom kinetic energyTop total energyBottom total energyFigure 4.4: Plot of the energy spectra of UCNs used for PENTrack simulations and calculations.The spectra were taken from the simulations of Ref. [6]. Solid lines show the kinetic energyspectrum, while dashed lines show the total energy (kinetic energy plus gravitational potentialenergy).4.3.1 General Features of the UCN PopulationSeveral of the discussions that follow in this and later chapters will depend on the geometry ofthe experiment and properties of the UCN population in the experiment. This section lists therelevant features.The cell geometry is shown in Figure 4.3. This geometry assumes a radius R = 0.25 mand electrode separation H = 0.16 m, which is the currently intended geometry in the 2C200configuration. Further details of the cell entrance and insulator trenches are shown in the figure.When discussing PENTrack simulations of any sensitivity effect, the simulation geometry isimplemented as shown in Figure 4.3.Two different coordinate systems are relevant when discussing systematics. When discussingsystematics for OC100, a coordinate system with the origin at the centre of the cell is used.When discussing systematics in 2C200, a coordinate system with the origin between the twocells, in the centre of the HV electrode, is used. In the double cell case, the bottom of the topcell is at z = 0.04 m, and the top of the bottom cell is at z = −0.04 m.Some example energy spectra of the UCN population have been extracted from the simula-tions of S. Sidhu [6]. These are shown in Figure 4.4. These energy spectra result in non-uniformheight distributions for the UCNs, shown in Figure 4.5. These spectra should be considered tobe representative of the energy spectra that can be expected in the cells, but are not necessarilyexact.The different kinetic energy spectra result in different mean velocities in the cells. In thetop cell the mean velocity is 3.632 ± 0.003 m/s, and in the bottom cell the mean velocity is884.3. Overview of Sensitivity EffectsFigure 4.5: Plot of the height distribution of UCNs seen in PENTrack simulations with theenergy spectra from Figure 4.4. A roughly linear distribution is observed, with UCNs slightlyfavouring lower heights.3.322 ± 0.004 m/s. The average wall-bounce frequency is 21.20 ± 0.03 Hz in the top cell, and15.32± 0.04 Hz in the bottom cell.These values may seem contrary to what would be expected: since the top cell is higher,more of the energy of those UCNs would presumably be used to climb to the greater height, sothe mean energy, and thus velocity, should be lower in the top cell. However, there is anothereffect that dominates. The UCN guides that connect the source to the top cell must first riseabove the cell before dropping down into the cell. The bottom of the guides are approximately25 cm above the centre of the central electrode, corresponding to an energy of ∼25 neV. Theadditional height that UCNs need to climb to reach the top cell therefore cuts out lower-energyUCNs, shifting the spectrum of the top cell towards higher energy. Similarly, the UCN guidesthat connect the source to the bottom cell must first drop below the cell before rising up intothe cell, with a similar height. The additional drop can increase the energy of higher-energyUCNs above the Fermi potential of the guides, potentially shifting the spectrum of the bottomcell towards lower energy. More about the guide configuration and spectra can be found in Ref.[6].The gravitational potential results in the UCN population’s centre of gravity being slightlylower than the centre of the cell. In the top cell, the offset is −1.7± 0.2 mm, and in the bottomcell it is −2.1± 0.2 mm.4.3.2 Magnetic Field DecompositionMuch of the discussion that follows will revolve around the magnetic fields present in theexperiment. In the absence of currents, the fields will satisfy Laplace’s equation for a scalar894.3. Overview of Sensitivity Effectsl m Πx Πy Πz0 -1 0 1 00 0 0 0 10 1 1 0 01 -2 y x 01 -1 0 z y1 0 −12x −12y z1 1 z 0 x1 2 x −y 02 0 −xz −yz z2 − 12(x2 + y2)3 0 38(x3 + xy2 − 4xz2) 38(x2y + y3 − 4yz2) z3 − 32z(x2 + y2)4 0 xz(32(y2 + x2)− z2) yz(32(y2 + x2)− 2z2) z4 − 3z2(x2 + y2)5 0 −52xz4 + 154 x3z2 −52yz4 + 154 y3z2 z5 − 5x2z3 − 5y2z3+154 xy2z2 − 516x5 +154 x2yz2 − 516x4y +158 x4z−58x3y2 − 516xy4 −58x2y3 − 516y5 +158 y4z + 154 x2y2zTable 4.1: Table of the harmonic polynomial basis functions for magnetic fields. The functionsshown are considered the most relevant for nEDM experiments at the 10−27 ecm level.magnetic potential, and can be described using the free-space harmonic polynomial basis [222].In this basis, the magnetic field is given by~B0 =∑l,mGl,m~Πl,m =∑l,mGl,m (Πx,l,m,Πy,l,m,Πz,l,m) , (4.11)where ~Πl,m is the polynomial vector of order l = 0, 1, 2... and m = −(l−1)..(l−1), and Gl,m arecoefficients describing the magnitude that a given basis function contributes to the magneticfield. A detailed description of this basis, including a list of all basis functions up to order l = 5,is given in Appendix B.4. Table 4.1 lists some of the most important basis functions for thenEDM experiment.4.3.3 Single Cell vs Double CellBefore looking at the details of the sensitivity contributions, a general feature of the double cellconfiguration compared to the single cell configuration should be noted. The use of a doublecell has advantages for both statistical and systematic uncertainty.The statistical gain from a double cell experiment is due to the fact that a larger numberof UCNs can be measured at once without having to make a much larger cell. Increasingthe cell radius has challenges with manufacturing, and it will also be seen that it increasessystematic effects. Increasing height has limitations due to the gravitational contribution toUCN energies and requires a higher voltage to provide the same field. Increasing the numberof cells is therefore a strategy to improve statistical sensitivity.The systematic gain from the double cell comes from the combination of the data from thetwo cells. There are two ways of looking at a double cell experiment:1. Measurements in each of the cells taken at different times with different HV polarities canbe paired together to form two separate measurements of the nEDM, dn,top in the top celland and dn,bot in the bottom cell, which are then averaged to produce a final value, or904.3. Overview of Sensitivity Effects2. Measurements in the two cells taken at the same time can be paired together to formtwo separate measurements of the nEDM, dn,HV+ at positive HV polarity and dn,HV− atnegative HV polarity, since the stacking of the cells results in one cell having a field Eand the other a field −E. These two measurements are then averaged to produce a finalvalue.The four dn measurements are combinations of four R measurements: Rtop,HV+ in the top cellwith positive HV, Rtop,HV− in the top cell with negative HV, Rbot,HV+ in the bottom cell withpositive HV, and Rbot,HV− in the bottom cell with negative HV. From the perspective of thefirst interpretation, the final result isdn =dn,top + dn,bot2=~〈ωHg〉top,HV+Rtop,HV+ − 〈ωHg〉top,HV−Rtop,HV−8E+ ~〈ωHg〉bot,HV−Rbot,HV− − 〈ωHg〉bot,HV−Rbot,HV+8E.(4.12)Using the second interpretation, the result isdn =dn,HV+ + dn,HV−2=~〈ωHg〉top,HV+Rtop,HV+ − 〈ωHg〉bot,HV+Rbot,HV+8E+ ~〈ωHg〉bot,HV−Rbot,HV− − 〈ωHg〉top,HV−Rtop,HV−8E.(4.13)By inspection, these are the same final result. The importance of these two interpretations andthe forms of their final cancellations is that1. Frequency shifts which are the same in one of the cells between the two HV polarities willbe cancelled, and2. Frequency shifts which are the same in the top and bottom cells with the same HV polaritywill be cancelled.In the single cell, only the first treatment, pairing different-polarity measurements, is possible,so only frequency shifts that are the same in the cell between the two polarities cancel. Thedouble cell therefore has an advantage in systematic uncertainty over the single cell.The double cell introduces one disadvantage. In order to perform the Ramsey measurementin the two cells simultaneously, the average values of the magnetic fields must satisfy a conditionso that the working points are valid in both cells. If ωn is the precession frequency with nogradient, then the actual precession frequencies will beωn,top = ωn + ∆ωTB/2ωn,bot = ωn −∆ωTB/2(4.14)where ∆ωTB is the difference between the top and bottom cells. The choice of working pointsin the cells will be offset by ∆ωTB/2 from where they should be. This modifies the statisticaluncertainty toσdn,stat ≈~2α| ~E|(TR + 4τpi )√Ntot(1 +pi28(s+∆ωTB∆ω)2)(4.15)914.3. Overview of Sensitivity EffectsWith an initial s = 0.10, with ∆ωTB/∆ω = 0.05 the increase in uncertainty is 1.5%, with∆ωTB/∆ω = 0.10 it is 3.6%, and for ∆ωTB/∆ω = 0.20 it is 9.7%. With TR = 200 s and τ = 2 s,∆ω ∼ 0.0078 s−1. The limit ∆ωTB/∆ω < 0.10 therefore requires that ∆ωTB/γn < 4.2 pT.This is a fairly strong requirement on the magnitude of vertical gradients. For l = 1, m = 0,∆ωTB/γn = G1,0∆H, where ∆H is the difference between the centres of the UCN distributionsin the top and bottom cells. Neglecting gravity, this is approximately the difference between thecentres of the cells, which is 0.20 m. The requirement becomes G1,0 < 20 pT/m. For l = 3, therequirement becomes G3,0 < 670 pT/m3. For higher order gradients, the magnitude required fora significant effect is large compared to the expected magnitude of the gradient, so no conditionis required.4.3.4 Redfield TheoryMajor contributions to both visibility and systematic shifts can be derived from Redfield’stheory of relaxation for nuclear magnetic resonance [223]. The application of this theory tonEDM experiments has been developed by a number of talented physicists [224–226], and thework in this section and the following sections is based heavily on that prior work.Very generally, the measurement of the nEDM can be assumed to occur in a time-varyingmagnetic field (Bx(t), By(t), Bz(t)). This time-variance can describe changes in the fields inreal time, but can also describe the observed changes in the field in the rest frame of a movingparticle. This field can be reformulated in terms of an average fieldB0 =√〈Bx(t)〉2 + 〈By(t)〉2 + 〈Bz(t)〉2 (4.16)and time-varying fluctuationsbi(t) = Bi(t)− 〈Bi(t)〉 , (4.17)which have zero average over time. The coordinate system is defined such that the average fieldis along the z-direction.From this starting point, Appendix B presents the derivation of the relaxation matrix ofa spin population using the density operator formulation of a spin population. The elementsof this matrix describe the frequency-shifts and relaxation rates of the population due to thefluctuating magnetic fields, and apply to both UCNs and Hg atoms. The element of this matrixdescribing the transverse relaxation rate Γ2,m = T2,m−1 isΓ2,m =12∫ ∞0cos(ω0t′)〈ωx(t′)ωx(0) + ωy(t′)ωy(0)〉dt′+12∫ ∞0sin(ω0t′)〈ωy(t′)ωx(0)− ωx(t′)ωy(0)〉dt′+∫ ∞0〈ωz(t′)ωz(0)〉dt′,(4.18)where ω0 = γB0, ωx = γbx, and 〈...〉 describes an ensemble average over the spin population.The forms of the quantities in these averages, in which a variable at an initial time is multipliedby a variable at a later time, is a correlation function, and so these quantities will often bereferred to as correlations, or autocorrelations when the variable at the two times is the same.The element that describes the frequency shifts isδω =12∫ ∞0cos(ω0t′)〈ωx(t′)ωy(0)− ωy(t′)ωx(0)〉dt′+12∫ ∞0sin(ω0t′)〈ωx(t′)ωx(0) + ωy(t′)ωy(0)〉dt′.(4.19)924.3. Overview of Sensitivity EffectsThese results apply for any magnetic field that can be described in the form above, under threeconditions:1. the fluctuating components Bi(t) are small relative to the average field B0,2. the timescale of the measurements are long compared to the correlation time τc, which isthe time at which 〈ωz(τc)ωz(0)〉 goes to zero,3. the system is statistically stationary.In the nEDM experiment there is a significant relativistic motional magnetic field appearing inthe rest-frame of a particle,~B~v× ~E =~E × ~vc2. (4.20)Combining the motional field with other magnetic fields leads to four terms for Γ2,m:Γ2,m,B2 =γ2∫ ∞0cos(ω0t′)〈bx(t′)bx(0) + by(t′)by(0)〉dt′, (4.21)Γ2,m,E2 =γ2E2c4∫ ∞0cos(ω0t′)〈vx(t′)vx(0) + vy(t′)vy(0)〉dt′, (4.22)Γ2,m,BE =γ2Ec2∫ ∞0sin(ω0t′)〈vx(t′)bx(0) + vy(t′)by(0)〉dt′, (4.23)Γ2,m,B2z =γ2∫ ∞0〈bz(t′)bz(0)〉dt′, (4.24)and three terms for δω:δωB2 =γ22∫ ∞0sin(ω0t′)〈bx(0)bx(t′) + by(0)by(t′)〉dt′, (4.25)δωE2 =E2γ22c4∫ ∞0sin(ω0t′)〈vx(0)vx(t′) + vy(0)vy(t′)〉dt′, (4.26)δωBE =Eγ2c2∫ ∞0cos(ω0t′)〈bx(0)vx(t′) + by(0)vy(t′)〉dt′. (4.27)The seven Equations 4.21-4.27 can describe much of the behaviour of the spins of UCNs andHg atoms that is relevant for an nEDM experiment. The depolarization rates play a major role indetermining the visibility α for the statistical sensitivity, while the frequency shifts describe themajority of systematic effects, including those which are anticipated to dominate the systematicuncertainty. It is therefore important to understand the physical meaning of these equations.While the full derivations are given in Appendix B, a qualitative interpretation is given here.Physically, these equations describe how variations in the magnetic field change the be-haviour of a spin population. The depolarization terms Γ2,m describe the dephasing of thespins, which can occur both due to field variation in the transverse plane and due to variationin the direction of the average precession axis (assumed to be the z direction). The frequencyshifts describe the accumulated change in precession frequency, and are only produced by vari-ations in the transverse plane.As mentioned above, the products in the averages give the correlation between quantitiesover time. These depolarization rates and frequency shifts therefore depend on how much the934.3. Overview of Sensitivity Effectsmagnetic field variation seen by the spins correlates with its previous values. Intuitively, whatthis means is that in order for a field variation to have some net effect, there must be sufficientnon-randomness so that changes accumulate rather than averaging to zero. A truly randomvariation in the field, for which the value of the field at any time is completely unrelated to thevalue at previous times, would cause neither depolarization nor frequency shifts, as the effectwould average to zero. The focus in calculating these effects in subsequent sections will thereforebe on the variations caused by magnetic field non-uniformity, for which this autocorrelation canbe significant, while considerations of things such as electrical noise in the magnetic coil currentsupply will be neglected as not being sufficiently correlated.The fact that these equations depend on correlations is also important for understandingwhy there are seven unique equations. The correlation is given by a product of the magneticfield at two different times, and the magnetic field can be described as the sum of two fields (themagnetic field measured in the lab frame and the relativistic motional field). The six equationslabelled as B2, E2, and BE arise from expanding the product and separating the terms. TheB2 terms are correlations of the lab frame magnetic field, the E2 terms are correlations of therelativistic field, and the BE terms are correlations between the relativistic field and the labframe field. The BE term, because it is linear in E, can lead directly to a frequency shift thatmimics an EDM, and is therefore responsible for the most significant systematic shifts.The presence of the sinusoidal terms in the integrals gives a dependence on the averageprecession frequency ω0. Since the interaction of a spin with a magnetic field depends on theorientation of the spin, this “main” precession impacts what effect the field variations can have.If ω0 is large relative to the variation of the field over time, then the spins will see effects ofequal magnitude but opposite sign, and the cumulative effect of the field variations will besuppressed. In order for an effect to be significant, the spins must be precessing slowly enoughthat there is time for a net effect to accumulate. This will lead to differing effects for UCNsand Hg atoms.A dependence on bz appears only in Equation 4.24, which has a uniquely different formcompared to the other equations. Because these variations occur in the direction of the mainprecession axis, this equation is simpler, lacking a sinusoidal term. Variations in the magneticfield in the z direction will cause instantaneous variations in the precession frequency of individ-ual spins. However, since ω0 is defined to be the precession frequency around B0, which is theaverage field seen by the entire population of spins, these variations cannot cause a shift in theaverage precession frequency. Still, because bz causes the precession frequencies of individualspins to vary, it does cause the spins to dephase, and this is what 4.24 describes.The depolarization rates will be discussed further in Section 4.3.5, and the frequency shiftsin Sections 4.3.6, 4.3.7, and 4.3.8.4.3.5 Magnetic DepolarizationThe statistical uncertainty of the Ramsey method scales inversely with the visibility, which islargely determined by the polarization of the UCN population when it is extracted into thespin analyzer. Other than wall depolarization, a population of UCNs can be depolarized dueto magnetic field inhomogeneity and fluctuations. There are two contributions to magneticdepolarization: intrinsic depolarization, which is described by Equations 4.21-4.24 and grav-itationally enhanced depolarization, which is a unique contribution due to the behaviour ofUCNs.944.3. Overview of Sensitivity EffectsIntrinsic DepolarizationFor UCNs, which behave adiabatically, the transverse contributions to the depolarization canbe expanded in terms of the fields using successive integration-by-parts [226], to arrive at theresultsΓ2,m,B2 =γn2[− 1ω02〈bx(0)b˙x(0) + by(0)b˙y(0)〉+1ω04〈bx(0)...bx(0) + by(0)...by(0)〉]Γ2,m,E2 =γn2E2c4[− 1ω02〈vx(0)v˙x(0) + vy(0)v˙y(0)〉+1ω04〈vx(0)...vx(0) + vy(0)...vy(0)〉]Γ2,m,BE =γn2Ec2[− 1ω0〈bx(0)vx(0) + by(0)vy(0)〉+1ω03〈bx(0)v¨x(0) + by(0)v¨y(0)〉],(4.28)where ˙ is used to denote a time derivative.All three of these expressions will contribute very little to the depolarization. In the case ofΓ2,m,B2 , bi will depend on position, and so b˙i will depend on velocity. Since the velocities andpositions at t = 0 should not be correlated, the average in this expression will therefore be zero.A similar argument applies to Γ2,m,BE , which depends explicitly on velocity. The Γ2,m,E2 termas written is difficult to interpret, since it depends on v˙i. The transverse velocities of UCNschange only in discrete steps when reflecting from a wall, so most UCNs have v˙x = v˙y = 0. Thisterm is also highly suppressed due to the factor of E2/c4, and previous detailed studies haveshown that this effect is not significant for UCNs [227]. Therefore all three of the transverseterms are expected to be small. Physically, this means that the variations of the magnetic fieldin the plane transverse to the precession axis do not significantly lead to an accumulation ofphase differences that would result in the population dephasing.There is a fourth depolarization term given by Equation 4.24. This describes the contribu-tion from variation in the z component of the magnetic field, and takes the formΓ2,m,B2z = γn2∫ ∞0〈bz(0)bz(t′)〉dt′. (4.29)This contribution is not expected to be small. The integral depends on the trajectories ofUCNs and can be evaluated using PENTrack simulations. For these simulations, UCNs werecreated inside a cell and allowed to evolve over time. Their trajectories were recorded, fromwhich the relevant features of the UCNs’ behaviour can be extracted. Figure 4.6 shows examplecalculations of the integrand over time extracted from such simulations. This example uses afield Πz,l,m(z) with l = 1 and m = 0, as defined in Table 4.1, corresponding to a linear gradientin z. The simulated correlations show a decay with time as the field value becomes uncorrelatedwith the initial value. A difference in shape for differing energy levels can also be observed,with a seemingly faster decay for higher energy.954.3. Overview of Sensitivity Effects0 0.5 1 1.5 2Time t (s)-1-0.500.511.522.5 1,0,z(0)1,0,z(t) 10-3[52,54] neV[92,94] neV0 0.1 0.20102010-4Figure 4.6: Simulated value of the autocorrelation of the magnetic field basis function Π1,0,z = zfor UCNs versus time. The field variation here is due solely to the motion of the UCNs.The autocorrelation is shown for two different energy bins in a cell with 2% diffuse reflection.Energy is reported as total energy, the sum of the kinetic energy and gravitational potentialenergy measured from the centre of the HV electrode, and is binned in 2-neV-width bins. Thecorrelation functions display an initial peak followed by a rapid decay to zero. The inset showsthe difference in the shape of the correlations in the initial peak for the two different energylevels.964.3. Overview of Sensitivity EffectsFigure 4.7: Simulated values of T2,m versus energy for a magnetic field of G1,0 = 1 nT/m. Theenergy spectrum is plotted on the second vertical axis for reference. T2,m is seen to increase withincreasing energy, consistent with the faster decay of the autocorrelation function seen in Figure4.6. T2,m is also seen to increase significantly with PL, also consistent with the expectation thatlarger probabilities of diffuse reflection cause UCN trajectories to be randomized more quickly,reducing the correlation over time.The difference in the shape of the correlation function for different energy levels could bedue to the different velocities, or due to the field differing at different heights. No studies toidentify the dominant effect have been performed, as it suffices to consider the depolarizationbinned by energy. Figure 4.7 shows T2,m versus energy for a gradient of G1,0 = 1 nT/m. Itis observed that T2,m increases gradually with increasing energy, which is consistent with thecorrelation function decaying more quickly with higher energy, as seen in Figure 4.6. Diffusereflection probability is also seen to play a major role, with higher diffuse reflection probabilitiesPL resulting in longer T2,m. This can be understood as higher PL causing the trajectories ofUCNs to be randomized more quickly, further reducing the correlation function when integratedover time.The total visibility after the precession time TR will be related to the intrinsic depolarizationrate byα(TR) = α(0)∫ Hmax0n(H)e−Γ2,m,B2z (H)TRdH, (4.30)where n(H) is the total energy spectrum. Figure 4.8 shows the visibility calculated from simu-lation versus the magnitude of gradient.Diffuse reflection probability is seen to play a significant role in the depolarization. ForG1,0 and G2,0, a more specular surface causes greater depolarization. This can be explainedby considering that diffuse reflections cause positions to become uncorrelated more rapidly,reducing the depolarization rate.For G1,1, the simulations suggest the opposite: a more specular surface sees less depolar-ization. It is not clear why this should be the case. One possible explanation is that, without974.3. Overview of Sensitivity Effects10-1 100 101Gl,m (nT/ml)0.750.80.850.90.9511.05Visibility G1,0, PL=0%G1,0, PL=10%G1,1, PL=0%G1,1, PL=10%G2,0, PL=0%G2,0, PL=10%Figure 4.8: Visibility after 150 s for different magnitudes of G1,0, G1,1, and G2,0, calculated usingintrinsic depolarization rates from PENTrack simulations. Results are shown for simulationswith diffuse reflection probabilities PL of 0% and 10%. Below the level of 1 nT/ml, the effectsare negligible with the exception of G1,0. However, the depolarization from G1,1 increases asPL increases, and could become a dominant effect in a cell with a high value of PL.diffuse reflections, UCNs can occupy orbits in which they effectively do not depolarize, as thevariation in precession frequency becomes periodic over the orbit. However, there is no obviousexplanation for why this would not be the same for G1,0 and G2,0. Since bz for the m = 0 modesdepend on z, while G1,1 depends only on y, this difference may be related to the contribution ofgravity to the distribution of UCNs in z. This is only speculative, and while this seems worthstudying further, the topic of depolarization is not a focus of this dissertation, so it is left tofuture work.The gradient with the greatest effect on visibility is G1,0. However, the condition for cellmatching discussed previously would restrict G1,0 < 20 pT/m, so it would not significantlycontribute under that condition. Higher order l-odd gradients would be similarly restricted,and their contributions will be smaller than that of G1,0.There is no cell-matching condition on G2,0, or other l even vertical gradients, as they aresymmetric for the two cells. It is observed that, with G2,0 < 1 nT/m2, the reduction in visibilitybecomes negligible. Contributions from higher order gradients will be increasingly small. If thevariation of the field from the central value (measured at the centre of the HV electrode) is keptsmall, such that∆Bz = |Bz(z = 20 cm)−Bz(z = 0 cm)| < 40 pT (4.31)then the even Gl,m contribution for l even and m = 0 will be kept below 1%.For m 6= 0, G1,1 < 1 nT/m would reduce the depolarization sufficiently to be negligible.Higher order gradients will contribute, but will also have increasingly small effects. Restrictingthe total variation inside the cells such that∆Bz = |Bz(x = 25 cm)−Bz(x = 0 cm)| < 250 pT (4.32)984.3. Overview of Sensitivity Effectswould meet the requirement for G1,1 while also rendering higher-order m 6= 0 contributionsnegligible.The TUCAN EDM apparatus has set a goal of ∆Bz < 140 pT within each cell using then× n coil array. This generously satisfies the condition for G1,1, but is not stringent enough tomake G2,0 negligible. This goal allows G2,0 < 2.25 nT/m2, and could therefore see a reductionin visibility of 1–2%, equivalent to an increase in statistical uncertainty of 1–2%. This could beimproved if more stringent magnetic field uniformity is achieved, which could be accomplishedusing dedicated Gl,m coils to tune specific contributing modes such as G2,0, but the gain instatistical uncertainty when considering only intrinsic depolarization would be small.It is interesting to compare these results qualitatively to those of Ref. [228], which reportson similar studies. In those studies, G1,1 was shown to have a lower T2,m, and thus would bethe dominant contribution, which is opposite to what is suggested by Figure 4.8. However,in Ref. [228] a much larger diffuse reflection probability was assumed in the cells, between20–100%. Considering that T2,m from G1,1 seems to decrease at higher PL, while T2,m fromG1,0 seems to increase, and that the curves overlap at PL=10% in Figure 4.8, it is likely thatG1,1 becomes dominant in a cell with large diffuse reflection probability. Real measurementsof the depolarization will therefore be needed to better determine the dominant contributionto intrinsic depolarization in the TUCAN EDM experiment and how best to achieve a highvisibility. Such studies could be readily accomplished in the OC100 version of the experiment,allowing improvements for the 2C200 version.Gravitationally-Enhanced DepolarizationAn additional contribution to depolarization occurs due to the strong role that gravity playsin the distribution of UCNs in the precession cells [228]. While UCNs of different energy willdepolarize at different intrinsic rates, they will also have different average heights, and willtherefore precess at different average rates due to vertical gradients. With a gradient G1,0, aUCN that is, on average, higher than another UCN by a distance ∆z will precess about a fieldthat differs by G1,0∆z, and so its precession frequency will differ by γnG1,0∆z. The differencein average precession rate is not captured by the previous description of depolarization, whichconsiders the fluctuation of the precession frequency away from the average.In general, over the precession time TR, UCNs with energy H will accumulate a phaseφ¯(H) = γn∫ TR0B0(x(t), y(t), z(H, t))dt. (4.33)The entire population will have an average phaseφ¯ =∫ Hmax0φ¯(H)n(H)dH (4.34)The difference of the individual phases from the average phase will cause depolarization. Thevisibility of the population due to this dephasing will beα(TR) =∫ Hmax0cos(φ¯(H)− φ¯)n(H)dH (4.35)In the presence of a vertical gradient of magnitude G1,0, the phase difference will beφ(H, TR)− φ¯ = γnG1,0(z¯(H)− 〈z〉)TR, (4.36)994.3. Overview of Sensitivity Effectswhere z¯(H) is the average z position of UCNs at energy H, and 〈z〉 is the average z of the entirepopulation of UCNs.For an l = 2 gradient of magnitude G2,0 the phase difference will beφ(H, TR)− φ¯ = γnG2,0(Πz,2,0(H)− 〈Πz,2,0〉)TR, (4.37)where Πz,2,0(H) is the average value of the G2,0 basis function for UCNs of energyH and 〈Πz,2,0〉is the average over the entire population.Figure 4.9 plots the simulated visibilities due to this gravitational depolarization. Gravitationally-enhanced depolarization is seen to contribute more strongly than intrinsic depolarization at agiven gradient. The condition G1,0 < 20 pT/m would be sufficient to make the first order con-tribution negligible. However, the condition that G2,0 < 2.25 nT/m2 allows for a significantsecond order contribution to depolarization.The calculated contribution from gravitational depolarization is seen to depend on PL inthe cell, with the visibility ranging from 0.92 for PL = 0% to 0.98 for PL = 10%. This can beunderstood by considering the behaviour of individual UCNs. When a diffuse reflection occurs,it can change the velocity component vz of a UCN in the z direction. It is obvious that this canchange the average height of that UCN: if vz was set to zero upon reflection from the bottomof the cell, then the UCN’s trajectory would become confined to the bottom of the cell (this isan extreme case, but illustrative of the general effect). By shuffling individual UCNs betweenaverage height levels, diffuse reflections can therefore reduce the accumulation of net phasedifferences between the individual UCNs. In other words, diffuse reflections appear to reducethe stratification of the UCN population into different well-defined average heights.The calculation for figure 4.9 was done on a per UCN basis, rather than with binning byH as the above equations suggest (this was done because binning was observed to reduce theoverall effect of gravitational depolarization). This means that the PL dependence is not anartifact of any averaging over parts of the population. Instead it must be due to an actualreduction in the variation of the average heights of individual UCNs from the average height ofthe population. This effect is interesting, but will not be discussed further here as it is outsidethe scope of this dissertation.The targets for the uniformity of the magnetic field in the TUCAN EDM experiment,achieved using the n × n coil array, in addition to the cell-matching condition achieved usinga dedicated G1,0 coil, will likely result in a visibility between 0.90 and 0.95. Consideringthe expected visibility from T2,w is 0.97, the final visibility when considering all sources ofdepolarization would be in the range of 0.87 to 0.92. This could be improved with the additionof a G2,0 coil to counteract that specific field mode, with the potential reduction in statisticaluncertainty being ∼10%. Because these results depend heavily on PL in the cells, whether ornot this is necessary will only become clear with real measurements of UCN depolarization.This is beyond the scope of this dissertation, and so the magnetic uniformity targets discussedhere will therefore be assumed to be unchanged in subsequent sections.4.3.6 BE Frequency ShiftsThe largest contribution to systematic shifts and uncertainty in the nEDM experiment comesfrom the BE frequency shifts of Equation 4.27. Because these shifts depend on the directionof the electric field, they will reverse with electric field direction, and produce a net frequencyshift when taking the difference in frequencies between the two polarities. In this way, the BEfrequency shifts introduce a false EDM signal.1004.3. Overview of Sensitivity Effects10-1 100 101Gl,m (nT/ml)0.850.90.9511.05Visibility G1,0, PL=0%G1,0, PL=10%G2,0, PL=0%G2,0, PL=10%Figure 4.9: Visibility after 150 s due to gravitational depolarization for different magnitudes ofG1,0 and G2,0, calculated using height distributions from PENTrack simulations. Results aregiven from simulations with diffuse reflection probabilities PL of 0% and 10%. Gravitationaldepolarization is seen to be a substantial effect compared to intrinsic depolarization.1014.3. Overview of Sensitivity EffectsThese shifts arise due to the combination of the lab-frame magnetic field with the motional~v× ~E field, and the shifts are therefore substantially different for slow-moving UCNs and for fast-moving co-magnetometer gas. UCNs are slow enough that they behave adiabatically. Underthat condition it can be assumed that ω0τc 1, and Equation 4.27 can be expanded in powersof ω0−1 to obtain the leading-order contributionδωBEn ≈ − γn2Ec2ω02(〈∂bx∂xvx2〉+〈∂by∂yvy2〉). (4.38)The co-magnetometer gas behaves non-adiabatically. Under that condition it can be assumedthat ω0τc 1, and Equation 4.27 can be expanded in powers of ω0 to obtain the leading-ordercontributionδωBEHg ≈ γHg2Ec2〈bxx+ byy〉. (4.39)A detailed derivation of these results is given in Appendix B.3.It is useful at this point to consider the relative magnitudes of these shifts. If the transversemagnetic field is due to a linear gradient ~B = G1,0(−x2 ,−y2 , z), then, assuming vx2 = vy2 =v2/3, the shift for the UCNs isδωBEn =γn2E3c2ωn2G1,0〈v2〉, (4.40)while the shift for Hg isδωBEHg =γHg2Ec2G1,0R22. (4.41)Taking the ratio of the two givesδωBEnδωBEn← Hg=γn2γnγHg2Ec2G1,0〈v2〉3Ec2ωn2G1,0R2=γnγHg2〈v2〉3ωn2R2, (4.42)where the Hg shift has been converted to a neutron shift by multiplication by γnγHg. Assumingsome typical values 〈v2〉 ≈ 9 m/s, ωn ≈ 180 rad s−1, R = 0.25 m, and γn/γHg ≈ 3.8 [229], theresult is δωBEn/δωBEn← Hg ≈ 0.01.Based on this result, the frequency shift for the UCNs is often neglected, while that of theHg is corrected. This is acceptable as long as the absolute magnitude of the frequency shiftis not too large. An absolute frequency shift giving a 10−27 ecm false EDM in the Hg wouldcorrespond to a 10−29 ecm shift in the UCNs. If the Hg effect is corrected to an uncertaintyof 5 × 10−28 ecm, then the UCNs’ contribution would increase this uncertainty by 0.02% —negligibly small. If the absolute Hg effect was 10−26 ecm in the Hg, this correction wouldbe 2%, which should not be neglected. If the absolute Hg effect was 10−25 ecm, then theUCNs would contribute the dominant part of the systematic uncertainty. The absolute fieldhomogeneity therefore does have some importance for systematic contributions. This topic willbe returned to in the subsequent section.The BE frequency shift in Hg in nEDM experiments has been studied extensively. In thefirst experiments to use a co-magnetometer, the focus was on the first order contribution of theG1,0 gradient field [230] and the contribution of small magnetic dipoles [231]. Starting with thePSI nEDM measurement, additional focus has been given to higher order Gl,m fields [222, 232].The formulation from Redfield theory can capture all frequency shift contributions. Thereare two major categories of fields to consider: spatially extended fields, which are measurableover a large volume, and spatially localized fields, which are measurable only in a small volume.1024.3. Overview of Sensitivity EffectsThese measures are relative to the experiment: an extended field can be measured by a sensor(in this case a CsM) located anywhere within the MSR, while a localized field can only bemeasured by a sensor that is sufficiently close to the source of that field. The remainder of thissection focuses on spatially-extended fields. The topic of spatially localized fields is outlinedbriefly in Section 4.3.9 and discussed in more detail in Chapter 7.Spatially Extended FieldsThe volume in which the neutron precession takes place should, in the absence of the centralregion, be free of any sources of magnetic fields. The magnetic field in this volume will be acombination of the shielded background field (from Earth’s magnetic field and other magnetsin the surrounding experimental hall) and the applied magnetic field from the coils.If the coils and other external sources are sufficiently distant from the volume of interest, thenthe field can be expanded as a polynomial and should be well-approximated by the free-spaceharmonic basis described in Section 4.3.2. As a simple example, consider the field produced byan infinite current-carrying wire at a distance r = r0 + r′ from the wire axis, where r0 is thedistance of the centre of the volume of interest from the wire, and r′ is a distance measuredfrom that centre. Assuming that r′ < r0, the field of the wire can be expanded as~B =µ0I2pirφˆ =µ0I2pi1r0 + r′φˆ≈ µ0I2r0pi(1− r′r0+(r′r0)2− ...)φˆ.(4.43)As long as r′ < r0, the field can be written as a polynomial. The accuracy of the basis dependson the ratio r′r0and the order of the expansion. The associated error when truncating to ordern is, from the definition of a geometric series,(r′r0)n+1. In the precession cell volume, whichhas a radius of 0.25 m, assuming that the wire is ∼1 m from the centre of the cells, a polynomialexpansion can be expected to describe the field at the edge of the cells accurately to 0.4% atn = 3 and 0.02% at n = 5.Assuming that the background field can be described by the harmonic basis, the BE con-tributions to systematics can be calculated. The false EDM signal is given bydfn =~4EγnγHg(δωBE(E)− δωBE(−E))=~2EγnγHgδωBE(E)=~γnγHg2c2〈bxx+ byy〉.(4.44)Whether a particular harmonic polynomial contributes a false signal depends on its symmetrywhen taken over the cell volume.All harmonic field terms ~Πl,m with m 6= 0 possess a rotational symmetry which cause theaverages in Equation 4.44 to evaluate to zero when taken over a rotationally symmetric volume.This symmetry can be seen most clearly in the full description of the basis in Appendix B.4, inwhich the scalar potentials are seen to contain factors of cos(mφ) or sin(|m|φ).This symmetry condition can be enforced regardless of where the cell is located. This issimply an observation that the harmonic basis must always be able to describe a free-space1034.3. Overview of Sensitivity Effectsfield in any coordinate system. As an example, consider ~B = G1,−2(y, x, 0) in some arbitrarycoordinate system and assume that the cell is located at an offset position (δx, δy, 0). The fieldscan be transformed into a coordinate system with origin at the cell centre with x′ = x − δx,y′ = y − δy. The resulting field in the new coordinate system isB1,−2 =G1,−2(y′ + δy, x′ + δx, 0)=G1,−2(y′, x′, 0) +G1,−2(δy, 0, 0) +G1,−2(0, δx, 0)=G1,−2~Π1,−2′ +G1,−2δy~Π0,1′ +G1,−2δx~Π0,−1′.(4.45)So the field can be transformed into a basis that is aligned with the cell centre, with the resultof any offset being increased contributions to other fields — in this case, constant x and y fields.This behaviour holds in general, which means that all m 6= 0 fields do not contribute BE shiftswhen the field is measured in a coordinate system that is appropriately aligned.For m = 0 terms in a single cell, only terms with l-odd will contribute. This is due to thepresence of z-odd factors in the even terms, which can be seen in Table 4.1. These cause thefrequency shifts to average to zero over the volume, assuming that z = 0 is at the centre of thecell. A similar argument as above makes it possible to always enforce this condition.In a double cell experiment this condition cannot be enforced, since the centres of the twocells cannot be placed in the same location. Therefore, each cell in a double cell experiment seesa contribution from l-even terms. However, if z = 0 is taken to be the midpoint between the twocells, then the two contributions will cancel when the results from the two cells are combined.In this symmetric case, the integrals over x and y will be the same in the two cells, while theintegral over the z-odd factors will introduce a sign-flip such that the false EDM signals willhave opposite sign between the two cells. From Equation 4.12 it follows that the false EDMswill cancel. As with the previous cases, this argument holds as long as the fields are measuredrelative to the appropriate coordinate system.Given these symmetry arguments, only l-odd m = 0 modes remain to contribute false EDMsignals. The plots in Figure 4.10 show calculated false EDM signals for these modes up to l = 7in the TUCAN EDM experiment geometry. These plots also show the variation in magneticfield that would be produced for a given gradient, plotted against the right axis. It can be seenthat the false signals are large even if the ∆Bz is small. With the goal of ∆Bz < 140 pT on thefield inside the cell, all gradients will contribute much larger systematic signals than desired.Achieving the systematic goal therefore requires additional tools to correct for the frequencyshifts.The variation in the cell does not give a complete picture of whether a gradient will con-tribute. A 3× 10−28 ecm signal from G7,0 would give σ(Bz) ≈ 20 pT, which could be present inthe cell. However, it would also correspond to a ∼0.5µT variation over a 1 m volume inside theMSR. Such a large gradient is not expected to be present in the MSR. A 3× 10−28 ecm signalfrom G5,0 would be present with an ∼8 nT variation over the same volume. Such a variationcould be present in the MSR volume. For that reason, the discussion can be limited to orderl ≤ 5.At this point, the BE frequency shifts from UCNs can be considered. Previously, it wasargued that these would be a factor or ∼100 smaller than the Hg shifts. In Figure 4.10, the G1,0field is the largest contribution, and with ∆Bz < 140 pT could reach 4.3×10−26 ecm before beingcorrected. It could therefore be expected that the UCNs contribute a 4.3×10−28 ecm systematicshift, which would be substantial. However, at the cell-matching condition G1,0 ≤ 20 pT/m, theuncorrected Hg contribution is 1.0 × 10−27 ecm, and the UCN contribution would be roughly1.0 × 10−29 ecm. Similarly, the cell-matching condition of G3,0 ≤ 670 pT/m3 would limit the1044.3. Overview of Sensitivity Effects100 101 102 103G1,0 (pT/m)10-2810-26d nf (ecm)100102B (pT)5 10-28 ecm3 10-28 ecmdnfBzBx,y(a)102 103 104G3,0 (pT/m3)10-2810-26d nf (ecm)100102B (pT)5 10-28 ecm3 10-28 ecmdnfBzBx,y(b)103 104 105G5,0 (pT/m5)10-2810-26d nf (ecm)100102B (pT)5 10-28 ecm3 10-28 ecmdnfBzBx,y(c)105 106 107G7,0 (pT/m7)10-2810-26d nf (ecm)100102B (pT)5 10-28 ecm3 10-28 ecmdnfBzBx,y(d)Figure 4.10: Calculated false EDM signals caused by BE frequency shifts for different mag-nitudes of gradients (a) G1,0, (b) G3,0, (c) G5,0, and (d) G7,0. The second axis and red linesshow the maximum magnetic field variation between any two points in the cell, with separateresults given for variation in Bx/By and Bz. Target systematic levels of 5 × 10−28 ecm and3× 10−28 ecm are shown by horizontal dashed lines.1054.3. Overview of Sensitivity Effects0.1 0.2 0.3 0.4 0.5R (m)10-3210-3010-2810-2610-24d nf (ecm)G1,0 = 10 pT/mG3,0 = 400 pT/m3G5,0 = 14500 pT/m5Figure 4.11: Calculated false EDM signals from BE frequency shifts for G1,0, G3,0, and G5,0versus cell radius. Magnitudes of the gradients were chosen to generate a 5× 10−28 ecm signalat R = 0.25 m.l = 3 UCN contribution to < 9 × 10−30 ecm. For l = 5, the ∆Bz < 140 pT condition limitsthe UCN contribution to 5.1 × 10−29 ecm, but a field of such magnitude is not expected inthe MSR. These effects will depend on what the final field looks like in the experiment, but aconservative estimate would be that BE frequency shifts from UCNs do not contribute morethan 5× 10−29 ecm.These systematics depend strongly on the size of the cells. Figure 4.11 shows the false EDMsignals versus cell radius. The magnitudes of the false signals are seen to increase greatly as R isincreased, with higher l modes becoming more significant. Since increasing neutron counts Ntotfavours increasing the radius, as discussed in Ref. [6], there is a trade-off between improvingstatistical uncertainty and the required precision of corrects to the frequency shifts. The valueof R = 0.25 m that was ultimately chosen was deemed to be a reasonable compromise betweenthese competing contributions.There are a number of methods proposed to remove these systematics. The obvious methodis to measure the gradients precisely enough to subtract the contributions from the final result.This is the goal of the Cs magnetometer array in the TUCAN EDM spectrometer, and therequirements on the precision of that system are discussed in the theses of W. Klassen [212, 214].In this case, the uncertainty on the measurement of each gradient translates linearly into asystematic uncertainty according to Figure 4.10. For example, to have a systematic uncertaintyfrom G3,0 of 5 × 10−28 ecm requires determining this mode with an uncertainty of 400 pT/m3on G3,0.With a measurement of the gradients, uncertainty in the coordinate system of the measure-ment can introduce systematic uncertainties from modes that do not have l-odd, m = 0. Thisis essentially the inverse of the argument presented in Equation 4.45. The magnetic field can1064.3. Overview of Sensitivity Effects102 103 104 105Gl,m (pT/ml)10-2910-2810-2710-26d nf (ecm)G 2,1, = 10 mmG 2,1, = 5 mmG 2,1, = 1 mmG 3,-2, = 10 mmG 3,-2, = 5 mmG 3,-2, = 1 mmBz,2,1 = 140 pTBz,3,-2 = 140 pTFigure 4.12: Calculated false EDM signals from BE frequency shifts for m 6= 0 modes in a cellthat is offset from the coordinate centre of the magnetic field by δx = δy = δ. Vertical linesindicate the size of gradient which would cause a 140 pT variation in Bz inside the cell.always be measured relative to an origin that enforces symmetry, but if the position of thatorigin is inaccurate, then the symmetry argument fails. In that case, ~Πl,ms for which the δωBEwas assumed to integrate to zero over the cell volume can have a non-cancelling effect.While many of the gradients do not produce large signals under these conditions, gradientssuch as G2,1 and G3,−2 can become significant. Figure 4.12 shows numerical calculations offalse EDMs from G2,1 and G3,−2 assuming gradients measured in a coordinate system offset byδx = δy = δ. These calculations show that l = 3 is not likely to be significant, as the magnitudeof the gradient that would produce a 10−28 ecm signal is large even with δ = 10 mm. TheG2,1 (and similarly G2,−1) gradients could be significant. With a 5 mm position uncertainty,G2,1 could produce a false signal as large as 2 × 10−28 ecm while still satisfying the overalluniformity goal. With a 1 mm position uncertainty, the contributions from each m 6= 0 modecan be reduced to 0.7× 10−28 ecm.Figure 4.13 shows similar calculations for l = 2, 4, m = 0 modes in a coordinate systemoffset vertically by δz. In a single cell, these false EDM signals are a result of the frequencyshift in the cell, while in the double cell they result from imperfect cancellation of the signalsbetween the two cells. The false signals are larger in the double cell configuration (because thepositions of the cells make these signals larger than in the centered single cell), with a 5 mmoffset producing a 2 × 10−27 ecm signal. An uncertainty of 1 mm in the position of the cellsresults in a 4× 10−28 ecm systematic uncertainty.This systematic contribution from uncertainty σO in the position of the origin of the coordi-nate system is dominated by the contribution from G2,0, and with σO = 1 mm this uncertaintyis larger than desired, potentially being a dominant systematic contribution at 4× 10−28 ecm.This could be solved by reducing σO, but it is likely easier to limit G2,0, which can be done1074.3. Overview of Sensitivity Effects102 103 104 105Gl,m (pT/ml)10-2910-2810-2710-26d nf (ecm)G 2,0, = 10 mmG 2,0, = 5 mmG 2,0, = 1 mmG 4,0, = 10 mmG 4,0, = 5 mmG 4,0, = 1 mmBz,2,0 = 140 pTBz,4,0 = 140 pT(a) Single cell102 103 104 105Gl,m (pT/ml)10-2910-2810-2710-26d nf (ecm)G 2,0, = 10 mmG 2,0, = 5 mmG 2,0, = 1 mmG 4,0, = 10 mmG 4,0, = 5 mmG 4,0, = 1 mmBz,2,0 = 140 pTBz,4,0 = 140 pT(b) Double cellFigure 4.13: Calculated false EDM signals from BE frequency shifts for l-even modes when thecells are vertically offset from the coordinate centre of the magnetic field by δz = δ. Verticallines indicate the size of gradient which would cause a 140 pT variation in Bz inside the cell.1084.3. Overview of Sensitivity Effectswith a dedicated Gl,m coil. Constraining G2,0 ≤ 0.5 nT/m2 would limit this contribution to1 × 10−28 ecm. Then, the total uncertainty with σO = 1 mm, including the smaller contribu-tions from m 6= 0 modes, would be limited to below 2× 10−28 ecm.Other than measuring and correcting the shifts from gradients, there are techniques thatattempt to perform the nEDM measurements in such a way as to cancel or suppress theseeffects. These techniques include the magic field [233], the dual co-magnetometer [234], and thecrossing-point analysis. The last of these is the only technique to have been implemented in anexperiment so far, and is currently intended to be used for the TUCAN EDM measurement.The Crossing-Point AnalysisThe crossing-point analysis attempts to perform the measurements in a way that allows thefalse EDM from some gradients to be interpolated to zero in the final analysis of the data. Thecrossing-point analysis was first implemented in the nEDM measurements at ILL [108], andcontinued to be used in the PSI nEDM measurement [114]. To perform this interpolation, it isnecessary to intentionally vary a gradient, which can be achieved using Gl,m coils, and to havesome measure of the gradient. The choice of this measure is different in a double cell experimentcompared to the single cell experiment. The implementation of the analysis in a single cell willbe discussed first, after which the refinements of the technique in a double cell experiment willbe introduced.The full derivation of the equations of the crossing-point analysis is given in AppendixC.1. The key to implementing the crossing-point analysis in the single cell experiment is theobservation that the UCNs and Hg atoms measure slightly different fields due to the strongercontribution of gravity to the distribution of UCNs in the cell. This means that there is a shiftin the ratio R related to a so-called “gravitational gradient”Gg = G1,0 +G3,0D3 +G5,0D5 + ..., (4.46)where Di are geometric coefficients calculated from the integration of the specific mode overthe cell volume. In general, these coefficients are not the same as those that appear whencalculating BE shifts. When all frequency shifts other than those depending on Gg are removed,the corrected frequency ratio becomesRc = γnγHg(1 + δg) =γnγHg(1 +Gg〈z〉B0,Hg), (4.47)where 〈z〉 is the UCNs average height in the cell and the subscript Hg in B0,Hg is used to makeclear that this is the field as measured by the Hg. This value is computed as an average Rcover the cycles used in one EDM sequence.5The false EDM from BE shifts can be rewritten partially in terms of Gg. The G1,0 shiftcan be encapsulated entirely by Gg, while the higher order terms will have some residual shiftnot related to Gg. If all shifts except those related to Gg and the true nEDM are removed fromthe measured nEDM, thendn,c =dn +~γnγHg8c2R2Gg=dn +B0,Hg〈z〉~γHg28c2R2(Rc − γnγHg)=dn ± k1 (Rc −R0) ,(4.48)5This is a key technical note, as it means that the BE frequency shifts themselves do not appear in thecalculation of the ratio, as half of the cycles will be taken at +E and the other half at −E.1094.3. Overview of Sensitivity Effectswhere the ± in the final line gives the orientation of B0.If the measurements can be tuned such that Rc = γn/γHg then the BE shift correspondingto Gg will be removed. This would require a very precise measurement of γn/γHg, but thisrequirement can also be avoided. The final necessary observation is that the remaining shiftterm depends directly on B0,Hg and will be reversed if B0 is reversed. This feature can beleveraged by performing half of the data-taking with +B0 and half with −B0. Combining thisreversal with varying an applied gradient will produce two data sets of dn,c and Rc which fit tostraight lines differing only by their sign. Those two lines will intersect at the point at dn,c = dn,where the contribution of Gg is zero. Hence the name crossing-point analysis, referring to thecrossing point of these two lines.The fitting procedure can be approached in different ways, either by fitting the common slopeas a parameter [108], or by extracting the slope in an auxiliary analysis [114]. In either case,error in the common slope can introduce an additional systematic uncertainty. A derivation ofsuch an error is given in Appendix C.2, with the resulting false EDM beingdfn = |σk1 |∑(±1)(R0 −Rc)n, (4.49)where σk1 is the error in the value of the slope, (±1) is the sign of B0, R0 = γn/γHg, and thesum is taken over the n EDM sequences used in the fitting procedure. This shift depends onthe statistics of the measured data. If Rc is distributed symmetrically about R0, then the sumevaluates to zero. A detailed estimate of this systematic requires studies of this distribution,and is outside the scope of this dissertation. As a rough estimate, it can be noted that in thePSI nEDM measurement, the slope was determined with an uncertainty of 7%, and an error of7 × 10−28 ecm was attributed [114]. With increased statistics to 10−27 ecm, the error in slopewould presumably drop below 1%, and the systematic below 10−28 ecm.An increase in statistical uncertainty also arises due to the use of the average R in thecalculation. For each measurement of dn,c, the increase in uncertainty, derived in AppendixC.2, isσ(dn,c)′σ(dn,c)=√1 +k12n∑Rc2i(σ(ωHg,i)2ωHg,i2+σ(ωn)2ωn2), (4.50)where n is the number of measurements binned into a determination of dn,c. From simulations,k1 ≈ 7 × 10−21 ecm, Rci ≈ 3.8. The neutron frequency measurement dominates, with anuncertainty of 4µHz per Ramsey sequence, or 2.8µHz per EDM sequence used to determinea value of dn,c. Setting n = 1, this results in only a 0.06% increase in uncertainty, which isnegligible.There is a major disadvantage to this analysis, which is that it depends directly on R, ratherthan depending only on the difference of R measured with two polarities. As a result, frequencyshifts that do not depend on E can shift the result and need to be corrected in R. In particular,this analysis must be concerned with shifts in R that are different for the two orientations ofB0.Consider a general shift of R, δ±, which takes different values for ±B0. Any such shift canbe rewritten as the sum of an average value δ¯± and a B0-dependent component ∆δ, such thatδ± =δ+ + δ−2± δ+ − δ−2= δ± ±∆δ. (4.51)This appears in the corrected nEDM asdn,c = dn ± k1(Rc − γnγHg− δ± ∓∆δ), (4.52)1104.3. Overview of Sensitivity Effectswhere B0,Hg = ±|B0,Hg| has been written explicitly. This can be rearranged asdn,c = (dn − k1∆δ)± k1(Rc −(γnγHg+ δ±)). (4.53)The average component simply shifts the value of R, but the B0-dependent component shiftsdn, so any shift that differs between the two magnetic field polarities will appear as a falseEDM. Such shifts are not related to BE shifts, and will be discussed in later sections.In the double cell configuration the crossing-point analysis can be implemented in a differentway, which mitigates the disadvantages of the single cell formulation. The derivation is given inAppendix C.3. Rather than using the ratio of UCN to Hg frequencies, the fields measured by thetwo co-magnetometers in the top and bottom cells can be combined to form a “co-magnetometergradient”GHg =B0Hg,top −B0Hg,bot∆HHg, (4.54)where ∆HHg is the difference between the centres of the co-magnetometers [115]. As with thesingle-cell crossing-point analysis, this gradient allows the false EDM signal to be rewritten asdfn =~γnγHg8c2R2(GHg −G3,0(D3,0 + C3,0) +G5,0(D5,0 + C3,0) + ...)=k2(GHg −G3,0(D3,0 + C3,0) +G5,0(D5,0 + C3,0) + ...) (4.55)where C l,0 and Dl,0 are geometric factors related to the measured field and BE shifts respec-tively. Note that GHg is a combination of different gradients, which includes the entirety of G1,0and some part of the higher-order gradients. The “remainder” of the higher order gradients isdescribed by the combination of C l,0 + Dl,0 and is described in more detail in Appendix C.3.Again, the BE systematic effects can be partially corrected by applying different gradients andthen interpolating the data to the point where GHg = 0.As with the single-cell crossing-point analysis, an error σk2 in the slope can translate to asystematic shift, derived in Appendix C.4, ofdfn = −σk2∑i=1..nGHgin, (4.56)where the sum is over the different values of GHg in the data set. However, k2 has minimalsources of error. Both γn and γHg have been measured with ppm uncertainty and can beneglected. The most uncertain part of k2 is the cell radius R. An uncertainty of 1 mm on theradius would yield σk2 = 4×10−19 ecm. The cell-matching condition sets GHg,max = ±20 pT/m.The average GHg is therefore not likely to be above 10 pT/m, and so the systematic contributionfrom the slope would not exceed 4× 10−31 ecm.The increase in the uncertainty on dn using this method, derived in Appendix C.4, isbounded byσ(dn)σ(dn,c)≤ 1n√1 +k22σ(dn,c)2(2σ(B0,Hg)2∆HHg2+σ(∆HHg)2∆HHg2GHg2,max). (4.57)The high precision of the co-magnetometers will render the σ(B0,Hg) term negligible, whilethe tight restriction of the cell-matching condition suppresses the ∆HHg term. The calculatedincrease of the total uncertainty is 9× 10−37 ecm, which is also negligible.1114.3. Overview of Sensitivity EffectsThe co-magnetometer gradient GHg is a cleaner measure of the gradient than R. Since thismeasure does not involve comparing UCNs to Hg, shifts between UCN and Hg frequencies willnot be relevant except when they induce a direct false EDM due to an E dependence. Whatwill be relevant is shifts between the two co-magnetometers. An uncorrected shift alters themeasured co-magnetometer gradient toGHg′ = GHg + δGHg, (4.58)where δGHg is a shift that is not related to the free space harmonics. Then, when GHg′ = 0,GHg = −δGHg, so that there will be a false EDMdfn = −k2δGHg. (4.59)Such a shift could be related to co-magnetometer systematics, discussed in Section 4.3.12, orto localized fields, discussed in Section 4.3.9.With this method it is also not necessary to reverse the magnetic field, since the true EDMwill be found at GHg = 0. Effects that reverse with the magnetic field therefore do not impactthe measurement. However, reversing the magnetic field remains valuable as a systematiccheck.6Both analyses do not fully compensate the BE shifts at l ≥ 3. The remaining contributionsmust also be measured and corrected, which can be achieved by the use of external magne-tometry. Figure 4.14 shows the residual systematic shifts caused by the third and fifth ordergradients in the single and double cells. These are modified compared to those in Figure 4.10.In the single cell case, the sensitivity to G3,0 is increased, while the sensitivity to G5,0 is de-creased. This relationship is reversed in the double cell. However, G3,0 is still expected to makethe largest contribution in both cases, and to be the largest systematic contribution overall.Given the expectation that the residual shifts from the gradients will be the dominantsystematic effect, the TUCAN EDM experiment assigns a large part of its systematic budget tothe uncertainty from correcting these effects. The target systematic uncertainty from correctingG3,0 will be 5 × 10−28 ecm, while from G5,0 it will be 3 × 10−28 ecm. From Figure 4.14, thisrequires an uncertainty on determining G3,0 less than 300 pT/m3 in a single cell, or less than1000 pT/m3 in a double cell, and an uncertainty on determining G5,0 less than 4500 pT/m5 ina single cell, or less than 3000 pT/m5 in a double cell. Achieving these targets will make itpossible to meet the overall experimental uncertainty.One final note should be made regarding the gravitational shift that was described as partof the single cell crossing-point analysis. In the single cell analysis, this gradient is tuned tozero, but in the double cell analysis it is not. Therefore, the UCNs can be expected to see someshift relative to the co-magnetometers. Ideally, this shift is independent of the electric fielddirection and therefore induces no systematic effect. However, because the l-odd Bz terms areodd in z, these shifts do correlate with the electric field when calculating dn from the two cellsas in Equation 4.13. The shifts do cancel when considering Equation 4.12, but not if there isan l-odd field that correlates with HV polarity reversal. The systematic associated with such acase isdfn =~4E(Gl,m,+ −Gl,m,−)γn((∫n,topΠz,l,m −∫Hg,topΠz,l,m)−(∫n,botΠz,l,m −∫Hg,botΠz,l,m)),(4.60)6The name crossing-point analysis would appear to not make sense for this analysis, given that it is no longerconstructed from two lines that cross at the desired point. However, the interpolation can be considered to be acrossing of the axis at GHg = 0, so the continued use of this terminology is not completely nonsensical.1124.3. Overview of Sensitivity Effects101 102 103 104 105Gl,0 (pT/ml)10-2810-2710-26d nf (ecm)5 10-28 ecm3 10-28 ecmG3,0G5,0(a)101 102 103 104 105Gl,0 (pT/ml)10-2810-2710-26d nf (ecm)5 10-28 ecm3 10-28 ecmG3,0G5,0(b)Figure 4.14: Calculated residual false EDM signals from BE frequency shifts for G3,0 and G5,0that remain after application of the crossing-point analysis in a (a) single cell and (b) doublecell configuration.1134.3. Overview of Sensitivity Effects10-1 100 101 102 103Gl,0,+-Gl,0,- (pT/ml)10-2910-2810-2710-26d nf (ecm)G1,0G3,0Figure 4.15: Calculated false EDM signal resulting from the gravitational shift between UCNsand Hg for a polarity-dependent gradient in a double cell configuration. The horizontal axis isthe magnitude of the different in gradient between the two polarities.where the integral subscripts describe the sampling by the UCNs and Hg atoms in either the topor bottom cell, and the Gl,m subscripts indicate HV polarity. Figure 4.15 shows calculated falseEDM signals for G1,0 and G3,0. Even with the requirement that these gradients are respectivelyless than 20 pT/m and 670 pT/m3 for cell-matching, it is still possible to have a systematiccontribution from the asymmetry in the gravitational shift between the two cells. These effectscan be reduced below 1×10−28 ecm by measuring the difference in the fields between polaritiesusing a combination of CsMs and offline magnetometry. The precision required on G1,0 is≤ 0.1 pT/m, while the precision required on G3,0 is ≤ 1000 pT/m3.The presence of a small systematic uncertainty associated with the gravitational shift isnotable as a disadvantage that arises from using the co-magnetometer gradient. Since thismethod improves other uncertainties associated with the use of the crossing point method inthe single cell (specifically with the determination of 〈z〉), this indicates some trade-off betweenthe methods. However, it does raise the possibility that analyzing a double cell experimentusing the gravitational gradient could be better in some ways, and in the final analysis of theTUCAN EDM experiment, all of these methods should be employed. The application of othermethods to the double cell is outside the scope of this dissertation, and will not be discussedfurther.1144.3. Overview of Sensitivity Effects4.3.7 B2 Frequency ShiftsThe second frequency shift to consider is the B2 shift described by Equation 4.25. The leading-order expansion of this shift gives, for gas atomsδωB2 ,Hg =γHg2ω0,Hg2∫ ∞0t′〈bx(0)bx(t′) + by(0)by(t′)〉dt′, (4.61)and for UCNsδωB2 ,n =γn22ω0,n〈bx(0)2 + by(0)2〉. (4.62)The t′ in the integral of δωB2 ,Hg suppresses this term relative to δωB2 ,n, since the correlation isonly large when t′ is small. The common interpretation of this is that the adiabatic UCNs aremoving slowly enough that they see the changing field due to transverse components, while thenon-adiabatic Hg atoms are moving so quickly that the transverse components are ignored.These shifts do not depend directly on E, and therefore do not directly appear as a falseEDM. However, there are conditions under which a false EDM can be caused by such a shift.First is the case where some transverse component of the B field correlates with the HVpolarity. This could result from some magnetization of some part of the environment e.g. inthe MSR walls. If this occurs, then a false EDM will occur in a single cell with magnitudedfn =~4Eγn22ω0,n∆〈bx(0)2 + by(0)2〉, (4.63)where ∆〈bx(0)2 + by(0)2〉 is determined by the difference of the fields in the two orientations.In the case of free space harmonics, the difference between the fields for the two polarities willbe in the magnitude Gl,m, so the false EDM can be written asdfn =~4Eγn22ω0,n(Gl,m2,+ −Gl,m2,−)〈Πx,l,m(0)2 + Πy,l,m(0)2〉= (Gl,m2,+ −Gl,m2,−)C l,m,(4.64)where C l,m is a geometrical factor.These effects can be corrected by measuring the field. An uncertainty σ(Gl,m) on measuringthe magnitude of a harmonic translates into a systematic uncertainty asσ(dfn) = 2C l,mσ(Gl,m)√(Gl,m2,+ +Gl,m2,−)= 2C l,mσ(Gl,m)√2 ¯Gl,m2 + ∆Gl,m2/2= 2√2C l,mσ(Gl,m) ¯Gl,m√√√√1 +(∆Gl,m22 ¯Gl,m2),(4.65)where ¯Gl,m2 is the mean value between polarities and ∆Gl,m2 is the difference between polarities.The dependence of this effect on the three parameters σ(Gl,m), ¯Gl,m2 and ∆Gl,m2 makesit somewhat difficult to analyze. To set requirements, it can be assumed that ¯Gl,m2 is thelargest value allowed by absolute field homogeneity ∆(Bz) < 140 pT. For m = ±(l + 1), thisrequirement does not apply since Bz = 0. To bound those values the requirement can insteadbe applied to ∆Bx and ∆By.1154.3. Overview of Sensitivity EffectsThere is a further complication of this effect due to the fact that 〈bx(0)2 +by(0)2〉 can coupledifferent harmonics together. In this casedfn = (Gl1,m1 ,+Gl2,m2 ,+ −Gl1,m1 ,−Gl2,m2 ,−)C l1,m1,l2,m2 , (4.66)and the uncertainty becomesσ(dfn) = 2C l,m ¯Gl1,m1¯Gl2,m2[(1 +(∆Gl2,m22 ¯Gl2,m2)2)2(σ(Gl1,m1)2 ¯Gl1,m1)2+(1 +(∆Gl1,m12 ¯Gl2,m2)2)2(σ(Gl2,m2)2 ¯Gl2,m2)2 ]1/2 (4.67)Figure 4.16 shows calculated uncertainties on the false EDM signal from B2 shifts. Theintegration is performed over a single cell in the double cell layout, with the coordinate centrein the centre of the central electrode. This tends to make the signals larger compared to acell-centre coordinate system. Results are shown up to l = 3, the reasoning being that thecontribution of higher order gradients becomes small (note that the l = 3 terms do not requireas high precision measurements of the field as the l = 1 and l = 2 terms. The results showthat the systematic uncertainty depends strongly on the difference between polarities. Withthe assumption that the average field is maximal under the field uniformity requirement, thereversing component cannot exceed 100%, as doing so would result in a field that greatly exceedsthe uniformity.7 The required precision for these fields can therefore be interpreted from the100% mark.There are 21 gradient terms up to order l = 3, all of which can contribute systematic shifts ofsimilar magnitude. There are also 22 contributing cross-terms. However, with the requirementon the absolute field variation, they cannot all contribute at the greatest level. If one termcontributes at the level in Figure 4.16 then, barring some particular cancellation between fields,the others generally cannot contribute, because their doing so would drive the variation abovethe requirement. Detailed analysis of the requirements is beyond the scope of this dissertation,but for the purpose of setting systematic targets, the worst such transverse contribution canbe taken as contributing a full 2× 10−28 ecm. In these calculations, the strongest effect seemsto come from G1,2. Although the cross-terms also appear to be quite substantial, it should benoted that this calculating assumes a maximal field for both G1,2 and G2,2, and so in the realexperiment the largest possible contribution would be up to a factor of four lower than what isshown.The systematic contribution from B2 shifts can therefore be limited, assuming up to a 100%field reversal, if G1,2 can be determined to 0.05 nT/m2. In this calculation G1,2 = 0.29 nT/m,so this requires only a 17.5% uncertainty on the determination of this mode. This is not a veryprecise measurement compared to the precision with which the modes required for BE shiftsneed to be measured, and should be easily achievable.In a double cell experiment, these effects are suppressed if the field is symmetric betweenthe two cells. The false EDM will becomedfn = dfn,top − dfn,bot. (4.68)7The requirements laid out here are somewhat stronger than they need to be. With a maximal average field,there cannot be any reversing component, because one polarity would have a larger field than allowed. Ratherthan complicating the details of the systematic requirements further, these looser assumptions result in simplerbut more stringent requirements.1164.3. Overview of Sensitivity Effects10-1 100 101 102 103 G1,m/2|G1,m| (%)10-2910-2810-2710-26(d nf) (ecm)(G1,0) = 0.05 nT/m(G1,0) = 0.1 nT/m(G1,0) = 0.5 nT/m(G1,2) = 0.05 nT/m(G1,2) = 0.1 nT/m(G1,2) = 0.5 nT/m(a) l=110-1 100 101 102 103 G2,m/2|G2,m| (%)10-2910-2810-2710-26(d nf) (ecm)(G2,( 1, 2)) = 0.05 nT/m2(G2,( 1, 2)) = 0.1 nT/m2(G2,( 1, 2)) = 1 nT/m2(G2, 3) = 0.05 nT/m2(G2, 3) = 0.1 nT/m2(G2, 3) = 0.5 nT/m2(b) l=210-1 100 101 102 103 G3,m/2|G3,m| (%)10-2910-2810-2710-26(d nf) (ecm)(G3,0) = 1 nT/m3(G3,0) = 5 nT/m3(G3,0) = 10 nT/m3(G3, 3) = 0.5 nT/m3(G3, 3) = 1 nT/m3(G3, 3) = 5 nT/m3(c) l=310-1 100 101 102 103Gl,m/2|Gl,m| (%)10-2910-2810-2710-26(d nf) (ecm)(G1/2,2)/2|G1/2,2| = 0.1(G1/2,2)/2|G1/2,2| = 1(G1/2,2)/2|G1/2,2| = 10(d) Cross-termsFigure 4.16: Calculated false EDM signals from B2 frequencies shifts in a single cell when thereis a difference in B2 with HV polarity reversal. The horizontal axis of each plot gives thenormalized asymmetry of the gradient which appears in the calculation. For (a), (b), and (c),both the largest and smallest contributions are shown. For (d) only the largest contribution,which involves l = 1,m = 2 and l = 2,m = 2, is shown, as the smallest contribution is well below10−29 ecm. Each line corresponds to a precision to which the relevant gradient is measured,given in absolute uncertainties for (a), (b), and (c), and relative uncertainties for (d).1174.3. Overview of Sensitivity Effects10-1 100 101 102 103Gl,m/2|Gl,m| (%)10-2910-2810-2710-26(d nf) (ecm)(G1/2,2)/2|G1/2,2| = 0.1(G1/2,2)/2|G1/2,2| = 1(G1/2,2)/2|G1/2,2| = 10(G2/3,0)/2|G2/3,0| = 0.1(G2/3,0)/2|G2/3,0| = 1(G2/3,0)/2|G2/3,0| = 10Figure 4.17: Calculated false EDM signals caused by B2 frequency shifts in a double cell whenthere is a difference in B2 with HV polarity. In the double cell experiment these shifts mostlycancel due to symmetry between the top and bottom cells, with the exception of a few termssuch as those shown here. The only such terms for which this is possible are cross-terms, andthe largest and smallest contributions are shown.In the case where the field changes with polarity, but is the same in the top and bottom cells,the double cell configuration will not see a systematic shift. Since the harmonics spans the fullspace of the cells, the double cell should lead to this systematic cancelling. Differences in thesampling due to different height spectra would lead to a small asymmetry, but this is not morethan a few percent difference.However, there are some terms that will not cancel in the double cell. This occurs in cross-terms when the product Πx,l1,m1Πx,l2,m2 or Πy,l1,m1Πx,l1,m1 is odd in z, in which case the topand bottom cell signals differ by a sign and add instead of cancelling. Such a sign reversaloccurs when one term has l = 1 or l = 3 and the other has l = 2. Since this is a cross-term, itdoes not make sense to assume the maximum possible value of the two fields. Instead, in thiscalculation, the mean values of the fields are assumed to be half of that which would create a∆Bz = 140 pT field variation over one cell.Figure 4.17 shows the contribution of such terms. In this case, the strongest contributionis from G1,2 crossed with G2,2. The systematic can be constrained below 1 × 10−28 ecm witha measurement precision of 20%. This is readily achievable and will likely be exceeded in theTUCAN EDM experiment.The B2 shifts have seen more significance in the crossing-point analysis in a single cell,where they will appear in R. In the previous section a difference in any such shift between thetwo B0 field orientations was shown to produce a false EDM. In the final ILL nEDM reanalysis,a G1,2 field was reportedly measured with∆Gl,m2|Gl,m| ∼ 1 between B0 polarity. For the B2 term1184.3. Overview of Sensitivity Effectsthis introduces a false EDM at the crossing point ofdfnR =|B0Hg|〈z〉~γHg28c2R2∆δωB2 ,nω0=|B0,Hg|〈z〉~γHg28c2R2γn22ω02(Gl,m2,+ −Gl,m2,−)〈bx(0)2 + by(0)2〉Gl,m2=|B0Hg|〈z〉EγHg22ω02c2R2dfn,HV ≈ 0.003dfn,HV,(4.69)where in this case the polarity-labelling +/− refers to B0 orientation, and dfn,HV is the falseEDM from B2 shifts due to a magnetic field asymmetry between HV polarities. This effectis apparently small compared to the direct effect on polarity reversal, and the correspondinguncertainty can be read off of Figure 4.16 with an appropriate multiplication by a constantfactor. In the double cell experiment, this contribution is not present, since the analysis nolonger relies on R.4.3.8 E2 Frequency ShiftsThe last of the frequency shifts is the E2 term, given by Equation 4.26. The expansion of thisshift gives, for the gas atomsδωE2 ,Hg = −γHg2E22c4ωHg〈x(0)2 + y(0)2〉 (4.70)and for UCNsδωE2 ,n =E2γn22ωnc4〈vx(0)2 + vy(0)2〉 (4.71)These shifts differ byδωE2 ,HgδωE2 ,n= −γHgγnωHgωn〈x(0)2 + y(0)2〉Hg〈vx(0)2 + vy(0)2〉n ≈ 7.9, (4.72)where the UCN velocity is taken from PENTrack simulations, so the dominant term is from theHg atoms.These shifts depend on E2 rather than E, and so do not produce a systematic shift directly.However, it is still possible for these terms to cause such a shift due to asymmetry in the fieldbetween the two HV polarities. If the magnitude of the field is different between the two HVpolarities, then there is a false EDM signaldfn =~4|E|γnγHgγHg2ω02c4R22(E+2 − E−2)=~γnγHgωHg4c4|E|R22E¯∆E,(4.73)where ∆E is the difference in the field between polarities, E¯ is the average field betweenpolarities, and |E| is the assumed magnitude used in the calculation of dn. This systematic canbe constrained below 2 × 10−28 ecm if ∆E is known with a precision of 1%, or 2 KV on thesupply.In the double cell, if the field magnitude changes with the polarity but is symmetric betweenthe cells, then this systematic will cancel by Equation 4.13. This cancellation will not hold ifthe fields are different in the two cells due to geometric differences. If the height of the cellsdiffered by 4 mm, then the field would differ by ∼2.5%. With the above requirement of 1%precision on ∆E, the 2× 10−28 ecm systematic is reduced to 5× 10−30 ecm.1194.3. Overview of Sensitivity Effects4.3.9 Spatially-Localized FieldsThe previous sections have considered the frequency shifts in an experiment with a magneticfield that is well-described by the harmonic polynomial basis. When the central region isinstalled, there are additional fields that could be present and which cannot be described bythe harmonic basis. This is because these fields are created by magnetic field sources in thecentral region, and therefore violate the free-space assumption.The most significant field of this type is due to material magnetization. Any parts of thecentral region could contain small ferromagnetic contamination. Highly paramagnetic partscould also generate a significant field in the 1µT field. These sources are considered to bemagnetic dipoles. The gradients created by a dipole produce a BE shift. They can also generateB2 shifts and direct B0 shifts that can induce false EDM signals. The topic of magnetic dipolesis discussed in detail in Chapter 7.Spatially-localized fields can also be generated by leakage currents through the insulatorsof the cells. A general leakage current will produce a field correlating the HV polarity. If aleakage current generates a field in the z direction, it will be measured by both UCNs and Hg.The Hg will partially compensate for such a field, but the gravitational distribution of UCNscan still produce some shift. Fields in the x and y directions generated by leakage currents canalso induce BE or B2 shifts. A detailed discussion of leakage currents is included in Section 6.A third field source exists due to currents driven by thermal gradients in the conductivematerials of the vacuum chamber and electrodes. This has been experimentally observed,but the source of these fields has yet to be confirmed. Section 7 reports some preliminaryobservations of thermal fields, but no detailed studies have been performed.4.3.10 Net Motional FieldsContributions of the motional field to frequency shifts were included through Redfield theory.In those derivations, it was assumed that the distribution of UCN velocities was random in thesense of velocities averaging to zero. If that assumption is violated, then the motional field cancause a frequency shift that is linear in E, independent of any magnetic field gradients. Forthis to happen, there must be a net motion causing the z-component(~v × ~Ec2)z=vxEy − vyExc2=vrEφ − vφErc2(4.74)to be non-zero on average. This expression is given in both Cartesian and cylindrical coordi-nates to highlight the possible net motions that can produce an effect. The Cartesian formrequires a combination of a linear drift velocity and a net electric field in the transverse plane.The cylindrical form requires either a radial drift velocity and a circulating electric field, or acirculating velocity and a radial electric field. Detailed studies of these net motions are includedin Chapter 64.3.11 AC FieldsIn addition to the E2 shift and leakage currents, the HV applied in the experiment has a poten-tial systematic effect due to alternating current (AC) ripple of the supply. A small oscillatingcomponent of the electric field will generate an oscillating magnetic field, which induces a fre-quency shift. In the worst case, if this field oscillates on resonance with the neutron or Hgfrequencies, the effect can be large. This is discussed further in chaper 61204.3. Overview of Sensitivity Effects4.3.12 Co-Magnetometer SystematicsOther than its substantial contribution to the BE frequency shifts, the co-magnetometer canintroduce systematic shifts from a number of other effects, which are highlighted here for com-pleteness. Detailed discussion of these effects are beyond the scope of this dissertation, but acumulative target of 2× 10−28 ecm uncertainty due to co-magnetometer effects is considered areasonable target.Co-Magnetometer EDMIf the co-magnetometer species has a non-zero EDM of its own, then it will introduce a frequencyshift that cannot be disentangled from the neutron EDM. The best measurement of the Hg EDMis dHg = (−2.20± 3.12)× 10−30 ecm [81], which contributes an uncertainty of 10−29 ecm.Light ShiftsA shift in the frequency of the Hg atoms can occur due to an interaction between the probelight and the atoms, known as a light shift [235]. This shift is proportional to the intensity ofthe light, and induces a false EDM signal if the intensity correlates with polarity or B0 direction(for the single cell crossing point). The details of such light shifts in Hg co-magnetometers arewell studied [236, 237]. The development of the Hg co-magnetometer for the TUCAN EDMexperiment includes the study of such light shifts and techniques to reduce them below thetarget level.Incoherent ScatteringThe UCNs can experience a spin-dependent interaction with the Hg nuclei, with the strengthof the interaction determined by the incoherent scattering length bHg199. This interaction canbe described as a pseudomagnetic field [114, 238]Binc =4pi~√3mnγnbHg199nHg199Pz,Hg, (4.75)where nHg199 is the number density of Hg atoms, and Pz,Hg is the polarization of the Hg spinsin the direction of B0. This effect appears only due to an imperfect pi/2 pulse applied to theHg, leaving a residual polarization in the direction of B0. It can therefore be controlled byincreasing the precision of the pulse.4.3.13 MiscellaneaThere are a few miscellaneous effects, with negligible uncertainty contributions, that mustnonetheless be considered in the systematic corrections.Johnson NoiseThe use of metallic electrodes to support the electric field in the experiment introduces someeffect of Johnson-Nyquist noise. This noise will change the field measured by both UCNs andHg atoms, and because the species sample the fields differently due to both their spatial andvelocity distributions, this could result in a frequency shift. It could also, in general, reducethe sensitivity of the magnetometry. A detailed study of this effect was carried out by the PSIn2EDM collaboration [239]. This study concluded that any effect of such noise is negligible1214.3. Overview of Sensitivity Effectsin an experiment aiming to measure at the 10−27 ecm level, as it will be reduced to below10−29 ecm over the course of the experiment.Earth’s RotationA frequency shift is introduced due to the experiment being performed in the Earth’s rotatingframe. The shift is simplyδω⊕ = ω⊕ sin(φTRIUMF), (4.76)where ω⊕ = 72.92106µrad/s is the Earth’s rotational frequency and φTRIUMF = 49.24731◦ isthe latitude of the TUCAN EDM experiment. While there are no conditions where this shiftcould correlate with electric field, it is relevant for the single-cell crossing-point analysis, whereit generates a shift in R depending on the orientation of B0. The shift isδ⊕ =± γnγHg(−ω⊕ sin(φTRIUMF)γn|B0| +ω⊕ sin(φTRIUMF)γHg|B0|)=± γnγHgω⊕ sin(φTRIUMF)|B0|(1|γn| +1|γHg|) (4.77)This effect can be corrected up to the precision of the Earth’s rotational frequency and thelatitude of the experiment. Earth’s rotational frequency is known to 10−11 rad/s [240], corre-sponding to an uncertainty of 6× 10−33 ecm. The latitude of the experiment can be identifiedto at least 0.01◦, corresponding to an uncertainty of 6 × 10−31 ecm. This effect is thereforenegligible after correction.Other EDM CouplingsAn EDM would interact with electric fields in exactly the same way as a magnetic dipolemoment. This means that there should be corresponding electric field versions of all of thedepolarization rates and frequency shifts that occur due to interaction of spins with magneticfields.Is is straightforward to consider these effects by analogy to the magnetic field. For example,the Hg atoms will see a motional ~v × ~B field, which will combine with Ex and Ey to produce“EB frequency shifts”. However, it can be determined by inspection that such effects will besmall, as they depend on the EDM. In the case of the EB shifts, by analogy to the BE shifts,they will depend on dHg2, and thus be extremely small. This will apply in general, and thusthese effects are neglected without further discussion here.4.3.14 The TUCAN EDM Systematics Budget and RequirementsBased on the information presented in the previous sections, a total systematic budget can bewritten down, in addition to several key requirements for statistical sensitivity.The cell-matching condition described in Section 4.3.3 requires that G1,0 < 20 pT/m andG3,0 < 540 pT/m3. Beyond this requirement, there is a general requirement for depolarization,described in 4.3.5 that ∆Bz < 140 pT in the cell volumes. Achieving the cell-matching con-ditions would likely require Gl,m coils to tune the G1,0 and G3,0 fields. It was further notedthat including such a coil for G2,0 could potentially improve visibility, as G2,0 may become thedominant mode contributing to depolarization, and such a coil may additionally be necessaryfor systematic considerations.1224.3. Overview of Sensitivity EffectsA few increases to statistical uncertainty were noted throughout the preceding sections, withthe significant effects being due to the use of working points and the use of the co-magnetometer.These amount to no more than a 5% increase in uncertainty. This does not include other effectsdue to stability of the magnetic fields, etc., which are beyond the scope of this dissertation.The total systematic uncertainty budget of the TUCAN EDM experiment is shown in Table4.2. These values are either derived in the previous sections of this chapter, or are detailed inlater chapters. The corresponding sections/chapters are listed in the table. In general, theseuncertainties are derived under some worst-case assumption and could be smaller in the realexperiment. The total systematic uncertainty of the TUCAN EDM experiment can be expectedto be ∼8× 10−28 ecm under the assumptions in this dissertation.A large portion of the systematic budget is due to the BE frequency shifts described inSection 4.3.6, either as direct contributions or through uncertainties associated with correctingthese shifts. The uncertainty values and requirements reported here assume that the crossing-point analysis is used as described in that section. In the case of the l = 3 and l = 5 BE shifts,the magnitudes of the uncertainties were chosen with the goal of keeping the total systematicuncertainty below 10−27 ecm, and the corresponding requirements on the precision at which thefields must be determined was extracted from Figure 4.14.1234.3.OverviewofSensitivityEffectsSystematic Section Single Cell Double Cell Requirement(10−28 ecm) (10−28 ecm)l = 1 Hg BE 4.3.6 0.0 0.0 Use of crossing-point techniquel = 3 Hg BE 4.3.6 5.0 5.0 σS(G3,0) ≤ 300 pT/m3, σD(G3,0) ≤ 1000 pT/m3l = 5 Hg BE 4.3.6 3.0 3.0 σS(G5,0) ≤ 4500 pT/m5, σD(G5,0) ≤ 3000 pT/m5UCN BE 4.3.6 0.5 0.5 General field uniformityCrossing point error 4.3.6 1.0 < 0.1 σ(〈z〉) ≤ 1% (single cell only)Cell position error 4.3.6 2.0 2.0 σO ≤ 1 mm, G2,0 ≤ 0.5 nT/m2Gravitational shift 4.3.6 < 0.1 1.0 σ(∆G1,0) ≤ 0.1 pT/m, σ(∆G3,0) ≤ 1000 pT/m3 (double cell only)B2 4.3.7 2.0 2.0 σ(Gl,m) ≤ 20%E2 4.3.8 2.0 < 0.1 1% uncertainty on ∆ENet Motion, rotational 4.3.10, 6.3 1.0 1.0 Er ≤ 14.45 KV/m, 2s wait time after fillingNet Motion, linear 4.3.10, 6.3 0.1 0.1 2s wait time after fillingLeakage currents 4.3.9, 6.4 3.4 2.1 Leakage current limited to 1 nA or lessAC E-fields 4.3.11, 6.4 < 0.1 < 0.1 ≤0.1% ripple, ≥1 GΩ inline impedanceParamagnetism 7.2 0.3 < 0.1 Al6061 vacuum chamber/electrodesDipoles 4.3.9, 7.4 2.2 2.2 250 dipoles with ≤ 1.26× 10−29 ecm/dipoleFeedthrough dipole 4.3.9, 7.4 1.0 0.1 Development of feedthroughFully reversing dipole 4.3.9, 7.4 2.0 2.0 Worst-case E-correlated dipole of 0.05 nAm2Thermal magnetic fields 7.5 < 0.1 < 0.1 No significant HV-correlated temperature change.Co-magnetometer 4.3.12 2.0 2.0 Not specifiedJohnson noise 4.3.13 < 0.1 < 0.1 None.Earth rotation 4.3.13 < 0.1 < 0.1 Correction of frequency shift.Total 8.5 7.8Table 4.2: Summary of systematic uncertainty contributions in the TUCAN EDM experiment, with expected size of the contributionsat the 1σ level. The uncertainties are conservative estimates given the listed requirements, with subscript S indicating requirements in asingle cell configuration and D in a double cell configuration, and assuming that any possible correction has been carried out. The listedsections are where these systematics are described and calculated. Values that are listed as < 0.1 are small enough that they do notcontribute significantly to the systematic uncertainty, and the actual values for these uncertainties can be found calculated in the text.124Chapter 5Development of a Prototype StorageCell5.1 A Prototype Storage CellThe subject of this chapter is the development and characterization of a prototype storagecell for the TUCAN EDM spectrometer. The purpose of the prototype cell is to validate theexpectations of the cell’s storage lifetime. The lifetime partially determines the maximum mea-surable UCN counts (and thus the statistical sensitivity) that the TUCAN EDM measurementcan achieve. This chapter will first describe the details of the mechanical design of the cell, andthen present the results of a series of storage lifetime measurements performed at the Japan Pro-ton Accelerator Research Complex (J-PARC) Material and Life Science Experimental Facility(MLF) in 2022 and 2023.The design of the prototype cell, henceforth referred to as the cell, is illustrated in Figure5.1. It consists of two flat aluminium plates that play the role of the “electrodes” of the cell,separated by an insulating ring, with an entrance on one of the electrodes. The entrance of thecell is sealed by a valve that is built into the “ground” electrode. Other than the valve, thecell volume is sealed by O-rings on the outside of the ring, which are radially compressed byclamping rings that bolt into the electrodes. The design of this sealing is discussed further inChapter 6. The metal surfaces of the cell are coated in NiP, while the insulator is coated withdeuterated polystyrene (dPS).The cell differs from those that will be used in the central region in a number of ways.Because the cell is not intended to be used with HV, it lacks any connection for a HV supply,the “electrodes” are not designed to sustain the HV, and there is no integration of the “ground”electrode into an insulating vacuum chamber. It is also not intended to be used with the Hg co-magnetometer, so it lacks both a Hg vapour valve and a window for the probe beam. However,for the purposes of measuring the storage lifetime, the internal geometry of the cell is intendedto be close to that used in the TUCAN EDM spectrometer, and so it includes the insulatortrenches and all surfaces are appropriately rounded. The trenches were made shallower thanthose in the concept model of Figure 4.2, since the central region geometry had not been finalizedat the time of the cell’s construction. Accordingly, the height of the cell is slightly larger, at128 mm instead of 122.4 mm. Similarly, the entrance cavity is smaller, with a depth of 12 mm,compared to 20 mm in the central region concept.Two parts of the cell involved substantial effort to design and build, and require some furtherelaboration. These are the cell valve and the insulator rings.5.1.1 Cell Valve DesignThe cell valve is a custom-built gate valve that seals the cell using the action of a single double-acting pneumatic actuator. The use of a custom valve was motivated by three factors that aredifficult to achieve with commercial valves. First, all parts in the valve must achieve a low level1255.1. A Prototype Storage CellValve actuatorPumping portUCN guide connectionCell valve"Electrode" platesInsulatorCELL HEIGHT 128.00 mmCELL 500.00 mmENTRANCE 85.00 mmTRENCHENTRANCE CAVITYR4.75 mmWIDTH 0.50 mmDEPTH 11.00 mmTRENCHSCALE 2 : 1ENTRANCE R10.00 mmPLUG R6.35 mm0.50 mmPLUGGED CAVITYHEIGHT 12.00 mmENTRANCE_CAVITYSCALE 1 : 1Figure 5.1: Diagrams of the prototype cell. The top view shows a 3D model of the cell withthe main features labelled. The bottom view shows a cut-through of the cell displaying majordimensions. Detailed views are given of the insulator trench and the entrance cavity with thevalve closed.1265.1. A Prototype Storage Cellof magnetism, a topic that is discussed in Chapter 7. Second, to reduce UCN losses any visibleslits in the valve need to be minimized. Third, it is desirable to minimize any cavity at theentrance to reduce systematic contributions.The valve mechanism is contained within a housing that mounts onto the ground electrodeof the cell. Inside the housing is a carriage that can be moved between two positions using apneumatic actuator. The mechanism of the carriage is illustrated in Figure 5.2. When actuatedinto the open position, an opening through the valve allows UCNs to pass into the cell. Whenactuated into the closed position, the actuator force is transmitted by the combination of aroller bearing and inclined plane to push a plug out of the carriage. The plug seals the entranceto the cell with an O-ring and partially fills the opening, leaving a cavity of 12 mm depth. Thiscavity is shown in the detailed view in Figure 5.1. When returning to the open position, springsassist with pulling the plug back into the carriage.Figure 5.2: Diagram of the cell valve mechanism. The top view shows the valve in the openposition, with the guide opening in place. The bottom view shows the closed position, withthe plug pushed out of the carriage and sealing the cell. The inset shows the details of theplug-pushing movement, which is facilitated by a roller bearing on an inclined plane.Any slits in the valve can play a major role in UCN storage, as well as the initial fillingof UCNs into the cell. In the open position, there are two 0.2 mm slits in the 85 mm guide.These slits are visible during filling of the UCN. In the statistics simulations, a 1 mm perfectly-absorbing slit was assumed in the valve [6], with the estimated losses being 2–3%. The 0.2 mmslits should further reduce these losses. When sealed, the plug is intended to have metal-to-metal contact with effectively no slit, so that there are no losses associated with the valve duringthe Ramsey measurement.The entrance cavity that is not filled by the plug is 12 mm deep, and 85 mm diameter. This1275.1. A Prototype Storage Cellleft-over cavity is a result of conflicting requirements in the design. The valve design assumeda thinner ground electrode, but in order to accommodate the various seals and fasteners, theground electrodes require a certain thickness, which ultimately resulted in the plug not fillingthe entire cavity. In the central region concept, the ground electrode thickness was furtherincreased, and the cavity depth became 20 mm.The primary concern from this cavity is the possible systematic effects. The cavity changesthe volume that is sampled by both UCNs and Hg. The calculations in Chapter 4 were per-formed without such a cavity. Calculations of BE frequency shifts show < 1% change associatedwith a cavity of such size (in fact, the cavity reduces these systematic effects due to the smallerradius volume having smaller bx and by components). Of greater concern for the cavity is thepresence of magnetic dipoles in the valve mechanism, which have a larger effect if they are closerto the volume sampled by the UCNs and Hg atoms. The depth of the cavity determines howclose such a dipole can be to this volume. The topic of dipoles is discussed further in Section7.4.The valve design can be modified to reduce the cavity. With this valve concept, a largervertical movement of the plug requires that the valve is made larger vertically, but it also mustbe made larger horizontally, since the actuation acts horizontally. This introduces challengeswith the footprint of the valve, but these challenges are not insurmountable. Such an alterationis beyond the scope of this work and will not be discussed further.5.1.2 Insulator CoatingsThe metal parts of the cell that are visible to UCNs have NiP coatings. Such coatings havebeen tested extensively by the TUCAN collaboration [5]. In the TUCAN EDM spectrometer theNiP coatings are intended to be replaced with diamond-like carbon (DLC) coatings to improvestorage lifetime and depolarization properties, which were still under development at the timethat the cell was built. The insulator rings must be made with non-conductive substrates andcoatings.Materials that are non-conductive and have low UCN losses are limited. In the early ILL andPNPI experiments, BeO was used (either as a coating onto SiO2 or in pure form) [103, 106].In the most-recent ILL experiment, uncoated SiO2 was used [108]. For the TUCAN EDMspectrometer, bare SiO2 was not considered due to its lower Fermi potential of 90.6 neV, inaddition to the limited availability of precise rings made from SiO2. BeO has largely beenabandoned as a material due to concerns about its toxicity as well as higher-than-expectedlosses in beryllium-containing materials [241, 242].The materials currently favoured for insulators in the various nEDM experiments are deuter-ated plastics (i.e. plastics in which the hydrogen content has been substituted for deuterium).Deuterated polystyrene (dPS), with V dPS = 160–170 neV, has been demonstrated as a goodUCN storage material [243–245], and has been used in the PSI nEDM experiment [114]. Thesecoatings can be applied to glass rings, but can also be applied to rings made from Rexo-lite, a cross-linked non-deuterated polystyrene [246]. Deuterated polyethylene (dPE), withV dPE = 204–224 neV, has also been explored as an improved material but has only been coatedonto glass substrates [244, 247]. The PSI nEDM experiment used deuterated polyethylene(dPE) as a coating for co-magnetometer windows due to its better optical properties with UVlight, but this coating has not yet been used as the coating of an entire insulator ring by anyexperiment.Because the performance of dPE and dPS had been well-established, they were consideredas possible materials for the TUCAN EDM spectrometer, with dPS coated onto Rexolite being1285.1. A Prototype Storage Cellthe baseline. However, while these deuterated materials are available in small amounts fornuclear magnetic resonance (NMR) applications, large coatings are not commercially available.To prepare the insulators for the TUCAN EDM spectrometer it was therefore necessary todevelop and set up the coating process at TRIUMF.Figure 5.3 shows the coating machine that was constructed for the application of dPScoatings. The coatings are applied by a rotational coating process. The ends of the insulatorring are sealed by clamping between two polytetrafluoroethylene (PTFE) plates and filled witha solution of dPS dissolved in deuterated toluene (toluene-d8). The ring is then rotated whilethe interior is flushed with nitrogen, which is provided by the boil-off from a dewar of liquidnitrogen (LN2). The nitrogen flushing removes the solvent vapour, driving evaporation anddepositing the dPS onto the ring.Teflon sealing plates MotorRotating hollowshaftInsulator (transparent)0.25 mNitrogen flushing lineNitrogen diffuserSolvent fill portFlushing exhaustFigure 5.3: Diagram of the deuterated-plastic coating machine. The insulator is clampedbetween two PTFE plates. The coating solution is filled into the clamped insulator usinga PTFE tube inserted through the fill port on the left. A motor continuously rotates theinsulator, allowing the solution to cover the entire surface. During the process, the coatingmachine is flushed with nitrogen to remove the solvent vapour, causing the solvent to evaporateand leave the plastic coating adhered to the insulator. The solvent vapour exits out the exhauston the right, which is connected to a hose that leads outside (not shown).A number of coating tests were performed with smaller rings prior to coating of the final ring.These tests were performed with different parameters: concentration and volume of solution,nitrogen flow rate, and rotation speed. The results were inspected visually with the goal offinding a procedure that produced a uniform coating.Uniformity of the coating was improved with a slower rotation speed. It was observed that,as the solvent evaporates, the solution becomes increasingly viscous. Beyond a certain point thehigh-viscosity solution is dragged along with the walls rather than pooling at the bottom, andthe dragged solution no longer spreads across the full width of the ring. However, too slow of arotation speed can cause the coating to fail due to friction of the shaft causing the machine tostop rotating. For the final coating of the cell insulator, the rotation speed was set to ∼1 rpm.1295.2. J-PARC BL05 UCN SourceA lower nitrogen flow rate is also better, as a high rate can cause the solution to evaporatetoo quickly. For the volume of the large insulator, a flow rate of 10–15 slm was used. Thevariation in the rate is due to the changing pressure in the LN2 dewar used to supply thenitrogen.Solutions with a mass concentration of 1%, 2%, and 4% dPS in toluene-d8 were tested.Due to the viscosity problem mentioned above, it was expected that a lower concentration ofdPS in solution could be better. However, the tests with 1% did not appear to produce bettercoatings compared to the higher concentrations. Ultimately, a ∼2% concentration was foundto produce reliable coatings and was used for the cell insulator. The solution used 2.5 g of dPS(molecular weight 2.120 × 106 g/mol, polydispersity index (PDI) 1.10) in 150 g of toluene-d8(99.6% deuterium content).Deuterated Polyethylene Coating AttemptsConsidering that dPE is reportedly superior for UCN storage [244, 247], it was desirable toalso attempt dPE coatings. Unlike dPS, dPE can only be dissolved at elevated temperaturesof ∼140◦C. To attempt dPE coatings, thermal insulation (two layers of melamine foam, innerlayer 2” thickness and outer layer 1” thickness) was mounted onto the frame. To raise thetemperature inside the insulation, the nozzle of a hot air blower was inserted through theinsulation. A number of thermocouples were also routed through the insulation to monitor thetemperature.The dPE solution could not be pre-made for this coating, as the temperature could notbe maintained outside the coating machine and during the transfer. Instead, the solid dPEpowder (for which no chemical analysis data was provided by the supplier) and solvent (o-xylene, chosen due to its high boiling point of 144◦C, and not deuterated due to the cost) werefilled into the sealed insulator together and the temperature of the coating machine raised to thedesired operation point. For the base material of the insulator ring, a material was needed thatcould withstand high temperature. While glass had previously been used, no supplier of 0.5 mdiameter glass rings with sufficient dimensional precision could be found. The material chosenfor testing was acetal, which has a melting temperature of 175◦C and a maximum operationaltemperature of 120◦C (although this value varies with supplier). While the thermal propertiesdid not appear ideal, it was thought that a coating could be possible without severe damage tothe ring.With this setup, it was possible to produce dPE coatings on small test insulators machinedfrom acetal. However, with the larger ring the temperature variation between the top andbottom of the insulator was also much larger, and it was not possible to uniformly reach thetarget temperature. The conclusion of those tests is that the dPE coatings appear to be possible,but would require a different insulator material and better temperature control to coat a full-sizering. Possible insulator materials would be high strength engineering plastics such as PEEKand Ultem, both of which have operating temperatures exceeding the required temperature fordPE coatings. Since no full-sized insulator was produced with a dPE coating, it will not bediscussed further.5.2 J-PARC BL05 UCN SourceThe cell prototype was assembled in late 2021, with the insulator being coated in early 2022. Bythis time, the prototype source from Chapters 2 and 3 had been decommissioned in preparation1305.2. J-PARC BL05 UCN Sourcefor the installation of the TUCAN source, but that installation had not been completed. Testingof the cell with UCN therefore required the use of another UCN source.The J-PARC MLF in Tokai, Japan is home to one of the most powerful spallation neutronsources in the world [248], alongside the Spallation Neutron Source at Oak Ridge National Labo-ratory (ORNL) in Oak Ridge, Tennessee [249] and the not-yet-operational European SpallationSource in Lund, Sweden [250]. J-PARC MLF currently operates 21 beamlines with a variety ofinstruments available for neutron experiments. Beamline 05 Neutron Optics and FundamentalPhysics (BL05) houses a UCN source based on the Doppler-shifting turbine concept that wasdescribed in 1.3.2 [4].Figure 5.4 shows a diagram of the turbine source, referred to as the Doppler shifter (DS).A cold neutron beam from the spallation source enters the DS and strikes a rotating neutronmirror. The speed and phase of rotation is tuned such that reflection from the mirror cancelsthe majority of the cold neutron velocity, with some of the resulting scattered neutrons beingshifted into the UCN energy range. Some of these UCNs exit out of a guide port and fromthere can be extracted into experiments.Figure 5.4: Diagram of the BL05 Doppler shifter UCN source. Cold neutrons from the MLFspallation source are reflected from a rotating mirror, the speed of which is tuned to cancel thevelocity of the neutrons. The diagram shows UCN extraction vertically out of the DS, but thereis also a second exit out of the page that was used for the cell measurements. This diagram wastaken from Ref. [4], published under the http://creativecommons.org/licenses/by/4.0/, and itsuse here is permitted under that license.The MLF spallation source provides cold neutron pulses at a rate of 25 Hz, and the DSreflects every third pulse, so that UCN production occurs every 120 ms [4]. At the exit to the1315.3. Energy Spectrum EstimationDS, a UCN detector sees approximately 90 cps (not all of which are technically UCNs, as somewill slightly exceed the potential of the guides). This count rate is significantly lower than therates seen in Chapter 3 (∼2000 cps after ∼10 m of guide), which limits the achievable precisionof any measurement. For storage lifetime measurements, uncertainties of 3–5% were achievablewith this source, which was deemed sufficient to judge the performance of the cell.The BL05 DS was used for characterizing the performance of the cell with UCNs duringtwo beamtimes in June 2022 and 2023. The characterization primarily involved measurementsof storage lifetime similar to that described in Chapter 3, but without the presence of He-II. However, there are some additional complications associated with the nature of the source.One major complication is the energy spectrum, which will be discussed in the following sectionbefore proceeding to the storage lifetime measurements.5.3 Energy Spectrum Estimation5.3.1 Experiment DescriptionAs with the experiments of Chapter 3, it is desirable to interpret storage lifetime results interms of the imaginary Fermi potential W of the materials used in the cell, which can thenbe compared between other measurements and to values assumed in statistics estimations.The relationship between storage lifetime and W is energy-dependent, and so making thisinterpretation requires some knowledge or assumption of the energy spectrum of the UCNs.For the BL05 DS, an auxiliary measurement of the energy spectrum was made to enable thisinterpretation, and this section will describe that measurement and the analysis.Due to the pulsed nature of the BL05 source, it is straightforward to measure an energyspectrum using time-of-flight (TOF). In this measurement, the time for UCN created in thesource to travel from the source to a detector is measured and converted to a velocity, whichcan then be converted to energy.The geometry of the experimental setup is shown in Figure 5.5. UCNs are created at a 1 cm3source volume inside the turbine. They exit through a straight guide that connects directly to a3He detector, of the same type described in Chapter 2. The detector’s foil window would blocksome lower energy UCN in this configuration, but the turbine exit was fitted with a similar Alfoil for both the TOF and storage lifetime measurements, so those UCNs do not exit the sourceand do not play any role.The geometry of the source volume is a possible source of error that has not been accountedfor in the remainder of the analysis. The 1 cm3 volume was described in Ref. [4] as a pseudo-point source in which the beam intersected the mirror. However, the same studies suggestedthat extractable UCNs were produced over a larger volume, potentially up to 3× 3× 6 cm3, asthe 3× 3 cm2 mirror could produce UCNs over a 170 mrad angular width as it rotates throughthe very cold neutron (VCN) beam. These details were not realized until after this analysishad been completed, and so it is possible that this could introduce some inaccuracy. This willbe discussed again after describing the analysis methods.5.3.2 Analysis MethodsThe calculation of the energy from the TOF is seemingly straightforward. The distance fromthe DS mirror centre to the detector is known, as is the timing of the mirror intersecting a coldneutron pulse. For each count in the detector, the elapsed time since the last pulse is measured.The distance and timing are used to calculate the energy. Background counts are measured1325.3. Energy Spectrum EstimationDoppler shifterUCN guidesPumping port 3He detectorEXIT GUIDE 72.06 mmUCN GUIDE 84.68 mmSQUARE GUIDE 41.00 mmSOURCE TO DETECTOR FOIL 501.60 mmDETECTOR FOIL THICKNESS 0.10 mmSOURCE FOIL THICKNESS 0.025mmPOINT SOURCEFigure 5.5: Diagrams of the geometry of the TOF measurement. The top view highlights thekey parts of the measurement. The lower view shows a section through the centre of the guidesand indicates the dimensions.1335.3. Energy Spectrum Estimationwith the DS mirror positioned out of the path of the beam and with zero rotation speed, andare subtracted from the TOF spectrum before converting to energy.Unfortunately, there are some limitations to calculating the energy in this way. First is thatthe TOF only measures the neutron velocities in the longitudinal direction (along the guideaxis), ignoring transverse velocities. The energy calculated with this method therefore onlygives the “longitudinal energy spectrum”. In addition, UCNs that diffusely reflect from theguide walls can travel a much longer distance to reach the detector than they would by a directpath, which invalidates the calculation. This further obscures the true energy of the ultracoldpart of the spectrum.If the beam is well-collimated, then the longitudinal energy spectrum may be close to thetotal energy spectrum. At the time of these experiments the angular distribution of the neutronsexiting the DS had not been measured. With the distribution unknown it is difficult, if notimpossible, to calculate the true total energy spectrum from the TOF measurement. To estimatethe energy spectrum and its uncertainty using the TOF measurement data, a Monte Carlomethod using PENTrack simulations was instead developed.To estimate the initial energy spectrum from the TOF measurement, the experiment wassimulated using PENTrack assuming a uniform initial energy spectrum. The simulated countswere then re-weighted by an initial energy function, with the function parameters being fitto obtain the closest match to the experimental TOF spectrum. Several different weightingfunctions were attempted, and are described in Table 5.1. The functions included in this list donot have any strong physical justification for being used. Normal functions were initially chosenbecause the TOF spectrum itself looked close to a skewed or double normal. Generalizations ofthe normal function such as the Weibull and generalized normal functions were then included inan attempt to improve the reproduction of the spectrum. The polynomial and radical functionswere included because they had been noted to fit extracted UCN spectra reasonably well atother sources. Several spectra that were attempted produced unrealistic initial energy spectra(such as a very large weight at zero energy), and were discarded from the final analysis.There are several additional unknown parameters in this measurement, and so these simu-lations were repeated while varying the corresponding unknown parameters. These parametersare described in Table 5.2. Two initial angular distributions were considered, which describethe extremes of what should be possible: either velocities that are directed entirely along theguide axis (a “collimated” angular distribution), or velocities that are isotropically distributed(a “uniform” angular distribution). The material properties considered are the stainless steelguide imaginary potential WSS and diffuse reflection probability PL,SS, and the real poten-tial V Al of the Al foils (both the DS exit and detector). For the guides, baseline values ofWSS = 0.05 neV and PL,SS = 5% were assumed based on previous measurements [3, 5], whilefor the foil the theoretical value of V Al = 54.1 neV was assumed as a baseline. Each simula-tion varies at most one of the material parameters. All material parameter configurations weresimulated with both angular distributions.The goal of this analysis is not to find the true initial energy spectrum. While it may bepossible to accomplish this, the lack of knowledge about the angular distribution makes sucha search challenging, as subtle variations in the distribution could have a significant impacton the measured TOF. The analysis also requires the assumption of a functional form for thespectrum, which is not guaranteed to be able to perfectly represent the spectrum.Instead, the analysis aims to build a collection of possible initial energy spectra. Noneof these spectra are likely to be the true energy spectrum, but, as they try to reproduce theTOF result, they should capture some of the features of the true energy spectrum (such asrelative amounts of faster or slower UCNs). When applied in the analysis of the storage lifetime1345.3. Energy Spectrum EstimationFunction name Functional formNormal (N) e−(x−p0)2/(2p21)Double normal* (DN) p0e−(x−p1)2/(2p22) + e−(x−p3)2/(2p24)Skewed normal (SN) e−(x−p0)2/(2p21)(1 + erf(p2(x−p0)√2p1))Quadratic* (P2) p0 + p1x+ p2x2Quartic* (P4) p0 + p1x+ p2x2 + p3x3 + p4x4Radical* (R) (x− p0)p1Weibull (W) p1p0(xp0)p1−1e−(xp0)p1Generalized Weibull (GW)(xp0)p2e−(xp0)p1Generalized normal (GN) e−((x−p0)2/(2p21))p2Skewed generalized normal (SGN) e−((x−p0)2/(2p21))p2(1 + erf(p3(x−p0)√2p1))Table 5.1: Weighting functions used in the simulation analysis of the total energy spectrumfrom the TOF experiment. Functions marked with a * were tested and found to generallyproduce unrealistic energy spectra, so they are not included in the reported results. Acronymsare given for each function for ease of labelling of results. The parameters pi are arbitrary andare not given any physical interpretation.Parameter ValuesAngular distribution Collimated: UCN initial velocitiespoint along the guide axisUniform: UCN initial velocitiesare isotropicGuide imaginary potential WSS 0.01/0.05/0.10/0.15/0.20 neVGuide diffuse reflection probability PL,SS 1/3/5/7/9/11/15%Foil real potential V Al 44.1/54.1/64.1 neVTable 5.2: Parameter values used in the simulation analysis of the total energy spectrum.Default values are bolded.1355.3. Energy Spectrum Estimationexperiments, each spectrum will result in a different interpretation (in the form of materialparameters), and the range of those interpretations should be broad enough that the trueparameter values are captured with some uncertainty that is representative of the uncertaintyin how to treat the initial spectrum.With the intent of this analysis in mind, the topic of the source volume mentioned at theend of the previous section should be addressed. Ignoring the more extended source volume of3 × 3 × 6 cm3 in favour of the 1 × 1 × 1 cm3 volume has differing effects when considering thedimension along the guide axis and the dimensions in the transverse plane.Along the guide axis, the use of a 1 cm length instead of 3 cm (corresponding to the widthof the mirror) has a direct effect on the TOF, as the UCNs produced closer to the detectorwill take less time to reach it, while those produced further away will take longer. With UCNsdistributed over 1 cm, the time taken to reach the detector for a given longitudinal kinetic energyhas a range of ±1%. If this distance is instead 3 cm, the range is ±3%. To compensate forthe narrower range in the 1 cm case, the fitting might be expected to produce a slightly widerenergy spectrum. However, it will be seen that this analysis yields a wide range of possibleenergy spectra, and the effect of the additional 2% uncertainty in the final analysis should notbe significant.In the transverse plane, the wider 3× 6 cm2 area (associated with the height of the mirrorand the distance swept out by the mirror) primarily alters the angles at which UCNs can beextracted from the DS. Accounting for the guide geometry but neglecting gravity, a 1 cm sourcewidth allows UCNs travelling at ±49.3◦ relative to the guide axis to leave the DS, while a 6 cmsource width allows UCNs travelling at ±70.4◦ to leave the DS. This is a significant difference,as UCNs travelling at 70.4◦ would take almost twice as long to reach the detector than thoseof the same energy travelling at 49.3◦. This would be somewhat mitigated by these large-angleUCNs having more wall reflections, increasing their overall loss probability, but this is a lessercorrection to the impact of the larger angular acceptance. The lack of these UCNs with lowerlongitudinal energy could lead to the analysis shifting towards lower total energy UCNs. If thesimulated energy spectra are systematically shifted towards lower total energy, then this wouldresult in the extracted values of WNiP and WdPS being higher in the final analysis. At thisstage, such an uncorrected systematic error is deemed to be acceptable, as larger values of theseparameters would be more conservative in estimating future experiment performance.5.3.3 ResultsFigure 5.6 shows the measured TOF spectrum after background subtraction. Despite back-ground having been subtracted, there are significant counts at 0 ms. This is due to “frameoverlap”, in which the slowest neutrons from one pulse arrive after the fastest neutrons fromthe next pulse. Because UCNs are produced every 120 ms, the TOF cuts off at that time, andthe data shows the sum of counts taken over many 120 ms periods. UCNs that have not reachedthe detector before 120 ms elapses are therefore counted in the next period. To illustrate this,the spectrum has been fit to a quadruple Gaussian, in which two of the peaks share the sameparameters as the other two, but are translated by 120 ms. The functional form of this fit isN(t) = p0e−(t−p1)2/2p22 + p3e−(t−p4)2/2p25+p0e−(t+120−p1)2/2p22 + p3e−(t+120−p4)2/2p25 ,(5.1)where pi are the fit parameters, and t is the measured TOF. The third and fourth terms representthe overlapping pulse, and seek to capture the leftover UCN.1365.3. Energy Spectrum Estimation0 20 40 60 80 100 120Time of flight (ms)00.050.10.150.20.250.30.35Normalized countsMeasured spectrumFitted spectrumVSS = 183 neVFigure 5.6: Plot of normalized counts vs TOF measured out of the DS. The data is fit toa function N(t) = N ′(t) + N ′(t + 120), where N ′(t) = p0e−(t−p1)2/2p22 + p2e−(t−p3)2/2p24 is adouble Gaussian. The addition of N ′(t+120) captures residual UCN that are left-over from theprevious pulse. The error bars are the statistical uncertainties of the counts. The vertical linemarks the TOF corresponding to a UCN with longitudinal energy equal to the Fermi potentialof stainless steel.This model is seen to produce a relatively good qualitative fit, although quantitatively thefit is poor (χ2/ndof ≈ 47, with the number of degrees of freedom ndof = 114). This qualitativebehaviour could be interpreted as there being two populations. One population travels alongfaster direct paths to reach the detector, while the other undergoes diffuse reflections and takeslonger to reach the detector. The TOF for a straight-line path to the detector with an energyequal to the Fermi potential of the stainless steel guides is marked on Figure 5.6, and it canbe seen that most of the counts in the larger peak have a shorter TOF than this level. Thesecounts cannot be true UCNs in this setup. This supports the two-population suggestion, sincethese neutrons with higher energies are less likely to reflect from the guide walls, and a diffusereflection is likely to cause them to be lost instead of detected.Since this fitting function is not a perfect fit, it is not used for any further analysis of thespectrum. In the simulations of the TOF measurement, a single pulse was simulated. The TOFis then calculated as the time of detection modulo 120 ms. For example, a UCN detected at124 ms would be taken to have a measured TOF of 4 ms. This calculation allows the simulatedTOF spectrum to capture the frame overlap.Figure 5.7 shows several representative fits of the TOF spectrum compared to the measureddata. Many of the fits do not reproduce the TOF well. There are two features of the TOFspectrum that are not well-represented by most of the fit results. The first feature is the shapeof the fast peak. Most fits do not produce as high of a peak as observed, and they also donot produce as narrow of a peak as observed. In particular, there is a range of 75–100 mswhere the simulations tend to see more neutrons than experimentally observed. The secondpoorly-represented feature is the flat region between 0–50 ms. Collimated angular distributions1375.4. Storage Lifetime Measurementgenerally fail to capture this region, although uniform angular distributions replicate it fairlywell. The reason for this difference could be that this region contains mostly diffusely-reflectedUCNs that are taking much longer to enter the detector than they would by a direct path. Sincethe collimated distribution will have fewer UCNs hitting the walls, it might fail to simulatethese indirect-path UCNs. This region could also contain many UCNs moving at larger anglesrelative to the guide axis, which are not simulated by the 1 cm3 source volume. This wouldalso be consistent with the collimated distribution failing to simulate this region, as the tightly-directed beam would be a much worse approximation of the real distribution of UCNs.Since the UCN cut-off for the steel guides is at ∼85 ms, the fast peak is less importantto reconstruct, since none of those counts can be UCNs and will therefore not appear in thestorage lifetime measurement. However, there is a small range of 85–100 ms where the increasedcounts in simulations correspond with a greater number of high-energy UCNs than expectedexperimentally. Since higher-energy UCNs are lost more quickly, these spectra could yield lowerW i values.The reconstruction of the flat region could also be important. If the observed neutrons areUCNs taking indirect paths, failing to reconstruct this region could also mean that the resultingspectra are higher-energy. However, since this failure is specific to the collimated distribution,it should not bias the final results of the storage lifetime analysis when considering the fullrange of spectra.There are a few results that replicate the TOF spectrum quite well, at least qualitatively.The SGN function with a uniform angular distribution, shown in subfigure (a) of Figure 5.7,appears to reproduce all of the features of the spectrum quite well.While many of the initial spectra produced by this analysis do not replicate the TOF well,there are some that do, and those spectra could be close to the true spectrum. However, thoseindividual spectra cannot be validated with the available data. Rather than assuming thatthe best-fitting spectra are representative of the true spectrum in the analysis of the storagelifetimes, all of the simulated initial spectra will be used to generate a broad collection of filledspectra inside the cell. This method should capture the uncertainty in the initial spectrum fromthe DS.Figure 5.8 shows the assortment of initial energy spectra obtained from the analysis. Manyof the spectra that use the normal, SN, W, and GW functions have a similar shape, with thegreatest difference due to whether a collimated or uniform distribution was used, with uniformdistributions tending to be shifted towards higher energies. The GN and SGN functions yieldsome spectra with very sharp peaks. It is worth noting that these peaks are above the UCNcutoff for the guides, so the role they play is not as extreme as they appear. In the UCN energyrange, shown in the inset of Figure 5.8, the spectra vary significantly in how flat they are, withthe sharper peaks producing flatter spectra that will weight lower energy UCN more strongly.5.4 Storage Lifetime Measurement5.4.1 Experiment DescriptionThe primary measurements of the cell characterization are a series of storage lifetime measure-ments. A total of seven measurements were performed over the two beamtimes in 2022 and2023. These are listed in Table 5.3, using the naming convention described in Chapter 2. Thedifferences between these experiments are outlined here briefly, with further details included inthe discussion of the results.The geometry of all experiments except TCN22-030-SL was the same, and is shown in Figure1385.4. Storage Lifetime Measurement0 20 40 60 80 100 120TOF (ms)00.010.020.030.040.050.06Normalized countsExperiment0.01 neV, 5%, 54.1 neV0.05 neV, 5%, 64.1 neV0.15 neV, 5%, 54.1 neV(a) Uniform, SGN0 20 40 60 80 100 120TOF (ms)00.010.020.030.040.050.06Normalized countsExperiment0.01 neV, 5%, 54.1 neV0.05 neV, 5%, 64.1 neV0.15 neV, 5%, 54.1 neV(b) Collimated, SGN0 20 40 60 80 100 120TOF (ms)00.010.020.030.040.050.06Normalized countsExperimentSkewed normalGeneralized WeibullGeneralized normal(c) Uniform, (0.01 neV, 5%, 54.1 neV)0 20 40 60 80 100 120TOF (ms)00.010.020.030.040.050.06Normalized countsExperimentSkewed normalGeneralized WeibullGeneralized normal(d) Collimated, (0.01 neV, 5%, 54.1 neV)Figure 5.7: Example fits of the simulated TOF spectrum, using different initial energy spectrafrom Table 5.1, to attempt to match the experimental data. The figures (a) and (b) showexamples using the SGN for the uniform and collimated angular distributions, with the legendentries labelling (WSS, PSS, VAl). Figures (c) and (d) show examples with some of the otherdistributions: SN, GW, GN.Name DescriptionTCN22-010-SL Storage lifetime with NiP-coated ring and cell valveTCN22-011-SL TCN22-010-SL repeated after bakingTCN22-012-SL TCN22-011-SL repeated after manually cleaningTCN22-020-SL Storage lifetime with dPS-coated ring and cell valveTCN22-030-SL Storage lifetime with dPS-coated ring and VAT valveTCN23-010-SL TCN22-010-SL repeated with new cleaning procedureTCN23-020-SL TCN22-020-SL repeated with new cleaning procedureTable 5.3: List of storage experiments performed in the 2022 and 2023 beamtimes.1395.4. Storage Lifetime MeasurementFigure 5.8: Simulated initial total energy spectra of neutrons from the DS obtained from fittingthe TOF experiment. Total energy is defined as the kinetic energy plus the gravitationalpotential energy measured relative to the bottom of the cell in the storage experiments. Theexit of the DS is 53 cm, or ∼55 neV, below the bottom of the cell. Red curves assume a uniformangular distribution from the source, while blue curves assume a collimated angular distribution.Beyond the angular distribution, no effort is made to distinguish the curves, which cover a widerange of parameter values from Table 5.2.1405.4. Storage Lifetime Measurement5.9. For these measurements, the cell was connected to the DS and to the 3He detector using arotary valve, which could be switched to connect the cell to the DS or to the detector. Theseexperiments differ only in the preparation of the surfaces inside the cell. The TCN2X-01Y-SLseries all used an Al ring coated with NiP rather than the dPS-coated Rexolite ring, which wasused in the TCN2X-02Y-SL series.The TCN22-030-SL experiment differed in its geometry by the replacement of the cell valvewith a short guide section and a commercial VAT valve (of the same kind used in the prototypesource and described in Chapter 3). This experiment was performed as a comparison forevaluating the performance of the cell valve. The altered geometry is shown in Figure 5.10.The three experiments TCN22-010-SL, TCN22-011-SL, and TCN22-012-SL differ due tosurface cleanliness. TCN22-010-SL was performed with the cell surfaces having been cleanedwith a degreaser (Alconox) and isopropanol prior to shipping to J-PARC. After measuringthe lifetime, the cell was baked to try and improve cleanliness, measured in TCN22-011-SL.Unfortunately, this made the cleanliness worse, and resulted in no UCNs being detected, and sothe cell was opened and manually cleaned again. The lifetime was remeasured in TCN22-012-SLbefore proceding with the other measurements.None of the measurements in 2022 produced a satisfactory storage lifetime result. It wasexpected that the storage lifetime would exceed 100 s based on the simulations of Ref. [6],with the caveat that the energy spectrum would be different at J-PARC, but the TCN22measurements ranged from 54–85 s. This was hypothesized to be due to contamination inthe cell, possibly from some components of the valve or from the o-rings used in the seals.Between 2022 and 2023 a new cleaning procedure was developed for the cell, which is detailedin Appendix D. The subsequent experiments TCN23-010-SL and TCN23-020-SL measured theimprovement due to the cleaning procedure.To measure the storage lifetime in any configuration, the cell was connected to the DS withthe cell valve open for a time tfill. Once this time had elapsed, the cell valve was closed andthe UCNs stored for a time t. During this time, the rotary valve was switched to connect theclosed cell to the detector and empty the guides of UCNs. After t had elapsed, the cell valvewas open and the surviving UCNs were counted.The filling time tfill was determined from a preliminary measurement of counts versus tfillwith a storage time of 15 s. The value of tfill was selected to be close to the time at which thecounts were saturating. In the 2022 measurements, tfill was chosen to be 80 s, while in the 2023measurements a value of 60 s was used. The difference was due to the low precision of thispreliminary measurement creating some uncertainty in when the counts were saturating. Thefilling time determination from 2023 is plotted in Figure 5.11.Storage times were varied from 15–270 s in 2022, and from 15–400 s in 2023. The lower limitwas due to the need to empty the guides between the cell and detector of any UCNs leftoverfrom the filling —shorter storage times would see these UCNs as additional counts. The upperlimit was chosen with the aim of measuring to at least 3τs to properly capture the storagelifetime. The increase in maximum storage time from 2022 to 2023 was due to an increase inthe measured storage lifetimes, which will be shown in the results.5.4.2 Pressure Dependent MeasurementsDuring the measurements it was observed that outgassing inside the cell was significant. Toreduce UCN losses, the cell does not have any internal opening for pumping, so when the cellis sealed during the storage period, the outgassing caused the pressure to increase. With dryroughing and turbomolecular pumps, and using the dPS-coated ring, the pressure measured at1415.4. Storage Lifetime MeasurementDS EXIT TO ROTARYVALVE 696.31mmCELL ELEVATION 533.88mmROTARY VALVE TO CELL 145.63mm DETECTOR DROP 296.10 mm3He detectorRotary valvePumping portCellDoppler shifterFigure 5.9: Diagrams of the geometry of the storage lifetime measurement with cell valve. Thetop view highlights the key parts of the measurement. The lower view gives the key dimensionsof the guide layout. The internal geometry of the DS and exit guide are as previously shownin Figure 5.5. All guides after the exit guide are are 84.68 mm ID, except for the rotary valve,which has a 60×60 mm square guide geometry. The plane of the cell, rotary valve, and detectorare offset by a distance of 712.81 mm from the DS point source, not shown in the lower figure.1425.4. Storage Lifetime MeasurementVAT valveVAT valve adaptersCell adapterFigure 5.10: Diagram of the modified experiment layout for TCN22-030-SL. In this layout thecell valve was removed and replaced with an adapter and VAT gate valve. This geometry hasthe same dimensions as used in the other storage experiments, except that the valve gate is176 mm below the bottom of the cell, creating a much larger cavity.0 50 100 150 200 250Fill time (s)01020304050607080Counts after 15s storageMeasured CountsFit N(1-e-t/ )A = 64.4 1.7 = 15.9 2.0 s2/ndf = 9.865/5Figure 5.11: Measured UCN counts after 15 s storage in the cell versus filling time, in theTCN23-010-SL configuration. A filling time of 60 s was selected from this data, as the countsare observed to be close to saturation after that time. A saturation curve is also shown fit tothe data. The results of this fit shows that the counts would be at ∼98% saturation after 60 s.1435.4. Storage Lifetime Measurementthe entrance to the cell could achieve 5 × 10−4 mbar, but pressure spikes up to 1 × 10−2 mbarwere observed after opening the cell valve following the longest storage times. This pressurecomplicates the storage lifetime measurements because the UCNs can be lost by interactionswith the gas inside the cell. Since a longer storage time will see a higher pressure than a lowerstorage time, this can also make the storage lifetime depend on the storage time, skewing theresults.To address the outgassing problem, some storage lifetime measurements were repeated withdifferent base pressures, achieved by changing the configuration of the vacuum pumping. Inthe high pressure configuration, only roughing pumps were used. In the intermediate pressureconfiguration, a single turbopump (connected to the cell valve) was turned on. In the lowestpressure configuration, a second turbopump (connected to the guide at the exit of the DS) wasadded. With this range of pumping configurations, storage measurements could be taken withaverage pressures inside the cell in the range of 1.6× 10−3 to 4.0 mbar.With no pumping connection on the interior of the cell, there is also no connection for apressure gauge to measure the pressure inside the cell. This means that the pressure during thestorage lifetime measurements was not directly measured. However, a pressure gauge (PfeifferPKR251 full range gauge) was connected to the pumping port of the cell valve. The pressuremeasured outside the cell should relate to the pressure inside the cell, and if that relationshipcan be determined then it is possible to estimate the pressure inside the cell during the storagelifetime measurements.To obtain this pressure calibration, an auxiliary measurement was performed, in which thetop plate of the cell was replaced by an uncoated aluminium plate with a connection for apressure gauge (another PKR251). Storage cycles were repeated with this setup but withoutUCN production to measure the pressure inside the cell. This was repeated for each of thepumping configurations.Figure 5.12 shows an example of the pressure data obtained from the auxiliary pressuremeasurement and illustrates the pressure values used in the analysis. The average pressureinside the cell, P¯c, taken between the valve closing and opening, is the desired value. Thisinternal pressure depends on the initial pressure in the cell, which is taken to be proportionalto the initial pressure outside the cell, Pcv,i. The internal pressure should also be relatedto the increase in pressure measured outside the cell when the cell valve is opened, labelled∆Pcv = Pcv,max − Pcv,min. When the valve opens, the built-up pressure inside is pumped out,which causes the pressure outside the cell to increase rapidly. The size of this pressure spikemust certainly depend on the pressure inside the cell. These three values are extracted for eachmeasured storage cycle, and used to fit a functionPc = a0 + a1Pcv,i + a2∆Pcv, (5.2)where a0, a1, and a2 are the fit parameters.Because of the need for the pressure calibration measurements, pressure-dependent storagelifetime measurements were only taken for TCN22-030-SL and TCN23-020-SL, which were thelast configurations in each beamtime. The calibration measurements were then performed beforefully dismantling the experiments. This avoided the need to swap in the plate with the pressuregauge connection between storage measurements.5.4.3 Analysis MethodsThe raw detector counts obtained in the storage experiments were pre-processed to subtractbackground, and re-scaled to account for fluctuations in beam current. The background was1445.4. Storage Lifetime Measurement4800 5000 5200 5400Time (s)10-1100101Pressure (Pa)Cell valve gaugeCell gaugeValve closedValve openPcv,minPcv,maxPcv,iFigure 5.12: A sample of pressure data versus time from the auxiliary pressure measurementused to calibrate pressure inside the cell during storage. The blue curve measures the pressureoutside the cell, while the red curve measures the pressure inside the cell. Vertical dashedlines indicate when the cell valve closed and opened. The values of interest for the calibration,described in the text, are indicated by green and purple horizontal lines.1455.4. Storage Lifetime Measurementcalculated from counts in all periods when UCNs would not be detected. The re-scaling wasdone using counts from a VCN detector, the location of which is shown in Figure 5.4 as the“neutron beam monitor”.When the effects of pressure are not significant, the processed counts vs storage time canbe fit with an exponential decay to obtain the storage lifetime. In measurements where thepressure inside the cell appeared to be significant, a more complicated fitting model is used.For a storage time ti and a pressure Pc,i in the cell, the counts areN(Pc,i, ti) =Nfillede−ti(τw−1+ΓPPc,i)=N0(τw−1 + τh−1 + ΓPPf)−1×(1− e−tfill(τw−1+τh−1+ΓPPf))e−ti(τw−1+ΓPPc,i).(5.3)Here, τw is the lifetime due to wall losses (and beta decay), τh is the lifetime associated withthe cell valve being open during filling (the “hole lifetime”), ΓP is a constant that describes thepressure-dependent lifetime, and Pf is the pressure during filling (which was calculated as thepressure measured at the pumping connection to the valve plus some offset determined fromthe pressure calibration data).This model describes the detected number of UCNs in two parts. The first part is theexpected filled number of UCNs Nfilled, which depends on the pressure during filling, and mustalso include τh to account for the cell valve being open. The second part is the exponential decayof Nfilled during the storage period, which depends on the pressure inside the cell. Fitting thismodel to the experimental data allows the wall lifetime to be extracted and analyzed withoutthe pressure contribution.The hole lifetime τh in the pressure correction was estimated from the fit of the fillingoptimization data in Figure 5.11. The lifetime τs,fill in the filling data should be τs−1,fill =τs−1,TCN23−020−010 + τh−1, where τs,TCN23−020−010 is the result of TCN23-020-010. From thisrelation, τh is estimated to be 18.0 ± 2.6 s. The contribution of the uncertainty of τh was ex-amined by re-doing the fit at τh − σ(τh) and τh + σ(τh). The effect of the uncertainty on thisparameter was negligible compared to the uncertainty of the overall measurement.There is a limitation to this pressure-dependent lifetime model. Since outgassing is causedby substances that are adsorbed or absorbed in a material, there may be wall losses that alsochange as the cell is pumped out for longer. This effect should be at least partially capturedby ΓP, but could also be partially included in τw. The data obtained is not sufficient toguarantee that wall contaminants are removed by the pressure analysis. This could result in anoverestimation of W .Fitting directly to pressure-dependent data was only possible for TCN22-030-SL and TCN23-020-SL, where such data was collected. These experiments, both of which used the dPS-coatedinsulator, had a large pressure effect. TCN22-020-SL also used the dPS-coated insulator andhad a large pressure effect, but the pressure was not intentionally varied, so it could not be fitin the same way. To correct this measurement, the same fitting model was used, but the valueof ΓP was set to the value obtained from TCN22-030-SL, which was 0.002± 0.0001 s−1Pa−1. Inthe other experiments, which used the NiP-coated ring, the pressures were observed to be rela-tively small. Based on the largest pressures seen compared to those in the pressure-dependentmeasurements, and the value of ΓP, the possible contribution from pressure appeared to be nolarger than 1 s, and an additional uncertainty of 1 s is therefore added to those measurements.Two separate methods of analyzing the storage lifetime in terms of W were applied forthese experiments. The first method uses a combination of the simulated energy spectra with1465.4. Storage Lifetime Measurementan analytical description of storage based on kinetic gas theory (this method will be labelled askinetic theory (KT)). The second method is based entirely on PENTrack simulations, similarto the methods of Chapter 3 (this method will be labelled as full simulation (FS)). The reasonfor the implementation of these two methods for analyzing these experiments was largely dueto the complexity of the spectrum analysis. The analytical approach allows some simplificationof the analysis and was initially performed to obtain quick results. The full simulation analysiswas then performed later as a cross-check on the results. The comparison of these two methodsis of general interest, so the results of both are included and discussed.The KT description is developed in Chapter 4 of Ref. [126], and has previously been usedin the analysis of guide experiments by the TUCAN collaboration [5]. It uses a set of equationsthat describe the number N of UCNs expected after a storage time t asN(t) =∫ HmaxHmindNdH e−t/τs(H)dH, (5.4)where H is total energy, so that Hmin and Hmax are the minimum and maximum total energypossible in the cell and dNdH is the energy spectrum. This integration is simply a sum over allUCNs in the filled energy spectrum, with an energy-dependent exponential decay applied.The energy-dependent storage lifetime τs(H) is given byτs−1(H) =√2H/mn4γ(H)∫ zmaxzminH−mngzH×∑idAidzµi(H−mgz)dz + τβ−1,(5.5)where z is the height in the cell, with zmin and zmax the minimum and maximum heightsrespectively, mn is the neutron mass, g is gravitational acceleration,dAidz is the differentialsurface area of the ith surface in the cell, µi is the UCN loss rate of the ith surface in the cell,and γ(H) is given byγ(H) =∫ zmaxzmindVdz√H −mngzH dz, (5.6)where dVdz is the differential volume. In Equation 5.5, the expression√2H/mn4γ(H)H−mngzH (5.7)is a calculation of the energy-dependent rate of wall collisions assuming that the UCNs are inmechanical equilibrium. The term√2H/mn is the maximum velocity of a UCN of energy H,and γ(H) is the effective real space volume available for UCNs when accounting for effects ofgravity. Factors of√H−mngzH scale quantities at z = 0 to arbitrary z. This wall collision rate isthen multiplied by the areas at a specific height, which are in turn multiplied by the loss ratefor those areas. The lifetime from beta decay, which has no energy or height dependence, isfinally added.The loss rate is written asµi(Ek) =2∫ pi/20(1−∣∣∣∣∣√Ek cos2(θ)−√Ek cos2(θ)− V i + iW i√Ek cos2(θ) +√Ek cos2(θ)− V i + iW i∣∣∣∣∣)× cos(θ) sin(θ)dθ,(5.8)1475.4. Storage Lifetime Measurementwhere here θ is the angle that a UCN makes with the normal to the wall, which is integratedover the full range. The detailed derivation of these equations can be found in Ref. [126].In the analysis of the cell experiments, the above equations are used to calculate N(t) atdifferent values of W i. Those calculated counts are then fit to a single exponential, which iscompared to the single exponential lifetimes obtained from fitting the experimental data. Thiscontrasts with the previous application of this model by the TUCAN collaboration, in whichN(t) was compared between experiment and model. Comparing lifetimes is less direct thancomparing the actual count data, but has some advantages. With a single measured value foran experiment, it becomes easier to visualize the behaviour when changing parameters in themodel. This is particularly useful for these experiments, which have both multiple materialparameters (WNiP and WdPS) and high uncertainty on the energy spectrum. Because theexperimental data was observed to fit well to a single exponential, this was judged to be anacceptable simplification of the modelling.It is worth emphasizing the assumptions of the KT approach. In particular, the constructionof Equation 5.5 for the lifetime includes a description of the frequency of wall collisions. Thisis based on modelling the UCNs as a gas according to kinetic gas theory, and so requires thatthe UCNs are in equilibrium for the description to be accurate. In Ref. [126] it is discussedthat the wall loss probability needs to be much less than the diffuse reflection probability inorder for this to apply. This is expected to be generally true for UCNs, but could be invalid incases of very high losses or extremely specular surfaces. Another simplification that has beenapplied in this analysis is the assumption of a cylindrical geometry. This neglects the trenchesof the cell, and was done to simplify the calculation.The FS method fully simulates the experiment using PENTrack. The simulation of eachmeasurement is divided into separate simulations for the filling and storage/detection stages,which makes it easier to vary the spectrum. The experiment is simulated for different W i valuesin the cell, and the simulated τs values are then compared to the experiment results as with theKT method.The storage and detection simulations for this analysis start with UCNs with a uniformenergy spectrum in the cell (with the spatial distribution weighted by phase space availability).The cell is kept closed for a storage time ti, and then opened and emptied into the detector.This is repeated for all of the storage times used in the experiment. The counts in the detectorfor a given storage time are binned by energy, and then re-weighted using each given estimatedenergy spectra. Finally, the total re-weighted counts versus storage time are fit to obtain thesimulation storage lifetime.Both the KT and FS methods require the energy spectrum of UCNs filled into the cell. Thisis different from the initial energy spectrum at the DS which is estimated by the TOF analysis.To transform an initial spectrum into a filled spectrum, additional PENTrack simulations of thefilling of the cell were performed. In such a simulation, a uniform initial spectrum is assumedat the DS and the UCNs are allowed to fill into the cell. At the end of the filling period,the spectrum of the filled UCNs are obtained. This is then re-weighted by the initial energyspectrum obtained from the TOF analysis. These simulations were repeated for each of theparameter sets in the TOF analysis.Additional parameters were considered in the filling simulations to account for changes tothe spectrum due to interactions with the cell surfaces during filling. In particular WNiP, PL,NiP,WdPS, and PL,dPS were considered. Including these parameters adds considerable complexityto the analysis, as any simulated filled spectrum should technically only be used to calculatethe storage lifetime with the exact parameters used to simulate the spectrum. If the fillingdepends on the cell spectrum, then the parameter fitting cannot vary those parameters without1485.4. Storage Lifetime Measurementalso varying the spectrum. To account for this, the filling simulations were repeated for eachvalue of the cell parameters that would be considered. Lifetimes at given values of WNiP andWdPS are then calculated only using those spectra that were filled into a cell with those sameparameters.The analysis should consider the uncertainty in both the initial and filled spectra. For theinitial spectrum, for which the fitted functions in Table 5.1 are used, the uncertainty of the fits isneglected. The reason for this is that the uncertainty in the fit parameters is small compared tothe variation between the different initial spectrum fit results. The initial spectrum uncertaintyis therefore captured by analyzing the lifetime for all of the fit results of the previous analysis.Uncertainty of the filling is included by calculating the√N(H) uncertainty on counts in aparticular energy bin. This uncertainty is then propagated when the total counts are calculatedfor each storage time.Both methods require some interpolation between calculated/simulated values to find aprecise intersection with the experimental results, as well as an uncertainty. This interpolationalso needs to capture the uncertainties of the calculated/simulated values. Both methods haveuncertainty from the simulated energy spectrum, and the FS method has additional uncertaintyfrom the simulations of the storage experiment. This introduces some additional complexity tothe interpolation.In general, to determine the best-fit value of a parameter x, a linear function y = p1x+ p0can be fit to the simulated values y. The intersection (yexp − p0)/p1 of this line with theexperiment result can then be taken to be the best-fit value. To calculate the error bars onthis parameter, the four lines y = (p1 ± σ(p1))x + (p0 ± σ(p0)) are calculated using the fituncertainties. The intersections of these lines with the 1σ bounds of the experimental valueyexp ± σ(yexp) are taken to describe the range of possible parameter values, with the minimumand maximum values found by this process corresponding to the 1σ bounds of the parameter.When the experiment contains multiple parameters, this process can be extended to fit themultivariate function y =∑i=1...n pixi + p0. When a multivariate function is used, unless thedataset is broad enough to fit multiple parameters, constraints must be applied to the valuesof some parameters.For these storage lifetime experiments, τs−1 should depend linearly on W i, so the fittingprocedure fits the function τ−1 = τs−10 +∑piW i. This will generally produce a better fit whenthe values of Wi span a large enough range for the non-linearity to be visible. To constrainthose experiments which depend on both WdPS and WNiP, values of WNiP from analysis ofthose experiments which did not include the dPS-coated ring are used as constraints. ForTCN22-020-SL, TCN22-012-SL is used as the constraining experiment, while for TCN23-020-SL, TCN23-010-SL is used.To further improve the quality of the fitting procedure, not all values of W i were used.Instead, a subset of values closest to the experimental τs were selected as the fit input. ForTCN2X-01Y-SL experiments, which depend only on WNiP, the three values of WNiP whichproduced the closest match to the experimental τs were used. For TCN2X-020-SL experiments,three values of WNiP that were closest to the value determined by analysis of the constrainingTCN2X-01Y-SL experiments were used, and for each of those WNiP values, the three values ofWdPS which were closest to the experimental τs were used, for a total of nine points used in theconstruction of the interpolated function.In the calculation of the uncertainty of the interpolation, the uncertainty σ(WNiP) fromthe constraining experiments should be included. The interpolated uncertainty calculation istherefore carried out using WNiP±σ(WNiP) as constraints. The final error bars then correspondto the minimum and maximum values determined by a scan over all combinations of parameter1495.4. Storage Lifetime MeasurementName τs (s) τw (s) χ2/ndofNiP, cell valveTCN22-010-SL 64.9± 2.5 64.9± 2.7 132/68TCN22-011-SL 0.0± 0.0 — —TCN22-012-SL 54.1± 1.5 54.1± 1.8 294/256TCN23-010-SL 137.5± 4.7 137.5± 4.8 120/104NiP/dPS, cell valveTCN22-020-SL 77.2± 1.5 83.9± 1.0 283/314TCN23-020-SL 111.3± 3.6 132.5± 2.8 196/199NiP/dPS, VAT valveTCN22-030-SL 76.3± 2.4 84.5± 1.1 404/334Table 5.4: Measured storage lifetimes in all cell experiments. The second column gives thelifetime from fitting the raw data to a single exponential. In the case of TCN2X-020-SL andTCN22-030-SL, these fits include only the data collected at the lowest achievable pressure. Thethird column gives the lifetimes with corrections for pressure effects. For TCN22-030-SL andTCN23-020-SL, the corrections are fit results to pressure-dependent data. For TCN22-020-SL,the correction is based on the fit parameters of TCN22-030-SL. For the rest the uncertainty hasbeen increased to account for a small pressure contribution. The fourth column gives the χ2and degrees of freedom of the fits after accounting for pressure effects.uncertainties at the 1σ level.The interpolation procedure is repeated independently for each possible estimated energyspectra, and the final results report the minimum and maximum values over the range of spectra,alongside the uncertainties determined by the described error propagation.5.4.4 ResultsThe measured storage lifetimes for each experiment are shown in Table 5.4. The measurementsTCN2X-01Y-SL all used the same configuration of a NiP-coated aluminium ring in place of thedPS-coated Rexolite ring. However, these results differed substantially, with the lowest lifetimemeasured as 0.0 s and the highest as 137.5± 4.7 s. The only difference between this set of fourmeasurements was in how the cell was cleaned prior to the measurements.TCN22-010-SL was the first measurement taken, with the cell having been assembled atTRIUMF and shipped to J-PARC. During the initial assembly the inner surfaces of the cellwere cleaned with Alconox and isopropanol. This yielded the longest NiP-only lifetime in 2022of 64.9± 2.5 s.Between TCN22-010-SL and TCN22-011-SL the cell underwent vacuum baking for approx-imately 12 hours at a temperature of ∼78◦C while still installed in the beamline. The tem-perature was limited by the heaters that were available for baking. The lifetime measured inTCN22-011-SL was 0.0 s, with no UCNs being detected after the baking. A visual inspectionof the interior of the guides connected to the cell showed that some liquid had condensed insidethe guides, shown in Figure 5.13.Between TCN22-011-SL and TCN22-012-SL, the guides and cell were manually cleaned.The guides were removed from the beamline and cleaned using acetone and ethanol. The cellcould not be removed from the beamline due to the need for a crane to do so, which was notavailable at short notice. Instead, the upper plate of the cell was removed, allowing the interiorsurfaces of the cell to be cleaned with ethanol. Acetone was not used for cleaning the cell due1505.4. Storage Lifetime MeasurementFigure 5.13: Photograph of a contaminated guide taken following TCN22-011-SL. Liquid isclearly visible on the interior surface in the lower left.to restrictions on the use of chemicals inside the beamline. With this cleaning the lifetime inTCN22-012-SL recovered to 54.1± 1.5 s.The results of the analysis of WNiP (discussed below) showed that in both TCN22-010-SLand TCN22-012-SL the NiP surfaces were unusually poor. Given the observations of a liquidcontaminant in TCN22-011-SL, it was hypothesized that the cell was being contaminated bysome insufficiently clean components in its assembly. Even in the cases where a non-zero lifetimewas measured, the presence of a lower concentration of contamination in the cell would stillresult in greater UCN losses and a lower lifetime.Between TCN22-012-SL and TCN23-010-SL, the cell was returned to TRIUMF, disassem-bled, and had (almost) all components cleaned using the new cleaning procedure described inAppendix D. The exception to this was the dPS-coated ring, which was gently but thoroughlycleaned by wiping with isopropanol. This was due to concerns over possibly damaging thecoating using the more intense procedure of Appendix D. Using this different procedure for thering was considered acceptable as it had only been installed into the cell after the baking in2022, and was therefore thought to have not been severely exposed to contamination. Afterbeing reassembled and shipped back to J-PARC, the lifetime was measured to be 137.5± 4.7 s.This significant improvement — more than a 100% increase in storage lifetime compared toTCN22-010-SL — can only be attributed to the preparation of the cell to a greater standard ofcleanliness. These results show the necessity of a stringent cleaning and assembly procedure forthe final version of the cell for the TUCAN EDM experiment. The results also indicate that,with complicated assemblies like the cell valve, it is not sufficient to clean only those surfacesthat UCNs will see. In TCN22-010-SL the UCN-exposed surfaces of the cell had been cleanedwith similar chemicals as those used in the procedure of Appendix D, but the lifetime was stillpoor compared to TCN23-010-SL. This can be explained by the migration of contaminants1515.4. Storage Lifetime MeasurementKT WNiP (neV) FS WNiP (neV)Experiment Uniform Collimated Uniform CollimatedTCN22-010-SL Min 0.23+0.02−0.02 0.29+0.02−0.02 0.16+0.02−0.02 0.19+0.02−0.02Max 0.36+0.03−0.03 0.53+0.06−0.06 0.25+0.03−0.03 0.28+0.05−0.04TCN22-012-SL Min 0.30+0.02−0.02 0.41+0.03−0.03 0.20+0.02−0.02 0.24+0.03−0.02Max 0.53+0.05−0.05 0.84+0.07−0.07 0.32+0.06−0.06 0.41+0.09−0.08TCN23-010-SL Min 0.066+0.004−0.004 0.072+0.007−0.007 0.056+0.006−0.007 0.063+0.006−0.007Max 0.103+0.006−0.006 0.110+0.007−0.007 0.086+0.010−0.009 0.090+0.010−0.009Table 5.5: Values of WNiP obtained from analysis of the three storage lifetime measurementsTCN22-010-SL, TCN22-012-SL, and TCN23-010-SL. Minimum and maximum values are takenover the range of estimated energy spectra, and are given separately for the uniform and colli-mated angular distributions, and for the kinetic theory (KT) and full simulation (FS) analyses.from elsewhere, such as the valve or the o-rings, onto the inner surfaces when the cell was undervacuum.8Figure 5.14 shows τ versus WNiP for the TCN2X-01Y-SL experiments. The extracted valuesof WNiP are shown in Table 5.5. The angular distribution of UCNs produced in the DS has asignificant impact on the value of τs for a given WNiP, due to the differences in energy spectrum.The lifetime at a given WNiP tends to be lower for spectra derived from a uniform distributioncompared to those derived from a collimated distribution. This means that the value of WNiPis lower in the uniform case than in the collimated case, and the true value will be lower if theangular distribution at the DS is closer to uniform than collimated.Figure 5.14 also clearly shows how the analysis method changes the interpretation. The FSmethod predicts lower lifetimes compared to the KT method. There are some possible expla-nations for this. One is that the PENTrack results also simulate the UCNs being transportedto the detector. This transport would have some energy dependence from wall losses and lossesin the foil, which could skew the measured lifetime. The significance of this has not been in-vestigated. Another explanation is that, in applying the KT method, detailed features suchas the insulator trenches were not included (in favour of a simple cylindrical geometry). Suchfeatures can result in higher loss rates due to increased wall hit frequencies for UCNs that enterthe trenches and are reflected between the trench walls. Since these features are modelled inPENTrack, their impact on the measurements is separated from the value of WNiP in the FSanalysis. This is an important consideration for predictions of future performance of the cellusing PENTrack simulations such as in S. Sidhu’s work [6]. Using the values of WNiP obtainedfrom the KT analysis in PENTrack simulations may predict a lower lifetime than what wouldactually be obtained. These discrepancies in the interpretation of storage lifetimes in terms ofWNiP require careful consideration when seeking to make accurate predictions.The measurements TCN22-020-SL, TCN22-030-SL, and TCN23-020-SL all used the dPS-coated ring. TCN22-030-SL replaced the cell valve with a commercial VAT valve. The VATvalve was intended only as a comparison for the cell valve, and the two showed similar perfor-mance, so a detailed analysis has not been performed on TCN22-030-SL. From 2022 to 2023,8While by no means an empirical observation, it is worth noting that when the cell was opened after pumping,there was a distinct organic odor that would gradually dissipate. This had been noticed prior to the 2022beamtime when leak-checking the cell, but it was not recognized at the time that this could indicate a problemwith the cell’s cleanliness.1525.4. Storage Lifetime Measurement(a) Uniform angular distribution(b) Collimated angular distributionFigure 5.14: Simulated storage lifetime of the cell versus WNiP for the TCN2X-01Y-SL config-uration. The horizontal bands show the experimental results with 1σ ranges. The blue curvesshow the lifetimes obtained by the kinetic theory (KT) analysis of the storage, while the redcurves show results obtained by the full simulation (FS) analysis of the storage.1535.4. Storage Lifetime MeasurementKT WdPS (neV) FS WdPS (neV)Experiment Uniform Collimated Uniform CollimatedTCN22-020-SL Min 0.03+0.02−0.02 0.02+0.02−0.02 −0.3+1.9−0.1 −0.8+1.3−1.0Max 0.07+0.01−0.01 0.06+0.01−0.01 0.05+0.06−0.05 −0.01+0.05−0.07TCN23-020-SL Min 0.047+0.004−0.004 0.054+0.005−0.005 0.03+0.02−0.02 0.03+0.02−0.02Max 0.073+0.006−0.006 0.079+0.008−0.008 0.09+0.02−0.02 0.09+0.03−0.02Table 5.6: Values of WdPS obtained from analysis of the two storage lifetime measurementsTCN22-020-SL and TCN23-020-SL. Minimum and maximum values are the smallest or largestvalues obtained over the range of estimated energy spectra. The values between 2022 and2023 agree for the kinetic theory (KT) analysis, suggesting that the dPS-coated ring performedsimilarly in the two years. However, the full simulation (FS) analysis shows anomalous negativevalues in the 2022 analysis. This is discussed further in the text. The uncertainty in the energyspectrum leads to a wide range of results, and the FS analysis additionally seems to have beenlimited by simulation uncertainty.the storage lifetime with the dPS-coated insulator also increased substantially, with a measuredwall lifetime similar to that obtained in TCN23-010-SL. Since nothing about the insulator waschanged, this is also attributed to the improved cleanliness of the cell.All three of these measurements saw significant effects from outgassing of the dPS-coatedring, which have been corrected. The greatest impact was on TCN23-020-SL, where the cor-rection exceeds 20 s. This pressure effect could have a significant impact on the TUCAN EDMexperiment. It is not possible to pump out the cell volume during the EDM measurement,as doing so would remove the co-magnetometer gas. In the ideal case, the outgassing can bereduced to a negligible level by long-term pumping or baking. This has yet to be investigated.Figures 5.15 and 5.16 show the storage lifetimes versus WdPS for the uniform and collimatedcases respectively. Since the results of these measurements also depends on the value of WNiP,these plots show lifetimes for a few different WNiP values. Simulations and calculations wereperformed for more values of WNiP than are shown, allowing a reasonable interpolation of theWdPS values. Table 5.6 shows the final values of WdPS.In the analysis of TCN22-020-SL, the KT results and FS results are seen to differ signifi-cantly. The KT analysis predicts what appear to be reasonable results, but the FS analysis inmost cases requires negative values of WdPS, albeit with large uncertainties. This raises twoquestions. First, why do the methods yield different results, and second, why does the FSmethod give non-physical values for this experiment?The differences in the methods are most clearly visible in the Figures 5.15 and 5.16. Itcan be seen that, when the value of WNiP is large, the PENTrack-simulated lifetimes have alow sensitivity to the value of WdPS, with flatter curves compared to the KT analysis. Thisimplies that the lifetime is dominated by WNiP, and that the KT analysis does not capture thisbehaviour. It could be that, with such high wall losses from the NiP surfaces, the assumptionsof the KT analysis do not produce a good description of storage. This is supported by the factthat the methods largely agree for TCN23-020-SL, where the losses on the NiP surfaces werereduced. A more detailed comparison of the differences in the methods could be carried outusing further PENTrack simulations to examine the distributions and behaviours of individualUCNs, but this is left to future work.The question of the negative values is less clear, and there are several different explanations1545.4. Storage Lifetime Measurement(a) Kinetic theory(b) Full simulationFigure 5.15: Simulated storage lifetime of the cell versus WdPS for the TCN2X-020-SL con-figuration using a uniform angular distribution. The horizontal bands show the experimentalresults with 1σ ranges. The differently coloured curves correspond to different WNiP coveringthe range of values from TCN2X-01Y-SL. Solid lines are the maximum over different energyspectra, while dashed lines are the minima.1555.4. Storage Lifetime Measurement(a) Kinetic theory(b) Full simulationFigure 5.16: Simulated storage lifetime of the cell versus WdPS for the TCN2X-020-SL config-uration using a collimated angular distribution. The horizontal bands show the experimentalresults with 1σ ranges. The differently coloured curves correspond to different WNiP coveringthe range of values from TCN2X-01Y-SL. Solid lines are the maximum over different energyspectra, while dashed lines are the minima.1565.4. Storage Lifetime Measurementfor these results. One possibility is that many of the spectra used are weighted towards toohigh energy. This would result in higher losses and could make it impossible to describe theexperiment using such spectra. A second possibility is that the assumed values of WNiP are toohigh. This could be the case if the NiP-coated ring was more heavily contaminated than theplates, in which case it is incorrect to assume the value of WNiP from TCN22-012-SL. Thirdly,this could be an artifact of the FS analysis resulting from a combination of the low sensitivityto WdPS and the precision of the storage lifetime results from PENTrack simulations, which islimited by computation time. While the third cause could be investigated by improving the FSprecision, examining the others would require further data (such as an improved measurementof the energy spectrum). Since the measurement was repeated in 2023, the 2022 results canlargely be regarded as preliminary, and so no further work has been done on examining thesepotential causes.The outcome of the analysis of TCN23-020-SL raises fewer questions. Again, some differenceis seen between the uniform and collimated assumptions for the KT analysis, although this is notvisible in the FS analysis. The FS results have larger uncertainties, which may obscure suchdifferences. These larger uncertainties are seemingly due to running the PENTrack storagesimulations with a limited precision to save computation time, so the results could be improvedby running further simulations. However, as the KT and FS results overlap, increasing theprecision of the FS method is not likely to change the conclusions at this stage.It is worth noting that, in the KT analysis of 2022 and 2023, similar results were obtainedfor WdPS. As was mentioned previously, the dPS-coated ring was only gently cleaned withisopropanol (to avoid damaging the coating), and so should have been in a similar state betweenthe two years. These results suggest that the performance of the dPS coating was similar, andthe difference in the measured lifetime between the two years was mostly due to the cleanlinessof the NiP-coated surfaces. Unfortunately, the results of the FS analysis are insufficient tosupport this due to the behaviour discussed above.The values obtained for WdPS can be compared to literature values. Previous measurementshave reported values of 0.048 ± 0.016 neV [244] and 0.024 ± 0.008 neV [245]. These values dooverlap the min/max ranges of the FS analysis of the 2023 data, so there is reasonable agreementwith other measurements. The KT analysis overlaps with the higher value in 2023, and withboth values in 2022. The maximum values in both the KT and FS analyses are higher thanthese literature values, which could indicate a worse coating, but the minimum values, includingthe uncertainty, cover lower values, so it is not possible to make a suggestion either way. At thispoint, it can be concluded that the dPS coating produced for the cell is most likely of similarquality to those produced by other groups.In the studies of the TUCAN EDM statistical sensitivity, a value of WdPS = 0.05 neV wasassumed. For the KT analysis, this value is on the low end of the parameter ranges, while forthe FS analysis, it falls closer to the middle. While tentatively promising that the the dPS-coating achieves a value similar to what was used in these studies, it should not be ignored thatthis analysis allows for significantly higher values. A value of WdPS = 0.10 neV would have asignificant impact on the statistical uncertainty. Estimating the real impact of such a differingparameter would require a full optimization, but a rough calculation based on 4.1 suggests thatthis could be up to a factor of two increase.The values of WNiP obtained here are less important, as the electrodes in the TUCAN EDMexperiment are intended to be DLC-coated, but they are still of some interest to compare.The values from the TCN23-010-SL experiment are similar to those obtained from a guideexperiment, reported on in Ref. [5]. The NiP-coating for the cell was produced by AdvancedSurface Tech., which produced guide #5 in Ref. [5]. This guide had WNiP = 0.069 ± 0.016,1575.4. Storage Lifetime Measurementobtained using the KT method. This overlaps with the ranges obtained from TCN23-010-SL,although the results here again allow for some significantly higher values. One notable differenceis that, for the cell, the aluminium surfaces were only electropolished after machining, whilethe guides were typically hand-polished to a very low roughness and then electropolished (withmeasured Ra values below 150 nm for SS guides and below 300 nm for NiP-coated Al guides, asreported in Ref. [5]). Whether there is an effect on τs due to only electropolishing is currentlyuncertain, as no direct studies have been performed.Much of the uncertainty in the outcome of these experiments is due to the limited infor-mation about the energy spectrum of the DS. This results in a wide range of possible valuesobtained from assuming different spectra. This range could be made more narrow by improvingthe understanding of the energy spectrum. At the time of writing, some further measurementsof the angular distribution of neutrons leaving the DS have been taken, but this data has yetto be analyzed. This is left as future work.The assumptions on the energy spectrum estimation could also introduce some bias. Somedeficiencies of the modelling of the TOF experiment were discussed in Section 5.3.2, and it wasargued that these would most likely bias the extracted W values to higher values. Additionally,the choice of weighting functions used in the analysis of the energy spectrum, listed in Table 5.1,could introduce some bias depending on how they were selected. The justifications for the chosenweighting functions were given in Section 5.3.2. It was also noted that several of the functionsthat were attempted were discarded from the final analysis as they seemed to produce unrealisticinitial spectra. These spectra generally showed larger weights at low energies (including zero)compared to high energies. This was thought to be generally inconsistent with observationsof UCN energy spectra and therefore an acceptable, although subjective, judgment. However,including these spectra would likely have pulled the range of results for WNiP and WdPS to evenhigher values, and so this potential bias should not be ignored. This can also be resolved infuture improved measurements by reducing the need to analyze the energy spectrum in thisway.158Chapter 6Electrical Fields in the CentralRegion DesignThis chapter describes the design of the central region with a focus on the electric field. Thereare two considerations: the ability to sustain the electric field, and its contribution to systematicuncertainties. The first part will be addressed in Section 6.1, which will review fundamentalconcepts of HV design in the context of the TUCAN EDM experiment, and in Section 6.2, whichwill show results of finite element method (FEM) simulations of the electric field in the centralregion. The second part will be addressed in Section 6.3, which will discuss the systematicimpact of field inhomogeneity, and Section 6.4, which will discuss other systematics related tothe supply of HV to the experiment.6.1 High Voltage DesignThe top and bottom surfaces of the cells in the central region act as electrodes, as indicated inFigure 4.2. With HV applied to the central electrode, and the other electrodes connected toground, an electric field is generated inside the cell volume. It is the interaction of this electricfield with the nEDM that will produce a frequency shift that the TUCAN EDM experiment seeksto measure. This frequency shift is directly proportional to the field magnitude E. Achievingthe desired statistical sensitivity of the experiment therefore requires maximizing E.Unfortunately, E cannot be made arbitrarily large. This is due to electrical breakdown,which refers to any process in which, at sufficiently large applied field, the insulating propertiesof a medium fail. Breakdowns can be transient, with a rapid discharge in the form of sparks,or sustained, with a constant or runaway current that prevents the applied voltage from beingraised further. While the second situation prevents the field strength from being increased,sparks are also to be avoided in an EDM experiment, as they can create temporary largemagnetic fields that interfere with the measurement.In the TUCAN EDM experiment, there are four different regions over which the high voltageis applied. Inside the cells is a gas mixture. Nominally this is a very low pressure of Hg vapour.Previous experiments reported number densities of 5×1010–1×1013 cm3 [236], so the pressureis not likely to exceed ∼10−6 mbar. However, it could also be filled with a buffer gas to improvehigh-voltage performance. The use of such a buffer gas was suggested in relation to the pastPSI experiment, where a He pressure of 1 × 10−2 mbar was reportedly considered, but it isunclear whether this was done in the real experiment [236]. Outside the cells is a vacuum,likely 10−4 mbar or better, which serves as an insulator between the HV electrode and thegrounded vacuum chamber. These regions are separated by a third region, the solid insulatorring, made of Rexolite [246]. The fourth region is the vacuum feedthrough that supplies the HVto the central electrode. This is intended to be a solid feedthrough, made of insulating materialssurrounding a central conductive pin. The feedthrough enters the vacuum chamber through aside flange, and, outside of the chamber, is shielded by a grounded aluminium connection tothe flange.1596.1. High Voltage Design10-3 10-2 10-1 100 101Pressure (mbar)10-2100102104Breakdown field strength (kV/cm) HeliumXenonHydrogenNitrogenOxygenFigure 6.1: Electric field strength at breakdown versus pressure for different gases, calculatedfrom Paschen’s law, with an electrode separation of 16 cm. Values of AP and BP for the differentgases were taken from Ref. [254], and a value of γse = 0.1 was used based on reported values inRef. [251].In gases, breakdown depends on both pressure and geometry. A common description ofbreakdown in gases is Paschen’s law, which describes the maximum electric field EB, or voltageVB, between two flat electrodes asEB =VB∆z=BPPlnAPP∆z − ln (ln (1 + γse−1)) (6.1)where P is pressure, ∆z is electrode separation, and AP, BP, and γse are material coeffi-cients [251]. The physical process behind Paschen’s law is the Townsend discharge process, andthe coefficients are related to this process. In the Townsend process, a free electron between theelectrodes (produced by, for example, a cosmic ray interaction), accelerated by the electric field,ionizes a gas particle. The electrons resulting from this ionization are also accelerated and ionizemore gas particles. The number of ionizations that occur is described by the first Townsendcoefficient αTE = APPe−BPP/E , where E is field strength. At the same time, the positive ionsare accelerated in the opposite direction and impact one of the electrodes, where they cause theemission of further electrons. The number of secondary electrons produced by this process isdescribed by the secondary electron emission coefficient γse. This avalanche of electrons fromionization and emission from the electrode leads to breakdown. The first Townsend coefficientdepends on the gas composition, while the secondary electron emission coefficient depends onthe material and preparation of the surface of the electrodes in addition to the gas [252, 253].Figure 6.1 plots several typical Paschen curves for some gases of interest, using AP andBP values taken from Ref. [254], a value of γse = 0.1 (reported to be typical of aluminiumelectrodes in Ref. [251]), and ∆z = 16 cm. At high pressures, breakdown is suppressed due to1606.1. High Voltage Designa small mean-free-path for electrons, which prevents them from gaining enough energy betweencollisions to ionize the gas. At low pressures, breakdown is suppressed due to the long mean-free-path for electrons, which prevents them from encountering any gas atoms or molecules toionize. The TUCAN EDM experiment expects to operate on the low pressure side of thesecurves, likely below the apparent asymptotic rise in field strength. This does not necessarilymean that the 12.5 kV/cm (for a 200 kV supply and 16 cm electrode separation) can definitelybe sustained inside the cell. Paschen’s law is constructed for planar electrode conditions, whilethe electrodes in the central have both inhomogeneities inside the cell region, which createlocalized higher fields [255], and have a curved geometry at the electrode edges, which causesdeviation from Paschen’s law at low pressures [256]. Further, Paschen’s law only describes theTownsend process, and other processes may dominate at low pressures.Outside the cells, and potentially inside the cells in the absence of a buffer gas, the pressureregion is far to the left of the curves in Figure 6.1. In this vacuum region, breakdown istypically attributed to field emission, in which the electric field becomes sufficiently intenseto remove electrons from the electrodes [257]. Breakdown in vacuum is typically observed atmuch lower fields (up to three orders of magnitude lower) than would theoretically be expectedfor field emission [258]. Substantial work has attempted to explain the occurrence of fieldemission in lower-than-expected fields. These explanations include surface imperfections causinghigh localized fields [259, 260], mechanical or work function changes at the surface [258], anddesorption of gases from the surface [261].Because of the complexity of modelling vacuum breakdown, the best guidance comes frompublished experimental data of similar electrode configurations. Experimental data showsthat in the vacuum region, for large electrode separations (on the order of centimetres), thebreakdown strength increases with increasing pressure until reaching the Paschen curve re-gion [262, 263]. This and other data [264] seems to generally indicate that, in high vacuum andwith flat electrodes at separations on the order of 15 cm, fields of up to 40 kV/cm are sustainablein high vacuum, with higher fields being possible at higher pressures. Geometry does play asignificant role in vacuum breakdown conditions. Several of the surfaces in the central regionare better described as spherical rather than planar, and studies have reported sustaining fieldsup to 54 kV/cm between spherical and planar electrodes at a 7.2 cm separation, dropping to40 kV/cm at 14.7 cm separation [265].In the low pressure, large electrode separation region that the TUCAN EDM experiment willoperate at, it is difficult to have a clear grasp on a condition for breakdown. This necessitatessome practical testing, and to that end, the TUCAN collaboration has constructed and operatedan HV test setup that has been used to evaluate the performance of model electrodes at asmall scale and 100 kV. Those tests are beyond the scope of this dissertation, and are as-yetunpublished, but the results have so far supported the feasibility of this design. Further testingwill be required before moving to 200 kV, and the first stage OC100 configuration could enablethis testing to be done at full scale before building the full 2C200 experiment. In the nextsection, the above value of 40 kV/cm will be taken as a rough guideline when examining theresults of electric field simulations.In a solid dielectric material, such as the insulator ring or feedthrough, the mechanisms ofbreakdown are somewhat less well understood than for gaseous media. Electrical breakdownin dielectrics is divided into two categories: intrinsic, in which the field is believed to becomesufficient to energize electrons in the material into the conduction band, and avalanche, in whichan external electron (e.g. emitted from the HV electrode) is accelerated to energies at which itcan trigger impact ionization in the material [251, 266]. Intrinsic breakdown depends only onmaterial properties, while the condition for avalanche breakdown, similar to the mechanisms1616.1. High Voltage DesignMaterial Dielectric Dielectricconstant strength (kV/cm)Rexolite 1422 [246] 2.53 2.0× 102Araldite CW 229 with 4.10 1.7× 102–2.0× 102Aradur HW 229-1 [274]Ethylene propylene rubber (EPR) [275] — 3.5× 102–4.0× 102Apiezon L Vacuum Grease [276] 2.3 3.0× 102Macor [277] 6.01 12.9× 102Table 6.1: Properties of select dielectric materials of relevance to the TUCAN EDM experiment.behind Paschen’s law, depend on geometry and electrode material [267, 268]. Material defectsalso play a role in breakdown in dielectrics, as they can form regions of higher field or differentmaterial properties [269, 270].There are other failure mechanisms for dielectrics, also referred to as breakdown, as theyultimately lead to the failure to support an electric field. These other mechanisms include elec-tromechanical failure, where the electrostatic forces between electrodes cause deformation ofthe supporting dielectric [251], and thermal failure, where dielectric losses or small conductivecurrents cause heating of the material [271, 272]. These mechanisms can trigger breakdownby a macroscopic structural failure, but can also change the properties of the dielectric, ulti-mately leading to an electrical breakdown [273]. Thermal breakdown is mostly relevant for ACfields, where dielectric losses occur, and so is not a significant concern for the TUCAN EDMexperiment. Electromechanical failure can be examined based on the expected force betweenthe electrodes and the mechanical properties of the insulator.The space between the electrodes is partially filled with dielectric, which increases the force.As an upper bound, for a plate capacitor of area Aelec completely filled with a solid materialof dielectric constant r the force is 0rAelecE2/2 [251]. For the TUCAN EDM experiment,Aelec ≈ 0.36 m2. This gives a ∼6.4 N force between the electrodes. The cross-sectional areaof the insulating rings is ∼0.016 m2, and so the stress is only 400 Pa. Rexolite has a yieldstrength of 55 MPa, and so there is no reason to expect any significant deformation due to theelectrostatic force.Material suppliers generally report typical dielectric breakdown strengths EB which havebeen obtained by standardized measurements. Properties for a number of materials used inthe TUCAN EDM apparatus are given in Table 6.1. However, because avalanche breakdownmechanisms are geometry and electrode dependent, these EB values can only be used as astarting point, and an accurate prediction of performance requires testing, which is beyond thescope of this work. Here, where the goal is to establish the validity of the overall central regiondesign, these typical values can be considered as maximum allowable fields.One additional consideration concerns the interfaces between different media. A locationwhere a conductor, dielectric, and vacuum (or gas) meet is known as a triple point. At a triplepoint, the difference in relative permittivity between the dielectric and vacuum/gas can resultin a highly enhanced electric field, increasing the chance of field emission [278, 279]. Triplepoints are therefore favoured locations for breakdown to occur, and the design of HV systemsrequires reducing fields at such points.Finally, it should be noted that the TUCAN EDM experiment operates in a potentiallydifferent magnetic field environment compared to the various studies discussed above. Sincebreakdown involves the movement of charges, a magnetic field can be expected to have some1626.2. Finite Element Simulations of the Electric Field500160 OR 122.47728025R40R40BCDR4.750.515DETAIL BSCALE 1 : 28520R6.35R11DETAIL CSCALE 1 : 2R10 0.5R4.759DETAIL DSCALE 1 : 2Figure 6.2: Schematic showing the key geometric features and dimensions of the cell for HVsimulations. All quantities are in units of mm.effect. Theoretical and experimental studies find that breakdown voltages are reduced by thepresence of a magnetic field [280, 281]. However, the magnetic fields considered in literatureare generally much larger (> 1 mT) than the 1µT field of the TUCAN EDM experiment. Infact, as the 1µT field is smaller than Earth’s field, which would be present in the studies ofbreakdown that have been cited, it might be expected that this magnetic environment will haveimproved HV performance. It therefore seems unlikely that the magnetic field will have animportant effect, and so it is neglected in the design of the central region for HV.6.2 Finite Element Simulations of the Electric FieldTo study the electric field in the central region, two simulations of the electric field using finiteelement methods have been developed. The first is a 3D simulation using Opera [282], whichattempts to capture all of the details of the geometry and their impact on the field. The secondis a simpler rotationally-symmetric 2D simulation using FEMM [283]. This 2D simulation isintended for use in PENTrack simulations which are described in Section 6.3.Figure 6.2 shows the intended dimensions of the cells in the 2C200 configuration. To sustainthe electric field, it is necessary to hide the triple point of the insulator and electrodes from1636.2. Finite Element Simulations of the Electric FieldHV PORT ID = 240 mm HV RECEPTACLE OD = 79.40 mmCORONA RING MINOR RADIUS = 35.00 mm ELECTRODE BLEND RADIUS = 40.00 mm VACUUM INNER HEIGHT = 374.80 mm (152.4 mm INSULATOR)VACUUM INNER HEIGHT = 382.42 mm (190 mm INSULATOR)VACUUM ID = 1077.00 mm Figure 6.3: Schematic showing the geometric features of the central region around the HVfeedthrough. The HV on the central electrode is supplied through a commercial feedthroughthat passes through a large port in the vacuum chamber. The feedthrough receptacle mates tothe central electrode inside a corona ring, intended to suppress breakdown at the interface ofthe receptacle and electrode.any large electric fields, which would otherwise have a high probability of breakdown. This isdone by adding trenches to the electrodes where the insulator is located. Since the walls ofthe trenches are at the same potential, the electric field in these regions is very low. It is alsonecessary to minimize any large changes in potential by rounding all edges, and the dimensionsof the various curvatures are indicated.Figure 6.3 shows critical dimensions of the vacuum chamber and the HV feedthrough. TheHV feedthrough is a commercial 225 kV feedthrough manufactured by Essex X-Ray, and consistsof a receptacle and a connector. This feedthrough is being customized by Essex to meet themagnetics requirements of the TUCAN EDM experiment, discussed further in Chapter 7. Thereceptacle is made of a two part epoxy resin (Araldite CW 229 with Aradur HW 229-1), whichinsulates an HV pin. The connector is molded around a cable using EPR insulation. The exactcomposition of the connector is not described, as it is proprietary information. The cut-throughof the feedthrough is therefore a rough geometric representation only, and in the simulation it1646.2. Finite Element Simulations of the Electric FieldFigure 6.4: Electric field map inside the top cell in the xz plane, simulated using Opera. Theentrance hole (closed by the cell valve plug) and insulator trench is indicated. The origin is atthe centre of the central electrode.is assumed to be made entirely of the Araldite CW 229 resin. The receptacle is mounted to aplate on a large port on the side of the vacuum chamber. Currently, this plate is assumed tobe made of Macor, in order to reduce the field at the entrance, but it is possible that it willinstead be made of aluminium depending on the final specifications for the feedthrough. Thepin through the centre of the feedthrough makes contact with the central electrode, resultingin a triple point which is concealed by a toroidal corona ring. Because the corona ring extendsbeyond the footprint of the HV electrode, it is generally closer to the ground surfaces insidethe vacuum chamber than any other HV component (except for the pin inside the feedthrough,where the use of dielectrics allows a larger field to be sustained). Because of this proximity, thisregion will see some of the highest electric fields, and will be the focus of much of the followingdiscussion of the simulation results with respect to vacuum breakdown.Figure 6.4 shows the field map inside the top cell in the 2C200 configuration, with an elec-trode separation of 16 cm. The peak fields are 21 kV/cm, around the trenches and entrance hole.Assuming only the low density of Hg atoms in this region, the pressure would be ∼10−6 mbar.The published data indicates that these fields are generally safe for such a pressure [262, 264],and so this region is not of concern.Figure 6.5 shows maps in this same configuration outside the cell, focused around thefeedthrough. In this region the fields are generally higher than in the interior of the cell. Atthe edge of the central electrode, field strengths reach 32 kV/cm, which should be acceptable.However, at the corona ring the maximum field reaches 43 kV/cm. Considering the publisheddata, it is possible that this is allowable. In Ref. [265], maximum fields up to 54 kV/cm weresustainable between spherical and planar electrodes with a gap length of 72 mm, and 40 kV/cmwith a gap length of 147 mm. The corona ring is slightly more than 90 mm away from any1656.2. Finite Element Simulations of the Electric Field0.3 0.35 0.4 0.45 0.5 0.55 0.6x (m)00.050.10.150.20.250.3z (m)102030405060708090100Electric field (kV/cm)FeedthroughCorona ring(43 kV/cm)Vacuum chamberGround electrode(a) Cut through vertical plane xz at y = 00.3 0.35 0.4 0.45 0.5 0.55 0.6x (m)00.050.10.150.20.250.3y (m)102030405060708090100Electric field (kV/cm)HV electrodeVacuumchamberCorona ringFeedthrough(b) Cut through horizontal plane xy at z = 0Figure 6.5: Electric field maps in the vacuum region of the central region, focused around thefeedthrough, simulated using Opera.1666.2. Finite Element Simulations of the Electric Fieldground surface in the central region geometry, so it is possible that a field up to 50 kV/cm inthis region will not cause breakdown.There is also an elevated region at the intersection of the feedthrough with the matingflange, where the field reaches 51 kV/cm. This elevated field motivated the inclusion of a Macorflange in the simulations. With a ceramic flange, this region becomes an intersection betweentwo dielectrics with much higher breakdown strengths, and with no conductor present for fieldemission. However, this may not be necessary dependent on the manufacturer specificationsof how to mount the feedthrough, which are still pending at the time of writing. Inside thefeedthrough, the fields are also much higher, with peak values of 210 kV/cm inside the connector.This would appear to be reasonable given the high breakdown strengths of the materials used.The exact behaviour of the internal fields of the feedthrough may also not be correct here, asthe internal composition of the connector is not modelled, so these results only serve to give asense that the feedthrough will perform to the supplier’s specifications.The field around the corona ring is the biggest challenge for the design. The corona ringis close to the vacuum chamber wall and the ground electrodes, and requires a curvature, so ittends to see the largest fields. Simulations of both a larger and smaller corona ring (40 mm or30 mm minor radius respectively), both showed increases in the field magnitude. The currentsize of the corona ring is therefore close to optimal for this geometry. If the field in thisregion causes breakdown, then it may be necessary to increase both the radius and heightof the vacuum chamber. Increasing the height would seemingly require either an increase inelectrode separation or an increase in the central electrode thickness. Increasing electrodeseparation would reduce the magnitude of the field and thus should be avoided. Increasing thecentral electrode thickness would be more acceptable, but it would have some impact on UCNtransport into the cells and the resulting energy spectra due to the gravitational potential. Itwould also slightly increase the distance over which the magnetic field must be made uniformfor depolarization and systematic considerations.A major question about the construction of the central region is that of the height of theinsulator. Due to limitations on the size of Rexolite stock material, the OC100 configurationassumes an insulator height of 15.24 cm, giving an electrode separation of 12.24 cm. This differsfrom the height of 19 cm, with an electrode separation of 16 cm, that is assumed for the 2C200configuration of the spectrometer. With a reduced electrode separation, the field inside the cellscould be made higher. While the smaller cells would have a lower volume and storage lifetime,and thus fewer total UCNs, the increase in electric field could make up for that and allow somegains in statistical sensitivity. However, changing the insulator height also changes the overallgeometry of the vacuum chamber and could result in regions of overly-large electric field thatcannot be sustained.Opera simulations can help to address the question of the feasibility of different heights.Figure 6.6 shows maps of the electric field with the insulator height reduced to 15.24 cm. Whilethese are largely unchanged from the 19 cm case, they do show a significantly elevated field onthe upper surface of the corona ring, of 50 kV/cm. It is very probable that such a field will resultin breakdown, given the published data on similar cases. This would rule out the possibilityof operating at 200 kV with a 15.24 cm insulator height (12.24 cm electrode separation). Thiselevated field is caused by proximity to the upper surface of the vacuum chamber, and so it ispossible that it could be reduced by shrinking the size of the corona ring. The voltage couldalso be reduced slightly to prevent breakdown while still allowing a gain in field magnitudeinside the cell. This is an optimization study that would require more dedicated studies of thestatistical sensitivity at different heights, and is left for future work.Ultimately, simulations of the electric field in the central region indicate that its suitability1676.2. Finite Element Simulations of the Electric Field0.3 0.35 0.4 0.45 0.5 0.55 0.6x (m)00.050.10.150.20.250.3z (m)102030405060708090100Electric field (kV/cm)FeedthroughCorona ring(50 kV/cm)Ground electrodeVacuum chamber(a) xz-plane y = 00.3 0.35 0.4 0.45 0.5 0.55 0.6x (m)00.050.10.150.20.250.3y (m)102030405060708090100Electric field (kV/cm)Corona ringHV electrodeFeedthroughVacuumchamber(b) xy-plane z = 0Figure 6.6: Electric field maps inside the central region around the HV feedthrough, with a15.24 cm insulator height, simulated using Opera.1686.3. Impact of Electric Field Inhomogeneityfor 200 kV operation is borderline. The peak field of 43 kV/cm on the corona ring is too highto say that it will not cause breakdown, but not high enough to say that it definitely will.Completing the design of the central region for 200 kV should therefore involve more dedicatedtesting, focused on breakdown in the critical areas that have been identified. However, thisgeometry should have no problems operating at 100 kV, and building out the OC100 version ofthe experiment with this geometry will enable 200 kV tests to be performed before moving tobuild the 2C200 version.6.2.1 Comparison of 2D and 3D Simulation ResultsIn the next section, a 2D, rotationally-symmetric field map simulated in FEMM will be usedin PENTrack simulations. It is useful to examine the difference between this map and the full3D simulations. The inhomogeneity of the field, given by the radial component, is of particularinterest. Figure 6.7 shows Ex in the cell for the two simulations. Large regions of the fieldinside the cell are qualitatively similar, but the presence of the corona ring does have an effect,shifting the sign of Ex in a large region that overlaps with part of the cell interior. Maps onthe opposite side of the cell, away from the corona ring (not shown), are qualitatively similarto the FEMM simulation.The difference between the two simulation maps can be made quantitative by calculating theaverage field components inside the cell. FEMM gives an average |Er| = 1.791 × 10−1 kV/cm,and |Ez| = 12.45 kV/cm, while Opera gives |Er| = 1.445×10−1 kV/cm, and |Ez| = 12.47 kV/cm.The FEMM simulation therefore overestimates the amount of inhomogeneity compared to theOpera simulation by 24%. Considering the appearance of the field in Figure 6.7, this can beexplained by the cancellation of some of the inhomogeneity in the Opera simulation due to theeffect of the corona ring. Using the FEMM simulation to simulate systematic effects in thefollowing section will therefore give an overestimate.6.3 Impact of Electric Field InhomogeneityIn addition to making the electric field magnitude large, the design of the central region mustalso consider the homogeneity of the electric field. The presence of small Ex or Ey terms hasthe potential to cause systematic frequency shifts which give rise to a false EDM signal, aspreviously described in Section 4.3.10. These systematic shifts will occur only when there is anet motion of the UCN population in the cells, which combines with the electric field terms inthe motional magnetic field ~v × ~E/c2. Since Ex or Ey terms will reverse with reversal of theelectric field, the frequency shift also changes sign, and this can result in a false EDM signal.This section presents a detailed analysis of two possible net motions, rotational and linear, thatcan lead to such a signal.6.3.1 Rotational v × EConsider a UCN which, as it repeatedly reflects from the cell walls, circulates around therotational axis of the cell with a velocity vφ. In the presence of a radial component of theelectric field Er =√Ex2 + Ey2, then this UCN will see a magnetic fieldBz± = γn(~v × ~E/c2)z=γnc2vφEr±, (6.2)1696.3. Impact of Electric Field Inhomogeneity(a) FEMM(b) OperaFigure 6.7: Map of the radial electric field component in the top cell, simulated using FEMMand Opera. The map is in the vertical xz plane at y = 0 and x > 0. The geometry differencesbetween the two simulations are clearly visible, as the FEMM simulation lacks the corona ring,as well as some features such as bolt holes (for clamping the insulator ring) on the centralelectrode. Significant differences are visible in the fields from the two simulations.1706.3. Impact of Electric Field Inhomogeneitywhere the superscript ± denotes the electric field orientation. This field results in a false EDMsignaldfalsev×E,r = −vφµn2Ec2(Er+ − Er−). (6.3)If the velocities of the UCN population are directed randomly, then on average this falseEDM signal will be measured to be zero. However, if the population has a net circulation, thenthe average will be non-zero and a false EDM signal will appear. Such a net motion could occurif the UCNs fill into the cell in such a way that they have a preferred direction. Whether thiswill actually occur is unknown, and may seem unlikely, but measuring the precise behaviourof the UCNs in a real cell is challenging, and the possibility of such a motion cannot be ruledout without such a measurement. A conservative consideration of the systematic uncertaintiestherefore requires that the worst case is assumed.If such a net rotational motion exists, it cannot be sustained indefinitely. When a UCNreflects diffusely from the walls of the cell the resulting direction of motion is randomized,which effectively removes the UCN from the net motion. Over time, this randomization willcause any net motion to decay with a characteristic time constant τr. This decay will suppressthe false signal. There are three different suppressing effects, based on the dynamics of the UCNsduring different periods of the experiment. The three suppression factors will be considered inturn.1. Filling-time suppression:UCNs are not filled into the cells instantaneously, but over an extended period. ManyUCNs will therefore spend significant time in the cell before the valve is closed and themeasurement period begins. During this time, any initial motion that resulted from thefilling will be decaying. This filling-time suppression depends on the distribution of timespent in the cell during this period. This distribution can be estimated using PENTracksimulations of the filling and is investigated in Section 6.3.1. For now, the filling-timesuppression is labelled as a factor F .2. Precession-time suppression:The net motion will decay throughout the precession period. The false signal will thereforebe suppressed by a time-averaged exponential decay1TR∫ TR0e−t/τrdt =τrTR(1− e−TR/τr) ≈ τrTR, (6.4)where the approximation assumes that the precession time TR τr. For these studies,TR = 150 s, and it will be seen when τr is computed that TR τr is satisfied.3. Delay-time suppression:Finally, before the precession starts there can be some short delay period td during whichthe net motion decays. Such a period is necessary to apply the spin-flip of the Hg co-magnetometer, but a longer delay can also be included specifically to allow the net motionto decay. Because the UCNs do not precess during this time, they accumulate no phase-shift, and thus the delay-time suppression is a simple exponential e−td/τr . In these studies,td = 4 s is used, which includes 2 s for the Hg spin-flip and 2 s of additional waiting time.Including these factors modifies the false signal to bedfalsev×E,r = FτrTRe−td/τrvφµn2Ec2(Er+ − Er−). (6.5)1716.3. Impact of Electric Field InhomogeneityValues of Er can be easily taken from the FEA simulations of the central region, but the value ofτr depends on the behaviour of the UCN population. To analyze this false signal it is thereforenecessary to either derive an expression for τr, or use PENTrack simulations to capture theUCN behaviour. Both methods are explored in the following sections.Decay Time DerivationIn modelling this false signal, there are two populations of UCNs: those that are part of thenet motion, and those that have undergone a diffuse reflection and are no longer part of thenet motion. Note that, although half of diffuse reflections will result in UCNs continuing totravel in the direction of net motion, there will be an equal number of diffusely-reflected UCNsthat travel in the opposite direction. Those opposing UCNs cancel the false signal contributionfrom those that do not change direction, so the false signal averaged over all UCNs that haveundergone at least one diffuse reflection should approach zero with a large number of UCNs.This reasoning is based on the Lambert model of diffuse reflection, in which forward or backwardreflections will be equally likely. A model with a different distribution of reflections could havea slower decay.The decay time τr is the time after which a fraction 1/e of the initial net motion populationwill remain in the population, meaning that those UCNs will have not undergone any diffusereflections. If the probability of a diffuse reflection is PL, then after a UCN reflects n timesfrom the walls, the probability that it will have not undergone a diffuse reflection is (1− PL)n.From this, the expected number of reflections that will occur after τr isn(τr) =−1ln (1− PL) . (6.6)The decay time then depends on the frequency with which the UCNs reflect from a surface.Figure 6.8 illustrates the trajectory of a UCN in the cell in the xy plane, assuming a specularreflection from the cylindrical wall formed by the insulator. The distance that a UCN musttravel between two wall collisions is 2R cos(θ), where R is the cell radius and θ is the angle ofincidence of the UCN with the wall. With velocity vxy, the frequency of reflections from thewalls isfw =vxy2R cos(θ). (6.7)Note that the velocity vxy in this equation is the velocity in the xy plane, which is not necessarilythe same as the azimuthal velocity vφ. This difference is illustrated in Figure 6.8.There is also a contribution from collisions with the electrodes. For simplicity of the calcu-lation and later simulations, collisions with the ceiling can be neglected. Assuming no initialvertical velocity, the frequency with which a UCN will reflect from the floor of the cell isfg =√g8zmax, (6.8)where zmax is the peak height. Combining these three equations leads to an expression for thedecay time of the net motionτr(vxy, zmax, θ) =n(τr)fw + fg=− 1ln 1− PL√8zmaxgcos(θ)C + cos(θ),(6.9)1726.3. Impact of Electric Field InhomogeneityFigure 6.8: Illustration of the path geometry of a neutron in the horizontal plane, being reflectedoff the perimeter wall of a cylindrical cell. A neutron traveling along a straight line trajectoryat an angle θ to the normal of the cell wall will travel a distance 2R cos(θ) between bounces.where C = vxy√2zmax/R√g. The decay time depends on the specific combination of maximumheight, velocity, and the angle of incidence with the cell walls for a given UCN. For a UCN ofvxy = 3 m/s, zmax = 0.16, and θ = 45◦, in a cell of R = 0.25 m and PL = 5%, this would giveτr = 1.7 s. If gravity is neglected entirely (such that fg = 0), this simplifies toτr(vxy, θ) =n(τr)fw=− 1ln 1− PL2R cos(θ)vxy.(6.10)For the same UCN as above, this would give τr = 2.3 s. This gravity-less form is a usefulcross-check for validating simulations.Any further calculation requires knowledge of the distribution of velocities (or, equivalently,energies), and of heights when including gravity. Simulation results for both the height andenergy distributions were shown in Chapter 4 in Figures 4.5 and 4.4. In the next section thesedistributions will be used to calculate the decay time and compared to PENTrack studies.Decay Time StudiesThe decay time can be studied in PENTrack by simulating a UCN population with an initialnet motion and examining the evolution of the trajectories of individual particles.There are two simulation geometries used in these studies. The “real” cell geometry isthe same as that which has been established in previous sections and chapters, with an innerdiameter of 0.5 m and electrode separation of 0.16 m, and including the cell entrance and in-sulator trenches. This geometry is as shown in Figure 6.2. The second “simplified” geometrymaintains the diameter and electrode separation but removes the entrance and trenches. Sincethese geometric features could enhance the number of wall collisions, the simplified geometry1736.3. Impact of Electric Field Inhomogeneityshould match more closely with analytical calculations. The simplified geometry results alsotypically set g = 0 m/s2, removing the effect of gravity and making it even easier to compareto the analytical results. In both simulations the UCNs are created in a 1-mm-thick volume,centred at y = 0, that covers the half of the xz plane with x > 0. All UCNs are created in thisplane, with velocities directed in the positive y direction, which is a state of maximum initialnet rotational motion.To reduce the time needed to collect sufficient statistics in these simulations, especially forthe simulations of the false signal in subsequent sections, both wall losses and beta decay aredisabled. The expected spectrum of the UCNs that would be detected in the TUCAN EDMexperiment, taken from the work of S. Sidhu [6], is used in order to account for the shaping ofthe spectrum by losses during the precession period. While disabling losses reduces the amountof computation time needed to obtain results, it does potentially introduce some error into thesimulation results. In reality, a UCN that enters the insulator trenches should have a highprobability of being lost due to the high rate of wall collisions in this confined geometry. Withlosses disabled, those UCNs will survive and continue to contribute to the simulation results.However, the high rate of wall collisions means that such UCNs should quickly experience adiffuse reflection and stop contributing to the net motion, so they should not skew the resultsin a significant way.The quantity of interest, vφ, is not something that can be output by PENTrack by default.It is the component of velocity in the xy plane that is tangential to the radial direction fromthe z-axis of the cell, or vφ = ωφ√x2 + y2, where ωφ is the angular velocity around the z-axis.This can be calculated asvφ =xvy − yvx√x2 + y2, (6.11)where (x, y, z) and (vx, vy, vz) are the position and velocity in Cartesian coordinates. Computingthe average vφ over time and fitting to an exponential decay should give a value for τ¯r.The definition of vφ reveals some subtlety in describing this systematic effect. UCNs donot travel along circular orbits, but instead taken straight-line paths between wall collisions.Because vφ depends on x and y, this means that vφ is constantly changing along the pathof a UCN. This has some impact on what is observed in simulations. At the start of thesimulations, all UCNs are in a plane, with velocities at 90◦ to the plane. This is a state ofmaximum tangential velocity, with all UCNs effectively synchronized. However, once the UCNsstart moving, the mean vφ over the population will immediately decay, even before there areany reflections. The high level of synchronization is never recovered, due to the different UCNsmoving at different speeds along paths of different lengths. So, the observed mean vφ will showan initial very fast decay due to desynchronization, followed by a slower decay due to diffusereflections. The slower decay is the parameter of interest, and can be extracted by neglecting thefirst few points in the fit. Alternatively, the mean vφ of a UCN can be calculated by averagingover the straight line trajectory, which givesv¯φ =vyx√R2 − x2 asinh(√R2 − x2x). (6.12)This equation will be used for calculations based on initial distributions of position and velocity.Figure 6.9 shows some examples of simulation data fit to exponential decay functions. Singleexponential decays are seen to fit well to the data, with the exception of the first 2–3 pointswhich show the decay of the initial synchronization.The value of τ¯r can also be estimated using the expressions derived above. Since the initialtrajectories have no vz, all UCNs start at their maximum height, and the initial heights and1746.3. Impact of Electric Field Inhomogeneity0 2 4 6 8 10Time (s)012345Mean v [m/s]PL = 1%, "real"PL = 6%, "real"Fit r = 4.77 0.01 sFit r = 0.930 0.005 sPL = 1%, "simplified"PL = 6%, "simplified"Fit r = 7.65 0.02 sFit r = 1.251 0.006 sFigure 6.9: Simulated decay of the azimuthal velocity vφ of a UCN population over time. Thepopulation begins in an initial state with all velocities pointing in the same direction. Resultsare shown for different values of the diffuse reflection probability PL, and for both a realistic cellgeometry including geometry features such as trenches, and a simplified cell geometry which isa featureless cylinder. Fitted exponential decays through the results give the reported decaytimes.1756.3. Impact of Electric Field Inhomogeneity0 0.01 0.02 0.03 0.04 0.05 0.06 0.07PL012345678Average decay time r (s)Distribution calc.Distribution calc. w/o gravityCollision rate calc.Collision rate calc. w/o gravityReal geometry sim.Simplified geometry sim. w/o gravityFigure 6.10: Mean τr of the net motion vφ versus diffuse reflection probability PL, calculatedusing different methods. Simulation results are obtained from fitting UCN track data, as shownin Figure 6.9. Simulation results are shown for both the real cell geometry and the simplifiedgeometry, with gravity disabled in the simplified case. The four calculation results use Equation6.9 when including gravity, or Equation 6.10 without gravity. The distribution calculation usesvelocity and position data taken from initial distributions used in the simulations. The collisionrate calculation uses average collision frequencies obtained from the simulations. Note that thetwo calculation methods overlap in the case without gravity.velocities provide the values needed to evaluate Equation 6.9.9 This gives a distribution ofdecay times, from which an average can be computed. Since the fit values are obtained fromfitting the mean vφ over time, the comparable average τr should be computed by fitting anexponential decay to the sum ∑i=1...ne−t/τr(vxy ,zmax,θ), (6.13)where the sum is over each neutron in the sample population in the simulations. This calculationis referred to as the “distribution calculation”.Alternatively, a rough calculation can be made by taking the average number of wall hits,as recorded by PENTrack, computing the wall hit frequency, and inserting that into Equation6.9. Since the collision rate of interest is that for specular paths, this is computed from asimulation with 0% diffuse reflection probability. As with the above computation based onthe initial distributions of velocity and height, τr is calculated for individual neutrons and theaverage taken from fitting to a sum of exponential functions. This calculation is referred to asthe “collision rate calculation”.Figure 6.10 shows mean decay times extracted using the three methods described above, forboth the real geometry and the simplified geometry without gravity. In the simplified case, both9Obviously the values of vxy and zmax can change when a UCN undergoes a diffuse reflection, but sincethe first diffuse reflection is the event of interest here, the initial values are the correct values to use for thiscalculation.1766.3. Impact of Electric Field Inhomogeneityanalytical calculations are seen to agree completely, meaning that the frequency calculation fromthe velocity distribution is correct. In the real case, these two calculations give very differentresults. This can be explained by the presence of the trenches, which would increase the observedcollision rate. Comparing simulation to calculation shows that the distribution calculationpredicts a longer decay time than the simulations in both cases, although the discrepancyseems to be larger for the real geometry compare to the simplified geometry. For the realgeometry, the collision rate calculation predicts a faster decay time. This could also be dueto the trenches skewing the collision rate to a higher value, which does not necessarily meanthat the net motion decays faster. A UCN that enters the trenches could remain there for theduration of the simulation and see a very large number of collisions, but after the first few itwill already have stopped contributing to the net motion, so the contribution of these UCNsmay not be accurately counted using the collision rate.A close study of the decay times reveals a number of interesting artifacts introduced bysimulation methods, which require caution when interpreting the results. One pleasing outcomeof the decay time simulations is that an analytical calculation using initial distributions appearsto serve as an upper bound on the decay time. Since a slower decay time will lead to a largerfalse signal, this introduces the possibility that such a calculation can be used to bound thefalse signal without requiring resource-intensive simulation. This will be further explored in thefollowing sections, which will present spin-tracking simulations of the false signal.Spin-Tracking SimulationsThe expected magnitude of the false EDM signal due to the rotational v × E effect can becalculated using the decay times extracted in the previous section, assuming some value ofEr. However, it is more accurate to simulate the exact interaction of the UCN trajectorieswith the electric field. This can be done using spin-tracking in PENTrack, which allows theprecession frequency of each UCN to be calculated over the duration of the simulation. Inthese studies, separate simulations are performed with an electric field map oriented in bothpolarities, as in the real experiment, and with no magnetic field gradients or other effects whichwould introduce an E-dependent frequency shift. The average precession frequency in each ofthe field orientations is computed, and the difference between these frequencies used to computethe false signal asdfalse =~(ωn,+ − ωn,−)4E, (6.14)where E is taken to be 12.5 kV/cm (this is slightly higher than the average value indicated bythe FEM maps, but the difference is less than 1%).These simulations use the same simulation geometry as in the previous section, with simu-lations performed for both the real and simplified geometries in the top cell. Results are alsoshown for the bottom cell, which uses the real geometry and is the mirror of the top cell acrossthe xy plane.The electric field is modelled using the 2D map from the FEMM simulation, which PENTrackassumes to be rotationally symmetric. The use of a 2D map here was motivated by minimizingthe memory required for the PENTrack simulations, as well as being computationally fasterthan a 3D map. In order to accurately represent the field, the map must be exported fromFEMM with a sufficiently high resolution. This is illustrated in Figure 6.11, which comparesthe magnitude of the false signal to the step size used in the map. It can be seen that themagnitude of the signal is stable at small step size, but changes as the step size becomes larger.1776.3. Impact of Electric Field Inhomogeneity10-1 100 101 102Field mesh stepsize (mm)00.511.522.533.54d false (ecm)10-24Simplified geometry, PL = 0%Real geometry, PL = 0%Figure 6.11: Magnitude of the simulated false signal from net rotational motion vs. the step sizeof the electric field map. Results are from PENTrack spin-tracking simulations with maximuminitial vφ. A dependence of the result on step-size is apparent.This stabilization occurs at a step size of 4 mm. In the remaining simulations, the smallest stepsize of 0.25 mm is used.10The accuracy of these simulations is also seen to depend on the resolution of the mesh usedfor the geometry. PENTrack uses geometries built using the STL file format, which models allshapes as collections of triangles. For these studies, geometries were modelled in Solidworksand then exported in STL format. When exporting these files, the user must set the allowedpositional and angular deviation of the STL representation from the modelled geometry. Theseparameters determine how much the positions and angles in the STL mesh can differ from thosein the Solidworks model, which is constructed with an exact geometry from primitive shapes.Smaller allowed deviations more accurately reconstruct the model, with the angular deviationbeing the more relevant of the two for reconstruction of small details. Figure 6.12 shows how thefalse signal changes when varying angular deviation, assuming an allowed positional deviationof 0.1 mm.The most interesting result in this meshing study is when assuming 0% diffuse reflection. Inthe real cell geometry, the signal is seen to change as the allowed angular deviation decreases, andnever reaches a stable value. This behaviour may also be present in the simplified geometry,where the signal also rises slightly with decreasing allowed angular deviation, although it is10In chapter 8, a 3D map of a magnetic field is used in PENTrack, with a step size of 2.5 mm. Runningsimulations with this map required 16 GB of memory, while the electric field map at 0.25 mm step size could berun with a “standard” 4 GB. The studies of the v×E effect require the simulation of a larger number of neutrons(compared to the simulations in chapter 8) to be able to see the false signal, and the DRAC clusters used forthese simulations account for larger memory requests in their job prioritization, so minimizing the computationaldemand did have some value. In hindsight, however, these simulations could have used a 3D map.1786.3. Impact of Electric Field Inhomogeneity100 1011.522.533.54d false (ecm)10-24Simplified geometry, PL = 0%Real geometry, PL = 0%100 101Allowed angular deviation [°]246810d false (ecm)10-27Simplified geometry, PL = 3%Real geometry, PL = 3%Figure 6.12: Magnitude of the simulated false EDM signal from net rotational motion vs.allowed angular deviation when exporting the STL geometry from Solidworks. Results arefrom PENTrack spin-tracking simulations with maximum initial vφ. Top: PL = 0%, bottom PL= 3%. An effect of the geometry resolution is visible for PL = 0%, but this vanishes for PL =3%.1796.3. Impact of Electric Field Inhomogeneitymuch less pronounced in this case. At 3% diffuse reflection this effect is no longer visible at theprecision of the simulations.Investigations of the UCN trajectories at 0% diffuse reflection showed that the net motiondecays with very long decay time, which should not be possible analytically. This impossibilitycan be understood from the vector form of Snell’s law, which describes specular reflection. Foran incoming velocity direction vˆ1, incident on a surface with normal nˆ, the direction of theoutgoing velocity must bevˆ2 = −2(nˆ · vˆ1)nˆ+ vˆ1. (6.15)The orbital motion is defined by the azimuthal component of the velocity, and since nˆ · vˆ1is a scalar, the motion can only decay from the geometry if a UCN reflects from a surfacethat has a normal with a non-zero azimuthal component. Analytically, the cell geometry canbe constructed from cylinders, horizontal planes, and toroids (for the curved surfaces of thetrenches, plug, and entrance). None of these shapes, as oriented in the cell geometry, haveazimuthal normal components. Thus this decay must be caused by the mesh.This meshing effect can be further explored by looking at PENTrack hit logs, which providethe surface normals for each collision of a UCN with a surface. If a normal vector (nx, ny, nz)at position (x, y, z) has no azimuthal component, then the quantityqn =nxx+ nyy√x2 + y2√nx2 + ny2(6.16)should be either 1, -1, or undefined (in the case where nx = ny = 0. Figure 6.13 plots dis-tributions of |qn| for 0.5◦ and 16◦ allowed angular deviation, for both the real and simplifiedgeometries. These results were calculated using logs of wall reflections from PENTrack simula-tions, which provide data on the normal at the location of a reflection. For the 0.5◦ cases, bothgeometries have very narrow distributions around 1, as expected. With 16◦, the distributionsbecome much more extended. This is the case for both real and simplified geometries, with theextent of the distribution being broader for the real geometry.A part of the meshes, at the insulator trench, is shown in Figure 6.14 for the 16◦ and 0.5◦angular deviations. The 16◦ case has a coarse mesh, with the triangles of all surfaces beingclearly distinguishable. The 0.5◦ case has a much finer mesh in comparison. Much of thisincreased density of the mesh is in the rounded edge of the trench, where many triangles areneeded to reproduce the surface to higher precision. This is consistent with the observation thatthe simplified geometry, without the trenches, has less deviation in its normal vectors comparedto the real geometry. It is apparently this surface, which is rounded in two directions, that isdifficult to reconstruct with high fidelity.When using a discretized mesh to represent curved geometries, this effect appears difficult toavoid. The 0.5◦ value is the lowest that Solidworks will allow to be exported. Fortunately, at 3%diffuse reflection probability, the effect of the mesh is not visible compared to the uncertaintyof the spin-tracking, and so the results should still be acceptable.Figure 6.15 shows the magnitude of the false signal, calculated using three different methodsfor comparison. The “spin-tracking” method uses the PENTrack simulations described in thissection to extract the frequency shift using spin-tracking. The “decay time fit”method usesEquation 6.5 with fitted mean decay times and velocities taken from the particle trackingsimulations used to produce Figure 6.19. The “distribution” method uses theoretical Equations6.9 and 6.10 to calculate τr using the initial positions and velocities for each UCN in a simulationdistribution, and then uses those τr values to calculate the false signal using 6.5. For thedistribution method, the azimuthal velocity vφ is calculated using Equation 6.12, which accounts1806.3. Impact of Electric Field Inhomogeneity0.998 0.9985 0.999 0.9995 1|qn|10-810-610-410-2100ProbabilityReal, 0.5°Real, 16°Simplified, 0.5°Simplified, 16°Figure 6.13: Distributions of |qn| for different allowed angular deviations of the STL mesh of thecells, calculated from hit logs of PENTrack simulations, and excluding undefined values wherenx = ny = 0. The quantity |qn| is expected to be exactly 1. With an allowed angular deviationof 0.5◦, the distributions for the two geometries are very narrow at |qn| = 1. The distributionsare seen to deviate when the allowed angular deviation is made larger, which causes reflectionsto deviate from specularity. The extent of the deviation is seemingly much broader in the caseof the real cell geometry compared to the simplified cell geometry.Figure 6.14: Visual comparison of the mesh at the inner edge of the insulator trench of the cell,with different allowed angular deviations. Left: 16◦, right: 0.5◦.1816.3. Impact of Electric Field Inhomogeneity0 0.01 0.02 0.03 0.04 0.05 0.06 0.07PL10-2810-2710-2610-25d false (ecm)Spin tracking, realSpin tracking, simplifiedDecay time fit, realDecay time fit, simplifiedDistribution, realDistribution, simplifiedFigure 6.15: Calculated magnitude of the false EDM signal caused by a rotational v ×E effectin the top cell, assuming maximum initial vφ, for different diffuse reflection probabilities PLof the cell surfaces. Results are shown for the real cell geometry, and for the simplified cellgeometry with gravity disabled. The magnitude is calculated using three methods: from thefrequency shift extracted from spin-tracking simulations, from the simulations of mean decaytimes in Figure 6.19, and from the analytical models of Equation 6.9 and 6.10 using initialvelocity and height distributions.for the variation of the tangential velocity over a straight line trajectory. In the decay time fitand distribution methods, the mean value of Er was computed from the FEMM map and usedin the calculation. The use of the mean value may not be strictly correct, as there could besome correlation between decay time and the path through the electric field, but no discrepancyis visible in the results.The results from the theoretical calculation and the spin-tracking simulations are seen toagree very well, while those from fitting the decay seem to predict a lower value. This is thecase for both the real geometry and the simplified geometry without gravity, so it does notappear to be an effect of geometry or trajectories. It therefore seems likely that this is due tothe averaging associated with fitting a mean decay time. The strong agreement between thetheory calculation and spin-tracking simulations provides confidence in calculating this effectusing a model and predicted initial distributions for the UCNs in the cells, which can reduce theneed for future spin-tracking simulations. As these spin-tracking simulations can take tens ofCPU-years to accumulate sufficient precision on the false EDM signal (when running on currentDRAC cluster hardware), this could save significant computing resources.Figure 6.16 shows the false signals from spin-tracking in the top and bottom cells. Nosignificant difference is seen between the two cells.The magnitude of the signal seen from these results is larger than desirable. Even at 6%diffuse reflection, the false signal is 4 × 10−28 ecm, which would make this effect a dominantsystematic. However, these calculations have assumed a very strict initial condition, with allUCNs being part of the net motion. The next section addresses that assumption.1826.3. Impact of Electric Field Inhomogeneity0.01 0.02 0.03 0.04 0.05 0.06Diffuse reflection probability10-2810-2710-2610-25False signal magnitude (ecm)Spin tracking, top cellSpin tracking, bottom cellFigure 6.16: Magnitude of the false EDM signal caused by a rotational v × E effect in the topand bottom cells, with maximum initial vφ. There does not appear to be any difference betweenthe two cells.Filling SuppressionUntil now, the calculations have assumed that UCNs start in the cell with maximum vφ, ne-glecting any suppression of the initial net motion due to decay during the filling into the cell. InEquation 6.5 this is equivalent to F = 1. Since the filling takes place over a significant periodof time, there will certainly be some fraction of the population that has already decayed out ofany initial motion when the measurement starts. It should therefore be expected that F < 1.This can also be examined using PENTrack simulations.These results were extracted from the simulation model of the TUCAN source connected tothe cells used by S. Sidhu [6] to simulate the filling of UCNs into the cells. The value of interestfor each UCN is the time between entering the cell and the cell valve being closed. During thatperiod of time, even if each UCN initially filled into the cell such that they had a consistentrotational motion, that motion will undergo exponential decay.Figure 6.17 shows the distribution of these times, in the top cell, for 2% and 6% diffusereflection probabilities. From these distributions it can be seen that many UCNs spend tens ofseconds in the cell before the filling process ends.For a given UCN, the suppression due to filling is given byFµν = e−tf/τr , (6.17)where tf is the time in Figure 6.17. Since a faster UCN will have more probability of exitingthe cell than a slower UCN, the distribution of tf is likely to be velocity dependent. Accuratelyestimating the filling suppression therefore requires calculating τr for each neutron in the filleddistribution and computing the resulting false signal with and without F . This computation isperformed using the theoretical equation for the false signal, which has now been seen to givegood agreement with spin-tracking simulations.1836.3. Impact of Electric Field Inhomogeneity0 5 10 15 20 25 30 35 40Time tF spent in cell since last entering (s)10-410-310-210-1ProbabilityPL = 2%PL = 6%25% quantile, PL = 2%25% quantile, PL = 6%Median, PL = 2%Median, PL = 6%75% quantile, PL = 2%75% quantile, PL = 6%Figure 6.17: Distribution of time tf spent in the cell prior to the end of the filling period,simulated in PENTrack for the top cell, for PL = 2% and PL = 6%. Quantiles at 25%, 50%,and 75% are indicated by vertical lines. Considering the 75% quantiles, which are 7.7 s forPL = 2% and 5.6 s for PL = 6%, and the decay time constants τr in Figure 6.9, it can beexpected that at least 25% of UCNs will have decayed out of net rotational motion during thefilling.1846.3. Impact of Electric Field InhomogeneityAt 6% diffuse reflection, there is an average F = 0.28. With td = 4 s, TR = 150 s, assumingthat UCNs have maximum vφ when entering the cell, and integrating over the distribution oftf in Figure 6.17, this gives an expected false signal of 1.1 × 10−28 ecm due to net rotationalmotion. Since the condition of maximum vφ is an extreme case, which is improbable to occurin a real experiment, this can be taken as an upper bound on the systematic shift from netrotational motion.Because this effect depends on the exact behaviour of the distribution of the UCNs inthe cell, it is not something that can be easily measured, and so the presence of this effectin the TUCAN EDM experiment is unknown. Furthermore, these calculations have assumedthe worst possible case, with UCNs filling the cell with a completely-ordered motion, which ishighly unlikely. This calculation therefore represents the largest such signal that could possiblybe present, and it is treated as a conservative systematic uncertainty.Final NotesThere are a few further points to discuss about this effect and the calculations that have beenperformed in this section.A major question is the value of the diffuse reflection probability that can be assumed inthe cell. Very highly polished guide surfaces might see values in the 3–5% range. The cellelectrodes should be well-polished for HV considerations, and therefore might have values inthat range. The insulator, however, is a machined plastic surface with dPS coated over it. Itseems likely that this surface would be reasonably rough, and so taking false signal values at a6% average diffuse reflection does not seem unreasonable. However, no attempt has yet beenmade to measure this roughness, and if the diffuse reflection probability is in fact much lower,the signal could be much larger.A second point is to recall the differences between the two FEA simulations. Consideringthat the FEMM simulation gives a greater value of Er compared to the more complete Operasimulation, the signal can be further reduced, and can reasonably be assumed to be less than10−28 ecm.Finally, this analysis has so far only looked at the effect in a single cell. Unfortunately,in this case the double cell configuration cannot be argued to give any cancellation. This issimply because the direction of any net motion in the two cells cannot be guaranteed to obeyany particular relation. In the worse case, each cell will produce the same signal, and whenaveraged this will be the same as in the single cell.6.3.2 Linear vxEThe other possible net motions occur if the centre of the UCN population shifts linearly overthe course of the measurement. This could occur either due to the UCNs preferentially fillinginto one part of the cell, or due to the UCNs changing in energy and redistributing within thegravitational potential.Linear Motion from FillingIf the UCNs fill into the cell such that their distribution at the start of the precession periodis not isotropic, then there will be some net motion as the distribution equilibrates to isotropythrough diffusion. Consider a population of UCNs in a cell, distributed such that the centre ofmass in the xy plane is at (0, δy) when the precession period starts, with the origin (0, 0) at thecentre axis of the cell. Over time the population will diffuse throughout the cell such that the1856.3. Impact of Electric Field Inhomogeneity-0.3 -0.2 -0.1 0 0.1 0.2 0.3x (m)00.511.52|E x| (kV/cm)z = 1 cmz = 5.5 cmz = 10 cmz = 14.5 cmFigure 6.18: Horizontal electric field magnitude |Ex| vs. position x inside the top cell, takenfrom Opera simulations of the central region. All results are in the symmetry plane of thecentral region at y = 0, which cuts through the HV feedthrough, with each line at a differentheight z measured from the central electrode surface. The range of x plotted is from −R to+R, and positive x is towards the corona ring. There is a clear asymmetry between the twohalves of the cell. This is especially noticeable for z = 5.5 cm and z = 10 cm, with |Ex| forx < 0 being larger (more negative) compared to values at x > 0.centre of mass is at (0, 0). This results in an average net velocity vy = −δy/T over the courseof the precession period.This net velocity vy will produce a motional magnetic field −vyEx/c2 = δyEx/TRc2. Withradial symmetry of the cell, there will be an equal number of UCNs that see ±Ex, and sothis motional field would be zero on average. However, the central region as a whole is notradially symmetric. The presence of the corona ring and HV feedthrough breaks the symmetricgeometry, and can result in a slightly different field in two halves of the cell. Figure 6.18 showsthe distribution of Ex taken from the Opera FEA simulations with +200 kV on the centralelectrode. Comparing the half of the cell nearest to the corona ring to the half furthest fromthe corona ring shows that there is a visible asymmetry in the field, with |Ex| being smallercloser to the corona ring.From the field map, the expected systematic shift can be calculated. The average of Exfrom the Opera field map, inside the cell volume, is −1.89 × 10−2 kV/cm. If UCNs filled onlyinto one half of the cell, but with an otherwise isotropic distribution, then they would have anaverage offset δy ≈ 0.10 m (based on an analytical calculation of the value of y which evenlydivides the volume occupied by the UCNs). The magnitude of the false signal will be|dfalsev×E,y| = vyµn2Ec2|(Ex+ − Ex−)|. (6.18)With the above values, this evaluates to 7.4× 10−29 ecm.This possible systematic will also be suppressed depending on how quickly the distributionapproaches an isotropic state, as any offset will be reduced in the time between the end of1866.3. Impact of Electric Field Inhomogeneity0 1 2 3 4 5Time (s)-0.06-0.04-0.0200.020.040.060.080.1Mean y (m)Top cellBottom cellFigure 6.19: Mean y coordinate of UCNs vs. time from PENTrack simulations, starting with allUCNs in the half of the cell with y > 0 and with an isotropic velocity distribution. The meany coordinate is seen to decay rapidly as the population shifts to equilibrium. No significantdifference is observed between the top and bottom cells.filling and start of precession. As with a net rotational motion in the previous section, thisdecay time of a linear offset can be examined in PENTrack simulations. Figure 6.19 shows themean y coordinate over time for UCNs initially filling the half of the cell with y > 0, assumingan isotropic velocity distribution. Within 0.5 s the population has a mean y of 0 with 10%uncertainty. The delay time before UCN precession started was td = 4 s in calculating theeffect of the net rotational motion. So, the effect is suppressed by at least a factor of 10, downto 7.4× 10−30 ecm. As with the rotational motion case, there is no guarantee that the offset inthe two cells has any relation, and so no further cancellation can be guaranteed.Linear Motion from HeatingOther than a shift in position due to uneven filling into the cell, there is another possiblesource of linear motion for a UCN population, known as small heating. Small heating is anobserved increase in the energy of UCNs at the neV level, which does not necessarily result inthose UCNs being lost. Significant study has gone into small heating in the context of neutronlifetime experiments, with the current explanation being that this is caused by UCNs scatteringfrom nanoparticles in motion on the surfaces [284–286]. Small heating of UCNs in an nEDMexperiment will cause the population to rise slightly against gravity, resulting in a linear motion.The expected linear motion from small heating can be estimated from the expected shiftin energy, using values taken from PENTrack simulations. Measured values of the probabilityof small heating on the order of 100 neV upon reflection from solid surfaces are consistentlyreported to be on the order of 10−7 [284, 287]. The average wall-bounce frequency in simulationsof the TUCAN EDM spectrometer is 21.2 Hz. Therefore, over 150 s of measurement, 0.03% ofUCNs may gain energy on the order of 100 neV, for an increase of 0.03 neV in the average1876.4. High-Voltage Systematicsenergy. The average total energy is 83.4 ± 0.1 neV in the top cell and 48.8 ± 0.1 neV in thebottom cell, so that the increase in energy would be 0.04% in the top cell and 0.06% in thebottom cell. The total energy is measured relative to the centre of the central electrode, so thegravitational potential energy should be proportional to the height measured from this point.In the bottom cell, the UCN population is expected to be at −202.1± 0.2 mm, and in the topcell 198.3 ± 0.2 mm. Assuming that the increase in energy simply increases this height, smallheating would result in a 0.13 mm rise in the bottom cell and 0.08 mm in the top cell. Thesmall heating velocity would be greatest in the bottom cell, and over 150 s the velocity wouldbe vsh = 8.7× 10−7 m/s.If the magnetic and electric fields are perfectly aligned with each other, and with the direc-tion of gravity, then this linear motion will result in a magnetic fieldBsh =vshErc2. (6.19)If Er differs between polarities, then this can produce a false EDM by a B2 frequency shift,with magnitudedfalse,sh =~4Eγn22ωnvsh2∆Er2c4, (6.20)where ∆Er2 is the difference in Er2 between the two electric field polarities. If Er differs by10%, then ∆Er2 ≈ 0.2|Er|2. This gives a false signal of 4.7 × 10−45 ecm and is completelynegligible.This linear motion can have a more direct effect if the electric and magnetic fields andgravity are misaligned such that they are not co-planar. This would occur if the vertical axesof both the B0 coil and the central region were tilted relative to the direction of gravity, butin such a way that the tilts are in different directions from each other. If the electric field ismisaligned from the direction of gravity by θgE , and the magnetic field is misaligned from theplane of gravity and the electric field by θgM , then this linear motion results in a magnetic fieldcomponent that points along the direction of B0, and reverses with E,Bsh =vshE sin(θgE) sin(θgM )c2. (6.21)The false EDM signal will bedfalse,sh =~2c2γnvsh sin(θgE) sin(θgM ).(6.22)If the misalignments are both 5◦, this gives a false signal of 4.4× 10−31 ecm. Alignment betterthan 5◦ should be easily achievable, and so small heating does not appear to be a significanteffect.6.4 High-Voltage SystematicsThe design of the central region with respect to the electric field is driven by increasing themaximum sustainable field magnitude while minimizing the inhomogeneity of the electric field,which drives systematic effects. This topic has largely been covered in the preceding sections.There are a few additional systematic effects which are also related to the use of high voltagein the EDM measurement. The first is due to leakage currents, which can be present in anyHV system. The second is due to instability of the HV supply. Both of these effects can inducemagnetic fields which could cause systematic shifts. These effects will be described in thissection.1886.4. High-Voltage SystematicsFigure 6.20: Two example leakage current paths in the bottom cell. The black cylindrical meshshows the insulator inner surface, while the blue and red lines give the current paths. The redline has an azimuthal path segment of φ = pi/8 at a height h = 8 cm above the bottom electrode.The blue line has an azimuthal path segment of φ = 2pi at a height h = 6 cm above the bottomelectrode. The incoming and outgoing current paths from and to the HV are indicated bydashed lines. Both paths have overlapping incoming paths (top), but unique outgoing paths(bottom).6.4.1 Leakage CurrentsLeakage currents are small electrical currents that flow in, or on the surfaces of, insulatingmaterials. While the electrical resistivities of insulating materials are very large, they are notinfinite, and this allows some small current to flow. The nominal volume resistivity of Rexoliteis 1.1× 1020 Ωm. With a 0.25 m radius and 1 cm wall thickness, insulator height of 19 cm, andan applied voltage of 200 kV, the expected leakage current through the insulator is only 1.5 fA.However, imperfections in the material will typically result in larger leakage currents. In theprevious ILL experiment, with a quartz insulator of 0.235 m radius and 1.5 cm wall thickness,with an insulator height of 15 cm, leakage currents of 1 nA were reported [208].The field generated by a leakage current can produce any of the frequency shifts describedin Chapter 4. When the HV polarity is reversed, the leakage current through a cell can also beexpected to reverse, and thus the field will be reversed. If the currents follow the same pathregardless of polarity, the BE shift will be the same (on polarity reversal the shift will pickup one negative sign from the electric field and one negative sign from the current) and willnot produce a false EDM signal. Because B2 shifts depend on the field components squared,electric field reversal on its own will also not induce a B2 shift. This leaves only a shift due togravitationally-weighted sampling by the UCNs compared to the Hg gas.The magnitude of the leakage current can be measured during the experiment, but the paththat it takes across the insulator is unknown. To evaluate the systematic effect, multiple possiblepaths should be considered, and the systematic uncertainty taken as the largest estimatedmagnitude of false EDM produced by any of those paths. In the results presented here, a1896.4. High-Voltage Systematicsspecific class of paths is examined. These paths are constructed (in the bottom cell) in thefollowing way:1. starting at the feedthrough, the current flows as a straight line to the opposite side of theinsulator,2. then flows upwards, on a straight line, stopping at some distance h above the lowerelectrode,3. then flows azimuthally around the insulator for some angle φ,4. then continues to flow upwards, stopping at the ground electrode,5. and finally flows out horizontally along a straight line to the outer edge of the vacuumchamber near the feedthrough.Figure 6.20 shows two examples of these paths. This class of paths should generally capturethe features of the possible current paths, which must travel from one electrode to the othervertically, but may also travel azimuthally around the insulator. Each path is characterized bya pair of parameters (h, φ).The field for each path inside the cell volume is computed using the Biot-Savart equation.The possible systematic effects can then be computed using the equations of Chapter 4. Themaps of the fields are generated with a stepsize of 2.5 mm, below which the calculation of thefalse signal appeared stable at the level of 5% — this computational precision is included as anerror bar in the results.Figure 6.21 shows the resulting gravitational false EDM resulting from leakage currentstaking paths of different h and φ. Results are shown for leakage currents following the fullpaths as described above as well as paths that do not include the line currents of step (1) andstep (5), i.e. the incoming and returning currents. The comparison of the full path to the partialpath is only of interest due to the unusual scatter of results seen with the full paths. This isapparently the result of the contributions of the incoming/outgoing currents, which see differentlevels of cancellation dependent on φ, but this does not affect the maximum value of the falseEDM signal. The signal increases with φ, which is to be expected, as increasing φ increases themagnitude of the Bz component of the field inside the cell. The maximum false signal occurswhen the loop is roughly 2 cm above the bottom of the cell, with a lower loop showing a smallersignal. This is apparently the point at which the loop creates the greatest non-uniformity inBz over z. For a 1 nA current, this maximum signal is at the level of 3.4× 10−28 ecm.The maximum signal from a leakage current is quite large, and could be a dominant system-atic. However, this is only for the very extreme path which involves a full loop around the cellat a specific height, for a 1 nA current. It seems more likely that the current would only travela short distance azimuthally, and it is also possible that the leakage current could be lower.More research into leakage current paths could further constrain this systematic, and the actualmagnitude will depend on the observed leakage current magnitudes in the real experimentalsetup.Since the leakage current path is expected to be the path of least resistance, it shouldremain the same for the two polarities, and so there it is reasonable to neglect the possibilityof different fields that would create other shifts. However, it could be the case that the leakagecurrent magnitude differs between two polarities, related in some way to the HV supply. Inthis case, a BE or B2 shift could appear. Calculating BE shifts for a current difference of 10%does not yield any false signals above 10−31 ecm, while for B2 shifts calculations did not yield1906.4. High-Voltage Systematics0 50 100 150 200 250 300 350 400Azimuthal travel (degrees)00.511.522.533.5False EDM signal (ecm)10-28h=8 cmh=6 cmh=4 cmh=2 cmh=0 cm(a) Full path0 50 100 150 200 250 300 350 400Azimuthal travel (degrees)00.511.522.533.5False EDM signal (ecm)10-28h=8 cmh=6 cmh=4 cmh=2 cmh=0 cm(b) Path without incoming or returning currentFigure 6.21: Calculated magnitude of the false EDM signal produced by a 1 nA leakage currenttravelling a distance φ around the insulator at a height of h from the bottom of the cell, for thebottom cell. Figure (a) shows the result for the full current path as described in the text, whilefigure (b) shows the result without the incoming and outgoing sections of the path (indicatedas dashed lines in Figure 6.20.1916.4. High-Voltage Systematics0 1 2 3 4 5 6 7Height of loop above bottom electrode (cm)00.511.52False EDM signal (ecm)10-28Figure 6.22: Calculated false EDM signal produced by a leakage current in the bottom cell,with no leakage current in the top cell, when combining data from the top and bottom cells.The leakage current travels a full loop in the bottom cell, at variable height measured from thebottom electrode.any false signals above 10−35 ecm, so the contribution of leakage currents is dominated by thegravitational shift.Operating in a double cell configuration has some advantage for dealing with leakage cur-rents. It is unlikely that leakage currents would travel along paths that result in exactly can-celling false EDM signals. However, a leakage current in one cell also creates a field in theother cell. This field results in a partial cancellation of the leakage current frequency shift,and averaging the top and bottom cells also reduces the false signal. Figure 6.22 shows thecalculated false signal versus loop height when combining data from two cells. In the doublecell configuration, the expected systematic error falls to 1.5× 10−28 ecm.These calculations have assumed a leakage current only in one cell, with current of 1 nA. Ifthe same current flowed through both cells, the total uncertainty would be increased by a factorof√2. This would put the systematic uncertainty at 2.1× 10−28 ecm, which is still reasonable.6.4.2 High Voltage InstabilityThe last topic of this section is the presence of an AC component in the high voltage supply.High voltage supplies typically specify some “ripple”, which is the AC component associatedwith the construction of the supply. In the most general case, each cell can be considered aspart of a serial resistor-capacitor (RC) circuit connected to an AC voltage supply, as shown inFigure 6.23. The cell is a parallel plate capacitor, which can be approximated with capacitance1926.4. High-Voltage SystematicsZ0ACCcellFigure 6.23: Diagram of an AC circuit including the cell with capacitance Ccell and a seriesresistance Z0.Ccell =0piRe2H, (6.23)where Re = 0.34 m is the radius of the electrodes neglecting the outer curvature, and H = 0.16 mis the electrode separation. It is common for an HV system to have a large inline resistor, whichlimits current when discharging. A value of Z0 = 1 GΩ could be typical.The current that flows in the circuit due to the AC voltage is determined by the totalimpedanceZ = Z0 +1iCcellωAC. (6.24)If the voltage isVAC(t) = |VAC| sin (ωACt) (6.25)then there is a currentI(t) =|VAC|Zsin (ωACt). (6.26)Inside the cell, this appears as a displacement current according to the Ampere-Maxwell equa-tion. Considering that such a displacement current must be proportional to the electric field,and the electric field is distributed uniformly inside the cell, the displacement current density isalso uniformly distributed, and so integrating the Ampere-Maxwell equation gives the magneticfield~BAC = −rµ0ωACR2∣∣∣∣ |VAC|Z∣∣∣∣ cos (ωACt)φˆ = −|BAC|r cos (ωACt)φˆ, (6.27)where r is the radial distance from the rotational axis of the cell, and R is the radius of thecell. This is a circulating transverse magnetic field. The calculation of the frequency shift isdescribed in Appendix E. The result isδωAC =γn2|BAC|2R2ωn4(ωn2 − ωAC2) ,(6.28)Notably, this does not depend linearly on E. In order for a systematic effect to appear, theremust be a difference in the ripple between the two electric field polarities, such that|VAC| = VAC,0 ±∆VAC. (6.29)1936.4. High-Voltage Systematics100 101 102 103 104Frequency (Hz)10-3410-3210-3010-28False EDM signal (ecm)VAC,0=0.001%VAC,0=0.01%VAC,0=0.1%Figure 6.24: Calculated false EDM signal resulting from an AC ripple in the high voltage supplyfor a single cell vs. frequency of the ripple. Each line is a different nominal ripple value on thesupply, as a percentage of the supply voltage, and the difference in ripple between polaritiesis assumed to be 50% of this value. The peak corresponds to the nominal neutron precessionfrequency ∼30 Hz.Then, the false EDM signal will bedfalse =~2Eµ02γn24R2∣∣∣∣ 1Z∣∣∣∣2 ωnωAC2(ωn2 − ωAC2)4|VAC,0δVAC| (6.30)At the time of writing, the TUCAN collaboration has two 100 kV power supplies. One is aGlassman WK Series 125 kV bipolar supply. This supply’s test data reports a ripple voltage of0.003% at the supply frequency (60 Hz), and 0.09% at the switching frequency (140 kHz). theother is a Gamma High Voltage Research 125 kV single polarity supply. This supply reports aripple voltage of 0.05%, at unspecified frequency. For 200 kV operation, the intention had beento purchase a 200 kV bipolar switching supply from FuG Elektronik, with a ripple specifiedat 0.001%, but this supply is no longer available and another supply has not been identified.These values give some sense that the expected ripple magnitude will be between 0.001% and0.1%. The expected difference of the ripple voltage between polarities is unknown, and wouldneed to be characterized, but logically should not be more than 50% of the reported ripple.Figure 6.24 shows the expected false EDM signal assuming a ±50% difference in nominalripple value between polarities, for different ripple values. High frequency shifts produce negli-gible values, well below 10−30 ecm. Low frequency values produce larger shifts, peaking at theneutron precession frequency, where the shift grows asymptotically. With a 60 Hz power gridfrequency, the shift would still be below 10−30 ecm.A large part of the suppression of this false signal is due to the inline impedance, whichserves to dampen the current. Figure 6.25 shows the worst case results with a ripple of 0.1%1946.5. Summary100 102 104 106Frequency (Hz)10-3410-3210-3010-28False EDM signal (ecm)Z0 = 1 kZ0 = 1 MZ0 = 1 GFigure 6.25: False EDM signal resulting from an AC ripple in the high voltage supply for asingle cell vs. frequency of the ripple. Each line is calculated assuming a different inline resistorvalue. The supply ripple is assumed to be 0.1% of the supply voltage, and the difference inripple between polarities is assumed to be 50% of this value. The peak corresponds to thenominal neutron precession frequency ∼30 Hz.and differential ripple of 50% for different values of Z0. If the inline impedance is 1 mΩ or lower,the shift at 60 Hz could reach 1.1× 10−29 ecm.In a double cell configuration, this systematic should be suppressed further, since the fre-quency shift depends on |BAC|2 and it will therefore be the same in the top and bottom cellsfor a given polarity. This suppression would hold up to the level of asymmetry between thetwo circuits. If one cell has a 10% larger impedance, it would see a frequency shift of 82% theother cell. It is reasonable to expect that the cells will be similar to this level, and so the shiftwould be suppressed to the low 10−30 ecm level for any reasonable choice of supply, making itnegligible for a double cell experiment.6.5 SummaryThis chapter considered a variety of topics related to the application of HV in the TUCANEDM experiment. The conclusions are summarized here for convenience.The first two Sections 6.1 and 6.2 were focused on evaluating the geometry of the centralregion against HV breakdown conditions using FEM simulations. The maximum field in thecurrent design of the central region was found to be 43 kV/cm on the corona ring. Based on acomparison to literature, it appears that such a field may be acceptable. However, this valueis not so low that a definitive conclusion can be drawn. These studies therefore lead to therecommendation that experimental HV studies need to be performed to validate the design of1956.5. SummarySystematic Section Single Cell σ Double Cell σ(10−28 ecm) (10−28 ecm)Net Motion, rotational 6.3 1.0 1.0Net Motion, linear 6.3 0.1 0.1Leakage Currents 6.4 3.4 2.1AC E-Fields 6.4 < 0.1 < 0.1Total 3.5 2.3Table 6.2: Summary of systematic uncertainties calculated in Chapter 6. All values were derivedwith some conservative assumptions on what signal might be present, which are described indetail in the indicated sections.the central region.The rest of this chapter looked at systematics related to the application of the electricfield. The different systematic uncertainties that have been calculated are summarized in Table6.2. The total systematic uncertainty is expected to be 3.5 × 10−28 ecm for a single cell and2.3× 10−28 ecm for a double cell. While not large compared to the main systematics that werediscussed in Chapter 4, these are not insignificant contributions and are included in the totalsystematic budget of Table 4.2.196Chapter 7Magnetic Considerations in theCentral Region DesignThis chapter deals with the central region design from the perspective of material magnetismand related effects. While Chapter 4 discussed numerous effects related to the ~B0 field, thosecalculations assumed that the measurement occurs in free space. This is not true when thecentral region is in place. One violation of this assumption, the possibility of leakage currents,was described in the previous chapter. In this chapter, additional effects related to materialproperties will be considered. Section 7.1 will review the different material properties that arerelevant to this discussion. Sections 7.2 and 7.3 will focus on how the central region may distortthe ~B0 and ~B1 fields. Section 7.4 will discuss issues related to ferromagnetic contaminationof the various components of the central region. Finally, Section 7.5 will consider the possibleimpact of thermally-induced magnetic fields.7.1 Materials and Magnetic FieldsAll materials exhibit a response to a magnetic field. While there are some isotopes, suchas helium-4, which, in their ground state, have paired nucleons and electrons and thus zeromagnetic moment, even these systems have non-cancelling responses to magnetic fields [288].Therefore, the presence of any material in an applied magnetic field can be expected to havesome effect.Most materials are classified as diamagnetic or paramagnetic. In diamagnetic materials anapplied magnetic field induces currents in bound [289] or free [290] electrons. These currentsgenerate fields that oppose the applied field. In paramagnetic materials unpaired electronsinteract with the applied magnetic field. The difference in energy levels of different quantum-mechanical states results in a net polarization of the electron spins [289, 291–293]. This leadsto a net field in the same direction as the applied field.The magnetic response of diamagnetic and paramagnetic materials can be described by alinear magnetic susceptibility χm such that the field inside the material is given by~B = µ0(1 + χm) ~H, (7.1)where ~H is the applied magnetic field strength, which is given by ~H =~Bµ0in the absence of thematerial. Diamagnetic materials have χm < 0, such that the induced field inside the materialopposes the applied field, while paramagnetic materials have χm > 0, such that the inducedfield amplifies the applied field. Magnetic susceptibility values for several materials of interestare given in Table 7.1.Diamagnetism and paramagnetism are generally weak effects which cannot be observedoutside of a lab, and so such materials are commonly referred to as non-magnetic. Furthermore,diamagnetism and paramagnetism is only observed in response to an applied magnetic field— when the field is removed, the induced currents of a diamagnetic material disappear, and1977.1. Materials and Magnetic FieldsMaterial χm ×10−5 Ref.Aluminium alloy 6061 (Al6061) 1.939 [294]Ti Grade 3 19.68 [294]Ti Grade 5 18.33 [294]Copper (OFHC) -0.98 [294]Rexolite -0.9173 [295],(a)Polyether ether ketone (PEEK) -0.9332 [295]Araldite Resin -1.0 (b)Macor -1.117 [294]Phosphor Bronze C51000 -0.586 [296]Table 7.1: Magnetic susceptibilities of materials of interest for the TUCAN EDM experiment,used in simulations of the magnetic field. Note that values of χm are unitless and given inthe SI convention, which differs from the CGS convention by multiplication by a factor of 4pi.(a) Published data on Rexolite could not be found, so the value is assumed to be the same asreported for polystyrene. (b) Published data on the Araldite resin mixture could not be found,so the value is assumed to be of a similar magnitude to those of other epoxy resins reported inRef. [294].paramagnetic materials return to thermal equilibrium. These properties contrast with those offerromagnetic materials, which can form strong permanent magnetic sources.Ferromagnetic materials, like paramagnetic materials, have unpaired electrons that caninteract with an applied magnetic field. The difference is that in a ferromagnetic material themagnetic fields of neighbouring magnetic moments are strong enough to spontaneously alignwith each other, creating localized regions of net magnetic field. At the same time, it is notenergetically favourable for all moments in a material to align. These competing effects lead tothe formation of domains. The moments within a domain are aligned, but adjacent domains donot necessarily align with each other [297, 298]. The application of a magnetic field causes theboundaries between domains to move, or the moments in a domain to realign [299, 300]. Thesechanges result in a net magnetic field, which tends to be much larger than that produced byparamagnetic effects due to the greater degree of alignment of the magnetic moments.In addition to their response to a magnetic field being strong, ferromagnetic materials canalso exhibit hysteresis. This occurs when, after a magnetic field has been applied and then re-moved, there is insufficient energy for the domains to return completely to an unaligned state.Ferromagnetic hysteresis is caused by a combination of anistropy in the structure of the ma-terial, interactions between the domains, and pinning of domain walls due to imperfections inthe material [301, 302]. Returning the material to a demagnetized state requires the carefulapplication of a field of appropriate size, sufficient to overcome the effects keeping the materialmagnetized without leaving the material in a different magnetized state. Ferromagnetic be-haviour is complex and history-dependent, and there is a substantial body of work attemptingto accurately model ferromagnetic hysteresis [303]. Fortunately, these details are not relevantto the work that will be discussed here. It is only relevant that ferromagnetic materials canbecome permanently magnetized, producing their own magnetic field that is independent froman applied field.Both the weak magnetic disturbances produced by paramagnetic and diamagnetic materialsin response to an applied field (the 1µT B0 field), and the strong magnetic fields produced byferromagnetic materials, are relevant to nEDM measurements. The calculations in Chapter1987.1. Materials and Magnetic FieldsMaterial σE (S/m) Ref.Al6061 2.506× 107 [304]Ti Grade 3 1.85× 106 [305]Ti Grade 5 5.62× 105 [306]Copper (OFHC) 5.917× 107 [307]Table 7.2: Electrical conductivities of materials of interest for the TUCAN EDM experiment,used in simulations of the ~B1 field.4 demonstrated that the uniformity of the magnetic field is the major source of systematicuncertainties in the TUCAN EDM experiment. If the fields produced by materials are largeenough, then they will certainly contribute to these systematic effects.7.1.1 Eddy CurrentsWhen an electrically conductive material is exposed to a time-varying magnetic field, there isanother kind of interaction that takes place. In this case, a circulating electric field is inducedin the material according to Faraday’s law∇× ~E = −∂~B∂t(7.2)The electric field drives currents, known as eddy currents, which produce a magnetic field inopposition to the applied field. The overall field satisfies∇2 ~H = µ0σE(∂ ~M∂t+∂ ~H∂t), (7.3)where σE is the electrical conductivity and ~M is the magnetization, given by ~M = χm ~H for amaterial with linear susceptibility. The strength of the eddy currents depends on how good ofa conductor the material is, and Table 7.2 contains values for several materials of interest.Eddy currents are relevant in an nEDM measurement when applying the ~B1 field thatrotates the spins. Any conductive materials present in the central region will respond to thisfield with their own field, distorting the overall ~B1 field seen inside the cells. The impact of thevacuum chamber, which encloses the entire central region, but which is intended to be insidethe B1 coils, is of particular concern.7.1.2 Thermal EffectsThe magnetic effects discussed above all relate to currents internal to the materials, either frombound or free electrons, and how those currents are altered or created by magnetic fields. Anyother processes which create currents inside materials will also create magnetic fields. Thereare two such processes related to thermal behaviour that are of potential interest for EDMexperiments: Johnson-Nyquist noise, and thermoelectric effects. The effect of Johnson-Nyquistnoise was discussed in Section 4.3.13 as being negligible for the target precision of the TUCANEDM experiment. This leaves only thermoelectric effects to discuss here.Thermoelectric effects have not yet been studied in detail in the context of an nEDMexperiment. Section 7.5 will discuss some preliminary observations. What has been empiricallyobserved is that conductive materials generate net magnetic fields (not noise) when heated, and1997.2. Para- and Diamagnetic Distortion of the B0 Fieldthat this appears to correlate with temperature gradients. Similar preliminary observationshave also been reported on by the n2EDM group [308].These observations appear to be consistent with currents caused by the thermoelectric See-beck effect [309–311]. If one part of a conductor is heated, then the electrons in that regionwill have greater thermal energy, and thus greater velocities. This results in an imbalance inthe diffusion of electrons between the hot region and the rest of the material, driving a netcurrent [312]. The electromotive force of the Seebeck effect is given byES = −SE∇T (7.4)where SE is the Seebeck coefficient. The net current leads to a buildup of charge, such that inthe steady state there is a voltage given by∇VS = SE∇T , (7.5)with the charge buildup driving a current that cancels that of the thermal diffusion. TheSeebeck effect is well-known as the mechanism that underlies the measurement of temperatureby a thermocouple. If two different materials with different values of SE are connected together,and the voltage across the connection measured, the resulting measurement will be dependenton the temperature of the joint due to the different potentials proportional to SE [313].There have been a number of published measurements of magnetic fields reportedly createdby thermal gradients [314, 315], as well as attempts to model these observations from the-ory [314, 316]. The proposed explanation is that the thermally driven current and electricallydriven current, although equal in magnitude, repel each other and follow different paths. Thisforms a current loop that generates a magnetic field. The published results have so far appearedconsistent with this explanation, but further study of this effect is needed.The TUCAN EDM experiment does not currently include a plan for thermal control ofthe magnetically-shielded room and central region. Environmental temperature changes couldcause thermal currents in the electrodes and vacuum chamber of the central region, producingadditional fields close to the cells. While such a field would be partially corrected by the co-magnetometer, there could still be some residual effect from gravity or from BE shifts. Forrandom temperature fluctuations these effects would average to zero, but could still increasestatistical uncertainty depending on the magnitude of the fluctuations. More concerning wouldbe a temperature change that correlates with the timing of the experiment, which would create aconsistent field and potentially a systematic shift. Discussion of this possibility will be returnedto in Section 7.5.7.2 Para- and Diamagnetic Distortion of the B0 FieldThis section will consider the effect of paramagnetism and diamagnetism from the materialsused in the central region. These materials will respond to the static ~B0 field and generatetheir own fields, distorting the applied field. The question to be addressed here is how muchthis distortion can be expected to contribute to systematic uncertainties, and which materialsare best for minimizing these effects.It may seem that this is no different from the consideration of field gradients in Chapter4. However, the strategies used for correcting frequency shifts may neglect these effects. Thepolynomial basis used to describe the field is derived under the assumption that the volumeof interest contains no magnetic sources. This assumption does not hold if the materials ofthe central region act as sources themselves. Furthermore, offline mapping without the central2007.2. Para- and Diamagnetic Distortion of the B0 Fieldregion in place could give inaccurate results if the central region modifies the field significantly.For these reasons, the impact of the material response is important for informing the approachto correcting frequency shifts associated with the shape of the field.These effects can be addressed using the same finite element simulation model used inChapter 6. Magnetic susceptibilities for each material used in the central region are set accordingto Table 7.1. The coatings on the electrodes are neglected, under the assumption that theyare too thin to have a large effect on the overall magnetic field. A uniform Bz = 1 µT fieldis applied, which is equivalent to turning on a distant set of magnetic coils that would, in theabsence of any materials, produce a uniform field. OPERA then solves for the actual field ateach point throughout the central region with those assumed linear magnetic susceptibilities.Maps of the resulting magnetic field inside the cells are extracted from these simulations to beused in calculating various systematic contributions.The focus of this section is on those parts of the central region with the greatest size, whichcan be expected to have the largest influence on the field. These are the vacuum chamber andthe electrodes. The baseline design for the central region assumes that these components willbe made of aluminium alloy 6061. Other materials considered for the vacuum chamber aretitanium and non-conductive plastics such as PEEK. Materials considered for the electrodesinclude titanium and copper. The motivation for considering these different materials is relatedto eddy currents, discussed in Section 7.3.There are many other parts of the central region for which material choices are not studiedhere. These include such things as the HV feedthrough, the insulator rings, parts of the cellvalve (bearings, springs, etc.), assorted fasteners and O-rings. The larger parts are included inthese studies, but their materials are set by other considerations (such as the feedthrough beinga commercial product). For smaller parts there are some different options, but because theseparts are small their effects should be small relative to the larger materials. Since including allsmall parts would greatly increase the complexity of the simulations, these components havebeen neglected. Section 7.2.1 includes some further discussion of the paramagnetism of theseparts.Figure 7.1 shows the effect of the central region with both the electrodes and a vacuumchamber constructed from Al6061. The distortion of the Bz component away from a uniform1µT field is small, and does not exceed 30 pT in this region, consistent with the magneticsusceptibilities on the order of 10−5. The Bx and By fields inside the cell are generally onlya few pT. The field shows some asymmetry in the y = 0 plane, which is due to the cell valveand feedthrough. Although these fields are very small compared to the magnitude of ~B0, theystill have the potential to create significant systematic effects due to BE, B2, or gravitationalfrequency shifts. Each of these possible systematics will be analyzed in turn.The false signal associated with the BE shifts are calculated by numerical integration of thefield maps from Opera to evaluate Equation 4.44. The results are given in Table 7.3 for threedifferent vacuum chamber materials: Al6061, Grade 5 titanium, and PEEK. Of the two metalchambers, Al6061 has a significantly lower false EDM signal compared to titanium. This isconsistent with the larger magnetic susceptibility of titanium resulting in a greater distortion ofthe field and thus larger Bx and By field components. Comparing Al6061 to PEEK, the PEEKchamber yields a signal over four times that of the Al6061 chamber. This seems to be contrary tothe magnetic susceptibility of PEEK being smaller in magnitude than that of Al6061. Becausethe geometry of the central region is complicated, it is difficult to determine the exact causeof this greater field. However, examining the field maps suggests that this greater false EDMis due to PEEK being diamagnetic, in combination with the presence of the electrodes, whichare always assumed to be Al6061. Since the internal field generated by paramagnetic and2017.2. Para- and Diamagnetic Distortion of the B0 Field-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4x (m)00.050.10.150.20.25z (m)-30-20-100102030Bz - B0 (pT)(a) xz-plane y = 0-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4x (m)00.050.10.150.20.25z (m)-20-1001020Bx (pT)(b) xz-plane y = 0-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4y (m)00.050.10.150.20.25z (m)-20-1001020By (pT)(c) yz-plane x = 0Figure 7.1: Maps of paramagnetic distortion of an applied Bz = 1 µT field in the top cell witha vacuum chamber made of Al6061, generated using Opera FEM simulations. Maps of Bx andBy give the absolute field value, while the map of Bz gives the deviation of the field from 1µT.2027.2. Para- and Diamagnetic Distortion of the B0 FieldChamber material Magnitude of false EDMfrom BE shift (ecm)Al6061 2.7× 10−29Ti Grade 5 4.7× 10−28PEEK 1.14× 10−28PEEK/Al6061 Mixture 8.3× 10−29PEEK (4× wall thickness) 6.44× 10−29Table 7.3: False EDM signal resulting from BE frequency shifts calculated from simulationsof the distortion of a uniform 1µT field due to magnetic susceptibility of the central regionmaterials. Each result considers a different vacuum chamber material. The first three results,labelled Al6061, Ti Grade 5, and PEEK, have chambers made entirely from these materials(excepting the electrodes, which are always Al6061). The PEEK/Al6061 mixture uses PEEKfor the chamber walls and Al6061 for the top and bottom plates. The final result also usesa PEEK chamber, but with the wall thickness increased by a factor of four, from 6.35 mm to25.4 mm.diamagnetic materials will be in opposite directions, more inhomogeneity of the field appearswhen mixing materials of these different types. With materials of the same type, such as with anAl6061 chamber and electrodes, the internal fields are in the same direction, and the resultingfields seem to have a greater degree of homogeneity. This hypothesis is seemingly supportedby the result obtained for a “mixed” vacuum chamber, in which the walls are made of PEEKand the top and bottom plates of Al6061, for which the false signal is less than that of a purePEEK chamber. In this configuration, the interface between PEEK and Al6061 is moved awayfrom the cells to the perimeter of the chamber, and the resulting field in the cells becomes moreuniform.One important consideration for the PEEK chamber is that the walls may need to be thickerdue to the lower strength and elasticity compared to Al6061 or titanium. With thicker wallsthe diamagnetic response of the PEEK could be stronger. Table 7.3 also shows the result whenincreasing the wall thickness by a factor of four, from 6.35 mm to 25.4 mm. Interestingly, thefalse signal is smaller with thicker chamber walls. The sign of the false signal also reversesfor this case. This again implies that there are competing contributions either from differentmaterials (the electrodes and the chamber) or from different parts of the geometry (such as thechamber walls and top flanges), which can result in a smaller signal depending on the geometry.The important conclusion here is that a significantly thicker chamber wall will not necessarilyresult in a larger signal, and it appears possible to keep the false signal from a PEEK chamberbelow 10−28 ecm.Table 7.4 gives the results of similar calculations of the BE shift with an Al6061 chamber anddifferent electrode materials: Al6061, Ti grade 5, and oxygen-free high-conductivity (OFHC)copper. These results show that there is some effect from different electrode materials, and thatAl6061 is the best electrode material for minimizing the effect of paramagnetic distortion whenusing an Al6061 vacuum chamber.Systematic effects from the B2 shift are calculated by numerical integration of the field mapsfrom Opera to evaluate the frequency shift given by Equation 4.62. TheB2 shifts associated withparamagnetism are found to be negligible. With a titanium vacuum chamber, the frequency shiftfrom B2 would correspond to an EDM of 1.46×10−31 ecm without considering any cancellations.Since B2 shifts only contribute systematic effects if they differ between HV polarities, or between2037.2. Para- and Diamagnetic Distortion of the B0 FieldElectrode material Magnitude of false EDMfrom BE shift (ecm)Al6061 2.7× 10−29Ti Grade 5 1.5× 10−28OFHC Copper 5.1× 10−29Table 7.4: False EDM signal resulting from BE frequency shifts calculated from simulationsof the distortion of a uniform 1µT field due to magnetic susceptibility of the central regionmaterials. Each result considers a different electrode material.B0 orientations in a single cell crossing-point analysis, this effect will be further suppressed.Since this effect depends on the magnitude of a uniform B0 field, neither of these conditionscan be expected to be fulfilled in a significant way. Any part of the field that reverses withHV polarity should be smaller than the 1µT field, and the field magnitude should not differsignificantly between the two orientations. Symmetry of the double cell configuration furthercancels this effect.Systematic effects from gravitational shifts are calculated by numerically integrating thefield maps to obtain average Bz values as measured by UCNs and Hg atoms. These resultsare then compared to find the residual frequency shift that is not compensated by the Hg co-magnetometer. The gravitational shifts also appear to be negligible. For a titanium vacuumchamber, the gravitational frequency shift in the top cell is 22.4 fT, which is substantial. How-ever, gravitational shifts contribute systematics only if they correlate with HV polarity. For thisshift to be significant, there would need to be a large part of the B0 field (on the order of 1%)that correlates with the HV polarity. Considering results reported in previous experiments, it isunlikely that this will be the case. In the most recent PSI measurement, the neutron frequencybefore compensation varied by roughly 0.4 mHz over 500 cycles, corresponding to a 13.7 pTvariation in the field seen by the UCNs [114]. If the B0 field varies by 13.7 pT correlated withpolarity, it would induce a paramagnetic shift of 7.42× 10−31 ecm in a titanium vacuum cham-ber. So it can be concluded that gravitational shifts associated with paramagnetic distortionshould not be significant.This analysis has focused on the distortion of a uniform 1µT field. Higher order ~Πl,mfields will also be distorted by the paramagnetic response of the central region. Since thesefields have different symmetries, they could produce non-cancelling effects in a double cellexperiment. However, the magnitudes of the l > 0 fields are restricted to be at least below1 nT over these volumes, for reasons relating to depolarization described in Section 4.3.5, so thesystematic effect of distorting such fields can be expected to be a factor of 1000 less than theeffect described here, and therefore should be negligible.It is possible to correct these effects if they can be calculated with sufficient confidence.If the distorted field can be mapped and the corresponding frequency shifts calculated, thenthe false EDM could be subtracted from the final experiment results. However, this could bequite challenging to accomplish given the localized nature and magnitude of the fields in Figure7.1. Simulations such as those performed here could allow these effects to be calculated andsubtracted, but that would require the simulations (and the assumptions of the simulations,such as the χm values) to be well-validated with measurements. Lacking a clear strategy, theseeffects are, for now, treated as an uncertainty and included in the systematic uncertainty budgetof Table 4.2.The false EDM from the BE frequency shift connected to the para- and diamagnetism of the2047.2. Para- and Diamagnetic Distortion of the B0 Fieldvacuum chamber seemingly has the potential to be a significant contribution, if not compensatedby field mapping, but this will be greatly mitigated in the double cell configuration. Thiscompensation can be illustrated by considering the field in the top cell at an arbitrary pointB(x, y, z) = B0zˆ + (Bx,p, By,p, Bz,p) , (7.6)where Bi,p are the field contributions from the paramagnetic (or diamagnetic) response of thematerials. There is a corresponding point in the bottom cell, mirrored about the xy-plane,which has magnetic fieldB(x, y,−z) = B0zˆ +(Bx′,p, By′,p, Bz′,p). (7.7)If the entire experiment (both geometry and B0) is flipped such that z → −z, then these pointsbecomeB(x, y, z) = −B0zˆ +(Bx′,p, By′,p,−Bz ′,p), (7.8)andB(x, y,−z) = −B0zˆ + (Bx,p, By,p,−Bz,p) . (7.9)Finally, if B0 is then reversed, the fields becomeB(x, y, z) = B0zˆ +(−Bx′,p,−By ′,p, Bz ′,p) , (7.10)andB(x, y,−z) = B0zˆ + (−Bx,p,−By,p, Bz,p) . (7.11)If the central region is symmetric across the xy-plane, then this final state after making thesetwo transformations is exactly the same as the original state, from which it can be concludedthat Bx,p = −Bx′,p and By,p = −By ′,p. Thus, the x and y components of the magnetic fieldresulting from paramagnetism or diamagnetism are reversed between the top and bottom cellsin a symmetric experiment. This means that the false EDM signals in the two cells will alsodiffer by their sign, and will cancel when data from the top and bottom cells are combined,even if these signals are not corrected by other analysis.The cancellation in the double cell holds up to the symmetry of the central region. Thissymmetry could be broken if, for example, the susceptibility of the central region varied betweenthe top and bottom electrodes. In this case the cancellation would only be partial. If it isassumed that the symmetry is maintained up to the level of 10%, then the signals in Table7.3 would be reduced by a factor of ten. From the perspective of these systematics, a vacuumchamber of Al6061 would be the best choice, with < 10−29 ecm systematic contribution, whiletitanium would be the worst, with a 5×10−29 ecm contribution. There are other considerationsin the choice of the vacuum chamber and electrode materials, which will be explored in the restof this chapter.These calculations are all based on the typical literature values of susceptibilities in Table7.1. Different values in the materials used in constructing the central region would change theresults, and, as noted above, variation in the susceptibilities between parts of the same materialwould be relevant for arguments of cancellation in the double cell. Such deviations from theassumed values could occur due to impurities or differences in microstructure, which could bepresent in the raw material or introduced during machining. This has not been studied, but willbe of future relevance for understanding this systematic effect. Paramagnetism could potentiallybe surveyed using the CsMs, which are capable of measuring fields of the magnitudes shown inFigure 7.1.2057.2. Para- and Diamagnetic Distortion of the B0 FieldBellows stand-inFigure 7.2: Picture of the geometry as implemented in the Opera model, indicating the simpli-fied model of the titanium bellows for the cell valve actuation. Other parts of this actuation,which can be seen in Figure 4.2, and are expected to be made of aluminium, are not included.The colours correspond to different groupings of parts and are not of any importance.7.2.1 Material Choice of Other ComponentsThe preceding analysis looked at the larger components of the central region, the electrodesand vacuum chamber, where there is some freedom to choose the material. The simulationsalso included some other substantial geometry features such as the insulating rings and HVfeedthrough, which have some contribution to these systematic calculations. The materials ofthese components are driven by other considerations, and there does not appear to be a largeparamagnetic contribution from them, so performing studies of possible material choices doesnot have any value at this stage. However, there are some components that were not includedto reduce the complexity of the simulations, but which could potentially contribute significantlyto these systematics.One section of the central region that was not included in the Opera model is the cell valveactuation. The majority of this actuation can be built using low-susceptibility materials suchas Al6061 or C51000 bronze (also known as phosphor bronze), and the parts are relatively smalland distant from the cells. The expected contribution is therefore small and these parts havebeen neglected in this work. However, one part that is potentially of interest is the vacuumfeedthrough, which transmits the linear motion of the actuator into the vacuum volume. Aswas shown in Figure 4.2, the feedthrough is currently assumed to be an edge-welded bellows.The material choices for such a part are limited, and it is likely that the least-magnetic materialthat can be used is titanium. Given that the calculations for the vacuum chamber have shown asignificant systematic effect from using titanium, there is some interest in simulating the effectof this component.The bellows geometry is difficult to simulate, as the bellows segments have thin walls of nothan 0.5 mm, and join with sharp edges. This makes it challenging to produce a good qualitymesh in Opera. To estimate the magnitude of the effect from the bellows, it has been simplifiedto a flanged tube with roughly the same dimensions and mass of material as what would beexpected in the real bellows. This simplified model has flanges of 63.5 mm outer diameter,19.05 mm inner diameter, and 5 mm thickness. The bellows section is 256.7 mm long, with aninner radius of 12.65 mm and thickness of 0.9 mm. The position of this simplified model, relativeto the other components in the Opera simulation, is shown in Figure 7.2.The calculated false EDM solely from the distortion of the field from the bellows is 1.6 ×2067.3. Eddy Currents and the B1 Field10−30 ecm. This is small compared to the effect from the vacuum chamber and electrodes, andwould be further suppressed in a double cell experiment due to symmetry.A second important consideration is the choice of fasteners. The central region will contain asubstantial number of bolts, and while the effect of a single bolt should be small, the cumulativeeffect of many bolts could be large. Steel fasteners cannot be used in the central region, as evenstainless steel, typically marketed as non-magnetic, is too magnetic to be used in the TUCANEDM experiment. Possible choices of fasteners will be discussed further in Section 7.4, withtitanium being one of the options. Here, it appears that titanium’s larger magnetic susceptibilitywould make it a less-than-ideal choice. However, these studies of the larger parts of the centralregion have shown that in a double cell experiment the effect of paramagnetism is significantlysuppressed if the central region is built symmetrically. With sufficient effort towards ensuringthat this symmetry is achieved, it should be possible to neglect considerations of paramagnetismin material choices for these smaller parts.7.3 Eddy Currents and the B1 FieldThis section will examine the impact of eddy currents in the TUCAN EDM experiment. Eddycurrents are relevant only for time-varying magnetic fields, and so are of interest for the ~B1field that rotates the neutron (and Hg) spins at the beginning and end of the precession period.When ~B1 is applied across the central region, conductive materials will generate eddy currentswhich act to counter the applied field. This will reduce the magnitude of the field seen insidethe cells, as well as introduce inhomogeneity that could prevent a complete rotation of the spins.This effect would appear as a reduction in the achievable visibility α of the UCN population,increasing the statistical uncertainty.The impact of eddy currents on the B1 has been studied using Opera simulations. Thesesimulations use the same model of the central region described in Chapter 6, with materialconductivities set according to Table 7.2. To correctly capture the behaviour of conductorsin oscillating fields, the mesh size must be selected based on the skin depth at the relevantfrequency, given by√2/(σEµ0(1 + χm)ωn). For Al6061 in a 30 Hz field, the skin depth is∼1.8 cm. The convention is to use a mesh size less than one third of the skin depth, and thiswas used to set the meshing of the Opera model.Opera simulations have been performed with different materials and different applied fields.The first sets of results, examining only the effect of different materials, uses a uniform oscillatingfield in the x direction applied at 30 Hz (approximately the neutron precession frequency in a1µT field). The second set of results uses fields generated by some possible coil geometries,also at 30 Hz, to study how the choice of coils might affect the conclusions. From the Operasimulations, maps of the ~B1 field are extracted at phases of zero and pi/2 rads, and then usedas input to PENTrack simulations.In the PENTrack simulations, a uniform ~B0 = (0, 0, 1) µT field is applied, and the populationof UCNs is created with spins polarized along this field. Following a 0.1s delay time, a ~B1 pulseof the form~B1(t) = ~B1,FEM(0) cos(ωnt) + ~B1,FEM(pi/2) sin(ωnt) (7.12)is applied for 2 s, followed by a 10 s precession time, followed by a second 2 s ~B1 pulse, allwith spin-tracking active. The initial delay time is necessary to allow the polarizations toalign with the ~B0 field, as they will otherwise align with the combined ~B0 and ~B1 fields. Ashorter precession time was used in these studies as it does not impact the visibility, since nodepolarization is included, and a shorter precession time reduces simulation time.2077.3. Eddy Currents and the B1 Field100 101 102B1 field map step size (mm)-0.3-0.2-0.100.10.2AsymmetryWP -/+WP -/-WP +/-WP +/+Figure 7.3: Calculated asymmetry of simulated Ramsey working points versus step size of themap of the B1 field. Working points are labelled according to the definition ω = ω0±∆ω2 (1± S).A dependence on the step size is visible above 5 mm.The visibility is examined by repeating this simulation for different magnitudes of the B1pulse, simulating the tuning of the spin-flip to find an optimal value. The output of thesimulations is the spin vector of each UCN, which is used to randomly assign a UCN to bedetected in an up or down state, with the probability of being in the up state given byP↑ = 1 + Sz2, (7.13)where Sz is the normalized component of the spin vector. The asymmetry of the simulatedpopulation is calculated in the normal fashion asA =N↑ −N↓N↑ +N↓. (7.14)In these studies, UCNs are simulated with no losses or depolarization, and so the counts do notsatisfy the conditions required to conform to Poisson statistics.11 Noting that A is simply themean of the spin state with each up-state neutron having value +1 and each down-state neutronhaving value −1, the uncertainty can instead be taken to be the standard deviation of the meanof the results. The visibility that can be achieved with a given ~B1 field is the maximum |A|.11The use of Poisson counting statistics requires that the number of events is large while the probability ofa “success” is small. If success is defined as a particular spin state, then these simulations do not satisfy therequirement on probability. If losses were included, then Poisson statistics can apply because the majority ofUCNs would be lost, and the probability of a simulated UCN ending in a particular spin state would becomesmall.2087.3. Eddy Currents and the B1 Field0 5 10 15 20 25 30Applied B1 (nT)-1-0.500.51Mean polarizationAnalyticalAl6061PEEKTi grade 3Ti grade 5PEEK/Al60610 50 100 150 200 250 300Applied B1 (nT), Al6061Figure 7.4: Simulated polarization of the UCN population in the top cell versus applied ~B1magnitude, for vacuum chambers of different materials. Results are calculated from the finalspin vectors of UCN populations of 1000 neutrons. The analytical result uses a uniform field,while the other results use field maps obtained from FEM simulations of the central region witha uniform applied external field.For these simulations to be accurate it is important to use a sufficiently dense field map.Figure 7.3 shows the results of a study of step size of the field map. In this study, PENTracksimulations were performed for each of the four working points of a Ramsey cycle, for field mapsof different step size. The calculated asymmetry of the population at each point can be seen tochange with step size. Below a step size of 5 mm, there are no significant variations visible atthe level of the simulation uncertainty. Based on these conclusions, a step size of 2.5 mm wasused in the main studies.Figure 7.4 shows the results of a study of different vacuum chamber materials: Al6061, Tigrade 3, Ti grade 5, PEEK, and a mixed chamber with a PEEK wall and Al6061 top andbottom flanges. In all of these simulations the electrodes are assumed to be Al6061. This studyshows that the choice of material has no significant impact on the maximum polarization, as inall cases the field magnitude can be tuned to achieve a complete spin flip with uncertainty lessthan 1%. However, different materials do require different applied fields. All cases require afield that is slightly larger than that predicted by an analytical calculation. The greatest fieldis required when using an Al6061, which needs a field more than 10× that of the analyticalcase. This is consistent with Al6061 having the largest conductivity, and thus the greatestsuppression of an applied field.Figure 7.5 shows the results of a study of different electrode materials: Al6061, Ti grade 3,Ti grade 5, and OFHC Cu. In all simulations a non-conducting PEEK vacuum chamber wasused. As with the vacuum chamber, the electrode material does not significantly impact thepolarization that can be achieved. There is a small effect on the required magnitude of theapplied field, but only by a few nT. Given that the electrodes are parallel to the direction ofthe applied ~B1 field, their minimal impact on the field is to be expected.2097.3. Eddy Currents and the B1 Field0 5 10 15 20 25 30Applied B1 (nT)-1-0.8-0.6-0.4-0.200.20.40.60.81Mean polarizationAnalyticalAl6061Ti grade 3Ti grade 5OFHC CuFigure 7.5: Simulated polarization of the UCN population in the top cell versus applied ~B1magnitude, for vacuum chambers of different materials. Results are calculated from the finalspin vectors of UCN populations of 1000 neutrons. The analytical result uses a uniform field,while the other results use field maps obtained from FEM simulations of the central region.In the studies so far, the applied ~B1 field has been assumed to be perfectly uniform. Thefinal study will examine how some possible coil geometries would change these results. The coilgeometries are shown relative to the central region in Figure 7.6. Each geometry consists of apair of coils on opposite sides of the central region. Two of the geometries are circular Helmholtzcoil pairs. Helmholtz pair #1 has a coil radius and separation of 70 cm, while Helmholtz pair#2 has a coil radius and separation of 95 cm. The first dimension was chosen to be as closeas possible to the central region, while the second was chosen to be as close as possible to thewalls of the magnetically-shielded room. The third coil geometry is a saddle coil pair, mounted1.5 cm from the outer surface of the vacuum chamber. The saddle has a height of 30 cm andan arc length of 53.7 cm (subtending an angle of 55◦). These three coils range from the closestand furthest that the B1 coil could be from the central region, with the expectation that themost distant coil pair (Helmholtz #2) would produce the most uniform field, while the closest(saddle) would produce the least uniform field. These simulations do not take into accountany interaction of the coil field with the MSR walls, which might be more significant for theHelmholtz coils.The effect of the different coil geometries has been examined assuming an Al6061 vacuumchamber and electrodes, as well as a non-conducting vacuum chamber and electrodes. Theresults are shown in Figure 7.7. These results show that, as with the material choices, there isvery little impact on the visibility of the simulated UCN population. The maximum polarizationwith Helmholtz #1 is 0.990±0.005, with Helmholtz #2 0.995±0.004, and with the saddle coils0.986± 0.005. The effect of the coils is therefore small. The saddle coil may be worse than theHelmholtz coils, but all of these results are close within the ranges of their uncertainties.Comparing the two plots in 7.7 it can be seen that the suppression of the ~B1 field from eddycurrents in the Al6061 chamber is ∼10 for the two Helmholtz coils, and ∼7 for the saddle coils.2107.3. Eddy Currents and the B1 FieldHelmholtz #1Helmholtz #2SaddleFigure 7.6: Picture of the Opera model with three different possible B1 coil geometries aroundthe central region. Not visible is the second saddle coil on the opposite side of the vacuumchamber.The saddle coils see a slightly lower suppression than the uniform and Helmholtz cases. Thisappears to be because the saddle coils are mounted at the vacuum chamber surface, which resultsin some of the chamber seeing the returning field. The eddy currents from the returning fieldact in the opposite direction from those of the applied ~B1 field, resulting in some cancellationof the suppression.These results suggest that practically any coil can be used, with the impact on the visibilitybeing no more than 2%. This is not to say that the coil geometry has no effect, but that theeffect is so small that it is not a cause for concern. The actual impact of different coils can beseen more clearly by looking at the distribution of final Sz values from PENTrack. Figure 7.8shows histograms of Sz from the simulation with the greatest visibility for each of the three coils.From this plot it becomes clear that there is an effect from the coil geometry. An analyticalfield yields an exact spin flip for all UCNs, while the three coils have wide distributions. Thesaddle coil has a much wider distribution than the two Helmholtz coils, which both have similardistributions. Despite these differing distributions, the majority of UCNs have Sz very close toa central value, and so the average effect is small even for the saddle coil.The choice of the central region materials, and of the coil geometries, seemingly has lessthan a 2% effect on the visibility of the UCN population, which in turn would increase thestatistical uncertainty by 2%. This may be reduced slightly by using coils that generate a moreuniform field, but the possible gain of 1% on the visibility is small. The greater effect is onhow large of a field needs to be applied, or on how large of a current is needed, which relatesto the detailed design of the B1 coil and is outside the scope of this work. However, there isone additional consideration related to the B1 coil and its interaction with the central region.This is the heating of the chamber due to the eddy currents, which will be returned to brieflyin Section 7.5.2117.3. Eddy Currents and the B1 Field0 0.05 0.1 0.15 0.2 0.25 0.3Coil current (amp-turns)-1-0.500.51Mean polarizationUniformHelmholtz #1Helmholtz #2Saddle #10 50 100 150 200 250 300 350 400 450Uniform B1 (nT)(a) Al6061 chamber and electrodes0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05Coil current (amp-turns)-1-0.500.51Mean polarizationUniformHelmholtz #1Helmholtz #2Saddle #10 5 10 15 20 25 30 35Uniform B1 (nT)(b) Non-conducting chamber and electrodesFigure 7.7: Simulated polarization of the UCN population in the top cell versus coil current forthree possible coil configurations, using field maps obtained from FEM simulations. The resultfrom applying a uniform field is also shown, as a comparison of the shape of the curves. Resultsare calculated from the final spin vectors of UCN populations of 1000 neutrons.2127.3. Eddy Currents and the B1 Field-1 -0.98 -0.96 -0.94 -0.92 -0.9Sz100101102103UCN countSaddleAnalyticalHelmholtz #1Helmholtz #2Figure 7.8: Histograms of the final Sz component of the spin vectors of simulated UCNs for~B1 fields using different coils. Data was taken from the simulation of each coil that showed thebest spin flip. Despite the coils having little effect on the overall visibility of the population,these histograms show that there is a significant physical effect from the coil choice.2137.4. Ferromagnetism and Magnetic Dipoles7.4 Ferromagnetism and Magnetic DipolesThe central region should ideally be free of materials that exhibit ferromagnetism. Whileit is possible to select materials that are nominally free of ferromagnetism, the possibilityof ferromagnetic contamination is unavoidable. This can result from traces of ferromagneticelements in bulk materials, or from ferromagnetic particulates that have adhered to, or beenembedded in, a material during manufacturing or handling. The high precision of the nEDMmeasurement requires magnetic properties to be examined at a level that is significantly beyondwhat commercial suppliers consider when manufacturing “non-magnetic” parts.This section examines small ferromagnetic contamination in parts intended or consideredfor use in the central region. First, the expected contributions of magnetic dipoles to sys-tematic uncertainties will be calculated. This will be followed by a description of preliminarymeasurements of ferromagnetic contamination of a variety of sample parts.7.4.1 Systematic Contributions from Magnetic DipolesAn object with magnetization ~M can be described in terms of bound currents. The volumecurrent is ~Jb = ∇× ~M , and the surface current is ~Kb = ~M × nˆ, where nˆ is the surface normal.Outside of this object, the magnetic vector potential can always be described as a multipoleexpansion~A(~r) =(µ04pi)(∑n1|~r|∫V ′( |~r′||~r|)nPn(rˆ · rˆ′) ~Jb(~r′)dV ′+∫S′( |~r′||~r|)nPn(rˆ · rˆ′) ~Kb(~r′)da′),(7.15)where the integrals are over the volume V ′ and surface S′ of the object, ~r′ is the location of theelement dV ′ or da′, and Pn is the Legendre polynomial of the first kind of order n. The positionvectors ~r′ and ~r are taken from the centre of the object, such that |~r| > |~r′|. This result can befound in any electromagnetism textbook [317]. The n = 0 term is identically zero, as it wouldrequire a non-zero divergence of the magnetic field, and the leading order non-zero term is thedipole term.When far from the object, the potential can be well-approximated using only the dipoleterm. This can be easily seen from the above equations, as when |~r| |~r′|, the higher-orderterms are suppressed by factors of( |~r′||~r|)n. Under this condition the magnetic field becomesB(~r) =(µ04pi) 1|~r|3(3(~m · ~ˆr)~ˆr − ~m), (7.16)where ~m is the magnetic dipole moment given by~m =12(∫V ′~r′ × ~Jb(~r′)dV ′ +∫S′~r′ × ~Kb(~r′)da′). (7.17)This is a reasonable approximation in many cases when considering ferromagnetic contam-ination. The scale of the cell is > 100 mm, while the size of a ferromagnetic inclusion in anotherwise non-ferromagnetic part, or a piece of dust that has adhered to some surface, wouldbe < 1 mm, and the higher order corrections would be below 1% of the field. Thus, even apiece of ferromagnetic dust on the inner surface of the cell can be reasonably treated as a mag-netic dipole. This approximation can also hold for low-level ferromagnetism in bulk material,if the parts are small enough and relatively distant from the cell. An example would be the2147.4. Ferromagnetism and Magnetic Dipolesbolts of the vacuum chamber, which are ∼300 mm from the inner surface of the cell. The boltsthemselves may be ∼ 15 mm diameter, and the leading order correction at the inner surfaceof the cell would be ∼ 5%, and even less in the rest of the cell. If the level of the systematicuncertainty introduced by the ferromagnetism of one such bolt is 10−29 ecm, a 5% correctioncan be neglected.Systematic contributions from dipoles have been a significant source of uncertainty in pre-vious experiments. In the previous ILL experiment, dipoles accounted for 6× 10−27 ecm uncer-tainty [113], while the more recent PSI measurement had 4 × 10−28 ecm due to dipoles [114].In the PSI case, these dipoles were noted to have not been present prior to the measurement,suggesting that some magnetization occurred during the course of data-taking.These systematic effects from dipoles are based on the BE shifts of the Hg co-magnetometer,and the calculation of the BE contributions of dipoles has received considerable attention [231,232]. There are additional systematic concerns associated with the inclusion of CsMs, as thepresence of a dipole close to one of the magnetometers will significantly alter the field seen inthat single location. This could greatly skew the extraction of the Gl,m values. The impact onthe CsMs is beyond the scope of this work and will be neglected.The BE shift has an exact form only in specific symmetric cases, but using numericalintegration to evaluate the general form of the false EDM from BE shifts, given by Equation4.44, it is straightforward to calculate the effect of arbitrary dipoles. These shifts are seen todepend on both the location and orientation of the dipoles. Ferromagnetic contamination couldbe present in any location throughout the central region, and given the nature of ferromagnetismthe orientation of these dipoles could be arbitrary. A study of these effects therefore involvesa broad parameter space. To make this problem more approachable, it is useful to first set atarget level for the contribution of an individual dipole and work backwards to determine howlarge of a dipole magnitude is allowed in a given location with arbitrary orientation.To set the systematic target, the expected number of dipoles nd is a useful number toestimate, as the total contribution will be√nddfalse, where dfalse is the contribution of a singledipole. One approach to arrive at nd is to consider the number of components of the centralregion, each of which can be individually surveyed and characterized with a dipole. The totalnumber of individual parts in the central region is ∼650.The count of nd can be reduced by considering only a single cell, which should be sufficient.With two cells, cancellation of ferromagnetic dipoles should be neglected, since the dipoles arehighly localized and cannot be expected to be distributed in a symmetric fashion. However,dipoles that are close to one cell are distant from the other, and so a dipole that contributesa significant systematic effect in one cell will contribute negligibly in the other. This makesit reasonable to neglect the contribution of ferromagnetic dipoles from the top cell valve tothe bottom cell, and vice versa. With this reasoning, ∼200 parts can be ignored in the countfor a single cell, and the contribution of ferromagnetic dipoles to a single cell and double cellexperiment should be largely the same. This would reduce the count of parts to the order of∼450.Finally, it will be seen that components located on the outer edge of the vacuum chamberwill have much smaller contributions because of their greater distance from the cells. So, a lessstringent limit can be set by considering only the components that are relatively close to a cell.This is taken to be parts that are within ∼12 cm of a cell, and includes the internals of theentirety of the electrodes, the cell valve, the Hg valve, and the various fasteners used to assemblethe cells. It does not include fasteners at the vacuum chamber wall, the HV feedthrough, orparts of the cell valve actuator, which includes roughly ∼200 parts (mostly fasteners for thevacuum chamber and connecting flanges). Thus, for a single cell, this leads to nd ≈ 250.2157.4. Ferromagnetism and Magnetic Dipoles0 0.1 0.2 0.3 0.4 0.5 0.6x (m)-0.100.10.20.3z (m)125102050100Allowed Magnitude (nAm2 )Figure 7.9: Map of the calculated magnitude of a magnetic dipole, located at each point inthe map, that will cause a false EDM signal of 1.26× 10−29 ecm. The origin of the coordinatesystem is at the centre of the central region, and the black region is the interior of the topcell, where no dipole should be present. This plot was generated by taking the minimum overdifferent maps for dipoles of different polar angles θ. Visible faint lines of lower magnitude arean artifact of the scan over angles, as a given θ will have locations for which the systematic iszero. This feature can be seen in single-angle maps included in Appendix F.1.This is only a rough order of magnitude estimate, but it should suffice for the purposes ofestimating target levels of magnetism. Even if there were another 50 dipoles, this would onlyincrease the total systematic effect by 10%. It is also likely that many parts will contributeless than the limits that will be set. So, the overall balance with these numbers should make itpossible to reach the following systematic targets.If the target systematic contribution of magnetic dipoles in close proximity to the cell is2×10−28 ecm, then the individual contribution from each of 250 dipoles can be 1.26×10−29 ecm.Figure 7.9 shows the corresponding dipole magnitude that would cause such a systematic. Sincemagnitude of the systematic effect for a dipole in a given location depends on the orientation,this map has been generated by taking the minimum allowed dipole over different angles. Themaps for individual angles can be seen in Appendix F.1.To limit the systematic of a single dipole to 1.26× 10−29 ecm, the magnitude of the dipoleshould be below 1.25 nAm2 on the inner surface of the cell. At the uppermost extent of the 12 cmenvelope, this limit increase to 3.4 nAm2, but this is such a small increase as to be essentiallynegligible for the purposes of setting limits. Directly on the side of the cell, the requirementis 2.2 nAm2, and this increases more rapidly with distance compared to the top and bottom ofthe cell, with the limit being 11 nAm2 at the edge of the 12 cm envelope (37 cm). While thiswould technically allow some parts of the electrodes to have a higher level of contamination,for the sake of practicality the 1.25 nAm2 limit will be imposed for all components in the 12 cm2167.4. Ferromagnetism and Magnetic Dipolesenvelope.A limit for parts outside of the 12 cm envelope, such as at the wall of the vacuum chamber,at r ≈ 0.5 m, is also needed. Ideally these parts would also meet the 1.25 nAm2 level, in whichcase their contribution would be negligible. However, for practical reasons it may be necessaryto allow parts in this region to be more magnetic. Using the same 1.26× 10−29 ecm target, thelimit increases substantially to 47 nAm2 at r ≈ 0.5 m. This would be acceptable for a relativelysmall number of dipoles, but if many components contributed at this level, they would becomea significant part of the systematic budget. If instead the allowed limit for parts at the outeredge of the vacuum chamber is reduced to 23.5 nAm2, then with 200 such dipoles the totalsystematic contribution from dipoles would only increase to 2.2× 10−28 ecm.For convenience, the three levels of allowable magnetic contamination will be labelled andreferred to throughout the next section. In addition, a fourth label is defined for parts intendedto be used in CsM assemblies, for which the acceptable dipole level has not been determined asit is beyond the scope of this dissertation. The four labels are:• Level A: 1.25 nAm2 for components within the ∼12 cm envelope surrounding a cell,• Level B: 23.5 nAm2 for general components at the outer wall of the vacuum chamber,• Level C: 47 nAm2 for special components at the outer wall of the vacuum chamber,• Level M: for components intended for use in assemblies with the CsMs, for which theacceptable dipole has not been calculated. Likely similar to or better than level A.7.4.2 Measurements of Ferromagnetic DipolesMeasurement DescriptionTo characterize the suitability of parts for use in the central region, preliminary measurements offerromagnetic contamination have been performed at the Physikalisch-Technische Bundesanstalt(PTB) in Berlin, Germany. PTB houses the Berlin Magnetically Shielded Room 2 (BMSR-2),one of the best magnetically shielded rooms in the world [318], which contains an array ofsuperconducting quantum interference devices (SQUIDs) for measuring magnetic fields with anoise level of ∼2 pT/√Hz at 0.01 Hz [319]. BMSR-2 and the SQUID array have been used in themeasurement of the Xe-129 EDM [84], as well as to characterize magnetic dipole contaminationin the PSI nEDM experiment [114]. In the long-term, there is a need to develop similar magneticsurveying methods at TRIUMF for the TUCAN EDM experiment, but at the timing of writingthese capabilities were not available, and the PTB facility offered the best option to explore thelevel of ferromagnetism that might be present in the construction of the central region.The 304 SQUID array consists of 19 modules inside a helium dewar. Each module contains16 SQUID sensors located at different heights throughout the module. For these measurements,only the lower-most layer of sensors is of interest, as they have the greatest sensitivity tomagnetic fields from samples at the bottom of the dewar. The layout of the array and of thelowermost layer of sensors in a module is shown in Figure 7.10, along with a photograph of thedewar inside BMSR-2.To measure the magnetic field of a sample part, it was placed in a plastic sample bag andheld as close as possible by hand to the outer surface of the bottom of the dewar. Each samplewas held only by the edges of the bag in order to avoid any contact between the sample andthe hand of the person performing the measurement, which would heat the sample and induceadditional magnetic fields. The sample was then moved from one side of the dewar to the other,2177.4. Ferromagnetism and Magnetic DipolesA BD EH I K L MN O P RS T UCF G50.4 mm⌀ 33.3 mmZ3Z2Z160°X1V1A1Figure 7.10: Pictures showing the measurement setup at PTB. Top: photo of the interiorof BMSR-2, with the helium dewar that contains the SQUID array in the centre. Bottom:schematic of the SQUID array. The left shows the arrangement of the 19 SQUID modulesinside the dewar. The right shows the layout of the lowermost six SQUID sensors in onemodule. The red sensors Z1, Z2, and Z3 are oriented with their axes pointing out of the page,so as to detect fields in the z direction. The green sensors X1, A1, and V1 are oriented withtheir axes pointing towards the centre of the module, so as to detect fields in the xy plane. Thegreen sensors are located with their centres 3.32 mm above the red sensors.2187.4. Ferromagnetism and Magnetic Dipoles10-1 100 101 102 103 104Dipole strength (nAm2)100101102103Maximum field (pT)Level ALevel BLevel C1 cm3 cm6 cm10 cmFigure 7.11: Calculated maximum field strength that would be measured for dipoles of differentstrength, at different distances from the dipole centre. The limits of 1.25 nAm2 at the innersurfaces of the cell and 47 nAm2 at the vacuum chamber radius are given as dashed lines.scanning across the sensor array. The scanning was repeated multiple times, and, for sampleswith significant size, it was also repeated for different orientations so that different parts of thesample would be captured at the same distance from the sensors.Each sample was scanned under two different conditions: magnetized and degaussed. Tomagnetize a sample, it was moved against a wooden plate with a rare earth magnet attachedto the far side from the sample. The estimated field seen by the sample is 30 mT. Since theTUCAN EDM experiment is located in an experimental hall that sees a stray field from theTRIUMF main cyclotron of 400µT, this level of magnetization should be larger than that seenby any components of the experiment during installation and operation. After the sample hadbeen measured in the magnetized state, it was then degaussed. This was done using a VeritySystem SV91M hard disc eraser. To degauss a sample, the sample was held close to the surfaceof the disc eraser and rotated, then slowly lifted out of the disc eraser’s field while continuingto rotate the sample. The magnetized configuration should represent the worst case, while thedegaussed configuration should represent the best case of what could be seen in the TUCANEDM experiment.Full sets of data were recorded for the lowest three layers of sensors (16 sensors per module).Hypothetically, this data could be used to try and map and fit dipoles in each sample. However,this detailed analysis has not yet been developed and will not be included in this work. Instead,a simplified analysis looks only at the maximum field detected from a given sample in any onesensor. Assuming that maximum field to the be maximum field of a dipole, the suitability ofthe different samples can be evaluated. The lowermost Z sensors are nominally located 27.5 mmfrom the bottom of the dewar. To allow for some uncertainty in the positioning of samples,a distance of 30 mm will be assumed to determine the level of magnetic field from a dipole.2197.4. Ferromagnetism and Magnetic DipolesFigure 7.11 shows the field that would be measured at this distance for a dipole of given size.The measured field would be 9.8 pT for level A, 166.5 pT for level B, 333 pT for level C.Two separate batches of samples were measured on different occasions. The first batchconsisted of 20 parts measured in June 2021, and the second consisted of 104 parts measured inSeptember 2023. In the first run, signals could only be distinguished at the 2 pT level, while inthe second run signals of 1 pT were visible. This noise is included as errors on the measurements.In order to interpret the data in terms of dipole magnitudes, an uncertainty on the positionof the sample should be included. While multiple passes over the array were made, and aneffort was made to scan different sides of each sample, it is still likely that the distance betweenthe centre of the magnetic source and the sensors will differ from the nominal distance. Forexample, many samples had thicknesses on the order of 10 mm, so a dipole embedded in thecentre of such a sample would be 5 mm further from the sensors than expected. To account forthis, an uncertainty of 5 mm is applied to the measurement distance. This is equivalent to ablanket 40% uncertainty on all measurements.Uncertainty on the orientation of the measured field was mitigated by using the X1, A1,and V1 sensors. Large signals in these channels indicated that the field was not oriented in thez direction, and the samples were rescanned with different rotations in these cases to maximizethe signals in the Z1, Z2, or Z3 sensors.ResultsResults for all samples are plotted and shown in this section. These results are organized bymaterial categories. Each sample is labeled with a unique description, while further details ofthe nature of each sample are given in Appendix F.2. In cases where a sample is intended tobe used in a specific location, it is additionally labelled with the corresponding level A, B, C,or M, while samples that were tested for general use have no such label. The measured fieldsare given for both the magnetized and degaussed states. In cases when no field was detectedabove the noise of the sensors, the plots show a downward-pointing arrow at the 1 pT level.This arrow is red when no field was observed in the degaussed state, and blue when no fieldwas measured in the magnetized or degaussed state. The three known levels are indicated forreference, as well as the saturation level of 1 nT.Results for aluminium samples are shown in Figure 7.12. Aluminium is seen to be one ofthe better-performing metals. While some samples exceeded level A significantly, those thatdid tended to have some property or history that would account for these observations.The worst of the aluminium samples were all commercial components. One sample wasa plate which had been anodized with a black dye, which could contribute to a greater levelof magnetism than the base material. The subject of pigmentation, which is present in manysamples, will be discussed further at the end of this section. A number of other poor-performingsamples are labeled as “actuator parts”. These were taken from a commercial pneumatic actua-tor which also contained steel and brass components. Although the parts had been disassembledand cleaned, there is the potential that material was transferred from these more-magnetic partsto the aluminium during prior operation of this actuator. An aluminium extrusion profile alsomeasured poorly. The explanation for this sample is less clear. The cut ends of the profilewould seem like an obvious culprit, but the greatest field was measured towards the middle ofthe profile. There is definitely the potential for contamination in the interior of such a part,and while an effort was made to clean all components it may not have been successful in thiscase.The rest of the aluminium samples were either screws or small parts that had been machined2207.4. Ferromagnetism and Magnetic DipolesUnknown alloy screwAl 2024 screw #1Al 2024 screw #2Al 2024 screw #3Al 2024 screw #4Al 2024 screw #5Al 2024 screw #6Al 2024 screw #7Al 2024 screw #8Machined Al 6061 #1 [A]Machined Al 6061 #2 [A]Machined Al 6061 #3 [A]Machined Al 6061 #4 [A]Machined Al 6061 #5 [A]Machined Al 6061 #6 [A]Machined Al 6061 #7 [A]Actuator part #1 [B]Actuator part #2 [B]Actuator part #3 [B]Actuator part #4 [B]Actuator part #5 [B]Al 6XXX Extrusion profile [B/M]Black anodized plate [M]100101102103Measured signal (pT)Level ALevel BLevel CSaturationDegaussedMagnetizedFigure 7.12: Measured magnetic fields of aluminium samples at a nominal distance of 27.5 mm.Downwards-pointing arrows indicate measurements with no observable signal above the sensornoise. The indicated levels are as defined in the text.at TRIUMF, and a few of these were slightly above the 9.8 pT level when magnetized. Thescrews were observed to have lower magnetism than the machined parts. Since the screws werealloy 2024 and the machined parts alloy 6061, this could suggest a difference between the twoalloys. However, no particular effort was given in the machining of these samples to ensure thatthey were not exposed to ferromagnetic materials from tooling or cutting fluid, so the greaterlevel of magnetism could be due to contamination introducing in the machining process.Overall, aluminium seems to be a suitable material to use in the construction of the TUCANEDM experiment. Further testing on different alloys could be of interest, as there is the potentialthat the most commonly used 6061 alloy is less suitable than others. The data also suggeststhat examining the machining process and whether there is any potential for contamination tobe introduced could be of interest.Results for titanium samples are shown in Figure 7.13. Almost all titanium samples mag-netized above level A, although the majority could be degaussed below this level. Almost allsamples remained below level B, even when magnetized, with only a few samples measuringslightly above. Grade 5 titanium, which is alloyed with vanadium and aluminium, appears tohave performed worse than grade 2, which is unalloyed. Titanium parts do seem to be accept-able at a distance from the cells, such as on the outer diameter of the vacuum chamber, but thesuitability near the cells is dependent on whether the magnetized or degaussed states are morelikely to be present in the TUCAN EDM experiment. A major potential usage of titanium atlevel A would have been screws in the cell valve and electrodes, but this may not be advisablegiven these results.Results for brass samples are shown in Figure 7.14. In most cases brass samples saturatedthe lower level of the SQUID array, regardless of whether the part was in the magnetized or2217.4. Ferromagnetism and Magnetic DipolesGrade 2 screw #1Grade 2 screw #2Grade 2 screw #3Grade 2 screw #4Grade 5 screw #1Grade 5 screw #2Grade 5 screw #3Grade 5 screw #4Grade 5 screw #5Unknown grade screw #1 [B]Unknown grade screw #2 [B]Unknown grade screw #3 [B]Unknown grade screw #4 [B]Grade 2 machined #1 [B]Grade 2 machined #2 [B]Grade 2 machined #3 [B]Grade 2 machined #4 [B]Grade 2 machined #5 [B]Grade 2 nut [B]100101102103Measured signal (pT)Level ALevel BLevel CSaturationDegaussedMagnetizedFigure 7.13: Measured magnetic fields of titanium samples at a nominal distance of 27.5 mm.Downwards-pointing arrows indicate measurements with no observable signal above the sensornoise. The indicated levels are as defined in the text.degaussed state. The consistency of these results shows that brass tends to have a high degreeof ferromagnetic contamination and should generally be avoided in nEDM experiments.The most interesting of the brass samples are two labeled “Essex brass clamp sample”.These samples were obtained from Essex X-Ray to be tested as representative samples of partsused in the construction of the non-magnetic HV feedthrough. The use of brass as a baselinematerial in the feedthrough was motivated by Essex having previously used brass, and thequestion of whether this is suitable is therefore of great interest. While the plotted data showssaturation of the sensors, it is only taken from the first layer in the array. Data was also recordedfor other sensors, located at 3–7 cm above the first layer, putting them at distances of ∼6 cmand ∼10 cm from any dipole source. The fields measured for a given dipole strength at thesedistances are also shown in Figure 7.11. The worst of the two samples was sample #1. Whenmagnetized, it showed a field of 900 pT at 10 cm, and when degaussed a field of 120 pT at 6 cm.The magnetized state corresponds to a dipole magnitude of 4.5µAm2. This is nearly 100 timeslevel C, and could therefore be expected to induce a false EDM of at least 1.26 × 10−27 ecm.The degaussed state would correspond to a dipole of 255 nAm2, which would cause a false EDMof 6.5× 10−29 ecm. Although this is larger than desired, a single such dipole contribution couldbe acceptable. The impact of using a brass component in the HV feedthrough is thereforehighly dependent on whether the magnetized or degaussed states are more probable. Sample#2 also performed significantly better than sample #1, and so it is possible that the level ofcontamination seen in sample #2 would not be present in the feedthrough. Further studies ofan as-built feedthrough will be needed to determine the overall suitability for use in 2C200.In the accounting of the systematics budget in Section 4.3.14, the feedthrough is treated asits own item with an independent contribution from all other dipole sources. In OC100, this is2227.4. Ferromagnetism and Magnetic DipolesBrass screw #1360 Brass screw #1360 Brass screw #2360 Brass screw #3360 Brass screw #4Brass screw set #1Brass screw set #2Brass screw set #3Brass screw set #4Brass pipe fitting #1 [B]Brass pipe fitting #2 [B]Brass pipe fitting #3 [B]Brass pipe fitting #4 [B]Essex brass clamp sample #1 [C]Essex brass clamp sample #2 [C]100101102103Measured signal (pT)Level ALevel BLevel CSaturationDegaussedMagnetizedFigure 7.14: Measured magnetic fields of brass samples at a nominal distance of 27.5 mm. Themajority of brass parts are at the saturation level of the SQUIDs.assumed to be at the level of 1×10−28 ecm, which is taken to be slightly above the contribution ofthe degaussed state of sample #1. At the time of writing, an Essex feedthrough using brass partsis expected to be purchased and used in OC100. In 2C200, it is assumed that the contributionof the feedthrough will be reduced to the single dipole limit of 1.26 × 10−29 ecm. This can beachieved in several ways. In the best case, surveys of the complete feedthrough purchased fromEssex would reveal that it performs better than the samples and remains degaussed duringan experiment. If not, the feedthrough may need to be replaced. Essex representatives hadpreviously indicated a willingness to use different materials, and replacing the brass componentswith a different material such as titanium would allow the possible systematic contribution tobe reduced. It is also worth noting that, with a larger dipole, the field becomes more easilymeasurable, and could therefore be captured by the CsMs or offline mapping, in which case thesystematic can be corrected. Although it would be preferable that this is not necessary, suchan analysis can also be studied in the OC100 configuration.Results for samples of copper alloys not including brass are shown in Figure 7.15. Thesesamples include silicon bronze, phosphor bronze, and copper beryllium. These materials areused in parts such as springs and retaining rings. The two bronzes are also available as materialsfor fasteners, although only silicon bronze screws were tested here. The silicon bronze samplesshowed magnetization above the level A even when degaussed, and so do not seem to be suitableclose to the cells. Phosphor bronze parts generally performed well, although in the magnetizedstate they tended to perform slightly above level A. Copper beryllium (CuBe) showed mixedresults. The CuBe retaining rings performed exceptionally well, but CuBe springs were amongthe worst of the copper alloy samples, and magnetized above level C. This difference could bedue to either a poor stock material used in the springs, or due to their manufacturing. These2237.4. Ferromagnetism and Magnetic DipolesSilicon bronze screw #1Silicon bronze screw #2Silicon bronze screw #3Silicon bronze screw #4Phosphor bronze spring #1 [A]Phosphor bronze spring #2 [A]Phosphor bronze spring #3 [A]Phosphor bronze spring #4 [A]Machined unknown bronze sample #1 [A]Machined unknown bronze sample #2 [A]Machined tin bearing bronze sample [A]CuBe spring #1 [A]CuBe spring #2 [A]Phosphor bronze spring #5 [A]Phosphor bronze spring #6 [A]Phosphor bronze spring #7 [A]CuBe retaining ring #1 [A]CuBe retaining ring #2 [A]CuBe retaining ring #3 [A]CuBe retaining ring #4 [A]Machined phosphor bronze sample #1 [B]Machined phosphor bronze sample #2 [B]Machined phosphor bronze sample #3 [B]Machined phosphor bronze sample #4 [B]Actuator CuBe retaining ring [B]100101102103Measured signal (pT)Level ALevel BLevel CSaturationDegaussedMagnetizedFigure 7.15: Measured magnetic fields of copper alloy samples at a nominal distance of 27.5 mm.Downwards-pointing arrows indicate measurements with no observable signal above the sensornoise. The indicated levels are as defined in the text, and parts labelled with those levels shouldfall below them.springs were noted to have ground ends which could very easily have embedded contamination.Copper alloys have a lot of potential for use as parts in the cell valve. Several springs arerequired in the valve, and these could be made from phosphor bronze, for which spring wireis readily available. Retaining rings are also used in the assembly to hold various shafts inplace, and the CuBe retaining rings appear to be suitable for this. Although phosphor bronzescrews were not tested, they are available, and these results indicate that they are potentiallya superior alternative to titanium screws for use in the cell valve. Further study is requiredto determine if they are mechanically suitable and to confirm the magnetic properties of suchscrews.Results for plastic samples are shown in Figure 7.16. Various nylon, acryolonitrile butadienestyrene (ABS), PTFE, and polylactic acid (PLA) samples were tested, and are intended to becomponents used outside the central region, such as in the mounting of the CsMs or in fieldmapping equipment. These parts generally performed well and saw no field in the degaussedstate. One exception was sample #1 of a magnetometer clamshell, which had some visible blackdiscoloration in the plastic. A few 3D-printed nylon samples also saw worse performance thanother nylon parts, with the notable difference being that these 3D-printed samples had a blackpigment, while the other nylon parts appeared to be naturally coloured. Several high-strengthengineering plastic samples were also tested: Ultem, Torlon, and PEEK. These materials weretested as they are of interest as candidate materials to replace the aluminium vacuum chamberwith a non-conductive vacuum chamber, or as an alternative material to Rexolite as an insulator2247.4. Ferromagnetism and Magnetic DipolesNylon screw #1Nylon screw #2Nylon screw #3Nylon screw #4Nylon screw #5PLA 3D-printed sampleNylon bearingPEEK screw #1PEEK screw #2PEEK screw #3PEEK screw #4PEEK screw #5Ultem sample [A]Torlon sample [A]PEEK sample [A]PTFE sample #1 [A]PTFE sample #2 [A]Rexolite PS-coated ring [A]Nylon 3D-printed sample #1 [B]Nylon 3D-printed sample #2 [B]ABS CsM clamshell #1 [M]ABS CsM clamshell #2 [M]ABS CsM clamshell #3 [M]ABS CsM clamshell #4 [M]100101102103Measured signal (pT)Level ALevel BLevel CSaturationDegaussedMagnetizedFigure 7.16: Measured magnetic fields of plastic samples at a nominal distance of 27.5 mm.Downwards-pointing arrows indicate measurements with no observable signal above the sensornoise. The indicated levels are as defined in the text, and parts labelled with those levels shouldfall below them.ring substrate. A raw sample of Ultem performed well, as did a sample of raw PEEK, but theTorlon sample, which contained a black pigment, performed poorly. In contrast to the sampleof raw PEEK, samples of PEEK screws were the worst of the plastic samples. The reason forthis is not clear, although the screws had a visibly darker colour than the raw sample, whichcould indicate a different composition or contamination.The most interesting plastic sample from the perspective of the central region is a samplenon-deuterated-PS-coated Rexolite ring. This was not the full-size ring used in the cell proto-type, but a smaller ring that had been coated in early tests of the coating process. This ringshowed some amount of magnetic contamination, consistent with a point source. This contam-ination magnetized to a fairly high level, but could be degaussed below the 9.8 pT level. Thisresult suggests that some piece of ferromagnetic material could have been present during thecoating of this sample and become embedded in the coating. The inner surface of the insulatoris as close as possible for a dipole to be to the interior of the cell, and so this contamination is aserious concern. While the coating process that was used in the preparation of this sample didinclude cleaning of the inside surface of the coating machine and insulator immediately priorto cleaning, no special effort was given to making sure that the rest of the coating machine wasparticularly clean. Some contamination could have been picked up, for example, in the fillingtube through which the coating solution is filled into the coating machine, or abraded fromsteel fasteners used in the clamping of the ring during assembly. Further development of thecoating process to reduce the potential for such contamination will be important for minimizingdipole-related systematic effects.Results for miscellaneous samples that do not fall into any of the previous categories are2257.4. Ferromagnetism and Magnetic DipolesViton O-ring sample #1 [A]Viton O-ring sample #2 [A]Ceramic-coated Al shaft [A]Ceramic roller bearing [A]Ceramic roller bearing [A]Frelon-coated Al bushing [A]Optical fibre sample #1 [B/M]Optical fibre sample #2 [B/M]Optical fibre sample #3 [B/M]PbSn solder sample #1 [B]PbSn solder sample #2 [B]Essex HV cable sample [C]NiP-coated Al Plate [A]100101102103Measured signal (pT)Level ALevel BLevel CSaturationDegaussedMagnetizedFigure 7.17: Measured magnetic fields from samples not in any of the broader metal or plasticcategories, measured at a nominal distance of 27.5 mm. Downwards-pointing arrows indicatemeasurements with no observable signal above the sensor noise. The indicated levels are asdefined in the text, and parts labelled with those levels should fall below them.shown in Figure 7.17. These include some optical fibre samples which could be used in thereadout of CsMs, or for optical limit switches on the cell valve actuation, and all of whichcould be magnetized. These samples were suitable for level B, but the effect of such magneticcontamination on the CsMs is outside the scope of this work. However, these results suggesta need for some caution in the assembly of the CsMs. It is worth noting that the plasticcasings of these fibre samples were black, and the results on plastics also suggested that blackpigmentation correlated with higher magnetization. Black Viton O-rings were also seen tocontain some magnetic contamination. Many O-rings are necessary in the current design of thecentral region for sealing the cell and the insulating vacuum, and so these observations do raisesome concerns. Only two samples of standard black Viton O-rings were tested here, and furtherwork should be done to compare to other O-ring materials and formulations.Several cell valve parts were included in the miscellaneous category: these include ceramicroller bearings, as well as a ceramic-coated shaft and a Frelon-coated bushing. One of theceramic bearings did magnetize to a higher level than expected, but the other was level A.These two bearings were slightly different models, and so this data could suggest that thebetter-performing model should be preferentially used in the cell valve. The Frelon-coatedbushing also magnetized slightly above level A. Since only single samples of each of these partswere tested, it would be useful to survey more samples in order to check the consistency of theseresults.One interesting sample was a NiP-coated aluminium plate. The prototype cell of Chapter 5used NiP coatings on all aluminium parts that would be seen by UCNs. Here, the NiP-coatedsample was seen to have a large amount of ferromagnetic contamination, with magnetized fields2267.4. Ferromagnetism and Magnetic Dipolesabove the saturation level and degaussed fields of 60 pT. This would suggest that NiP is notsuitable for use in the cell. The intention for the TUCAN EDM experiment is to coat theelectrodes with DLC rather than NiP, and these results would also suggest that any of thevalve parts need a similar coating if they will be seen by UCNs. However, this was only a singlesample, and may not be representative of the coating in general.The final set of samples were again related to the HV feedthrough. This included a sample ofa standard HV cable from Essex X-Ray, as well as samples of PbSn solder that would be used inthe construction of the feedthrough. The solder did not magnetize to an observable level. Thecable sample, which would be outside the vacuum chamber, exceeded level C when magnetized,but degaussed below level B. As with the results for the brass parts, further study of an as-builtfeedthrough will be most valuable in determining the suitability of this component.These sample surveys are extremely useful in understanding the potential systematic contri-butions of ferromagnetism from components of the central region. The majority of samples, evenfor materials such as plastics, could be magnetized to some level. When considering the sensi-tivity of a 10−27 ecm nEDM experiment, there is seemingly no such thing as “non-magnetic”.The 1.25 nAm2 dipole limit that has been suggested here level A parts is very challenging tomeet. However, these measurements have identified materials such as aluminium, some copperalloys, and ceramics that seem to be able to reach this requirement. In order to achieve thissystematic target, there remains a significant amount of work to be done to build upon thethese studies. It will be necessary to determine how well-degaussed a part can remain, andevery part that will be used in the central region’s construction will need to be surveyed. WhilePTB’s facilities were vital for these first studies, this future work will require the developmentof a dedicated surveying setup at TRIUMF.A major outstanding question from these studies is what materials to use for fasteners. Ti-tanium was originally intended, and these results show that if these parts remain in a degaussedstate during the experiment, this could still be acceptable. However, the magnetized state oftitanium screws tended to be significantly above the limit. Aluminium 2024 screws showedgood performance, but whether these are mechanically suitable is not certain. Since the bulkof the central region will be made of aluminium, using aluminium fasteners introduces a riskof galling if threading into tapped holes. One option would be to use aluminium screws withheli-coil inserts made of a material such as phosphor bronze. Phosphor bronze by itself may bea good candidate material for screws, but samples were not included in these measurements.Ceramics are also likely to be suitable, but were similarly not studied. This is an area wheremore work will need to be done to source and test appropriate parts.A final topic of interest is the tendency of black-pigmented materials to show a greater levelof magnetization. This was the case for a single anodized aluminium part, as well as for variousplastics and elastomers. The single anodized part is hardly a significant sample to concludethat there is a problem, but it is interesting to note that one common method of impartingblack pigment to aluminium involves the use of cobalt [320, 321]. Since elemental cobalt is fer-romagnetic, this could be a source of magnetism in black-anodized aluminium. In plastics andrubbers, the most likely black pigment is carbon black. There is no reason to suspect that thiscarbon would significantly increase the ferromagnetism of a sample (although graphite exhibitsferromagnetism [322], carbon black is structurally different). However, chemical analyses haveshown the presence of heavy metal impurities in carbon black, with iron appearing to be par-ticularly abundant [323]. While all of this is speculative, it cannot be denied that these surveysshow a correlation between the presence of black pigmentation and higher levels of magnetism.Since the application of a pigment to any material requires the introduction of some additive,the material composition of which may be unknown, it seems prudent to avoid pigments in2277.4. Ferromagnetism and Magnetic Dipolesgeneral.7.4.3 Fully Reversing DipolesThe preceding discussion and measurements focused on BE shifts from magnetic dipoles. Thefield from a dipole will also create gravitational and B2 frequency shifts. These will causesystematic effects if the dipole orientation correlates with the electric field direction. The B2shifts also create a systematic effect in a single cell crossing point analysis even if not correlatedwith E, but this effect was shown to be small relative to a direct effect in Section 4.3.7, so it willnot be discussed further. The worst case for an E-correlated dipole would be a fully reversingdipole (FRD), which reverses direction whenever the direction of the electric field is reversed.This FRD is a different object than the dipoles considered for BE shifts. In the previousanalysis, a random distribution of many dipoles was assumed to be present in the experiment.These dipoles were static relative to electric field, although they might appear or reorient atsome arbitrary time in the experiment. In contrast, an FRD would need to be present in theexperiment consistently, with a dynamic orientation dependent on the ~E. While static dipolescan be measured prior to the experiment, as was done at PTB, an FRD could only be measuredin the experiment, as it would have to be a feature of the assembled spectrometer as a whole.One possible source of an FRD would be a paramagnetic material interacting with a mag-netic field that correlates with the E field. Such a magnetic field is created by a leakage current.The induced dipole of a sphere of radius a and magnetic susceptibility χm in a magnetic fieldB is~m =4pia333χmµ0(3 + χm)~B. (7.18)Leakage currents are assumed to be less than 1 nA in the TUCAN EDM experiment. Thefield of a line current of 1 nA at a distance of 1 mm is 0.2 pT. A sphere of a = 0.5 mm, even ifχm → ∞, would have a dipole magnitude of only 0.25 fAm2 caused by a leakage current field.This will be shown to be negligibly small. In general, for an a = 0.5 mm object with infinite χmto have an induced dipole at the limit imposed for BE shifts (1.25 nAm2) it would need to seean E-correlated field of at least 1µT. Since this is the magnitude of B0, a paramagnetic FRDof significant magnitude can be ruled out.It appears that an FRD would need to result from ferromagnetism to be strong enough tohave a significant effect, but in such a way that the act of reversing the HV polarity wouldcause the material to re-magnetize in a consistent fashion. The ILL experiment did considerthe possibility of a HV-correlated dipole near the feedthrough through the magnetic shielding,which they suggested would be caused by fluctuations in the B0 coil current supply correlatedwith the HV switching [113]. Since the magnetic shielding is itself highly ferromagnetic, thismakes sense as a possible location for an FRD. The PSI nEDM experiment also estimatedan uncertainty from imperfect compensation of the gravitational shift, but did not report anyspecific studies searching for dipole-like contributions to this uncertainty, and it could also berelated to leakage currents [114].There is a magnetic field that appears inside the cell during ramp-up of the HV, which couldmagnetize an object correlated with the electric field direction. From the Ampere-Maxwellequation, the magnitude of this field at a distance r from the central axis of the cell isBHV =12c2dEdtr (7.19)At a distance of r = 0.25 m, this gives a field of only 1.7 pT when ramping to 200 kV, assumingthat the ramping happens over 1 s. Since the time to ramp the HV up and down would take much2287.4. Ferromagnetism and Magnetic Dipoles0 0.1 0.2 0.3 0.4 0.5 0.6r (m)-0.100.10.20.3z (m)10-3410-3310-3210-3110-3010-29False EDM (ecm/nAm2)Figure 7.18: Calculated magnitude of the false EDM signal produced by B2 frequency shiftsfrom a magnetic dipole of 1 nAm2 that reverses with the electric field at different locationsaround the top cell. Results are calculated with the dipole oriented along the z-axis. The blackarea is the interior of the top cell, with the origin at the centre of the central electrode.longer than this, the actual field would be even smaller. Considering that degaussed samplesat PTB were initially in Earth’s field, at several tens of µT, before being taken into BMSR-2,a 1.7 pT field could only account for magnetization of some very small dipoles, potentially afactor of 106 smaller than those measured at PTB in the degaussed state.Breakdowns causing a rapid discharge could produce much larger instantaneous fields. How-ever, breakdowns should not be a regular occurrence, so while they may cause magnetizationof ferromagnetic samples, and could explain the appearance of dipoles during the course of thePSI nEDM experiment, they should not be a source of FRDs.An FRD at the magnetic shielding could certainly appear in the TUCAN EDM experimentthrough some unforeseen coupling of the HV supply to the MSR. The existing published datadoes not rule out small FRDs inside or near the cells, but it is not obvious what physicalmechanism would cause them. Unlike the ordinary static ferromagnetic dipoles, it seems thatan FRD would have to be an isolated object, such as a rogue piece of highly ferromagneticmaterial that can be magnetized by relatively small fields. With that in mind, in calculatingthe possible contribution from FRDs, it makes more sense to consider a single FRD.Figure 7.18 shows the calculated false EDM from B2 shifts of one FRD of 1 nAm2 in a singlecell. This effect is seen to be negligible relative to the BE shifts of static dipoles. This meansthat if the ferromagnetism in the experiment is limited to the level needed for BE shifts, B2shifts from FRDs will not be able to contribute.Figure 7.19 shows the calculated false EDM from gravitational shifts for an FRD of 1 nAm2.These calculations include the partial cancellation of the field from the Hg co-magnetometer,and are shown for both a single and double cell. The cancellation effect of a double cell is2297.4. Ferromagnetism and Magnetic Dipolesminimal. The BE shifts limited dipoles near to the cell to a magnitude of 1.25 nAm2. Theseresults indicate that an FRD at that level would induce a systematic effect close to 5×10−27 ecmin the worst case. Ruling out the contribution of a single FRD to 2 × 10−28 ecm by limitingferromagnetic dipoles would therefore require a limit of 0.05 nAm2.One conclusion that can be drawn from these calculations is that dipoles measured in pre-vious experiments inside or near the cells do not appear to be FRDs. PSI reported measuringmultiple dipoles with a strength of 100 nAm2 at the conclusion of their measurement [114]. Ifa dipole of such magnitude was an FRD, then it would likely produce a systematic shift ofat least 10−26 ecm. Since the PSI nEDM collaboration estimated their systematic uncertaintyfrom imperfect compensation of gravitational shifts to be only 7.5 × 10−28 ecm, and did notreport a large 10−26 ecm shift, it seems that their measured dipoles were not fully reversing.However, this does not necessarily rule out the existence of smaller fully reversing dipoles thatcould neither be detected by their surveying, nor were large enough at the 10−26 ecm levelto cause a noticeable signal in the experiment. If the imperfect compensation uncertaintywas attributed entirely to the presence of an FRD, the required magnitude would be between2–20 nAm2 depending on the location.The PTB results show that, in contrast to static dipoles, attempting to limit possible con-tributions from FRDs by limiting the ferromagnetism of parts used in the experiment is likelynot possible at the level desired for a 10−27 ecm experiment. Materials are simply not thatnon-magnetic. However, the question remains as to whether FRDs would actually be presentat this level, as there is not a clear mechanism for them to be created. If FRDs do appear,then it should be possible to distinguish them from a true EDM using dedicated magneticmeasurements.It should be relatively easy to correct an FRD towards the outside of the vacuum chamberor the walls of the MSR. From Figure 7.19, a 2×1028 ecm false signal from an FRD at the outeredge of the vacuum chamber would need a magnitude of at least 4 nAm2. At a distance of 3 cm,such a dipole would produce a field close to 30 pT. Because an FRD must appear consistentlythroughout the experiment, it should be possible to observe a source that reverses directionin magnetic surveys following different HV operation, assuming the surveys are of sufficientprecision. It is also possible that an FRD at the walls would be captured and corrected by theCsMs. Because it would be relatively distant from the cells and CsMs, the dipole field might besubsumed into the Gl,m expansion of the field around the central region. Further considerationof these possible correction methods are beyond the scope of this work, but it seems very likelythat an FRD at the walls will be easily identifiable if one does appear.An FRD inside the cells likely would not be visible to the planned external mapping or CsMs,as the magnitude of the field decreases too rapidly with increasing distance. However, dedicatedoffline studies may still allow an FRD to be found. This would require placing magnetometersinside the cell and measuring the change of field with the HV polarity. A 1 nAm2 FRD wouldproduce a 9.8 pT at 3 cm distance. A single CsM currently has a reported 3.9 pT sensitivity over150 s [213], so this should be detectable. However, with a 3.9 pT uncertainty, the uncertaintyon the FRD magnitude would be 0.5 nAm2, which would still leave an uncertainty of 1.5 ×10−27 ecm. In the worst case, reducing the uncertainty to 2×10−28 ecm would require mappingthe source with a precision of ∼370 fT at a distance of 3 cm. This calculation assumes only asingle measurement of the FRD field. With simultaneous measurements at several locations byseveral magnetometers, and with longer or repeated sampling, this precision requirement wouldbe loosened. The required precision of mapping an FRD would also depend strongly on thelocation. Performing such a measurement is a challenging prospect, but could be developedand attempted if a signal appears in the experiment in order to constrain a contribution from2307.4. Ferromagnetism and Magnetic Dipoles0 0.1 0.2 0.3 0.4 0.5 0.6r (m)-0.100.10.20.3z (m)10-3010-2910-2810-2710-2610-25False EDM (ecm/nAm2)(a) Single cell0 0.1 0.2 0.3 0.4 0.5 0.6r (m)-0.100.10.20.3z (m)10-3010-2910-2810-2710-2610-25False EDM (ecm/nAm2)(b) Double cellFigure 7.19: Calculated magnitude of the false EDM signal from uncompensated gravitationalfrequency shifts produced by a magnetic dipole of 1 nAm2 that reverses with electric field, atdifferent locations around the cells. Results are shown with the dipole axis oriented along thez-axis. Black areas represent the interior of the cells, with the origin at the centre of the centralelectrode.2317.5. Thermal Currentsan FRD.It is difficult to estimate a systematic contribution from FRDs prior to building and oper-ating the experiment. The lack of a clear mechanism to create an FRD at a level which wouldcause a significant systematic shift means it is possible that no such entity will appear in theTUCAN EDM experiment. However, in the case that a signal appears above the level of theother systematics, it will be necessary to search for and rule out FRDs. Doing so at a level of2× 10−28 ecm appears to be challenging but not impossible with the magnetometry tools beingdeveloped by the TUCAN collaboration. The systematics budget of Table 4.2 therefore allowsa 2× 10−28 ecm contribution from FRDs.7.5 Thermal CurrentsThe effect of thermally-induced magnetic fields is a large unknown in the TUCAN EDM ex-periment. Such fields were observed during the measurements at PTB. In this section, theseobservations are reported, and the possible impact of thermal fields on the TUCAN EDMexperiment is discussed.7.5.1 Observations at PTBDuring the measurements of ferromagnetic contamination taken at PTB, there were a numberof cases where samples were accidentally heated during measurement. These included partsthat were in contact with a person’s hand while scanning across the sensor array, as well aslarger parts (machined titanium shafts) that were apparently heated by friction while slidingon a wooden surface. In these measurements, higher fields were measured, and those sampleswere remeasured when it was realized that this heating was apparently interfering with themeasurements.Figure 7.20 shows the difference in observed magnetic field for samples when in contact witha heat source and when not in contact with a heat source. It is observed that the heating ofconductive parts can create significant magnetic fields. In these measurements, the effect ofheating easily raised the field magnitude above the 9.8 pT level, with several samples exhibitingthermally-driven fields 2–3 times that level.Interpreting these results requires an understanding of the temperatures involved. No prepa-rations had been made for temperature measurements, so it is unknown exactly what temper-ature change these samples saw. Considering that a human hand is ∼10◦C higher than roomtemperature [324], the temperature gradients produced in these samples are potentially quitelarge. However, the measurements were fast, typically taking less than a minute, the partswere only resting in the measurement-takers hand rather than being tightly held, and the con-tact between hand and sample was not direct, as all samples were kept in plastic sample bagsduring the measurements. So, the overall temperature change of the material is probably notthat large. One of the few studies of thermal magnetic fields reported measuring a gradient-dependent field of 16.5 pT/(K/m) in aluminium, measured at 2.5 cm [315]. This is a similardistance to what was used in the PTB measurements, and so a rough comparison can be made.The measured field for actuator part #4, when held in a person’s hand, was 25 pT. This partwas a small piece of 3 cm along its longest dimension and 4 mm along its shorted dimension. As-suming 16.5 pT/(K/m), this would suggest that the field was induced by a temperature changeof between 6–45 mK.It does not seem practical to suggest that the central region must be made stable at the levelof mK, but it is also not likely to be necessary. The fields induced by temperature changes could2327.5. Thermal CurrentsGrade 2 machined #1 [B]Grade 2 machined #2 [B]Grade 2 machined #3 [B]Grade 2 machined #4 [B]Grade 2 machined #5 [B]Actuator part #4 [B]Actuator part #5 [B]Machined phosphor bronze sample #1 [B]Machined phosphor bronze sample #2 [B]Machined phosphor bronze sample #3 [B]Machined phosphor bronze sample #4 [B]Brass pipe fitting #3 [B]Brass pipe fitting #4 [B]Grade 2 nut [B]100101102103Measured signal (pT)Level ALevel BLevel CSaturationDegaussedHeatedFigure 7.20: Measured magnetic fields of samples that were exposed to some heat source ofunknown temperature. Measurements when samples were degaussed and not heated are shownfor comparison.2337.5. Thermal Currentsinduce small shifts in the Hg frequency on a per-fill basis, but if these shifts are due to randomtemperature fluctuations then there is no reason to believe that they will have a net effect.Rather, these will be a contribution to the statistical fluctuations associated with a changingmagnetic field. However, if there is a temperature change that correlates with the timing of theexperiment, then such a field could appear consistently and contribute a systematic shift. Apossible source of such a temperature change is discussed in the following section.7.5.2 Heating of the Central RegionIn Section 7.3, the effect of eddy currents on the B1 field was described. This effect was thedistortion and suppression of the field, which was concluded to not be significant in determiningthe choice of material of the central region. However, there is a secondary effect of these eddycurrents. Since any conductive material has some resistance, the flow of these currents dissipatessome energy as heat. This heating by eddy currents will therefore cause the temperature ofthe central region to increase. Since the B1 pulses happen as part of the experiment, parts ofthe central region that are heated will see a consistent heating effect, and therefore a consistentthermal magnetic field.A sense of the magnitude of this heating can be obtained using the results of the FEMsimulations of Section 7.3. From the simulation of eddy currents, a current density J is obtained,which can be converted to heating power per unit volume as P/V = J2/σE, where σE is theconductivity of the material. Since the eddy currents will follow the time-dependence of B1,this calculation must also be time-averaged. As in Section 7.3, the time-dependent quantity isextracted from the simulations asJ(t) = JFEM(0) cos(ωnt) + JFEM(pi/2) sin(ωnt). (7.20)The time-average of the current density squared is thenJ2 =J2FEM(0) + J2FEM(pi/2)2. (7.21)Neglecting heat transfer, P/V can be converted to a temperature change over time using thespecific heat capacity cm and density ρ of the materialdTRdt=J2σEρcm. (7.22)For Al6061, cm = 0.896 J/gK and ρ = 2.7 × 106 g/m3 [304]. The ~B1 pulse lasts for 2 s, so thetemperature change is given by 2dTRdt . Since heat transfer by conduction away from the heatedregion will reduce the temperature, neglecting heat transfer gives a maximum temperatureexpected from the eddy current heating.Assuming that the vacuum chamber and electrodes are Al6061, the temperature changeinduced by an optimized ~B1 pulse from a saddle coil is shown in Figure 7.21. This calculationshows that the maximum temperature increase is 0.45 nK. The saddle coil is the worst casefor heating, as the conductors are placed closest to the vacuum chamber. This results in thematerial seeing the higher local fields around the conductors, instead of the lower field in thecentre of the coils. Results for the Helmholtz coils, which are relatively far from the chamber,show even less heating.The results of the previous section suggested that temperature changes on the order of mKcould create significant fields. Given that this simulated result is six orders of magnitude lower,2347.6. Summary0 0.1 0.2 0.3 0.4 0.5 0.6 0.7x (m)00.050.10.150.20.250.3z (m)00.10.20.30.4T (nK)Figure 7.21: Calculated temperature change of the central region induced by an optimized ~B1pulse using a saddle coil mounted on the vacuum chamber at a distance of 1.5 cm. The origin isat the centre of the central region. The only visibly heated part on this scale is the outer wallof the vacuum chamber adjacent to the coil and above the HV feedthrough.it appears that the heating of the vacuum chamber from eddy currents is not likely to create asignificant field. This result supports the use of an aluminium vacuum chamber in the TUCANEDM experiment, and suggests that a non-conducting vacuum chamber may not be strictlynecessary.The eddy current heating is the most obvious and unavoidable temperature change, as itis caused by the experiment itself, but other changes due to environmental temperature vari-ation are still possible. However, as long as the experiment is performed in such a way thatenvironmental temperature changes do not correlate with the sequence of the measurements,there should not any systematic effect due to thermal magnetic fields. The more interestingcontribution of these fields would be to statistical uncertainty from increased field fluctuations.Although these would be partially corrected by the co-magnetometer, there would still be grav-itational shifts that may be uncorrected, and if significant these could increase the statisticaluncertainty of the experiment. This could motivate the use of a non-conducting vacuum cham-ber to minimize such fields. This can best be studied by building an aluminium vacuum chamberand attempting to measure thermal fields using the OC100 experiment. This topic is thereforebeyond the scope of this dissertation and is left for future work.7.6 SummaryThis chapter considered various topics of material magnetism related to the construction ofthe central region of the TUCAN EDM experiment. The conclusions are summarized here forconvenience.Most of the work in this chapter related to calculations of different systematic uncertaintiesassociated with the choice of materials in the central region. The results of those calculationsare summarized in Table 7.5. The total systematic uncertainty from these effects is expectedto be 3.2× 10−28 ecm for a single cell and 3.0× 10−28 ecm for a double cell. These are includedin the total systematic budget of Table 4.2.2357.6. SummarySystematic Section Single Cell σ Double Cell σ(10−28 ecm) (10−28 ecm)Paramagnetism 7.2 0.3 < 0.1Dipoles 7.4 2.2 2.2Feedthrough dipole 7.4 1.0 0.1Fully reversing dipole 7.4 2.0 2.0Thermal magnetic fields 7.5 < 0.1 < 0.1Total 3.2 3.0Table 7.5: Summary of systematic uncertainties calculated in Chapter 7. All values were derivedwith some conservative assumptions on what signal might be present, which are described indetail in the indicated sections.A large part of these systematic uncertainties are from ferromagnetic contamination. Thesurveys carried out at PTB and discussed in Section 7.4 have lead to the identification ofsuitable materials for use in the central region. These are aluminium, phosphor bronze, andceramics. For a few parts, such as the bellows of the cell valve actuation, titanium will likelyneed to be used, but this is acceptable for these parts due to their number and location. Onematerial choice that remains uncertain is for O-rings, which were seen to be more magneticthan desired. This is suspected to be due to black-coloured additives introducing ferromagneticcontamination, and so it may be possible to find a better choice by searching for O-rings ofdifferent formulations (especially “raw” O-rings).Section 7.3 looked at how the B1 field interacts with the central region using a combination ofOpera simulations of the field and PENTrack simulations of the UCN spins. These simulationsshowed small effects of no more than 2% reduction in the visibility of the population due todistortion of the B1 field by eddy currents. The larger effect was the suppression of the field,which requires a larger field to be applied. For an Al6061 vacuum chamber the suppressionby a factor of 7–10 depending on the coil design. While a different vacuum chamber materialcould reduce this, the use of Al6061 is preferred due to its suitability from the perspective offerromagnetic contamination.236Chapter 8ConclusionsThe TUCAN collaboration aims to achieve a measurement of the nEDM with an uncertaintyof 10−27 ecm. This requires the construction of both a high-yield UCN source, to provide thenecessary counting statistics, and a state-of-the-art nEDM spectrometer, to control systematicuncertainties to levels below the statistical reach.The first part of this dissertation presented some of the efforts connected to the constructionof the TUCAN source, which is being commissioned at the time of writing. This source,described in Chapter 2, will use He-II at a temperature of ∼1.1 K to convert cold neutrons,produced by a dedicated spallation target with D2O and LD2 moderators, into UCNs. Prior tothe construction of the TUCAN source, a prototype source was operated at TRIUMF between2017–2019. During the operation of the prototype source, many measurements were performedto better understand the production, storage, and transport of UCNs.A key set of measurements from this period, described in Chapter 3, relates to the storagelifetime of UCNs in He-II. This storage lifetime depends on the temperature T of the He-II, andcan be modelled as τHe−II−1 = BHeT 7, where BHe is a parameter related to the interaction ofUCNs with phonons in He-II. Given the extremely strong T 7 dependence, the value of BHe hasa direct impact on the ability to extract a large quantity of UCNs from a source. In designingthe TUCAN source, a conservative value of BHe = 0.016 s−1K−7 was assumed, based on priormeasurements from other groups [6, 189]. Chapter 3 presented the results of measurementsof the temperature dependence and an analysis of these results using the PENTrack softwarepackage for Monte Carlo UCN simulations. This analysis yielded a 95% confidence interval of(0.010 ≤ BHe ≤ 0.014) s−1K−7, with a best fit value of 0.011 s−1K−7. These results validatedthe modelling of the expected performance of the TUCAN source [116], and improve uponthe previous literature range of (0.004 ≤ BHe ≤ 0.016) s−1K−7. Additional results in Chapter3 examined the time-dependence of the storage lifetime, and suggested a gradual inflow ofcontamination into the source during the 2018 experimental campaign, which caused the storagelifetime to decrease over time. This will be mitigated in the TUCAN source through the use ofa vacuum separation foil.The second part of this dissertation described work towards the development of the TUCANEDM spectrometer. The spectrometer implements Ramsey’s method of separated oscillatingmagnetic fields to measure the precession frequency of neutrons in the presence of electric fieldsof different orientations, so as to measure a shift in the frequency associated with a non-zeronEDM. Chapter 4 gave an overview of the basic measurement theory and the componentsrequired to implement the measurement.A major part of the work presented in this dissertation has revolved around the calcula-tion of possible systematic uncertainties in order to inform the design of the TUCAN EDMspectrometer. Section 4.3 gave a detailed overview of the effects contributing to the achievablestatistical and systematic uncertainty of the spectrometer, which forms a basis for the workdescribed in subsequent chapters. The calculations of systematics throughout this dissertationconclude a predicted systematic uncertainty of 0.78 × 10−27 ecm at the 1σ level, leading to a1.27×10−27 ecm total uncertainty for the TUCAN EDM experiment assuming 1.00×10−27 ecm237Chapter 8. Conclusionsstatistical uncertainty. Table 4.2 at the end of Section 4.3 describes the 21 systematic contri-butions that have been considered, and the requirements necessary to constrain the estimateduncertainties. The greatest of these two effects are due to magnetic field gradients, contributing0.5×10−27 and 0.3×10−27 ecm after correction. The aim of the calculations of the other effectshas been to demonstrate how they can be constrained at or below the level of 0.2× 10−27 ecm,and this has generally been achieved.The work on the TUCAN EDM spectrometer described in this dissertation was largelyfocused on the design of the central region. This is the volume in which UCNs are storedduring the Ramsey measurement. Chapters 5, 6, and 7 examined the design of the central regionfrom the perspective of UCN storage, electric field generation, and magnetic field contributionsrespectively.The construction and characterization of a prototype storage cell, similar to those that willbe used in the TUCAN EDM spectrometer to contain the UCNs, was described in Chapter 5.This cell is a cylindrical volume formed by two flat NiP-coated top and bottom plates, betweenwhich a dPS-coated electrically insulating ring is located. A custom-built gate valve, integratedinto one of the plates, serves to seal the UCNs inside a cell during the nEDM measurement.The development of the dPS coating of the insulating ring, described in Section 5.1.2, wasa major outcome of this work. The dPS coating is required in order for the cell to have ahigh UCN storage lifetime, which directly impacts the achievable statistical uncertainty of thenEDM measurement. Electrically insulating, UCN-compatible coatings are not readily availablecommercially, and so the coating process for dPS needed to be developed in-house at TRIUMF.To characterize the prototype cell for UCN compatibility, two beamtimes were carried outat J-PARC in 2022 and 2023. During these beamtimes, the storage lifetime of UCNs in thecell was measured. The first measurements resulted in a storage lifetime of (83.9± 1.0) s, aftercorrecting for pressure, which was deemed to be lower than expected and desired. This lowlifetime was suspected to be from contamination of the cell, and between 2022 and 2023 a newcleaning procedure for UCN-handling components, described in Appendix D, was developedand applied. This lead to a significant increase in the storage lifetime, measured in 2023 to be(132.5± 2.8) s. The performance of the cell valve was also validated during these experiments,and seen to not significantly impact the storage lifetime.The rest of Chapter 5 attempts to interpret the storage lifetime results of the cell in termsof the imaginary Fermi potential WdPS of the coating. This was attempted due to possibledifferences in the energy spectrum of the J-PARC source and the future TUCAN source, whichwould lead to the cell having a different storage lifetime at TRIUMF. There was considerableuncertainty on the energy spectrum at J-PARC, resulting in considerable uncertainty in theextraction of WdPS. Two analyses were performed. One analysis, based on kinetic theory, gavea range of (0.043 ≤WdPS ≤ 0.087) neV, while the other, based on a full particle-tracking MonteCarlo simulation of the experiment using PENTrack gave a range of (0.01 ≤WdPS ≤ 0.12) neV,both at the 1σ uncertainty level. These ranges include the value of 0.05 neV that was assumed inthe statistical optimization of the TUCAN EDM experiment, but leave considerable uncertaintyas to whether the coating performs as expected. This will be addressed in future work, firstwith an improved determination of the J-PARC source energy spectrum, which was carried outwhile this dissertation was being written, and then with planned measurements of the cell atTRIUMF with the TUCAN source in 2025.Chapter 6 looked at the design of the central region as it related to HV breakdown andsystematic effects related to the electric field and HV system. This was done using FEMsimulations of the electric field in Opera and FEMM, and PENTrack simulations of UCN spinsin an inhomogeneous electric field. Comparison of literature on vacuum breakdown to the238Chapter 8. Conclusionssimulated fields suggests that the design of the central region can support the 200-kV voltagedesired for the TUCAN EDM experiment, which allows an electric field of 12.5 kV/cm in thecells. However, there were noted to be some areas with larger fields than desired in the insulatingvacuum chamber. This can likely only be addressed with HV testing. With a planned stagedapproach of the build of the TUCAN EDM experiment, starting at a lower 100 kV, there is thepotential to carry out such tests before moving to the 200 kV build.Two significant systematic contributions were studied in Chapter 6. The first is due tothe relativistic magnetic field ~v × ~E/c2 arising from ensemble motions of the UCNs during themeasurement. These studies showed that the low level of electric field inhomogeneity in thecells allows these contributions to be constrained to the level of 0.1 × 10−27 ecm with veryconservative assumptions on the presence of net motion. The second significant effect is dueto leakage currents between the plates of the cells. These were found to be constrainable to0.21× 10−27 ecm, with the requirement of an online measurement of the leakage current to thelevel of 1 nA.Interactions of the central region with the applied magnetic fields, in the form of para-/diamagnetism and eddy currents, was examined in Chapter 7. This was done using the resultsof FEM simulations of the magnetic fields and PENTrack simulations of UCN spins. Theseresults lead to the conclusion that the vacuum chamber of the central region can be safely madeout of aluminium, although with the caveat that there may be some effects due to temperaturechanges in the conductive material driving net currents. These thermal effects are predicted tonot be significant as a systematic uncertainty, but may introduce additional statistical uncer-tainty that has not yet been studied.Systematic contributions from small ferromagnetic contamination, modelled as magneticdipoles, was also studied in Chapter 7. The core of these studies was a set of measurementscarried out at PTB, in which sample parts and materials were measured with a high-precisionarray of SQUIDs with noise at the 1-pT level, to examine what level of magnetic contaminationwas present. A comparison of these measured fields to calculations of the systematic contribu-tions of dipoles lead to recommendations on which materials would be suitable for which partsof the central region. In the critical region close to the cells, which includes the cell valves, thebest materials were found to be aluminium, phosphor bronze, and ceramics. Some materialsthat were thought to be sufficiently non-magnetic, such as titanium, are now recommendedagainst, although still acceptable at greater distances from the cells. Further work needs tobe done on identifying suppliers of components made of the appropriate materials needed tobuild the central region. assuming this can be done to the level described in this work, thesystematic contribution of magnetic contamination should be limited to 0.22× 10−27 ecm. Themeasurements at PTB have also revealed that, at the level of a 10−27 ecm nEDM measurement,practically all materials have some notable level of ferromagnetic contamination. This moti-vates the need to develop magnetic surveying at TRIUMF with similar precision, and planningof how to accomplish this is underway.Taken together, Chapters 5, 6, and 7 have covered the key physics considerations in thecentral region related to achieving a measurement with low 10−27 ecm uncertainty. The dataand conclusions contained within these chapters will inform the detailed design and build of thecentral region for the TUCAN EDM spectrometer. Although there remains some uncertaintyin the outcome of this work, it appears that it is possible to build the central region to achievethe 10−27 ecm goal of the overall experiment.239Chapter 8. Conclusions8.0.1 Final RemarksThe work presented in this dissertation is all towards achieving the TUCAN EDM experiment’sgoal of a 10−27 ecm uncertainty on measuring the nEDM, and none of the results here suggestthat this cannot be achieved. Of course, there are many parts of this effort that are not partof this dissertation. Key among those is the completion of the TUCAN source, which needsto provide a sufficiently large number of UCNs to achieve the statistical reach. The historicaloverview of nEDM measurements in Chapter 1 showed that statistics have always been thelimiting factor for UCN measurements, and whether the TUCAN collaboration’s goal can bereached truly hinges on the performance of the source.The goal of measuring the nEDM with greater precision is to improve the understanding ofCP violation in the underlying laws of physics. A number of connections between this type ofsymmetry violation and open questions were highlighted in Chapter 1: baryogenesis, extensionsto the Standard Model, and the strong CP problem. Measuring a non-zero nEDM would pointto the discovery of new CP-violating physics that could shed light on these questions. On theother hand, a zero measurement would improve bounds on possible CP-violating contributionsto the nEDM.A non-zero measurement of the nEDM at the 10−27 ecm level would be a result of greatsignificance. However, it would not necessarily lead to a clear picture of CP-violating physicsbeyond the Standard Model. The effective field theory description in Chapter 1 showed thatthere are a number of different possible CP violating contributions to the EDM to differentsystems. This motivates attempts to measure the nEDM, as it is a somewhat unique system.However, it also means that a single measurement is not sufficient to constrain the variousdifferent contributions. Many measurements of different EDMs will be necessary to move theunderstanding of CP violating physics forwards.At the same time, a single non-zero measurement of the nEDM is likely not sufficient to es-tablish the discovery of new CP-violating physics. The calculations of different systematics thatwere done for this dissertation show that this measurement is extremely sensitive to many pos-sible effects. There is no guarantee that what has been considered here accounts for everythingthat could appear in the final measurement. A non-zero measurement would demand valida-tion. Fortunately, there is a sizable nEDM community, with multiple collaborations workingon independent measurements. While these different collaborations can be seen as competing,they should also be seen as parts of a larger collective effort.It is equally likely that the outcome of the TUCAN EDM experiment will be anothermeasurement of zero, with a tighter bound on the nEDM, rather than a non-zero measurement.To some extent, this would be business-as-usual for the nEDM community, which has beenmeasuring zero for more than seven decades. However, considering the great effort that hasbeen invested towards reaching 10−27 ecm, this likelihood raises the question of how much moreprecise these measurements can be made.In the event of a zero or non-zero measurement, improving precision beyond the 10−27 ecmlevel will become the next goal. This would be to either to search for an nEDM at a lower level, orto improve the precision on the 10−27 ecm nEDM. The goal of the TUCAN EDM experiment isto collect data over a 2–3 year period to achieve the 10−27 ecm statistical sensitivity. Continuingto run the experiment after this point would allow the precision to be improved, and after 10total years a ∼6× 10−28 ecm statistical uncertainty could be achieved (this precision could alsobe reached in a shorter time with improvements to the TUCAN source). However, if the tallyof systematic uncertainties in this dissertation is correct, then systematic contributions wouldbecome dominant. This would require the development of new techniques to reduce systematic240Chapter 8. Conclusionsuncertainty further. Fortunately, there are some efforts underway to develop new techniques,which were mentioned in Chapter 4: the magic field [233] and the dual co-magnetometer [234].If the technical challenges of these techniques could be overcome, then a future version of theTUCAN EDM experiment (or one of the other existing nEDM experiments) could potentiallyachieve a measurement at ∼7× 10−28 ecm.Pushing towards the low 10−28 ecm level will be an even greater challenge. For room-temperature nEDM experiments such as the TUCAN EDM experiment, this would requiredeveloping new UCN sources, capable of providing a factor of 100 increase in UCN counts.The proposed source in Ref. [144] claims a possible increase in counts by a factor of ∼10over what the TUCAN source expects to achieve. The feasibility of a source beyond this doesnot appear to have been studied, and so the possibility of building such a thing is unclear.Attempts to measure the nEDM at greater precision may therefore need to move away from thecurrent focus on room temperature UCN experiments. The various efforts to develop cryogenicUCN measurements [120], to revive cold neutron beam measurements [121], and to pursue newtechniques with crystal scattering [123] could all potentially lead to a new generation of nEDMmeasurements at greater precision.The measurement of the nEDM, and of EDMs in general, is an area with great potentialto improve the understanding of fundamental physics. The road to achieving this goal is long,and the efforts of the TUCAN collaboration, and by extension this dissertation, are one stepalong this road. There is much more work that remains to be done.241Bibliography[1] S. Ahmed, T. Andalib, M. J. Barnes, C. B. Bidinosti, Y. Bylinsky, J. Chak, M. Das,C. A. Davis, B. Franke, M. T. 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URLhttps://link.aps.org/doi/10.1103/PhysRev.100.1191.274Appendix ADerivations of Ramsey’s MethodThis appendix presents derivations of the key results of Ramsey’s method of separated oscillatingfields. These results can be found in many existing publications. The purpose of this appendix isto present a complete collection of the derivations of the key results for using and understandingRamsey’s method. In all publications of these results that the author has seen, some level ofdetail in the derivations had been neglected, such that a complete understanding of the resultscould not be obtained. In this appendix, these results are derived from first principles with thegoal of rectifying this observed deficiency and aiding future researchers in understanding thecomplete picture.The first result is the spin flip probability, originally derived by Ramsey [124]. The secondresult derives, from the spin flip probability, the approximate equation used for the centralfringe, for which a nearly-complete derivation could also be found in Ref. [325]. The thirdresult is the statistical uncertainty on the precession frequency as can be obtained from Ramsey’smethod. The derivation for the statistical uncertainty is presented in two different ways. Thefirst is by inverting the equation for the central fringe to solve for the precession frequency.The second is by calculating the Hessian matrix for fitting the central fringe equation. Thefirst method has been used often in student theses [6], while the second method, which has astronger formal basis, has not been seen by the author in any publications.A.1 Probability of Spin FlipA.1.1 Rotation Transformation ResultsThese derivations will make use of the transformation into a rotating frame of frequency ω aboutthe z axis. Several important results for the application of this transformation are given here.The derivations of these transformation results can be found in any graduate-level quantummechanics text [66].For a Hamiltonian Hˆ and time-dependent wavefunction Ψ, the transformation isHˆ′ =ei(ωt+φ)σz/2Hˆe−i(ωt+φ)σz/2,ψ′(t) =ei(ωt+φ)σz/2ψ(t),(A.1)where σz is the Pauli matrix and φ is an arbitrary phase. The transformation applied to thespin operators isei(ωt+φ)σz/2Sˆxe−i(ωt+φ)σz/2 =Sˆx cos(ωt+ φ)− Sˆy sin(ωt+ φ),ei(ωt+φ)σz/2Sˆye−i(ωt+φ)σz/2 =Sˆy cos(ωt+ φ) + Sˆx sin(ωt+ φ),ei(ωt+φ)σz/2Sˆze−i(ωt+φ)σz/2 =Sˆz.(A.2)If Ψ is written in terms of the eigenfunctions of a spin 1/2 system ψ↑ and ψ↓, such thatΨ(t) = C↑(t)ψ↑ + C↓(t)ψ↓, (A.3)275A.1. Probability of Spin Flipthen the transformation of the state can be expanded asψ′ =ei(ωt+φ)σz/2(C↑(t)ψ↑ + C↓(t)ψ↓)=(C↑(t)ei(ωt+φ)/2ψ↑ + C↓(t)e−i(ωt+φ)/2ψ↓).(A.4)A.1.2 Hamiltonian in Rotating FrameThe system of interest is that of a spin 1/2 system (in this case a neutron) in the presenceof a uniform magnetic field ~B0, oriented arbitrarily along the z-direction, and a perturbingoscillatory field ~B1, here circularly-polarized in the xy plane. The Hamiltonian for ~B0 is simplyHˆ0 = −γnSˆ · ~B0 = −γnSˆzB0 = ωnSˆz, (A.5)where Sˆ is the vector of spin operators, γn the gyromagnetic ratio of the neutron, and ωn =−γnB0. This system will have eigenstates ψ↑ and ψ↓ corresponding to alignment or anti-alignment of the spin along B0. In general, the system will be in a stateΨ(t) = C↑(t)ψ↑ + C↓(t)ψ↓. (A.6)The Hamiltonian of the perturbing ~B1 field will have a Hamiltonian given byHˆ1 =− γnSˆ · ~B1 = −γn(Sˆx Sˆy Sˆz)B1 cos(ω1t+ φ)B1 sin(ω1t+ φ)0=− ~γnB12(0 ei(ω1t+φ)e−i(ω1t+φ) 0)=~Ω2(0 ei(ω1t+φ)e−i(ω1t+φ) 0),(A.7)where Ω = −γnB1.The time-dependent Schrodinger equation for this system is thusi~∂∂tΨ(t) =(Hˆ0 + Hˆ1)Ψ(t)=i~(C˙↑(t)C˙↓(t))=~ωn2(C↑(t)−C↓(t))+~Ω2(ei(ω1t+φ)C˙↓(t)e−i(ω1t+φ)C˙↑(t)),(A.8)where the notation ˙ denotes a time derivative.This can be simplified by applying the transformation into the rotating frame. The Schrodingerequation becomesi~∂∂tΨ′ =i~(∂∂t(ei(ω1t+φ)σz/2)Ψ + (ei(ω1t+φ)σz/2)∂∂tΨ)=(−~ω1σz2ei(ω1t+φ)σz/2Ψ + (ei(ω1t+φ)σz/2)(Hˆ0 + Hˆ1)Ψ)=(−~ω12σz + ei(ω1t+φ)σz/2(Hˆ0 + Hˆ1)e−i(ω1t+φ)σz/2)Ψ′=− ~ω12σzΨ′ + ei(ω1t+φ)σz/2(~ωn2σz + Hˆ1)e−i(ω1t+φ)σz/2Ψ′=− ~(ω1 − ωn)2σzΨ′ + ei(ω1t+φ)σz/2Hˆ1e−i(ω1t+φ)σz/2Ψ′.(A.9)276A.1. Probability of Spin FlipTo simplify the form of Hˆ1, is it convenient to return to the original definition, written interms of sin and cos:Hˆ1 = −γn(B1 cos(ω1t+ φ)Sˆx +B1 sin(ω1t+ φ)Sˆy). (A.10)Using this definition and the transformations A.2, the transformation of Hˆ1 becomesei(ω1t+φ)σz/2Hˆ1e−i(ω1t+φ)σz/2 =− γnB1(cos(ω1t+ φ) cos(ω1t+ φ)Sˆx− cos(ω1t+ φ) sin(ω1t+ φ)Sˆy+ sin(ω1t+ φ) cos(ω1t+ φ)Sˆy+ sin(ω1t+ φ) sin(ω1t+ φ)Sˆx)=− γnB1Sˆx=ΩSˆx.(A.11)So, the Schrodinger equation in the rotating frame isi~∂∂tΨ′ = −(ω1 − ωn)SˆzΨ′ + ΩSˆxΨ′ = (−λSˆz + ΩSˆx)Ψ′. (A.12)This can be written out in terms of the individual spin components, giving a system of linearequationsC˙↑′=iλ2C↑′ − iΩ2C↓′C˙↓′=− iλ2C↓′ − iΩ2C↑′,(A.13)where the C ′ are the coefficients in the rotating frame.A.1.3 Solving the EquationThe time-dependence of the perturbing field has been removed in the rotating frame, and itis possible to find an exact solution to this first-order linear system of equations. The generalsolution isC↑′(t) = c1 cos(at) + c2 sin(at)C↓′(t) = c3 cos(at) + c4 sin(at),(A.14)wherea =√λ2 + Ω22. (A.15)The initial conditions require c1 = C↑′(0) = C↑(0)eiφ/2 and c3 = C↓′(0) = C↓(0)e−iφ/2. Ap-plying these conditions to the general solutions and substituting into the original system ofequations then givesc2 =i2a(λC↑(0)eiφ/2 − ΩC↓(0)e−iφ/2)c4 =− i2a(λC↓(0)e−iφ/2 + ΩC↑(0)eiφ/2).(A.16)277A.1. Probability of Spin FlipThis can be transformed back into the lab-frame by applying the rotating frame transformationin reverse. Combining everything, the solutions areC↑(t1) =e−i(ω1t1+φ)/2(C↑(0)eiφ/2 cos(at1)+i2a(λC↑(0)eiφ/2 − ΩC↓(0)e−iφ/2) sin(at1)),C↓(t1) =ei(ω1t1+φ)/2(C↓(0)e−iφ/2 cos(at1)− i2a(λC↓(0)e−iφ/2 + ΩC↑(0)eiφ/2) sin(at1)).(A.17)This is the result for a single pulse of the B1 field, with the duration of the pulse labelled as t1.In Ramsey’s method, this is followed by a period of free precession, with a duration TR. Thesolution for this period of free precession is straightforward, as it is simply the solution withthe Hamiltonian Hˆ0 alone, which isC↑(t1 + TR) =C↑(t1)e−iωnTR/2,C↓(t1 + TR) =C↓(t1)eiωnTR/2.(A.18)This is followed by a second B1 pulse of duration t2. A necessity of Ramsey’s method is thatthe two pulses are in phase, such that the second pulse has a phase φ = ω1(t1 + TR). Includingthis phase givesC↑(t1 + TR + t2) = e−i(ω1t2+ω1(t1+TR))/2(C↑(t1 + TR)eiω1(t1+TR)/2 cos(at2)+i2a(λC↑(t1 + TR)eiω1(t1+TR)/2 − ΩC↓(t1 + TR)e−iω1(t1+TR)/2)sin(at2)),(A.19)andC↓(t1 + TR + t2) = ei(ω1t2+ω1(t1+TR))/2(C↓(t1 + TR)e−iω1(t1+TR)/2 cos(at2)− i2a(λC↓(t1 + TR)e−iω1(t1+TR)/2 + ΩC↑(t1 + TR)eiω1(t1+TR)/2)sin(at2)).(A.20)To arrive at a final, simplified solution for the Ramsey method, each term in the parenthesesmust be expanded. The initial conditions C↑(0) = 1 and C↓(0) = 0, corresponding to an initiallypolarized spin up state, and φ = 0, to set the initial phase of the perturbing field, must alsobe applied. Additionally, the condition that t1 = t2 = τ must be imposed. Given that this isa two state system, and the desired quantity is the probability of spin-flip, it is only necessaryto fully calculate one of the coefficients, and apply the condition C↑C↑∗ = 1−C↓C↓∗ to obtainthe other probability. It is somewhat easier to calculate C↓.278A.1. Probability of Spin FlipThe first term isC↓(t1 + TR)e−iω1(t1+TR)/2 cos(at2)=C↓(t1)eiωnTR/2e−iω1(t1+TR)/2 cos(at2)=ei(ω1t1+φ)/2(C↓(0)e−iφ/2 cos(at1)− i2a(λC↓(0)e−iφ/2+ ΩC↑(0)eiφ/2) sin(at1))eiωnTR/2e−iω1(t1+TR)/2 cos(at2)=− eiω1t1/2(i2aΩ sin(at1))eiωnTR/2e−iω1(t1+TR)/2 cos(at2)=− ei(ωn−ω1)TR/2 i2aΩ sin(aτ) cos(aτ).(A.21)The second term is− i2aλC↓(t1 + TR)e−iω1(t1+TR)/2 sin(at2)=− i2aλC↓(t1)eiωnTR/2e−iω1(t1+TR)/2 sin(at2)=− i2aλei(ω1t1+φ)/2(C↓(0)e−iφ/2 cos(at1)− i2a(λC↓(0)e−iφ/2+ ΩC↑(0)eiφ/2) sin(at1))eiωnTR/2e−iω1(t1+TR)/2 sin(at2)=− i2aλei(ω1t1)/2(− i2aΩ sin(at1))eiωnTR/2e−iω1(t1+TR)/2 sin(at2)=ei(ωn−ω1)TR/2(i2a)2λΩ sin2(aτ).(A.22)The third term is− i2a(ΩC↑(t1 + TR)eiω1(t1+TR)/2) sin(at2)=− i2a(ΩC↑(t1)e−iωnTR/2eiω1(t1+TR)/2) sin(at2)=− i2a(Ωe−i(ω1t1+φ)/2(C↑(0)eiφ/2 cos(at1) +i2a(λC↑(0)eiφ/2− ΩC↓(0)e−iφ/2) sin(at1))e−iωnTR/2eiω1(t1+TR)/2)sin(at2)=− i2a(Ωe−i(ω1t1)/2(cos(at1)+i2aλ sin(at1))e−iωnTR/2eiω1(t1+TR)/2) sin(at2)=− e−i(ωn−ω1)TR/2 i2a(Ω(cos(at1) +i2aλ sin(at1))sin(at2)=− e−i(ωn−ω1)TR/2(i2a(Ω cos(aτ) sin(aτ)) +(i2a)2λΩ sin2(aτ)).(A.23)279A.1. Probability of Spin FlipSumming the three terms gives− ei(ωn−ω1)TR/2 i2aΩ sin(aτ) cos(aτ)+ ei(ωn−ω1)TR/2(i2a)2λΩ sin2(aτ)− e−i(ωn−ω1)TR/2(i2a(Ω cos(aτ) sin(aτ)) +(i2a)2λΩ sin2(aτ))=−(ei(ωn−ω1)TR/2 + e−i(ωn−ω1)TR/2) i2aΩ sin(aτ) cos(aτ)+(ei(ωn−ω1)TR/2 − e−i(ωn−ω1)TR/2)( i2a)2λΩ sin2(aτ)=− 2 cos((ωn − ω1)TR/2) i2aΩ sin(aτ) cos(aτ)+ 2i sin((ωn − ω1)TR/2)(i2a)2λΩ sin2(aτ)=iΩa(−12cos(λTR/2) sin(2aτ) +λ2asin(λTR/2) sin2(aτ)).(A.24)Finally, this can be substituted back into the full equation, resulting inC↓(TR + 2τ) =eiω1(2τ+TR)/2iΩa(−12cos(λTR/2) sin(2aτ)+λ2asin(λTR/2) sin2(aτ)).(A.25)This is the original result presented by Ramsey. The spin-flip probability is then|C↓(TR + 2τ)|2 =eiω1(2τ+TR)/2e−iω1(2τ+TR)/2 iΩa−iΩa×(−12cos(λTR/2) sin(2aτ) +λ2asin(λTR/2) sin2(aτ))2=(Ωa)2sin2(aτ)(λ2asin(λTR/2) sin(aτ)− cos(λTR/2) cos(aτ))2.(A.26)The probability of no spin flip is|C↑(TR + 2τ)|2 =1− |C↓(TR + 2τ)|2=1−(Ωa)2sin2(aτ)(λ2asin(λTR/2) sin(aτ)− cos(λTR/2) cos(aτ))2.(A.27)A.1.4 A Linearly Oscillating FieldThe previous derivation assumes that the pulsed field is applied as a circularly rotating field. Inpractice it is often easier to apply a linear oscillating field. The solution for a linearly oscillating280A.1. Probability of Spin Flipfield can be obtained by recognizing that such a field is simply a sum of two circular fieldsB1 cos(ω1t+ φ)00 = 12B1 cos(ω1t+ φ)B1 sin(ω1t+ φ)0+B1 cos(−ω1t− φ)B1 sin(−ω1t− φ)0 . (A.28)The perturbing Hamiltonian becomesHˆ1 =− γnSˆ · ~B1 = −γn2(Sˆx Sˆy Sˆz)B1 cos(ω1t+ φ) +B1 cos(−ω1t− φ)B1 sin(ω1t+ φ) +B1 sin(−ω1t− φ)0=− ~γnB14(0 ei(ω1t+φ) + e−i(ω1t+φ)ei(ω1t+φ) + e−i(ω1t+φ) 0)=~Ωl2(0 ei(ω1t+φ) + e−i(ω1t+φ)ei(ω1t+φ) + e−i(ω1t+φ) 0),(A.29)where Ωl = −γnB12 , to distinguish from Ω used in the circularly rotating case. Note the addi-tional factor of 1/2 compared to the circularly rotating case.The time-dependent Schrodinger equation isi~∂∂tΨ(t) = (Hˆ0 + Hˆ1)Ψ(t) = i~(C˙↑(t)C˙↓(t))=~ωn2(C↑(t)−C↓(t))+~Ω2((ei(ω1t+φ) + e−i(ω1t+φ))C˙↓(t)(ei(ω1t+φ) + e−i(ω1t+φ))C˙↑(t)).(A.30)As before, the solution proceeds by applying the transformation to the rotating frame. Thetransformed Schrodinger equation has the same form as for for the circularly rotating field casei~∂∂tΨ′ = −~(ω1 − ωn)2σzΨ′ + ei(ω1t+φ)σz/2Hˆ1e−i(ω1t+φ)σz/2Ψ′. (A.31)The difference is in the transformation of Hˆ1. Writing Hˆ1 in terms of sin and cos, there is aterm that is the same as the circularly rotating case,(Hˆ1)1= −γn2(B1 cos(ω1t+ φ)Sˆx +B1 sin(ω1t+ φ)Sˆy), (A.32)which will transform in the same way. However, there is now an additional term(Hˆ1)2= −γn2(B1 cos(ω1t+ φ)Sˆx −B1 sin(ω1t+ φ)Sˆy). (A.33)Applying the transformation to this term givesei(ω1t+φ)σz/2(Hˆ1)2e−i(ω1t+φ)σz/2=− γn2B1(cos(ω1t+ φ) cos(ω1t+ φ)Sˆx− cos(ω1t+ φ) sin(ω1t+ φ)Sˆy− sin(ω1t+ φ) cos(ω1t+ φ)Sˆy− sin(ω1t+ φ) sin(ω1t+ φ)Sˆx)=Ωl(0 ei2(ω1t+φ)e−i2(ω1t+φ) 0).(A.34)281A.2. The Central Fringe and Asymmetry EquationThus the Schrodinger equation in the rotating frame isi~∂∂tΨ′ = (−λSˆz + ΩSˆx)Ψ′ + Ωl ~2(0 ei2(ω1t+φ)e−i2(ω1t+φ) 0)Ψ′, (A.35)or, written out in terms of the individual spin components,C˙↑′= iλ2C↑′ − iΩl2C↓′ − iΩl2C↓′ei2(ω1t+φ),C˙↓′= −iλ2C↓′ − iΩl2C↑′ − iΩl2C↑′e−i2(ω1t+φ).(A.36)Unlike the circularly rotating case, the time-dependence of the coefficients has not been com-pletely removed with the linearly oscillating field. The extra term oscillates at a frequency 2ω1.If this frequency is large relative to the precession frequency, that is |2ω1| ωn, then thisadditional term oscillates rapidly, and will intuitively have only a small effect on the result asit averages to zero over a long time scale. For measurements around resonance, where ω1 ≈ ωn,this is seen to be a not-unreasonable approximation in practice (where the perturbing fieldstrength is small relative to the holding field). Neglecting the exponential terms allows Ram-sey’s method with a linearly oscillating field to be formulated by simply replacing the pulsedfield strength in the previous derivation Ω→ ΩlFor completeness, it should be noted that the use of a linearly oscillating field introduces afrequency shift. The Bloch-Siegert shift has an approximate magnitude given by [326]∣∣∣∣ ω12l16ωn∣∣∣∣ . (A.37)The existence of this shift is largely irrelevant for an nEDM experiment. Since this shift doesnot correlate with electric field, it is removed in the comparison of electric field polarities, butits existence may be of interest when performing other resonance measurements.A.2 The Central Fringe and Asymmetry EquationA.2.1 The Central FringeIn an nEDM measurement, Equations A.26 and A.27 are not directly used. The experimentis performed at what is referred to as the central fringe, the centre of which has the highestprobability of a spin flip occurring. This is achieved when ωn = ω1, and so the approximate formof the central fringe can be derived by assuming λ Ω (where Ω can be for either the circularor linear case depending on the implementation of the experiment). Under this assumption,a ≈ Ω/2, and λ/2a ≈ 0, so the spin flip probability is|C↓(TR + 2τ)|2 =(Ωa)2sin2(aτ)(λ2asin(λTR/2) sin(aτ)− cos(λTR/2) cos(aτ))2=4 sin2(Ωτ2)(cos(λT2)cos(Ωτ2))2= sin2 (Ωτ) cos2(λT2)=12sin2 (Ωτ) (1 + cos (λT )) .(A.38)282A.2. The Central Fringe and Asymmetry EquationTo achieve a complete spin-flip from the two ~B1 pulses, each pulse is tuned such that Ωτ = pi/2,in which case the probability of a spin-flip at the central fringe is|C↓(TR + 2τ)|2 = 12(1 + cos (λT )) . (A.39)The probability of no spin-flip at the central fringe is then|C↑(TR + 2τ)|2 =1− 12(1 + cos (λT ))=12(1− cos (λT )) .(A.40)This result naively predicts that the width of the fringe, where λT = pi, is simply pi/T . Thevalidity of this assumption was explored by D. May [325], who presented a derivation of a moreaccurate, although not exact, expression. Note that the width of the central fringe ∆ω is givenby the value of λ corresponding to the first zeros of |C↑(TR + 2τ)|2. These zeros satisfy∆ω2asin(∆ωTR/2) sin(aτ) = cos(∆ωTR/2) cos(aτ). (A.41)Around the central fringe, Ω = pi/2τ , and, with the assumption that a ≈ Ω/2, aτ ≈ pi/4, sothis condition becomes∆ω sin(∆ωTR/2) = 2a cos(∆ωTR/2) =pi2τcos(∆ωTR/2), (A.42)which simplifies to∆ω =pi2τcot(∆ωTR/2). (A.43)Since the central fringe can be approximated as a cosine, the zeros should have ∆ωT ≈ pi. Thelinear expansion of a cotangent around pi/2 iscot (x) = cot(pi2)− csc2(pi2)(x− pi2)=pi2− x. (A.44)Applying this expansion to the condition for ∆ω leads to∆ω ≈ pi2τ(pi2− ∆ωTR2), (A.45)which can be solved to obtain∆ω ≈ piTR +4τpi. (A.46)The equations of the central fringe can then be written|C↓(TR + 2τ)|2 =12(1 + cos(piω1 − ωn∆ω)),|C↑(TR + 2τ)|2 =12(1− cos(piω1 − ωn∆ω)).(A.47)These are only approximate statements, but they have been shown to be very accurate [325].283A.2. The Central Fringe and Asymmetry EquationA.2.2 The Asymmetry EquationIn the nEDM experiment, information about the probabilities is found by counting UCNs inthe two spin states. If the total number of UCNs detected is N↑↓, then the expected counts inthe two states areN↓ =N↑↓2(1 + cos(piω1 − ωn∆ω)),N↑ =N↑↓2(1− cos(piω1 − ωn∆ω)).(A.48)Fluctuations in N↑↓ between different batches of UCNs will impact the measurement of thefrequency, so it has become typical to analyze the asymmetryA =N↑ −N↓N↑ +N↓= −N↑↓ cos(pi ω1−ωn∆ω)N↑↓= − cos(piω1 − ωn∆ω).(A.49)In practice, total polarization cannot be maintained throughout the experiment. Some fractionof the UCNs will be depolarized — their spins will be randomly flipped. Further, there will besome efficiency of correctly detecting the spins. Assuming a fraction fP maintains polarization,and a fraction fD is correctly detected, then the count equations becomeN↓ =(1− fP)N↑↓2+fPfDN↑↓2(1 + cos(ω1 − ωn∆ω))+fP(1− fD)N↑↓2(1− cos(ω1 − ωn∆ω)),N↑ =(1− fP)N↑↓2+fPfDN↑↓2(1− cos(ω1 − ωn∆ω))+fP(1− fD)N↑↓2(1 + cos(ω1 − ωn∆ω)),(A.50)and the asymmetry becomesA = −fP(2fD − 1) cos(piω1 − ωn∆ω)= −α cos(piω1 − ωn∆ω), (A.51)where α is referred to as the visibility, and includes all effects that change the amplitude of thefringe. If the probability of correctly identifying the state is different for the different states(e.g. spin up is counted correctly more often than spin down), the equation becomes morecomplicated. Assuming state-dependent values for fD we getN↓ =(fD↓ + 1− fD↑)(1− fP)N↑↓2+fPfD↓N↑↓2(1 + cos(piω1 − ωn∆ω))+fP(1− fD↑)N↑↓2(1− cos(piω1 − ωn∆ω)),N↑ =(fD↑ + 1− fD↓)(1− fP)N↑↓2+fPfD↑N↑↓2(1− cos(piω1 − ωn∆ω))+fP(1− fD↓)N↑↓2(1 + cos(piω1 − ωn∆ω)),(A.52)284A.3. Statistical Uncertaintyand the asymmetry becomesA =− (fD↑ − fD↓)2− fP(fD↑ + fD↓ − 1) cos(piω1 − ωn∆ω)=A0 − α cos(piω1 − ωn∆ω).(A.53)This is the general equation for the central fringe of the Ramsey method in the nEDM experi-ment.Note that an implicit assumption has been made here that the overall detection efficienciesare the same for each spin state. If N↓ is multiplied by an efficiency ↓, and N↑ by an efficiency↑, then the asymmetry does not cleanly reduce to a single cosine. This could be the case insimultaneous spin analysis if one of the detectors has a much lower detection efficiency than theother. Further discussion of these kinds of effects which modify the shape of the central fringeis not within the scope of this thesis, and will not be explored here.A.3 Statistical UncertaintyThis section shows two derivations of the estimated statistical uncertainty on the precessionfrequency determined using Ramsey’s method. The first of these is a commonly-seen derivationusing basic error propagation on the asymmetry equation. The second is based on calculatingthe error in a non-linear least squares fitting of the asymmetry equation. The inclusion of thesetwo methods is due to doubts about the formal validity of the first method, although bothmethods are shown to produce the same result.A.3.1 Asymmetry InversionThe first approach to estimate the statistical uncertainty from Ramsey’s method is to solve theasymmetry Equation A.53 for ωn, resulting inωn = ω1 −∆ω arccos A0 −Aα. (A.54)From this equation, the uncertainty in the frequency can be calculated using error propagation.The values of ω1 and A are the measured values, and the four parameters ωn, ∆ω, A0, and αare obtained by fitting. The uncertainty of ωn isσωn2 =(∂ωn∂ω1)2σω12 +(∂ωn∂A)2σA2=σω12 +∆ω2pi2α2(1− (A0 −A)2α2)−1σA2.(A.55)The first term is simply the uncertainty on the applied frequency. The second term comes fromthe uncertainty on the UCN counts used to calculate the asymmetry. Assuming that the countsobey Poisson counting statistics, we obtain from the definition of the asymmetryσA2 =(∂A∂N↑)2σN↑2 +(∂A∂N↓)2σN↓2=4N↓2(N↓ +N↑)4σN↑2 +4N↑2(N↓ +N↑)4σN↓2=4N↓2(N↓ +N↑)4N↑ +4N↑2(N↓ +N↑)4N↓ =4N↑N↓(N↓ +N↑)3.(A.56)285A.3. Statistical UncertaintyThe measurement is performed close to A = A0, where the slope of the fringe (and thus thesensitivity to changes in the frequency) is largest. In this region the approximation N↓ ≈ N↑ ≈N↑↓/2 can be used, where N↑↓ = N↑ + N↓ is the number of neutrons detected in either spinstate in one fill. Under these assumptions, σA2 = 1Nupdown , and thusσωn2 = σω12 +∆ωn2pi2α2N= σω12 +(σωncounting)2. (A.57)From this equation, the standard counting statistics uncertainty can be read off asσωncounting =1(TR +4τpi )α√N↑↓. (A.58)Note that this is only valid close to A ≈ A0, and measurements far from this point will have adifferent uncertainty due to the mismatch of N↑ and N↓. This topic has been further exploredin the thesis of M. Miller [327], which is unpublished at the time of writing.In practice, the measurement is taken at a set of four “working points” that do not satisfythe condition A = A0 exactly. The working points are instead chosen to satisfyωi = ωn ± ∆ω2(1± s), (A.59)where s is the separation of the working points and is a number between zero and one. Thechoice of working points can be evaluated by, instead of setting A = A0, substituting for A.This leads to the resultσωn2 =σ2ω1 +∆ω2pi2α2(1− cos2(λ∆ω))−1σA2=σω12 +∆ω2pi2α2(1− cos2(pi2(1± s)))−1σA2≈σω12 +∆ω2pi2α2(1 + cos2(pi2(1± s)))σA2≈σω12 +∆ω2pi2α2N(1 + cos2(pi2(1± s))),(A.60)where the third line is obtained by a Taylor expansion of 1/(1 − x). The counting statisticscontribution becomesσωncounting ≈ 1(TR +4τpi )α√N↑↓(1 +12cos2(pi2(1± s)))1/2≈ 1(TR +4τpi )α√N↑↓(1 +pi2s28),(A.61)where again a Taylor expansion is applied to the square root. The additional term dependingon s represents the increased uncertainty when measuring away from A = 0.To determine the nEDM two frequency measurements are required, and so the uncertainty286A.3. Statistical Uncertaintybecomesσdn,stat =~4E√σωn2↑ + σωn2↓ =~√24Eσωn=~√24E12pi(TR +4τpi )α√N↑↓(1 +pi2s28)=~2√2αE(TR +4τpi )√N↑↓(1 +pi2s28)=~2αE(TR +4τpi )√2N(1 +pi2s28).(A.62)Note that 2N↑↓ is the combined number of detected neutrons across a pair of frequency mea-surements. Further, if M measurements are taken and averaged, this simply introduces a factorof√M into the denominator. So, the equation can be rewritten with Ntot = 2MN↑↓ being thetotal number of neutrons detected across the entirety of data taking, givingσdn,stat =~2αE(TR +4τpi )√Ntot(1 +pi2s28). (A.63)This is the standard literature formula predicting the uncertainty due to counting statisticsin an nEDM measurement using Ramsey’s method. This is often quoted without the workingpoint contribution, givingσdn,stat =~2αE(TR +4τpi )√Ntot. (A.64)A.3.2 Hessian EstimationThe previous derivation contains an assumption that the precession frequency is extracted bya simple calculation that is equivalent to inverting the asymmetry equation and plugging invalues for frequency and asymmetry. In practice, four differing asymmetry measurements areused to obtain a single frequency measurement, as well as fitting the other parameters.A more formal approach to deriving the uncertainty is to consider a least squares fit of theasymmetry equation to the four working points. The parameter variance obtained from thefit can then be estimated by calculating the Hessian matrix. The Hessian H is related to thecovariance matrix by C = H−1, so it is possible to estimate the expected error in the fit of theasymmetry equation to four points by computing the elements of the Hessian Hkl and invertingthe matrix.In general, the elements of the Hessian matrix are defined byHkl =12∂2χ2∂ak∂al=N∑i=11σ2i[∂y(xi;~a)∂ak∂y(xi;~a)∂al− [yi − y(xi; a)] ∂2y(xi;~a)∂al∂ak], (A.65)where ~a is the vector of fit parameters, (xi, yi) the measured data, and σi is the uncertainty ofthe ith data point. In practice, the second term is ignored. If the fit is good, then [yi− y(xi; a)]should be uncorrelated with the model and therefore this term should cancel when summed.The Hessian matrix can therefore be obtained by calculating the first derivatives ∂y(xi;~a)∂ak .For the application of Ramsey’s method to the nEDM measurement, y = A, x = ω1, andthe parameters are A0, α, ωn, and ∆ω. The values of ω1 are assumed to be set with sufficientprecision to have negligible uncertainty.287A.3. Statistical UncertaintyThe first step is to evaluate the derivatives of A with respect to each parameter. Thederivative with respect to A0 is∂A(ωi;~a)∂A0=∂∂A0(A0 − α cos(piωi − ωn∆ω))= 1. (A.66)The derivative with respect to α is∂A(ωi;~a)∂α=∂∂α(A0 − α cos(piω1 − ωn∆ω))=− cos(piωi − ωn∆ω).(A.67)The derivative with respect to ωn is∂A(ωi;~a)∂ωn=∂∂ωn(A0 − α cos(piω1 − ωn∆ω))=− pi∆ωsin(piωi − ωn∆ω).(A.68)The derivative with respect to ∆ω is∂A(ωi;~a)∂∆ω=∂∂∆ω(A0 − α cos(piω1 − ωn∆ω))=− pi(ωi − ωn)∆ω2sin(piωi − ωn∆ω).(A.69)Now the products must be evaluated. For the diagonal elements(∂A(ωi;~a)∂A0)2= 1, (A.70)(∂A(ωi;~a)∂α)2= cos2(piωi − ωn∆ω), (A.71)(∂A(ωi;~a)∂ωn)2=( pi∆ω)2sin2(piωi − ωn∆ω), (A.72)(∂A(ωi;~a)∂∆ω)2=pi2(ωi − ωn)2∆ω4sin2(piωi − ωn∆ω). (A.73)For the off-diagonal elements∂A(ωi;~a)∂A0∂A(ωi;~a)∂α= − cos(piωi − ωn∆ω), (A.74)∂A(ωi;~a)∂A0∂A(ωi;~a)∂ωn= − pi∆ωsin(piωi − ωn∆ω), (A.75)∂A(ωi;~a)∂A0∂A(ωi;~a)∂∆ω= −pi(ωi − ωn)∆ω2sin(piωi − ωn∆ω), (A.76)288A.3. Statistical Uncertainty∂A(ωi;~a)∂α∂A(ωi;~a)∂ωn=pi∆ωcos(piωi − ωn∆ω)sin(piωi − ωn∆ω), (A.77)∂A(ωi;~a)∂α∂A(ωi;~a)∂∆ω=pi(ωi − ωn)∆ω2cos(piωi − ωn∆ω)sin(piωi − ωn∆ω), (A.78)∂A(ωi;~a)∂ωn∂A(ωi;~a)∂∆ω=pi2(ωi − ωn)∆ω3sin2(piωi − ωn∆ω). (A.79)The last step to obtain the elements of the Hessian is to evaluate the sums, using the fourworking points. Here, the same approximations as in the previous section are applied to obtainσi = σA =1N↑↓ . For the working points, recall the definitionωi = ωn ± ∆ω2(1± s). (A.80)Assuming that S is small, higher-order terms in S can be neglected. However, it is necessaryto keep terms up to second-order to both preserve invertibility of the Hessian and capture thecontribution of S to the uncertainty. In generalcos(±pi2(1± s))≈ ∓spi2, (A.81)sin(±pi2(1± s))≈ ±(1− s2pi28). (A.82)Care must be taken in this derivation to keep the meaning of the ± clear, given that it appearstwice. A ± or ∓ symbol directly in front of S is independent from such a symbol appearing infront of a 1 or an entire term, but all ± and ∓ symbols in front of S are dependent with eachother, and all non-S-dependent symbols are dependent with each other. With that in mind,the sums can be evaluatedFor the diagonal elements4∑i=1(∂A(ωi;~a)∂A0)2=4∑i=11 = 4, (A.83)4∑i=1(∂A(ωi;~a)∂α)2=4∑i=1cos2(piωi − ωn∆ω)=4∑i=1cos2(pi2(1± s))≈4∑i=1(∓pis2)2= 2pis,(A.84)289A.3. Statistical Uncertainty4∑i=1(∂A(ωi;~a)∂ωn)2=4∑i=1( pi∆ω)2sin2(piωi − ωn∆ω)=( pi∆ω)2 4∑i=1(±(1− s2pi28))2=4∑i=1( pi∆ω)2(1− s2pi24+s4pi464)=4( pi∆ω)2(1− s2pi24),(A.85)4∑i=1(∂A(ωi;~a)∂∆ω)2=4∑i=1pi2(ωi − ωn)2∆ω4sin2(piωi − ωn∆ω)=pi2∆ω44∑i=1∆ω24(1± s)2(1− s2pi28)2=pi24∆ω24∑i=1(1± 2s+ s2)(1− s2pi24+s4pi464)=pi2∆ω2(1 + s2 − s2pi24).(A.86)For the off-diagonal elements,4∑i=1∂A(ωi;~a)∂A0∂A(ωi;~a)∂α=−4∑i=1cos(piωi − ωn∆ω)=−4∑i=1(∓spi2) = 0,(A.87)4∑i=1∂A(ωi;~a)∂A0∂A(ωi;~a)∂ωn=−4∑i=1pi∆ωsin(piωi − ωn∆ω)=− pi∆ω4∑i=1(±(1− s2pi28)) = 0,(A.88)4∑i=1∂A(ωi;~a)∂A0∂A(ωi;~a)∂∆ω=−4∑i=1pi(ωi − ωn)∆ω2sin(piωi − ωn∆ω)=− pi∆ω24∑i=1±∆ω2(1± s)(±(1− s2pi28))=− pi∆ω24∑i=1∆ω2(1± s)(1− s2pi28)=− 2pi∆ω(1− s2pi28),(A.89)290A.3. Statistical Uncertainty4∑i=1∂A(ωi;~a)∂α∂A(ωi;~a)∂ωn=4∑i=1pi∆ωcos(piωi − ωn∆ω)sin(piωi − ωn∆ω)=pi∆ω4∑i=1(∓spi2)(±(1− s2pi28)) = 0,(A.90)4∑i=1∂A(ωi;~a)∂α∂A(ωi;~a)∂∆ω=4∑i=1pi(ωi − ωn)∆ω2cos(piωi − ωn∆ω)sin(piωi − ωn∆ω)=pi∆ω24∑i=1∆ω2(1± s)(∓spi2)(1− s2pi28) = −s2pi2∆ω2,(A.91)4∑i=1∂A(ωi;~a)∂ωn∂A(ωi;~a)∂∆ω=4∑i=1pi2(ωi − ωn)∆ω3sin2(piωi − ωn∆ω)=pi2∆ω34∑i=1(±∆ω2(1± s))(1− s2pi28) = 0.(A.92)Finally, combining all of these results, the Hessian matrix isH ≈ 1σA24 0 0 − 2pi∆ω(1− pi2s28)0 2pis 0 −pi2s2∆ω20 0 4 pi2∆ω2(1− pi2s24)0− 2pi∆ω −pi2s2∆ω20 pi2∆ω2(1 + s2 − pi2s28) . (A.93)The inversion of this Hessian is time-consuming and the form of the resulting matrix is compli-cated. However, for the purposes of the nEDM experiment, only the third diagonal term of thecovariance matrix, which gives the uncertainty in determining the frequency ωn, is of interest.Since there are no off-diagonal terms in the third row or column, the Hessian is block diagonal,so the corresponding entry in the covariance matrix can be obtained as the reciprocal of H33.The result isσωn2 =σA2 ∆ω24pi21(1− pi2s24)=1N↑↓14pi2pi2(TR +4τpi)2 (1 + pi2s24)=1α2(TR +4τpi)24N↑↓(1 +pi2s24).(A.94)Finally, we arrive at the resultσωn =1α(TR +4τpi)√4N↑↓(1 +pi2s28). (A.95)291A.3. Statistical UncertaintyThis differs from the result of the previous section only by the presence of an extra factor offour in front of the number of neutrons per-fill N↑↓. Since the previous derivation assumedfitting to a single measurement, and this derivation assumes fitting to four measurements, theresults are in fact consistent. Therefore it can be concluded that fitting to the complete modelwith four parameters does not impact the uncertainty on the determined frequency.A.3.3 Including the Co-MagnetometerTo compensate for fluctuations in the magnetic field between fills of UCNs into the cell, ameasurement of the magnetic field is taken using a Hg magnetometer. This is introduced intothe asymmetry equation asA =A0 − α cos(pi2∆ω(ωn − ωi)〈ωHg,i〉ωHg,i)=A0 − α cos(pi2∆ω(ωnωHg,i− ωiωHg,i)〈ωHg,i〉)=A0 − α cos(pi2∆ω(R− ωiωHg,i)〈ωHg,i〉).(A.96)where ωHg,i is the frequency measured during a single fill, and 〈ωHg〉 is the average frequencyover the four working points. It is shown in Equation 4.4 in the main text that a single R isappropriate for all four working points, and so the four fit parameters become R, A0, α, andpi〈ωHg〉2∆ω .Applying the inversion method to derive the uncertainty yieldsR = 2∆ωpi〈ωHg〉 arccos(A0 −Aα)+ωiωHg,i. (A.97)The total uncertainty on R is thenσR2 =σωn2〈ωHg〉2 +σωi2ωHg,i2+σωHg2ω12iωHg,i4. (A.98)Taking ωHg,i ≈ 〈ωHg〉 and ωi ≈ 〈ωi〉 ≈ ωn givesσR2 ≈ 〈ωi〉2〈ωHg〉2(σωn2〈ωi〉2 +σωi2〈ωi〉2 +σωHg2〈ωHg〉2)=〈ωi〉2〈ωHg〉2σωn2〈ωi〉2(1 +σωi2σωn2+σωHg2σωn2〈ωi〉2〈ωHg〉2)≈R2σωn2ωn2(1 +σωi2σωn2+σωHg2σωn2R2).(A.99)So the relative increase in uncertainty from the use of a co-magnetometer is found to beσRR ≈σωnωn√1 +σωi2σωn2+σωHg2σωn2R2. (A.100)292Appendix BRedfield Theory in an nEDMExperimentThis appendix presents detailed derivations for frequency shifts and relaxation rates in thenEDM experiment. These results have previously been reported in several papers [222, 226].However, those papers did not include all steps necessary for complete derivations. In some casesthis is because those steps require brute force calculation, but on occasion a mathematical orphysical argument was found to be difficult to follow. The purpose of this appendix is thereforeto present a complete set of derivations of these results, from first principles, such that a readercan fully understand how these results are arrived at without having to repeat the work.B.1 Relaxation MatrixThe physical system of interest is a large population of spins, with a collective state whichchanges over time due to the form of the magnetic field. For an nEDM experiment, there aretwo separate populations: the neutrons and the co-magnetometer atoms. Both of these havespin 1/2, and so this description will focus on that specific value of spin. These spin populationscan be described using the density operator of NMR, which can be written in terms of Paulimatricesρˆ = 1 + ρ1,0σz + ρ1,1σ+ + ρ1,−1σ−, (B.1)where 2σ± = σx ± iσy, and ρ1,1 = ρ1,−1∗.The diagonal elements of the density operator describe the probabilities of measuring thespin states. The off-diagonal elements are referred to as the “coherences”, and contain infor-mation about the probability of measuring different results in different bases than the diagonalbasis. For example, in a basis of σz, a population that is an equal mix of up and down spinscan be described by (0.5 00 0.5), (B.2)while a population that has been prepared in a σx superposition can be described by(0.5 0.50.5 0.5). (B.3)Both of these populations have the same probability of measuring the two spin states, despitethe fact that they are fundamentally different. The information about the difference is containedin the coherences.As the spins orbit in the cell, they experience the spatially-varying fields as small time-293B.1. Relaxation Matrixvarying fields. A general field ~B(t) = (Bx(t), By(t), Bz(t)) can be re-written asbx(t) =Bx(t)− 〈Bx(t)〉 ,by(t) =By(t)− 〈By(t)〉 ,bz(t) =Bz(t)− 〈Bz(t)〉 ,B0 =√〈Bz(t)〉2 + 〈Bx(t)〉2 + 〈By(t)〉2,(B.4)such that the perturbing fields have an average of zero, and B0 is the observed average field. Itis worth acknowledging that this formulation ignores the time-variation of the z-component ofthe fields as a spin moves along the z-axis. The z-variation results in localized changes in theLarmor frequency, but since the nEDM experiment is carried out over a long period of time,these changes average out.Using the above description of the magnetic fields, the Hamiltonian for this problem isHˆ =− γB02σz − γbz2σz − γbx(t)′2σx − γby(t)′2σy=ω02σz +ωz2σz +ωx2σx +ωy2σy = Hˆ0 + Hˆ1(t).(B.5)It is possible to rewrite Hˆ1(t) in the same Pauli basis as the density matrix, by defining2b = ωx + iωy, (B.6)in which caseHˆ1(t) = b∗σ+ + bσ− + czσz, (B.7)where 2cz = ωz. The time evolution of the density operator is given by∂ρˆ∂t= −i[Hˆ0 + Hˆ1(t), ρˆ]. (B.8)The dependence on Hˆ0 can be removed by working in the interaction picture of quantummechanics, in this case also known as the rotating frame. We apply the transformationOˆ → eiHˆ0tOˆe−iHˆ0t, (B.9)whereeiHˆ0t =(e−iω0t/2 00 eiω0t/2). (B.10)In the interaction picture, the time evolution of the density operator becomes∂ρˆ∂t= −i[Hˆ1,I(t), ρˆ]= −i [e−iω0tb∗σ+ + eiω0tbσ− + czωz, ρˆ] . (B.11)An approximate form for ρˆ can be found by successive substitution. Integrating the aboveexpression yieldsρˆ(t) = ρˆ(0)− i∫ t0[Hˆ1,I(t′), ρˆ(t′)]dt′. (B.12)Since this expression for ρˆ(t) is recursive, it can be substituted into itself. Doing so twice, andkeeping only those terms up to the double integral, givesρˆ(t)ρˆ(0)− i∫ t0[Hˆ1,I(t′), ρˆ(0)]dt′−∫ t0∫ t′0[Hˆ1,I(t′),[Hˆ1,I(t′′), ρˆ(0)]]dt′′dt′.(B.13)294B.1. Relaxation MatrixThe next highest-order term would be a triple integral with three nested commutators, whichwould therefore scale with (Hˆ1,I)3. Neglecting this term is therefore acceptable if the perturbingfield are small.The frequency shifts and relaxation rates will be captured by the time derivative of ρˆ.Differentiating the above approximation gives∂ρˆ∂t= −i[Hˆ1,I(t), ρˆ(0)]−∫ t0[Hˆ1,I(t),[Hˆ1,I(t′′), ρˆ(0)]]dt′′. (B.14)The first term is zero when averaged over an ensemble. Considering Equation B.12, ρˆ(0) can bereplaced with ρˆ(t), introducing a higher order error. As before, this neglected term depends ona higher power of the Hamiltonian and can be neglected when the perturbing fields are small.Redefining t′′ = t− τ leads to the definition of the relaxation tensor Γ∂ρˆ∂t=∫ t0[Hˆ1,I(t),[Hˆ1,I(t− τ), ρˆ(t)]]dτ = Γρˆ(t). (B.15)From this, the evolution of the elements of ρˆ can be obtained. To indicate dependence on (t−τ)as opposed to t the notation b′, b′∗, cz ′ will be used. The expansion of the full commutationrelation can be split into four parts, such that[Hˆ1,I(t),[Hˆ1,I(t− τ), ρˆ(t)]] =[b∗σ+ + bσ−, [b′∗σ+ + b′σ−, ρˆ(t)]]+[b∗σ+ + bσ−,[cz′σz, ρˆ(t)]]+[czσz,[b′∗σ+ + b′σ−, ρˆ(t)]]+[czσz,[cz′σz, ρˆ(t)]].(B.16)The first step is the evaluation of the two inner commutations. The first of these is[b′∗σ+ + b′σ−, ρˆ(t)]=[(0 b′∗e−iω0(t−τ)b′eiω0(t−τ) 0),(1 + ρ1,0 ρ1,1ρ1,−1 1− ρ1,0)]=(b′∗e−iω0(t−τ)ρ1,−1 b′∗e−iω0(t−τ)(1− ρ1,0)b′eiω0(t−τ)(1 + ρ1,0) b′eiω0(t−τ)ρ1,1)−(b′eiω0(t−τ)ρ1,1 b′∗e−iω0(t−τ)(1 + ρ1,0)b′eiω0(t−τ)(1− ρ1,0) b′∗e−iω0(t−τ)ρ1,−1) (B.17)=(b′∗e−iω0(t−τ)ρ1,−1 − b′eiω0(t−τ)ρ1,1 −2b′∗e−iω0(t−τ)ρ1,02b′eiω0(t−τ)ρ1,0 b′eiω0(t−τ)ρ1,1 − b′∗e−iω0(t−τ)ρ1,−1.). (B.18)Then second of these is[czσz, ρˆ(t)] =[(cz′ 00 −cz ′),(1 + ρ1,0 ρ1,1ρ1,−1 1− ρ1,0)]=(cz′(1 + ρ1,0) cz ′ρ1,1−cz ′ρ1,−1 cz ′(ρ1,0 − 1))−(cz′(1 + ρ1,0) −cz ′ρ1,1cz′ρ1,−1 cz ′(ρ1,0 − 1))=(0 2cz′ρ1,12cz′ρ1,−1 0).(B.19)295B.1. Relaxation MatrixNow the four parts of the commutation can be evaluated in turn. First we have[b∗σ+ + bσ−,[b′∗σ+ + b′σ−, ρˆ(t)]]=[(0 b∗e−iω0tbeiω0t 0),(b′∗e−iω0(t−τ)ρ1,−1 − b′eiω0(t−τ)ρ1,1 −2b′∗e−iω0(t−τ)ρ1,02b′eiω0(t−τ)ρ1,0 b′eiω0(t−τ)ρ1,1 − b′∗e−iω0(t−τ)ρ1,−1)]=(2b′b∗e−iω0τρ1,0 b′b∗e−iω0τρ1,1 − b∗b′∗eiω0(2t−τ)ρ1,−1bb′∗eiω0τρ1,−1 − b′beiω0(2t−τ)ρ1,1 −2bb′∗eiω0τρ1,0)−( −2bb′∗eiω0τρ1,0 b∗b′∗e−iω0(2t−τ)ρ1,−1 − b′b∗e−iω0τρ1,1b′beiω0(2t−τ)ρ1,1 − bb′∗eiω0τρ1,−1 2b′b∗e−iω0τρ1,0)=(4Re[b′b∗eiω0τ ]ρ1,0 2b′b∗e−iω0τρ1,12bb′∗eiω0τρ1,−1 −4Re[b′b∗eiω0τ ]ρ1,0)−(0 2b′∗b∗e−iω0(2t−τ)ρ1,−12b′beiω0(2t−τ)ρ1,1 0).(B.20)The terms of the second matrix oscillate at 2ω0, and so they can be neglected to second order.The first matrix captures the interesting effects of spatial variations in the field. The diagonalterms correspond directly to relaxation of the longitudinal state, while the off-terms includeboth frequency shifts and relaxation of the transverse states.The second commutation is[b∗σ+ + bσ−,[cz′σz, ρˆ(t)]]=[(0 b∗e−iω0tbeiω0t 0),(0 2cz′ρ1,12cz′ρ1,−1 0)]=(2b∗cz ′ρ1,−1e−iω0t 00 2cz′bρ1,1eiω0t)−(2cz′bρ1,1eiω0t 00 2cz′b∗ρ1,−1e−iω0t)=(2b∗cz ′ρ1,−1e−iω0t − 2cz ′bρ1,1eiω0t 00 2cz′bρ1,1eiω0t − 2cz ′b∗ρ1,−1e−iω0t).(B.21)The third commutation is[czσz,[b′∗σ+ + b′σ−, ρˆ(t)]]=[(cz 00 −cz),(b′∗e−iω0(t−τ)ρ1,−1 − b′eiω0(t−τ)ρ1,1 −2b′∗e−iω0(t−τ)ρ1,02b′eiω0(t−τ)ρ1,0 b′eiω0(t−τ)ρ1,1 − b′∗e−iω0(t−τ)ρ1,−1)]=(czb′∗e−iω0(t−τ)ρ1,−1 − b′eiω0(t−τ)ρ1,1 −2czb′∗e−iω0(t−τ)ρ1,0−2czb′eiω0(t−τ)ρ1,0 −czb′eiω0(t−τ)ρ1,1 + czb′∗e−iω0(t−τ)ρ1,−1)−(czb′∗e−iω0(t−τ)ρ1,−1 − b′eiω0(t−τ)ρ1,1 2czb′∗e−iω0(t−τ)ρ1,02czb′eiω0(t−τ)ρ1,0 −czb′eiω0(t−τ)ρ1,1 + czb′∗e−iω0(t−τ)ρ1,−1)=(0 −4czb′∗e−iω0(t−τ)ρ1,0−4czb′eiω0(t−τ)ρ1,0 0).(B.22)The meaning of the second and third commutations will be returned to momentarily.296B.2. Relaxation RatesThe fourth commutation is[czσz,[cz′σz, ρˆ(t)]]=[(cz 00 −cz),(0 2cz′ρ1,12cz′ρ1,−1 0)]=(0 2czcz′ρ1,1−2czcz ′ρ1,−1 0)−(0 −2czcz ′ρ1,12czcz′ρ1,−1 0)=(0 4czcz′ρ1,1−4czcz ′ρ1,−1 0).(B.23)This corresponds to an additional contribution to the transverse relaxation.Combining all of these expressions leads to the following results for the time-dependence ofthe density matrix. The coefficients of ρ1,0, ρ1,1, and ρ1,−1 are the elements of the relaxationtensor Γ:∂ρ1,0∂t=ρ1,0∫ t04Re[eiω0τ b′∗b]dτ + ρ1,−1∫ t02b∗cz ′e−iω0tdτ− ρ1,1∫ t02cz′beiω0tdτ,(B.24)∂ρ1,−1∂t=ρ1,−1(∫ t02eiω0τ b∗b′dτ −∫ t04czcz′dτ)− ρ1,0∫ t04czb′eiω0(t−τ)dτ,(B.25)∂ρ1,1∂t=ρ1,1(∫ t02e−iω0τ bb′∗ −∫ t04czcz′dτ)− ρ1,0∫ t04czb′∗e−iω0(t−τ)dτ.(B.26)In the relaxation tensor, Γaaaa terms represent population relaxation, while Γabab terms repre-sent decoherence. The imaginary part of Γabab is a frequency shift, while the real part is thetransverse relaxation.The additional terms of the form Γaaab, Γaaba, Γabaa, Γbaaa are argued to not contributesignificantly. The argument, known as the secular approximation, is that these terms rapidlyoscillate relative to the integration time. The basic outline of the argument is to consider thatthe relaxation tensor terms are small perturbations, and to write ρa,b = e−iωa,btρa,b,I , whereρa,b,I gives the part of the density matrix that is changing due to interactions with the magneticfield. If this is substituted into the above expressions, the cross-terms acquire a dependence ondifferences between frequencies. These cross-terms will therefore rapidly oscillate and can beignored on long time scales.B.2 Relaxation RatesThe longitudinal relaxation rate is given byΓ1,m = Re[∫ t04eiω0τ b′∗b]dτ, (B.27)297B.2. Relaxation Ratesand the transverse relaxation rate by the real part ofΓ2,m = Re[∫ t02eiω0τ b∗b′dτ −∫ t04czcz′dτ]. (B.28)The first terms are the same, except for a factor of 2, but the transverse relaxation sees anadditional contribution from variations in the Bz field. Taking the real part of the first term ofthe transverse relaxation, we haveRe∫ t02eiω0τ b∗b′dτ=12Re∫ t0(cos(ω0τ) + i sin(ω0τ))(ωx(t)− iωy(t) (ωx(t− τ) + iωy(t− τ)) dτ=12Re∫ t0(cos(ω0τ) + i sin(ω0τ))(ωx(t)ωx(t− τ) + ωy(t)ωy(t− τ)−iωy(t)ωx(t− τ) + iωx(t)ωy(t− τ))dτ=12∫ t0cos(ω0τ) (ωx(t)ωx(t− τ) + ωy(t)ωy(t− τ)) dτ+12∫ t0sin(ω0τ) (ωy(t)ωx(t− τ)− ωx(t)ωy(t− τ)) dτ.(B.29)The second term is simplyRe[−∫ t0ωz(t)ωz(t− τ)dτ]=∫ t0ωz(t)ωz(t− τ)dτ. (B.30)It is necessary to take an average over the population, givingΓ2,m =12∫ t0cos(ω0τ) 〈ωx(t)ωx(t− τ) + ωy(t)ωy(t− τ)〉 dτ+12∫ t0sin(ω0τ) 〈ωy(t)ωx(t− τ)− ωx(t)ωy(t− τ)〉 dτ+∫ t0ωz(t)ωz(t− τ)dτ.(B.31)One assumption that can be applied to simplify this result is that the system is statisticallystationary. This means that the value of 〈ωi(t)ωj(t− τ)〉 does not depend on the exact time,but only on the difference in time, and so we can write 〈ωi(t)ωj(t− τ)〉 = 〈ωi(τ)ωj(0)〉. This isa reasonable assumption to make: while individual UCNs or gas atoms change their positionsand velocities, the overall distribution is not changing during the measurement.A second simplifying assumption is that t τc, where τc is the correlation time — the timethat it takes for 〈ωi(t)ωj(t− τ)〉 to go to zero. Under this assumption we can take t→∞. Thisassumption is supported by simulations, where the correlations are observed to go to zero withina fraction of a second, compared to the much longer measurement time > 100 s. Logically, aftera handful of wall reflections the position of a particle is sufficiently randomized such that it nolonger correlates with the original position (and by extension the fields and thus the frequenciesare no longer correlated). These two assumptions lead to the resultΓ2,m =12∫ ∞0cos(ω0τ) 〈ωx(τ)ωx(0) + ωy(τ)ωy(0)〉 dτ+12∫ ∞0sin(ω0τ) 〈ωy(τ)ωx(0)− ωx(τ)ωy(0)〉 dτ +∫ ∞0ωz(τ)ωz(0)dτ.(B.32)298B.2. Relaxation RatesIf the total field is the combination of the motional field ~v × ~E/c2 and a magnetic field ~B thenthe transverse terms can be expanded to obtain three separate transverse-field contributions(assuming a vertical electric field, the z term will not see a contribution from the motionalfield). For the first term this givesγ22∫ ∞0cos(ω0τ)〈(bx(τ) + vy(τ)Ec2)(bx(0) + vy(0)Ec2)+ (by(τ)− vx(τ)Ec2)(by(0)− vx(0)Ec2)〉dτ=γ22∫ ∞0cos(ω0τ)〈bx(τ)bx(0) + vy(τ)vy(0)E2c4+ bx(0)vy(τ)Ec2+ bx(τ)vy(0)Ec2+ by(τ)by(0) + vx(τ)vx(0)E2c4− by(0)vx(τ)Ec2− by(τ)vx(0)Ec2〉dτ,(B.33)and for the second term it givesγ22∫ ∞0sin(ω0τ)〈(by(τ) + vx(τ)Ec2)(bx(0) + vy(0)Ec2)−(bx(τ) + vy(τ)Ec2)(by(0) + vx(0)Ec2)〉dτ=γ22∫ ∞0sin(ω0τ)〈by(τ)bx(0) + vx(τ)vy(0)E2c4+ bx(0)vx(τ)Ec2+ by(τ)vy(0)Ec2−bx(τ)by(0) + vy(τ)vx(0)E2c4+ by(0)vy(τ)Ec2+ bx(τ)vx(0)Ec2〉dτ.(B.34)These expressions include three terms involving correlations between x and y components,C1 = 〈bx(0)vy(τ) + bx(τ)vy(0)− by(0)vx(τ)− by(τ)vx(0)〉,C2 = 〈by(τ)bx(0)− bx(τ)by(0)〉,C3 = 〈vx(τ)vy(0) + vy(τ)vx(0)〉.(B.35)It is often assumed that these terms vanish, since x and y components should not correlate.This argument should be approached with some caution.Correlations of xy, vxy, and vyx should vanish. In simulations of UCNs it can be observedthat these correlations do not exhibit any structure that would integrate to a non-zero value.Therefore, C3 = 0 is a valid conclusion.However, in C1 and C2 the field components bx and by can be functions of all three co-ordinates, and so the correlations in C1 and C2 may contain terms of the form xx, vxx, yy,or vyy. Such terms can give a non-zero contribution when integrated. However, for the fieldsconsidered in the nEDM experiment, both C1 and C2 appear to always evaluate to zero. This isa feature of the harmonic polynomial basis, described in Section B.4, which describes magneticfields in free space. This basis has symmetry which results in the terms in C1 and C2 cancellingor otherwise evaluating to zero.To support this assertion, consider as an example a field ~B1,−2 = (y, x, 0), which is expectedto be one of the stronger field terms in the nEDM measurement. The correlation C2 isC2 = 〈by(τ)bx(0)− bx(τ)by(0)〉= 〈x(τ)y(0)− y(τ)x(0)〉, (B.36)299B.2. Relaxation Rateswhich can be expected to evaluated to zero. The correlation C1 isC1 = 〈bx(0)vy(τ) + bx(τ)vy(0)− by(0)vx(τ)− by(τ)vx(0)〉,= G1,−2〈y(0)vy(τ) + y(τ)vy(0)− x(0)vx(τ)− x(τ)vx(0))〉.(B.37)The individual yvy and xvx correlations will certainly be non-zero. However, they can beexpected to have equal contributions, and so the sign difference results in a cancellation.This is not a general proof of the result, and such a proof has not been constructed, butthis cancellation appears to hold where necessary. The majority of basis terms are not relevantbecause they mix coordinates, and terms of the form e.g. yvx will not give a non-zero contribu-tion, because y does not correlate with vx. This means that only terms that are purely in onecoordinate are relevant. Further, only odd powers are relevant, as terms of the form x2nvx willhave a zero average contribution when integrated over a symmetric geometry. By inspection ofthe basis functions, it appears that when by ∝ y2n+1, it is also the case that bx ∝ x2n+1, with thesame proportionality, and so a cancellation will occur as in the above example. Since the basiscan describe any field in free space (requiring only that sufficiently many terms are included),it follows that this property holds for all magnetic fields, as long as the region being integratedover does not contain any field sources. With this argument in mind, these cross-terms can bediscarded.Discarding the cross-terms leads to the results for the depolarization rate due to the trans-verse fieldsΓ2,m,B2 = γ2∫ ∞0cos(ω0τ) 〈bx(τ)bx(0) + by(τ)by(0)〉 dτ, (B.38)Γ2,m,E2 =γ2E2c4∫ ∞0cos(ω0τ) 〈vx(τ)vx(0) + vy(τ)vy(0)〉 dτ, (B.39)Γ2,m,BE =γ2Ec2∫ ∞0sin(ω0τ) 〈vx(τ)bx(0) + vy(τ)by(0)〉 dτ. (B.40)This is a general result, and applies for any species. It can be further simplified for the case ofUCNs.UCNs are expected to behave adiabatically, such that the correlation-time τc is large relativeto the precession period, with ω0τc 1. The above expressions can therefore be expanded inpowers of 1/ω0. This is done by repeatedly applying integration by parts, using the expres-sions [226]∫ ∞0sin(ω0τ)f(τ)dτ =[− cos(ωτ)ω0f(τ)]∞0+1ω0∫ ∞0cos(ω0τ)df(τ)dτdτ,∫ ∞0cos(ω0τ)f(τ)dτ =[sin(ωτ)ω0f(τ)]∞0− 1ω0∫ ∞0sin(ω0τ)df(τ)dτdτ.(B.41)300B.2. Relaxation RatesThis leads to the final expressionsΓ2,m,B2 =γ2[− 1ω02〈bx(0)b˙x(0) + by(0)b˙y(0)〉+1ω04〈bx(0)...bx(0) + by(0)...by(0)〉],Γ2,m,E2 =γ2E2c4[− 1ω02〈vx(0)v˙x(0) + vy(0)v˙y(0)〉+1ω04〈vx(0)...vx(0) + vy(0)...vy(0)〉],Γ2,m,BE =γ2Ec2[− 1ω0〈bx(0)vx(0) + by(0)vy(0)〉+1ω03〈bx(0)b¨x(0) + by(0)b¨y(0)〉].(B.42)The contribution from ωz can be similarly expanded with the inclusion of (v × E)/c2Γ2,m,B2z =∫ ∞0ωz(τ)ωz(0)dτ=γ2∫ ∞0(bz(τ) + vx(τ)Ey(τ)c2− vy(τ)Ex(τ)c2)×(bz(0) + vx(0)Ey(0)c2− vy(0)Ex(0)c2)dτ=γ2∫ ∞0bz(τ)bz(0) + bz(τ)(vx(0)Ey(0)c2− vy(0)Ex(0)c2)+ bz(0)(vx(τ)Ey(τ)c2− vy(τ)Ex(τ)c2)+(vx(τ)Ey(τ)c2− vy(τ)Ex(τ)c2)(vx(0)Ey(0)c2− vy(0)Ex(0)c2)dτ.(B.43)Here, the electric field contribution comes from a small transverse inhomogeneity, which wasneglected in the calculation of the xy terms in favour of the large z component of the field. Thespecific form of this inhomogeneity will determine whether these terms contribute. If Ex and Eyare radially symmetric, then the terms will cancel on average. If there is a non-symmetric Ex orEy, such as a constant due to misalignment of the electrodes relative to the magnetic field, thenthere could be non-zero contributions. However, Ex/c2 and Ey/c2 must ultimately be small inthe nEDM experiment compared to E/c2, so these terms must be negligible compared to thecontributions from the transverse components.Therefore, the practical contribution to depolarization of UCNs from fields in the z directionis simplyΓ2,m,B2z = γ2∫ ∞0〈bz(τ)bz(0)〉. (B.44)In the main text it is argued that, in fact, the majority of the depolarization comes from Γ2,m,B2z ,with the other terms being comparatively weak.301B.3. Frequency ShiftsB.3 Frequency ShiftsThe frequency shift is given byδω = Im[∫ t02eiω0τ b∗b′dτ −∫ t04czcz′dτ]. (B.45)Taking the second term to be purely real, only the first term, dependent on the transverse fieldcomponents, is of interest. Expanding this givesIm∫ t02eiω0τ b∗b′dτ=12Im∫ t0(cos(ω0τ) + i sin(ω0τ))(ωx(t)− iωy(t) (ωx(t− τ) + iωy(t− τ)) dτ=12Im∫ t0(cos(ω0τ) + i sin(ω0τ))(ωx(t)ωx(t− τ) + ωy(t)ωy(t− τ)− iωy(t)ωx(t− τ) + iωx(t)ωy(t− τ))dτ=12∫ t0cos(ω0τ) (ωx(t)ωy(t− τ)− ωy(t)ωx(t− τ)) dτ+12∫ t0sin(ω0τ) (ωx(t)ωx(t− τ) + ωy(t)ωy(t− τ)) dτ.(B.46)As with the relaxation rate, an ensemble average is taken. The same assumptions are alsoapplied: that (1) the system is statistically stationary and (2) that the measurement time islong compared to the correlation time. This simplifies the result toδω =12∫ ∞0cos(ω0τ) 〈ωx(τ)ωy(0)− ωy(τ)ωx(0)〉 dτ+12∫ ∞0sin(ω0τ) 〈ωx(τ)ωx(0) + ωy(τ)ωy(0)〉 dτ.(B.47)Introducing the motional field will again result in three contributions. The cos term gives12∫ ∞0cos(ω0τ)〈ωx(τ)ωy(0)− ωy(τ)ωx(0)〉dτ=12∫ ∞0cos(ω0τ)〈(bx(τ) + vy(τ)Ec2)(by(0)− vx(0)Ec2)− (by(τ)− vx(τ)Ec2)(bx(0) + vy(0)Ec2)〉dτ=12∫ ∞0cos(ω0τ)〈bx(τ)by(0)− bx(τ)vx(0)Ec2+ by(0)vy(τ)Ec2− vx(0)vy(τ)E2c4− bx(0)by(τ) + bx(0)vx(τ)Ec2− by(τ)vy(0)Ec2+ vy(0)vx(τ)E2c4〉dτ,(B.48)302B.3. Frequency Shiftsand the sin term gives12∫ ∞0sin(ω0τ) 〈ωx(τ)ωx(0) + ωy(τ)ωy(0)〉 dτ=12∫ ∞0sin(ω0τ)〈(bx(τ) + vy(τ)Ec2)(bx(0) + vy(0)Ec2)+ (by(τ)− vx(τ)Ec2)(by(0)− vx(0)Ec2)〉dτ=12∫ ∞0sin(ω0τ)〈bx(τ)bx(0) + bx(0)vy(τ)Ec2+ bx(τ)vy(0)Ec2+ vy(0)vy(τ)E2c4+ by(τ)by(0)− by(0)vx(τ)Ec2− by(τ)vx(0)Ec2+ vx(0)vx(τ)E2c4〉dτ.(B.49)As in the case of the depolarization rates, the cross-terms bibj , vivj , bivj , where i 6= j will notcontribute in the ensemble average. Additionally, some terms can be combined by observingthat, under the condition of being statistically stationary,〈bi(τ)vj(0)〉 = 〈bi(0)vj(−τ)〉 = −〈bi(0)vj(τ)〉. (B.50)The first equality is simply the application of stationarity. The second is that correlations ofthis form are odd, which requires some additional thought. This property can be demonstratedusing proof by contradiction. Assume that this equality is not true, such that〈bi(0)vj(−τ)〉 6= −〈bi(0)vj(τ)〉 = −〈bi(−2τ)vj(−τ)〉, (B.51)where the equality is simply stationarity again. Then〈bi(0)vj(−τ)〉+ 〈bi(−2τ)vj(−τ)〉=〈(bi(0) + bi(−2τ))vj(−τ)〉 6= 0.(B.52)This means that, when τ = 0, 2〈(bi(0))vj(0)〉 6= 0. This would mean that velocities and magneticfields are initially correlated, but if the distribution of velocities through space is isotropic, thiscannot be the case. This leads to the conclusion that 〈bi(0)vj(−τ)〉 = −〈bi(0)vj(τ)〉.The final result is the three frequency shift termsδωB2 =γ22∫ ∞0sin(ω0τ)〈bx(0)bx(τ) + by(0)by(τ)〉dτδωBE =Eγ2c2∫ ∞0cos(ω0τ)〈bx(0)vx(τ) + by(0)vy(τ)〉dτδωE2 =E2γ22c4∫ ∞0sin(ω0τ)〈vx(0)vx(τ) + vy(0)vy(τ)〉dτ.(B.53)For frequency shifts, it is important to consider the cases for both adiabatic particles (UCNs)and non-adiabatic particles (Hg atoms). In the adiabatic case, the Equations B.41 can be303B.3. Frequency Shiftsapplied. For the B2 shift,δωB2 =γ22∫ ∞0sin(ω0τ)〈bx(0)bx(τ) + by(0)by(τ)〉dτ=γ22[− cos(ωτ)ω0〈bx(0)bx(τ) + by(0)by(τ)〉]∞0+γ22ω0∫ ∞0cos(ω0τ)ddτ〈bx(0)bx(τ) + by(0)by(τ)〉dτ,=γ22ω0〈bx(0)2 + by(0)2〉+ γ22ω0∫ ∞0cos(ω0τ)〈bx(0)b˙x(τ) + by(0)b˙y(τ)〉dτ,=γ22ω0〈bx(0)2 + by(0)2〉+ γ22ω0[sin(ωτ)ω0〈bx(0)b˙x(τ) + by(0)b˙y(τ)〉]∞0− γ22ω02∫ ∞0sin(ω0τ)ddτ〈bx(0)b˙x(τ) + by(0)b˙y(τ)〉dτ=γ22ω0〈bx(0)2 + by(0)2〉 − γ22ω02∫ ∞0sin(ω0τ)〈bx(0)b¨x(τ) + by(0)b¨y(τ)〉dτ=γ22ω0〈bx(0)2 + by(0)2〉 − γ22ω02[− cos(ωτ)ω0〈bx(0)b¨x(τ) + by(0)b¨y(τ)〉]∞0− γ22ω03∫ ∞0cos(ω0τ)ddτ〈bx(0)b¨x(τ) + by(0)b¨y(τ)〉dτ=γ22ω0〈bx(0)2 + by(0)2〉 − γ22ω03〈bx(0)b¨x(0) + by(0)b¨y(0)〉− γ22ω03∫ ∞0cos(ω0τ)〈bx(0)...bx(τ) + by(0)...by(τ)〉.(B.54)For the BE shift,δωBE =Eγ2c2∫ ∞0cos(ω0τ)〈bx(0)vx(τ) + by(0)vy(τ)〉dτ=Eγ2c2[sin(ω0τ)ω0〈bx(0)vx(τ) + by(0)vy(τ)〉]∞0− Eγ2c2ω0∫ ∞0sin(ω0τ)ddτ〈bx(0)vx(τ) + by(0)vy(τ)〉dτ=− Eγ2c2ω0∫ ∞0sin(ω0τ)〈bx(0)v˙x(τ) + by(0)v˙y(τ)〉dτ=− Eγ2c2ω0[− cos(ω0τ)ω0〈bx(0)v˙x(τ) + by(0)v˙y(τ)〉]∞0− Eγ2c2ω02∫ ∞0cos(ω0τ)ddτ〈bx(0)v˙x(τ) + by(0)v˙y(τ)〉dτ,=− Eγ2c2ω02〈bx(0)v˙x(0) + by(0)v˙y(0)〉− Eγ2c2ω02∫ ∞0cos(ω0τ)〈bx(0)v¨x(τ) + by(0)v¨y(τ)〉dτ.(B.55)304B.3. Frequency ShiftsFor the E2 shift,δωE2 =E2γ22c4∫ ∞0sin(ω0τ)〈vx(0)vx(τ) + vy(0)vy(τ)〉dτ=E2γ22c4[− cos(ω0τ)ω0〈vx(0)vx(τ) + vy(0)vy(τ)〉]∞0+E2γ22c4ω0∫ ∞0cos(ω0τ)ddτ〈vx(0)vx(τ) + vy(0)vy(τ)〉dτ=E2γ22ω0c4〈vx(0)2 + vy(0)2〉+E2γ22c4ω0∫ ∞0cos(ω0τ)〈vx(0)v˙x(τ) + vy(0)v˙y(τ)〉dτ.(B.56)For each of these shifts, it can be argued that the remaining integral terms will go to zero bythe Riemann-Lebesgue lemma [226]. For the B2 shift, the third order term will be suppressedrelative to the leading order term by a factor of 1/ω02. So, the leading terms for each of theshifts are the dominant contributions.In the non-adiabatic case, ω0τc 1, so the desired expansion is in powers of ω0 rather than1/ω0. This expansion can be performed using the opposite form of integration by parts [226],∫ ∞0sin(ω0τ)f(τ)dτ =[sin(ω0τ)∫ τ0f(τ ′)dτ ′]∞0− ω0∫ ∞0cos(ω0τ)∫ τ0f(τ ′)dτ ′dτ,∫ ∞0cos(ω0τ)f(τ)dτ =[cos(ω0τ)∫ τ0f(τ ′)dτ ′]∞0+ ω0∫ ∞0sin(ω0τ)∫ τ0f(τ ′)dτ ′dτ.(B.57)The results can be further simplified by noting that, in this limit, sin(ω0τ) ≈ ω0τ . This gives,305B.3. Frequency Shiftsfor the BE termδωBE =Eγ2c2∫ ∞0cos(ω0τ)〈bx(0)vx(τ) + by(0)vy(τ)〉dτ=Eγ2c2[cos(ωτ)∫ τ0〈bx(0)vx(τ ′) + by(0)vy(τ ′)〉dτ ′]∞0+Eγ2c2ω0∫ ∞0sin(ω0τ)∫ τ0〈bx(0)vx(τ ′) + by(0)vy(τ ′)〉dτ ′dτ=Eγ2c2[cos(ωτ)〈bx(0)x(τ) + by(0)y(τ)〉]∞0+Eγ2c2ω0∫ ∞0sin(ω0τ)〈bx(0)x(τ) + by(0)y(τ)〉dτ=− Eγ2c2〈bx(0)x(0) + by(0)y(0)〉+Eγ2c2ω0∫ ∞0sin(ω0τ)〈bx(0)x(τ) + by(0)y(τ)〉dτ=− Eγ2c2〈bx(0)x(0) + by(0)y(0)〉+Eγ2c2ω02∫ ∞0τ〈bx(0)x(τ) + by(0)y(τ)〉dτ.(B.58)For the B2 term,δωB2 =γ22∫ ∞0sin(ω0τ)〈bx(0)bx(τ) + by(0)by(τ)〉dτ=γ2ω02∫ ∞0τ〈bx(0)bx(τ) + by(0)by(τ)〉dτ.(B.59)306B.4. Harmonic DecompositionFor the E2 termδωE2 =γ2E22c4∫ ∞0sin(ω0τ)〈vx(0)vx(τ) + vy(0)vy(τ)〉dτ=γ2E22c4[sin(ω0τ)∫ τ0〈vx(0)vx(τ ′) + vy(0)vy(τ ′)〉dτ ′]∞0− γ2E22c4ω0∫ ∞0cos(ω0τ)∫ τ0〈vx(0)vx(τ ′) + vy(0)vy(τ ′)〉dτ ′dτ=γ2E22c4[sin(ωτ)〈vx(0)x(τ) + vy(0)y(τ)〉]∞0− γ2E22c4ω0∫ ∞0cos(ω0τ)〈vx(0)x(τ) + vy(0)y(τ)〉dτ=γ2E22c4ω0∫ ∞0cos(ω0τ)〈vx(τ)x(0) + vy(τ)y(0)〉dτ=γ2E22c4ω0[cos(ω0τ)∫ τ0〈vx(τ ′)x(0) + vy(τ ′)y(0)〉dτ ′]∞0+γ2E22c4ω02∫ ∞0sin(ω0τ)∫ τ0〈vx(τ ′)x(0) + vy(τ ′)y(0)〉dτ ′dτ=− γ2E22c4ω0〈x(0)2 + y(0)2〉+γ2E22c4ω02∫ ∞0sin(ω0τ)〈x(τ)x(0) + y(τ)y(0)〉dτ= −γ2E22c4ω0〈x(0)2 + y(0)2〉+γ2E22c4ω03∫ ∞0τ〈x(τ)x(0) + y(τ)y(0)〉dτ.(B.60)For both the E2 and BE terms, the remaining integral contributions are small, since they areproportional to ω0τc, so the leading terms dominate.B.4 Harmonic DecompositionCalculating the contribution of magnetic fields to depolarization and frequency shifts in thenEDM requires a form of the magnetic fields expected in the experiment. The “main” field (theproduct of the residual background field and the field applied by the coils) can be approximatedas obeying Maxwell’s equations in free space, from which a polynomial basis can be derived.With no currents or time-varying electric fields, Maxwell’s equations read~∇ · ~B = 0, (B.61)~∇× ~B = 0. (B.62)The requirement on the curl is equivalent to requiring that the magnetic field can be writtenas the gradient of a scalar potential function~B = ~∇φB. (B.63)307B.4. Harmonic DecompositionSubstituting the scalar potential into the equation for the divergence gives Laplace’s equation∆φB = 0. (B.64)The spherical harmonics of Laplace’s equation are given byφBl,m(r, θ, φ) = rlYl,m(θ, φ) =√2l + 1pi(l −m)!(l +m)!rlPml (cos(θ))eimφ, (B.65)where Pml are the associated Legendre polynomials. The current standard treatment in theliterature of nEDM experiments is to split the real and imaginary parts and re-normalize thepolynomials such that [222]Σl,m(r, θ, φ) =(l − 1)!(−2)|m|(l + |m|)! cos(mφ)rlP|m|l (cos(θ)), for m ≥ 0,Σl,m(r, θ, φ) =(l − 1)!(−2)|m|(l + |m|)! sin(|m|φ)rlP|m|l (cos(θ)), for m < 0.(B.66)The magnetic field basis is thenΠx,l,m =∂∂xΣl+1,m,Πy,l,m =∂∂yΣl+1,m,Πz,l,m =∂∂zΣl+1,m.(B.67)This basis is tabulated up to l = 3 in Table B.1, with l = 4 and l = 5 tabulated separately inTables B.2 and B.3. A magnetic field is then described as~B =∑l,mGl,m~Πl,m, (B.68)where the coefficients Gl,m give the magnitude of each contribution and are in units of T/ml.In the case of a uniform field, G0,0 gives the value of the z-component. In the presence of alinear gradient, G1,0 describes that gradient, etc.308B.4. Harmonic Decompositionl m Πx Πy Πz0 -1 0 1 01 0 0 0 10 1 1 0 01 -2 y x 01 -1 0 z y1 0 −12x −12y z1 1 z 0 x1 2 x −y 02 -3 2xy x2 − y2 02 -2 2yz 2xz 2xy2 -1 −12xy −14(x2 + 3y2 − 4z2) 2yz2 0 −xz −yz z2 − 12(x2 + y2)2 1 −14(3x2 + y2 − 4z2) −12xy 2xz2 2 2xz −2xz x2 − y22 3 x2 − y2 −2xy 03 -4 3x2y − y3 x3 − 3xy2 03 -3 6xyz 3(x2z−y2z) 3x2y − y33 -2 −12(3x2y + y3 − 6yz2) −12(x3 + 3xy2 − 6z2) 6xyz3 -1 −32xyz −14(3x2z + 9y2z−4z3) 3yz2 − 34(x2y + y3)3 0 38(x3 + xy2 − 4xz2) 38(x2y + y3 − 4yz2) z3 − 32z(x2 + y2)3 1 −14(9x2z + 3y2z−4z3) −32xyz 3xz2 − 34(x3 + xy2)3 2 −x3 + 3xz2 −3yz2 + y3 3(x2z−y2z)3 3 3(x2z−y2z) −6xyz x3 − 3xy23 4 x3 − 3xy2 −3x2y + y3 0Table B.1: Free-space magnetic field polynomial basis up to order l = 3.309B.4.HarmonicDecompositionl m Πx Πy Πz4 -5 4xy(x2 − y2) y4 − 6x2y2 + x4 04 -4 12x2yz − 4x3z − 12xy2z 4x3z − 12xy2z 4x3y − 4xy34 -3 xy(12z2 − 3x2 − y2 6x2z2 − 34x4 − 32x2y2 12x2yz − 4y3z−6y2z2 + 45y44 -2 2yz(2z2 − 3x2) 2xz(2z2 − x2 − 3y2) 2xy(6z2 − x2 − y2)4 -1 xy(x2 + y2 − 83z2) z4 − 43x2z2 − 83y2z2 4yz3 − 83x2yz − 83y3z+18x4 + 34x2y2 + 58y44 0 xz(32(y2 + x2)− z2) yz(32(y2 + x2)− 2z2) z4 − 3z2(x2 + y2)+38(x4 + y4) + 34x2y24 1 z4 − 4x2z2 − 43y2z2 xy(12(x2 + y2)− 83z2 x(4z3 − 83(x2z + y2z) + 18x4+12x3z + 34x2y2 + 18y44 2 4xz(x2 − z2) 4yz(y2 − z2) 6z2(x2 − y2) + y4 − x44 3 6x2z2 − 54x4 − 6y2z2 = 12xyz2 + x3y + 3xy3 4x3z − 12xy2z+32x2y2 + 34y44 4 5x3z − 12xy2z 4y3z − 12x2yz y4 + x4 − 6x2y24 5 x4 + y4 − 6x2y2 4xy(y2 − x2) 0Table B.2: Free space magnetic field polynomial basis functions of order l = 4310B.4.HarmonicDecompositionl m Πx Πy Πz5 -6 6x4y − 10x2y3 x5 − 10x3y2 05 -5 20xyz(x2 − y2) 5z(y4 + x4 − 6x2y2) y+5x4y − 10x2y35 -4 30x2yz2 − 5x4y − 10y3z2 + y5 10x3z2 − 30xy2z2 + 5xy4 20xyz(x2 − y2)5 -3 5xyz(5z2 − x2 − 13y2) 252 x2z3 − 54x4z − 156 x2y2z 752 x2yz2 − 54x4y − 56x2y3−756 y2z3 + 2512y4z −756 y3z2 + 516y55 -2 358 yz4 − 754 x2y2 − 254 y3z2 358 xz4 − 254 x3z2 − 754 xy2z2 xyz(352 z2 − 25x2 − 25y2)−2516x4y − 516y5 − 158 x2y3 − 516x5 − 2516xy4 − 158 x3y25 -1 52xyz(x2 + y2 − 2z2) z5 − 52x2z3 − 152 y2z3 5yz4 − 152 x2yz2 − 152 y3z2+58x4z + 258 y4z + 154 x2y2z +58x4y + 58y5 + 54x2y35 0 −52xz4 + 154 x3z2 + 154 xy2z2 −52yz4 + 154 y3z2 + 154 x2yz2 z5 − 5x2z3 − 5y2z3− 516x5 − 58x3y2 − 516xy4 − 516x4y − 58x2y3 − 516y5 +158 x4z + 158 y4z + 154 x2y2z5 1 38z5 − 212 x2z3 − 72y2z3 −7xyz3 + 12xy3z + 12x3yz 158 xz4 − 212 x3z2 − 212 xy2z2+58x4z + 18y4z + 34x2y2z +18x5 + 18xy4 + 14x3y25 2 5yz4 − 10x3z2 + 1516x5 −58x2y3 + 516x4y 10z3(x2 − y2) + 5z(y4 − x4)−1516xy4 + 58x3y2 −5yz4 + 10y3z2 − 1516y55 3 10x2z3 − 254 x4z 5xyz(x2 + 3y2 − 4z2) 10x3z2 − 54x5+152 x2y2z − 10y2z3 + 154 y4z +52x3y2 − 30xy2z2 + 154 xy45 4 52xy4 + 5x3y2 10y3z+5x2y3 + 52x4y 5z(y4 + x4 − 6x2y2)+10x3z2 − 32x5 − 30xy2z2 −32y5 − 30x2yz25 5 5z(x4 + y4 − 6x2y2) 20xyz(y2 − x2) x5 − 10x3y2 + 5xy45 6 x5 − 10y2x3 + 5y4x y5 − 5yx4 + 10y3x2 0Table B.3: Free space magnetic field polynomial basis functions of order l = 5311Appendix CDerivations of the Crossing PointMethodThe crossing point method originated in the ILL nEDM experiments [108, 113] and was alsoused in the PSI nEDM measurement [114], for which the method was extended to higher ordergradients. The application of this method to a double cell has also first been proposed forthe PSI n2EDM experiment [115]. The main results of the corrected nEDM calculation cantherefore be found in the associated papers of these experiments. The use of this method isimportant for the TUCAN EDM experiment, and so this appendix gives a re-derivation of theseresults for completeness. However, the derivations of the uncertainties associated with usingthe method have not been seen elsewhere by the author.C.1 Single Cell Crossing Point EquationsThe construction of the crossing-point method in a single cell begins with the observation thatUCNs and Hg atoms sample the field in the cell differently. To fifth order, the average field,in a cylindrical coordinate system with vertical position z and radial position r seen by eitherspecies will beB0 = 〈Bz〉 =∑Gl,m〈Πz,l,m〉= G0,0 +G1,0〈z〉+G2,0(〈z2〉 − 12〈r2〉)+G3,0(〈z3〉 − 32〈z〉〈r2〉)+G4,0(〈z4〉 − 3〈z2〉〈r2〈+38〈r4〉)+G5,0(〈z5〉 − 5〈r5〉〈z3〉+ 158〈r4〉〈z〉).(C.1)For convenience, this calculation is performed in a coordinate system with z = 0 at the centreof the cell. For Hg atoms, which are isotropically distributed through the cell, the average fieldisB0,Hg = G0,0 +G2,0(H212− R24)+G4,0(H480− 3H2R224+3R424)+ ..., (C.2)where H is the electrode separation and R the cell radius. For UCNs, the result depends onthe form of height distribution. Figure 4.5 shows a simulated height distribution of UCNs filledinto the cell, from which it can be seen that the distribution is approximately linear. Assuminga distribution of the form ax+ b, the UCNs will measure a fieldB0,n = G0,0 +G1,0〈z〉+G2,0(H212− R24)+G3,0(3H220− 3R24)〈z〉+G4,0(H480− 3H2R224+3R424)+G5,0(3H4112− 3R2H28+15R424)〈z〉+ ....(C.3)312C.2. Single Cell Crossing Point UncertaintyThe two species see different fields, with the general pattern being that the UCNs see acontribution from l-odd terms, which depends on their average height. In Equation 4.4 thiswould appear as an additional shift to the ratio R, with the formδg =1B0,Hg(G1,0〈z〉+G3,0(3H220− 3R24)〈z〉+G5,0(3H4112− 3R2H28+15R424)〈z〉)=Gg〈z〉B0,Hg.(C.4)On its own, this shift is not relevant to an nEDM experiment, as it does not depend on Eand will therefore cancel in the calculation of dn. However, the presence of this shift turnsR into a measure of the l-odd gradients and can be used to achieve partial cancellation ofthe corresponding BE frequency shifts. The false EDM from BE shifts can be written outanalytically, up to fifth order, asdfalse =~γnγHg8c2R2(G1,0 −G3,0(R22− H24)+G5,0(5R416− 5R2H212+H416))=~γnγHg8c2R2(Gg +G3,0(R24+H210)−G5,0(5R416+R2H224+H428)).(C.5)Since R depends on Gg, it is possible to tune R such that Gg = 0, and thus remove some ofthe contribution from BE shifts (most notably, the first order shift is completely removed). Ifthe remaining shifts are removed from R, thenRc = γnγHg(1 + δg) =γnγHg(1 +Gg〈z〉B0,Hg). (C.6)If all shifts except for Gg and the true nEDM are removed from the measured nEDM, thendn,c = dn +~γnγHg8c2R2Gg = dn +B0,Hg〈z〉~γHg28c2R2(Rc − γnγHg). (C.7)C.2 Single Cell Crossing Point UncertaintyThe greatest potential source of error in the crossing point is due to the determination of theslopek1 =B0,Hg〈z〉~γHg28c2R2, (C.8)in the modeldn,c = dn + k1(Rc − γnγHg). (C.9)This is most easily illustrated by considering k1 as a value that has been measured throughsome auxiliary analysis, such as by comparing ratios R to gradients extracted from field maps313C.2. Single Cell Crossing Point Uncertaintyto determine ∂R∂Gg [114]. Consider an error σk1 in the measurement such that the measured valueis k1′ = k1 + σk1 . The measured values dn,c obey Equation C.9, but are assumed to obeydn,c = dn + k1′ (Rc −R0) , (C.10)where R0 = γnγHg . To obtain dn and R0, the measured data can be modified asdn,c′ = dn,c − k1′Rc, (C.11)so that the fit model isdn,c′ = dn − k1′R0. (C.12)This is now a simple linear model, with dn,c′ as the y variable and k1′ as the x variable, withk1′ = ±|k1′| depending on the direction of B0. Fitting the model will yielddˆn =∑wi(dn,c′,i + k1′,iR0)∑wi, (C.13)where wi = 1/σ2i . Assuming σi is a constant, this simplifies todˆn =∑(dn,c′,i + k1′,iR0)n, (C.14)where n is the number of data points. Inserting the definitions of the measured values givesdˆn =∑(dn,c − k1′Rc + k1′,iR0)n=∑(dn + k1 (Rc −R0)− k1′Rc + k1′,iR0)n=∑(dn + k1 (Rc −R0)− (k1 + σk1)Rc + (k1 + σk1)R0)n=∑(dn + σk1(R0 −Rc))n=dn + |σk1 |∑(±1)(R0 −Rc)n.(C.15)From this result, it can be seen that an error in the determination of the slope produces asystematic shift dependent on the statistical behaviour of the data.There is some additional increase in statistical uncertainty from the use of the crossing-pointanalysis. The modified dn,c′ will have an uncertaintyσ(dn,c)′ = σ(dn,c)2 + k12σ2Rc . (C.16)To attribute a single Rc to a measurement of dn,c there must be some binning of data. Sets ofn neutron frequency measurements are paired together, and the variance isσ2Rc =1n∑σ2Rc,i=1n∑Rc2i(σ(ωHg,i)2ωHg,i2+σ(ωn)2ωn2).(C.17)Plugging this into the above Equation C.16 and rearranging givesσ(dn,c)′σ(dn,c)=√1 +k12n∑Rc2i(σ(ωHg,i)2ωHg,i2+σ(ωn)2ωn2). (C.18)314C.3. Double Cell Crossing Point EquationsC.3 Double Cell Crossing Point EquationsThe basis of the crossing-point analysis in the double cell configuration is to construct a “co-magnetometer gradient” by combining the co-magnetometer measurements in the top and bot-tom cells asGHg =B0,Hg,T −B0,Hg,B∆HHg, (C.19)where ∆HHg is the difference between the centres of the co-magnetometers [115]. For thisdiscussion, a coordinate system with z = 0 at the centre point between the two cells is mostconvenient. In this coordinate system the top cell has h1 ≤ z ≤ h2, and the bottom cell has−h2 ≤ z ≤ −h1. In the top cellB0,Hg,top = G0,0 +G1,0(h1 + h2)2+G2,0((h22 − h213(h2 − h1) −R24)+G3,0((h42 − h414(h2 − h1) −32(h1 + h2)R22)+G4,0(h52 − h515(h2 − h1) − 3(h21 − h223(h2 − h1)R22+3R424)+G5,0(h62 − h616(h2 − h1) − 5(h42 − h414(h2 − h1)R22+158R43(h1 + h2)),(C.20)and in the bottom cellB0,Hg,bot = G0,0 −G1,0 (h1 + h2)2+G2,0((h22 − h213(h2 − h1) −R24)−G3,0((h42 − h414(h2 − h1) −32(h1 + h2)R22)+G4,0(h52 − h515(h2 − h1) − 3(h21 − h223(h2 − h1)R22+3R424)−G5,0(h62 − h616(h2 − h1) − 5(h42 − h414(h2 − h1)R22+158R43(h1 + h2)).(C.21)While l-even terms are the same, the l-odd terms have opposite sign. Using ∆HHg = (h1 +h2),GHg =(G1,0 +G3,0(h42 − h412(h22 − h21)− 32R2)+G5,0(h62 − h613(h22 − h21)− 5 h42 − h414(h22 − h21)R2 +54R4)= G1,0 +G3,0C3,0 +G5,0C4,0,) (C.22)where C3,0 and C5,0 label the geometrical coefficients for each of the gradients. As with thesingle cell case, this is compared to the false EDM from the BE shifts. Ignoring the even BE315C.4. Double Cell Crossing Point Uncertaintyterms, which are known to cancel in the double cell, the false EDM signal isdfalse =~γnγHg8c2R2(G1,0 −G3,0(R22− h32 − h31h2 − h1)+G5,0(h52 − h51h2 − h1 +53h32 − h31h2 − h1R2 − 5R432))=~γnγHg8c2R2(G1,0 −G3,0D3,0 +G5,0D5,0),(C.23)where again D3,0 and D5,0 are simply the geometrical coefficients for each of the gradients, whichare notably not the same as C3,0 and C5,0. As with the single-cell crossing-point analysis, thiscan be rewritten asdfalse =~γnγHg8c2R2(GHg −G3,0(D3,0 + C3,0) +G5,0(D5,0 + C3,0) + ...). (C.24)So theBE shifts can be partially corrected by applying different gradients and then interpolatingthe data to the point where GHg = 0.C.4 Double Cell Crossing Point UncertaintyIn the double cell, the data is now dn,c, GHg, which follows the modeldn,c = dn +~γnγHg8c2R2GHg = dn + k2GHg. (C.25)The calculation of dn, assuming constant σ, is simplydn =∑(dn,c − k2GHg)n. (C.26)As with the single cell case, an error σk2 in k2 will lead to a systematic shift dependent on thedistribution of the datadnf = −σk2∑GHgn. (C.27)The statistical uncertainty isσ2 = σ(dn,c)2 + k22σ2GHg , (C.28)where the variance in the co-magnetometer gradient isσ2GHg =2σ(B0,Hg)2∆HHg2+σ(∆HHg)2∆HHg2GHg2, (C.29)where σ(B0,Hg) is the co-magnetometer uncertainty, assumed to be equal for the two cells. Thecontribution from the determination of ∆H is proportional to GHg, which will range from somenegative value to some positive value, and therefore cannot be made approximately equal for alldata points. To place an upper bound on the uncertainty, an assumption that GHg ≈ GHg,maxcan be made. The total variance of each point is thenσ2 ≤ σ2dn,c′,i + k22(2σ(B0,Hg)2∆HHg2+σ(∆HHg)2∆HHg2GHg2max). (C.30)Assuming equal uncertainties for each point, the increase in uncertainty becomesσ(dn)2σ(dn,c)2≤ 1n√1 +k22σ(dn,c)2(2σ(B0,Hg)2∆HHg2+σ(∆HHg)2∆HHg2GHg2max). (C.31)316Appendix DUCN Cleaning ProcedureD.1 PrefaceThis appendix describes the cleaning procedure used to prepare the cell for the experimentsTCN23-010-SL and TCN23-020-SL of chapter 5.This procedure was based on the UHV-cleaning procedure in TRIUMF internal document44430 originally authored by Mel Good and Dimo Yosifov. The procedure was modified forlarger parts that could not be cleaned with an ultrasonic bath. Other modifications were madeto adapt the procedure to the specific concerns of cleaning parts for UCN handling.D.2 CleaningD.2.1 Small PartsSmall parts includes anything that can be cleaned in a benchtop ultrasonic bath, such asfasteners or internal components of valves. These parts are not typically visible to UCN butoccupy the same volume. The concern for these parts is that any contamination could migrateto UCN-visible surfaces.Parts should be cleaned in small batches of the same material. Different materials shouldnot be mixed in ultrasonic baths, as they can react with each other. Parts should not be cleanedas assemblies, as there is a risk of cleaning solution becoming trapped between mating surfaces.Required Items• Ultrasonic bath• Simple Green Extreme Aircraft and Precision Cleaner• Citranox Acid Cleaner• Deionized water supply• Isopropanol (reagent grade)• Clean glass beakers (sufficient to contain parts)• Toothbrushes (manual or electronic)• Metal trays (sufficiently deep to submerge parts)• Gloves• Optional: Heat lamp• Forceps/tweezers (for removing parts from solutions)317D.2. Cleaning• Lint-free wipes (ISO class 5)• Clean storage containersNotes• Clean gloves should always be worn when handling parts. When putting on gloves, donot touch any part of the glove that will touch a part. Gloves become dirty as soon asyou touch anything that isn’t a part (a door handle, a benchtop, etc.). You will need tochange gloves often to prevent cross-contamination. If a glove becomes damaged, replaceit immediately.• The formulation of the Simple Green Extreme Aircraft and Precision Cleaner is specificallyintended to be used for degreasing aluminium parts, and seems to work well for othermaterials. Using other formulas in an ultrasonic bath with aluminium tends to causevisible oxidation/damage to the parts.• Citranox Acid Cleaner is intended to be used for cleaning oxides and other inorganiccontaminants. It is observed to work well for removing rust, etc. Its use is strictlyoptional and only for the case where some inorganic contamination is suspected.• Beakers of cleaning solution should ideally be fresh for each batch of parts, but re-use isnot strictly forbidden. Solutions should not be reused too often, and should not be reusedfor different materials. The final deionized water bath should be fresh.• For any new material, always test cleaning on a sample before moving on to importantparts. Softer materials can be especially vulnerable to ultrasonic cleaning. Zinc-bearingalloys such as brass can respond especially poorly to ultrasonic cleaning.• The ultrasonic bath times of ∼5 minutes listed here are shorter times intended for alu-minium, coppers, bronzes, etc. Steel parts can be cleaned for longer if desired.• It is preferable that parts are not placed directly into the ultrasonic bath. Instead, theyshould be placed into solution-filled glass beakers. The beakers are placed in the bath.This avoids cross-contamination from the bath water to the cleaning solution. The bathshould be filled so that the level of the solution in the beakers is at or below the level ofthe bath. Larger parts can be suspended directly in the bath, but care must be takenthat the bath solution is clean.• To remove parts from a solution, use tweezers or forceps. These should be made from amaterial that does not react with the solution, and stainless steel is ideal. As with alltools, they should be thoroughly pre-cleaned before you start cleaning. This can be donewith acetone and a clean cloth, or with Simple Green and the ultrasonic bath. Betweenuses, clean these tools with isopropanol to remove cleaning solution residue.• Long drying times add an increased risk of oxidation for many parts (especially when wetwith deionized water). Drying can be sped up by using a heat lamp, or by blotting witha clean-room wipe (ISO 3 or better).318D.2. CleaningProcedure1. Read all the notes above. They contain important information.2. Ultrasonic Preparation:(a) Fill the ultrasonic bath with a mixture of water and a bit of Simple Green (thishelps the ultrasonic action). Tap water is better here, also to improve the ultrasonicaction.(b) Run the bath for 15 minutes with nothing inside.3. Manual Degreasing: the purpose of this step is to remove any very-heavy contaminationor machining residue prior to ultrasonic cleaning.(a) Fill a tray with a solution of 20% Simple Green and warm water (tap water is fineat this stage).(b) Place some of the parts to be cleaned in the bath and allow to soak for ∼5 minutes.(c) After soaking, scrub parts with toothbrushes in the bath.(d) Rinse the parts thoroughly with tap water, then rinse them with deionized waterand transfer them into a clean beaker.4. Optional Ultrasonic Acid Cleaning: this is an optional step that can be carried outin the case that you suspect inorganic contamination(a) Place the parts in a beaker containing a solution of 20% Citranox and deionizedwater.(b) Place the beaker in the ultrasonic bath. Run the bath for 5 minutes.(c) Remove the parts from the beaker and rinse thoroughly with deionized water.5. Ultrasonic Degreasing: this step will clean organic contamination(a) Place the parts in a beaker containing a solution of 20% Simple Green ExtremeAircraft Precision Cleaner and deionized water.(b) Place the beaker in the ultrasonic bath. Run the bath for 5 minutes.(c) Remove the parts from the beaker and rinse thoroughly with deionized water.6. Final Rinse and Drying: this step will reduce any traces of cleaning solution(a) Place the parts in a fresh beaker of deionized water.(b) Place the beaker in the ultrasonic bath. Run the bath for 5 minutes.(c) Remove the parts from the beaker and rinse again with deionized water.(d) Place the parts on clean wipes to air-dry. A heat lamp can speed up this process.(e) Once dry, transfer parts into a clean container until assembly.D.2.2 Large PartsLarge parts include anything that cannot go in the ultrasonic bath and can only be cleanedmanually. Generally, this covers surfaces that will be exposed to UCNs.Each large part should be cleaned individually. Parts should not be cleaned as assemblies,as there is a risk of cleaning solution becoming trapped between mating surfaces.319D.2. CleaningRequired Items• Simple Green Extreme Aircraft Precision Cleaner• Citranox Acid Cleaner• Deionized water supply• Acetone• Isopropanol (reagent purity)• Brushes (e.g. kitchen dish brushes)• Bottle brushes• Two large baths (sufficient to submerge parts, e.g. paddling pool)• Gloves• Lint-free wipes (ISO class 5)• Clean-room bagsNotes• Clean gloves should always be worn when handling parts. When putting on gloves, donot touch any part of the glove that will touch a part. Gloves become dirty as soon asyou touch anything that isn’t a part (a door handle, a benchtop, etc.). You will need tochange gloves often to prevent cross-contamination. If a glove becomes damaged, replaceit immediately.• Lower-grade (ISO 5) clean wipes can be used for cleaning the parts in the degreasingstep. A final wiping of surfaces with a high-grade wipe (ISO 3) and isopropanol shouldbe performed in a clean-room prior to assembly to remove dust.• For large parts, the optional Citranox cleaning was done as a first step. It isn’t confirmedthat this step is useful, and if so when to do it in the order of cleaning. Since largeparts could not be ultrasonically-cleaned, it was decided to do this step first so that anyresidue of the Citranox would be removed in the submerged cleaning step (this was lessof a concern for small parts, which would be ultrasonically cleaned after the Citranoxcleaning).• The bath and rinsing of parts may require a lot of deionized water. The main controlroom should be alerted if you will be drawing a large quantity of deionized water.Procedure1. Read all the notes above. They contain important information.2. Bath Preparation:(a) Clean out both baths.(b) Fill one bath with a mixture of deionized water and 20% Simple Green.320D.2. Cleaning(c) Leave the other bath empty. This bath will be used for rinsing.3. Optional Acid Cleaning: this is an optional step that can be carried out in the casethat you suspect inorganic contamination(a) Prepare a solution of 20% Citranox and deionized water(b) Pour some solution onto a fresh wipe, or onto the surface to be cleaned (this canmore thorough, but messier). Wipe the surface thoroughly with the solution. Repeatuntil part has been completely wiped and there is no visible residue on the wipe.(c) Pour deionized water onto a fresh wipe, or onto the part. Wipe the surfaces thor-oughly with the deionized water until all residue of the cleaning solution is gone.4. Submerged Cleaning: the purpose of this step is to remove any very-heavy organiccontamination or machining chips.(a) Submerge the part in the Simple Green bath.(b) Scrub the part with brushes. Pay special attention to holes and other tight spaces— use bottle brushes.(c) Remove the part from the bath.(d) Hold the part in the second bath and rinse with deionized water until all soap residueis removed.5. Degreasing: the purpose of this step is to thoroughly clean the surface of any organiccontaminants.(a) Prepare a solution of 20% Simple Green and deionized water.(b) Optional: Wipe the surface using acetone and clean wipes. This step can be in-cluded if heavy organic contamination is suspected, as acetone may be better atremoving such contamination. However, the acetone will leave its own residue, sothis step should be done before the following steps.(c) Wipe the surface thoroughly using the Simple Green solution and clean wipes. Con-tinue to pay attention to holes and other small features where contamination couldbe trapped. Clean brushes can be used here again to clean these features.(d) Wipe the part thoroughly using deionized water and clean wipes until no soap residueremains.(e) Wipe the part thoroughly with isopropanol.6. Drying:(a) If thoroughly wiped with isopropanol, the part should be dry as the isopropanol willhave evaporated rapidly.(b) If the part is not dry, allow to air-dry in a clean environment.(c) Once dry, bag the part in a clean-room bag until assembly.321D.2. CleaningD.2.3 O-RingsO-rings should ideally be purchased pre-baked and come in individually sealed bags. If not,then they should be cleaned and baked prior to use. Contamination of o-rings can be a majorsource of contaminants in vacuum volumes, either due to the manufacturing process or fromdirty handling.For the cell, the two large O-rings that seal the insulator ring were purchased pre-baked.Smaller O-rings throughout the cell valve were not, and so they were cleaned and baked priorto use.At the time of preparing the cell, no suitable vacuum oven was available at TRIUMF. ForO-ring baking, a makeshift oven was assembled. This was done by attaching a short UCNguide with a KF pumping connection to a Pfeiffer HiCube pumping station, and wrapping theguide with a flexible heater tape. Thermocouples were attached to the guide to monitor thetemperature.Required Items• Isopropanol (reagent purity)• Vacuum oven• Gloves• Lint-free wipes• Clean-room bags• Optional: Bottle of clean nitrogenNotes• Clean gloves should always be worn when handling parts. When putting on gloves, donot touch any part of the glove that will touch a part. Gloves become dirty as soon asyou touch anything that isn’t a part (a door handle, a benchtop, etc.). You will need tochange gloves often to prevent cross-contamination. If a glove becomes damaged, replaceit immediately.• Caution should be applied in the choice of vacuum oven for baking. The history of theoven should be well-known, as depending on how the oven has previously used it couldintroduce more contamination to the O-rings. This is especially important to consider ifthe oven has been used with oily pumps. An oven used for vacuum baking should haveonly been used with dry pumps and clean piping.Procedure1. Read all the notes above. They contain important information.2. Pre-cleaning: the purpose of this step is to remove macroscopic contamination beforebaking(a) Using a lint-free wipe, wipe each O-ring using isopropanol. Repeat until wipe comesaway clean.322D.3. Assembly(b) Note that when wiping a black O-ring, the wipe will likely come away black. This isapparently loose elastomer that is removed by the wiping, and it may take severalwipes before significant material stops appearing.3. Baking: the purpose of this step is to removed absorbed contaminants from the O-ring.(a) Place each O-ring into the oven. O-rings should have sufficient space to not be incontact with each other.(b) Seal and pump the oven down to base pressure.(c) Heat the oven to 100◦C.(d) Allow the O-rings to bake for at least 12 hours.(e) Turn off the oven.(f) Optional: Vent oven with clean nitrogen. The oven can also be vented to air, butnitrogen venting will reduce moisture absorbed into the O-rings and can improveperformance in vacuum when the O-rings are used.(g) Remove O-rings and store in sealed bags until used.D.3 AssemblyHaving exerted a great effort to clean all parts to a high standard, it is critical that the assemblyprocess does not undo all of that work. While the exact procedure will be different for anyassembly, there are certain guidelines that should be followed to ensure that the assembly doesnot reintroduce contamination.• It is strongly recommended that assembly is performed by two people. One person shouldtouch only clean parts, while the other provides assistance with tool handling.• Assembly should be done in a cleanroom in order to minimize contamination.• As with the cleaning procedures, clean gloves should always be worn when handling parts.When putting on gloves, do not touch any part of the glove that will touch a part. Glovesbecome dirty as soon as you touch anything that isn’t a part (a door handle, a benchtop,etc.). You will need to change gloves often to prevent cross-contamination. If a glovebecomes damaged, replace it immediately.• All tools to be used in the assembly should be prepared and cleaned beforehand. Partsof tools that will contact parts of the assembly should be cleaned by wiping with acetoneuntil there is no visible residue, and then wiped with isopropanol. Ideally, a set of “cleantools” should be maintained that are only handled in the cleanroom by gloved hands, butthis cannot be guaranteed in shared facilities.• When finishing assembles, a final wiping of UCN-visible surfaces with an ISO 3 wipe andisopropanol should be done to remove any dust that may have entered the assembly.• Assemblies should be finished in a “closed” state. For the cell, this means that all sealsare tightened, a clean blank flange is secured to the cell valve opening, and a clean KFblank is secured to the pumping port. With other assemblies, similar blank flanges shouldbe prepared and used to seal the internal volume. In the worst case, foil can be used tocover openings, but care should be taken that the foil is cleaned to vacuum standards andthat it is properly secured over the opening.323D.3. Assembly• When installing assemblies, both the assemblies and the existing installation should re-main sealed until just prior to installation. Installation should also be done with cleangloves, and caution should be applied regarding touching of O-rings and connectingflanges.324Appendix EDerivation of the Frequency Shiftfrom AC RippleThis appendix derives the frequency shift associated with a circulating magnetic field induced byan oscillating voltage signal in the supply voltage of the nEDM experiment. This frequency shiftcan be calculated using the results of Ramsey [328], but the construction of the perturbationmatrix is somewhat involved.The AC voltage signal produces a magnetic field~BAC = −r|BAC| cos (ωACt)φˆ, (E.1)where r is the radial distance from the rotational axis of the cell, and φˆ is the azimuthal anglein cylindrical coordinates. Noting that φˆ = −yr xˆ+ xr yˆ, the Hamiltonian of this perturbation isHˆAC = −γn~S · ~BAC = γn~2|BAC| cos (ωACt)(Sx Sy Sz)−yx0=γn~2|BAC| cos (ωACt)(0 −y − xi−y + xi 0).(E.2)Ramsey’s result is for a perturbation of the form(0 ~b2ei(ω2+θ)t~b2e−i(ω2+θ)t 0), (E.3)and so applying this result to the above Hamiltonian requires some gymnastics. Starting bysubstituting for the cosineHˆAC = γn~2|BAC|2(eiωAC + e−iωAC)(0 −y − ix−y + ix 0)=γn~2|BAC|2(− y(0 eiωACe−iωAC 0)− y(0 e−iωACeiωAC 0)+x(0 −ieiωACie−iωAC 0)+ x(0 −ie−iωACieiωAC 0)).(E.4)Then, noting that i = eipi/2 and −i = e−ipi/2HˆAC = γn~2|BAC|2(− y(0 eiωACe−iωAC 0)− y(0 e−iωACeiωAC 0)+x(0 ei(ωAC−pi/2)e−i(ωAC−pi/2) 0)+ x(0 e−i(ωAC+pi/2)ei(ωAC+pi/2) 0)).(E.5)325Appendix E. Derivation of the Frequency Shift from AC RippleFrom this result, it is clear that BAC is a superposition of four perturbations of the type inRamsey’s result. The frequency shift in Ref. [328] isδω =(2b2)22(ωn − ω2) . (E.6)In the AC field case, (2b2)2 is either (γn|BAC|)2x2/4 or (γn|BAC|)2y2/4, with ω2 = ±ωAC. Thisleads to the final resultδωAC =γn2|BAC|2r28(ωn − ωAC) +γn2|BAC|2r28(ωn + ωAC)=γn2|BAC|2r2ωn4(ωn2 − ωAC2) ,(E.7)or, averaging over r2δωAC =γn2|BAC|2R2ωn4(ωn2 − ωAC2) .(E.8)326Appendix FAdditional Material onFerromagnetic DipolesF.1 Single-Angle Dipole PlotsThis appendix shows plots of allowable dipole magnitudes for single values of the polar angleθ. Since the geometry is rotationally symmetric, all dipoles are taken to be entirely in the zrplane. These plots illustrate the cancellations of the systematic effects at specific locations andangles. Figure 7.9 was generated by taking the minimum allowed magnitude over these angles,and there are some faint artifacts in this figure related to the resolution of the angular scan.F.2 PTB Samples Detailed ListThe table F.1 summarizes more of the details of the samples tested at PTB, focusing on sizesand suppliers.327F.2. PTB Samples Detailed List0 0.1 0.2 0.3 0.4 0.5 0.6r (m)-0.100.10.20.3z (m)10-1010-910-810-710-610-5Allowed Magnitude (Am2)(a) θ=0◦0 0.1 0.2 0.3 0.4 0.5 0.6r (m)-0.100.10.20.3z (m)10-1010-910-810-710-610-5Allowed Magnitude (Am2)(b) θ=30◦0 0.1 0.2 0.3 0.4 0.5 0.6r (m)-0.100.10.20.3z (m)10-1010-910-810-710-610-5Allowed Magnitude (Am2)(c) θ=60◦0 0.1 0.2 0.3 0.4 0.5 0.6r (m)-0.100.10.20.3z (m)10-1010-910-810-710-610-5Allowed Magnitude (Am2)(d) θ=90◦Figure F.1: Map of the calculated magnitude of a magnetic dipole, located at any given pointin the region surrounding the top cell, that will cause a false EDM signal of 1.26× 10−29 ecm,with dipole orientations of θ=0◦, 30◦, 60◦, and 90◦, with φ = 0◦. The origin of the coordinatesystem is at the centre of the central region, and the black region is the interior of the top cell,where no dipole should be present.328F.2. PTB Samples Detailed List0 0.1 0.2 0.3 0.4 0.5 0.6r (m)-0.100.10.20.3z (m)10-1010-910-810-710-610-5Allowed Magnitude (Am2)(a) θ=-30◦0 0.1 0.2 0.3 0.4 0.5 0.6r (m)-0.100.10.20.3z (m)10-1010-910-810-710-610-5Allowed Magnitude (Am2)(b) θ=-60◦0 0.1 0.2 0.3 0.4 0.5 0.6r (m)-0.100.10.20.3z (m)10-1010-910-810-710-610-5Allowed Magnitude (Am2)(c) θ=120◦0 0.1 0.2 0.3 0.4 0.5 0.6r (m)-0.100.10.20.3z (m)10-1010-910-810-710-610-5Allowed Magnitude (Am2)(d) θ=150◦Figure F.2: Map of the calculated magnitude of a magnetic dipole, located at each point in themap, that will cause a false EDM signal of 1.26×10−29 ecm, with dipole orientations of θ=-30◦,-60◦, 120◦, and 150◦, with φ = 0◦. The origin of the coordinate system is at the centre of thecentral region, and the black region is the interior of the top cell, where no dipole should bepresent.329F.2.PTBSamplesDetailedListSample name DescriptionABS CsM clamshell #1 [M] Southwest Sciences, ∼8 cm, black discolorationABS CsM clamshell #2 [M] Southwest Sciences, ∼4 cmABS CsM clamshell #3 [M] Southwest Sciences, ∼2 cmABS CsM clamshell #4 [M] Southwest Sciences, ∼2 cmPLA 3D-printed sample White PLA cylinder, 3D-printed at TRIUMFNylon screw #1 M4, flathead, unknown supplierNylon screw #2 M6, socket, unknown supplierNylon screw #3 M4, socket, part of CsM assemblyNylon screw #4 M4, assembly, part of CsM assemblyNylon nut Nut, unknown supplierUnknown alloy screw Aluminium, M4, socket, unknown supplierBrass screw #1 M6, socket, unknown supplierNylon bearing Nylon races, unknown ball bearings, unknown supplierOptical fibre sample #1 [B/M] Glass sheathed in black plasticOptical fibre sample #2 [B/M] Glass sheathed in black plasticOptical fibre sample #3 [B/M] Glass sheathed in black plastic, with lens attachmentBlack anodized plate [M] Black-anodized plateNylon 3D-printed sample #1 [B] Black-coloured 3D-printed optical limit switchNylon 3D-printed sample #2 [B] Black-coloured 3D-printed optical limit switchUltem sample [A] Raw material sample, transparent amber, McMaster-CarrTorlon sample [A] Raw material sample, black, McMaster-CarrPEEK sample [A] Raw material sample, off-white, McMaster-CarrViton O-ring sample #1 [A] G90 Viton O-ring, blackViton O-ring sample #2 [A] G90 Viton O-ring, blackPEEK screw #1 M8, 30mm, McMaster-CarrPEEK screw #2 M8, 30mm, McMaster-CarrPEEK screw #3 M8, 30mm, McMaster-CarrPEEK screw #4 M8, 30mm, McMaster-CarrPEEK screw #5 M8, 30mm, McMaster-CarrPhosphor bronze spring #1 [A] 15 mm, Victory SpringPhosphor bronze spring #2 [A] 15 mm, Victory SpringPhosphor bronze spring #3 [A] 15 mm, Victory Spring330F.2.PTBSamplesDetailedListPhosphor bronze spring #4 [A] 15 mm, Victory SpringSilicon bronze screw #1 M8, 30 mm, McMaster-CarrSilicon bronze screw #2 M8, 30 mm, McMaster-CarrSilicon bronze screw #3 M8, 30 mm, McMaster-CarrSilicon bronze screw #4 M8, 30 mm, McMaster-Carr360 Brass screw #1 M8, 30 mm, McMaster-Carr360 Brass screw #2 M8, 30 mm, McMaster-Carr360 Brass screw #3 M8, 30 mm, McMaster-Carr360 Brass screw #4 M8, 30 mm, McMaster-CarrAl 2024 screw #1 M8, Phillips, 30 mm, McMaster-CarrAl 2024 screw #2 M8, Phillips, 30 mm, McMaster-CarrAl 2024 screw #3 M8, Phillips, 30 mm, McMaster-CarrAl 2024 screw #4 M8, Phillips, 30 mm, McMaster-CarrAl 2024 screw #5 M8, Hex, 30 mm, McMaster-CarrAl 2024 screw #6 M8, Hex, 30 mm, McMaster-CarrAl 2024 screw #7 M8, Hex, 30 mm, McMaster-CarrAl 2024 screw #8 M8, Hex, 30 mm, McMaster-CarrGrade 2 screw #1 M8, 30 mm, McMaster-CarrGrade 2 screw #2 M8, 30 mm, McMaster-CarrGrade 2 screw #3 M8, 30 mm, McMaster-CarrGrade 2 screw #4 M8, 30 mm, McMaster-CarrBrass screw set #1 M6. 4 mm, McMaster-CarrBrass screw set #2 M6, 4 mm, McMaster-CarrBrass screw set #3 M6, 4 mm, McMaster-CarrBrass screw set #4 M6, 4 mm, McMaster-CarrUnknown grade screw #1 [B] Ti, M6, 4 mm, MisumiUnknown grade screw #2 [B] Ti, M6, 4 mm, MisumiUnknown grade screw #3 [B] Ti, M6, 4 mm, MisumiUnknown grade screw #4 [B] Ti, M6, 4 mm, MisumiMachined Al 6061 #1 [A] Hg valve body KF, machined at TRIUMFMachined Al 6061 #2 [A] Hg valve body KF, machined at TRIUMFMachined Al 6061 #3 [A] Hg valve body internal, machined at TRIUMFMachined Al 6061 #4 [A] Hg valve body plate, machined at TRIUMF331F.2.PTBSamplesDetailedListMachined Al 6061 #5 [A] Hg valve body block, machined at TRIUMFMachined unknown bronze sample #1 [A] Hg valve bushing, machined at TRIUMFMachined unknown bronze sample #2 [A] Hg valve bushing, machined at TRIUMFMachined tin bearing bronze sample [A] Hg valve pin, machined at TRIUMFMachined Al 6061 #6 [A] Hg valve internal gate, machined at TRIUMFPTFE sample #1 [A] Hg valve shaft, cut at TRIUMF, unknown stockPTFE sample #2 [A] Hg valve shaft, cut at TRIUMF, unknown stockActuator part #1 [B] Actuator end block, McMaster-CarrActuator Al and brass part #1 [B] Actuator bearing block, McMaster-CarrActuator part #2 [B] Actuator internal piston, McMaster-CarrActuator part #3 [B] Actuator cylinder, McMaster-CarrGrade 2 machined #1 [B] Actuator shaft, machined at TRIUMFGrade 2 machined #2 [B] Actuator retaining rod, machined at TRIUMFGrade 2 machined #3 [B] Actuator retaining rod, machined at TRIUMFGrade 2 machined #4 [B] Actuator retaining rod, machined at TRIUMFGrade 2 machined #5 [B] Actuator retaining rod, machined at TRIUMFActuator part #4 [B] Actuator internal half-ring, McMaster-CarrActuator part #5 [B] Actuator internal half-ring, McMaster-CarrMachined phosphor bronze sample #1 [B] Actuator internally-threaded nut, machined at TRIUMFMachined phosphor bronze sample #2 [B] Actuator internally-threaded nut, machined at TRIUMFMachined phosphor bronze sample #3 [B] Actuator internally-threaded nut, machined at TRIUMFMachined phosphor bronze sample #4 [B] Actuator internally-threaded nut, machined at TRIUMFActuator CuBe retaining ring [B] McMaster-CarrBrass pipe fitting #1 [B] NPT elbow, unknown supplierBrass pipe fitting #2 [B] NPT elbow, unknown supplierBrass pipe fitting #3 [B] Polyflo NPT adapter, unknown supplierBrass pipe fitting #4 [B] Polyflo NPT adapter, unknown supplierGrade 2 nut [B]Essex brass clamp sample #1 [C] Sample from Essex X-RayEssex brass clamp sample #2 [C] Sample from Essex X-RayEssex HV cable sample [C] Sample from Essex X-RayAl 6XXX Extrusion profile [B/M] Sample from Essex X-RayPbSn solder sample #1 [B] ∼3 cm long332F.2.PTBSamplesDetailedListPbSn solder sample #2 [B] ∼3 cm longRexolite PS-coated ring [A] Machined and coated at TRIUMFDLC coating sample #1 [A] ∼1 cm coated silicon(?) waferDLC coating sample #2 [A] ∼1 cm coated silicon(?) waferNi coating sample ∼1 cm coated silicon(?) waferCs magnetometer cell [M] Blown glass, Southwest SciencesGrade 5 screw #1 M8, 20 mm, Allied TitaniumGrade 5 screw #2 M8, 20 mm, Allied TitaniumGrade 5 screw #3 M8, 20 mm, Allied TitaniumGrade 5 screw #4 M8, 20 mm, Allied TitaniumGrade 5 screw #5 M8, 20 mm, Allied TitaniumCeramic-coated Al shaft [A] PBC Linear, machined to length at TRIUMFCeramic roller bearing #1 [A] 2 cm, Boca BearingCeramic roller bearing #2 [A] 4 cm, Boca BearingFrelon-coated Al bushing [A] PBC LinearMachined Al 6061 #7 [A] 1× 2 cm, machined at TRIUMFCuBe spring #1 [A] Acxess Spring, 4” long, 0.5” OD, ground endsCuBe spring #2 [A] Acxess Spring, 4” long, 0.5” OD, ground endsPhosphor bronze spring #5 [A] Acxess Spring, 1.6” long, 0.7” ODPhosphor bronze spring #6 [A] Acxess Spring, 1.6” long, 0.7” ODPhosphor bronze spring #7 [A] Acxess Spring, 1.6” long, 0.7” ODCuBe retaining ring #1 [A] ∼1.5 cm, McMaster-CarrCuBe retaining ring #2 [A] ∼1.5 cm, McMaster-CarrCuBe retaining ring #3 [A] ∼1.5 cm, McMaster-CarrCuBe retaining ring #4 [A] ∼1.5 cm, McMaster-CarrNiP-coated Al Plate [A] 14 cm diameter, machined at TRIUMFTable F.1: Details of the samples tested at PTB.333
Thesis/Dissertation
eng
Physics
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Picker, Ruediger
Madison, Kirk W., 1969-
Studies of ultracold neutron dynamics and systematic effects for the TUCAN EDM experiment
Text