Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Evetts, Nathan
Doctor of Philosophy - PhD
University of British Columbia
The Einstein Equivalence Principle (EEP) has never been directly examined with an antimatter test body. To address this, ALPHA is planning to measure the Earth’s gravitational ﬁeld using antihydrogen atoms as test masses. The experiment calls for the careful release of antiatoms from a magnetic trap and requires precise characterization of the magnetic ﬁelds that are used. This thesis considers the need for magnetic ﬁeld characterization from the perspective of Nuclear Magnetic Resonance (NMR) as a magnetic sensing technique. A variety of theories from statistics and condensed matter physics are assembled into a framework for understanding and predicting the performance of solid-state NMR magnetometers. This framework is applied to two types of NMR sensors that were developed for ALPHA’s gravity experiments. Data acquired while commissioning the apparatus provide experimental conﬁrmation of the theoretical precision predicted for each sensor type. A room temperature sensor based on ¹H NMR in natural rubber exhibits a precision of about 270 nT at 1 T. A cryogenic sensor based on ²⁷Al NMR in microparticulate aluminium displays a precision of about 34 µT at 25 K and 1 T. These NMR data are then used to estimate the experimental constraint that should result from upcoming measurements of the gravitational ﬁeld (g¯) experienced by antihydrogen atoms in the lab. Improvements to the expected constraint on g¯ are sought through advances in state-of-the-art solid-state NMR magnetometry. A broad survey of candidate materials has been conducted in search of NMR samples that promise better precision at low temperature. Yttrium-bismuth is proposed as a sample with the potential to improve sensor precision by a factor of ∼7 over the use of aluminium. Further gains in performance are predicted through the introduction of broadband (or “delay line”) probes. A brief review and assessment of these probes is presented.
Solid-State Nuclear Magnetic ResonanceMagnetometry at Low Temperaturewith application to antimatter gravity experiments byALPHAbyNathan EvettsB.Sc., The University of British Columbia, 2010M.Sc., The University of British Columbia, 2015A 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)March 2021c© Nathan Evetts 2021The following individuals certify that they have read, and recommend to theFaculty of Graduate and Postdoctoral Studies for acceptance, the disserta-tion entitled:Solid-State Nuclear Magnetic Resonance Magnetometry at Low Temperaturewith application to antimatter gravity experiments by ALPHAsubmitted by Nathan Evetts in partial fulfilment of the requirements forthe degree of Doctor of Philosophyin PhysicsExamining Committee:Michael Hayden, Department of Physics, SFUResearch SupervisorCarl Michal, Department of Physics & Astronomy, UBCAcademic Co-SupervisorStefan Reinsberg, Department of Physics & Astronomy, UBCUniversity ExaminerAndrew MacFarlane, Department of Chemistry, UBCUniversity ExaminerWerner Heil, Institute of Physics, Johannes Gutenberg Universität MainzExternal ExaminerAdditional Supervisory Committee Members:Mark Halpern, Department of Physics & Astronomy, UBCSupervisory Committee MemberMarcel Franz, Department of Physics & Astronomy, UBCSupervisory Committee MemberWalter Hardy, Department of Physics & Astronomy, UBCSupervisory Committee MemberiiAbstractThe Einstein Equivalence Principle (EEP) has never been directly exam-ined with an antimatter test body. To address this, ALPHA is planningto measure the Earth’s gravitational field using antihydrogen atoms as testmasses. The experiment calls for the careful release of antiatoms from a mag-netic trap and requires precise characterization of the magnetic fields thatare used. This thesis considers the need for magnetic field characterizationfrom the perspective of Nuclear Magnetic Resonance (NMR) as a magneticsensing technique.A variety of theories from statistics and condensed matter physics are as-sembled into a framework for understanding and predicting the performanceof solid-state NMR magnetometers. This framework is applied to two typesof NMR sensors that were developed for ALPHA’s gravity experiments. Dataacquired while commissioning the apparatus provide experimental confirma-tion of the theoretical precision predicted for each sensor type. A roomtemperature sensor based on 1H NMR in natural rubber exhibits a preci-sion of about 270 nT at 1 T. A cryogenic sensor based on 27Al NMR inmicroparticulate aluminium displays a precision of about 34µT at 25 K and1 T. These NMR data are then used to estimate the experimental constraintthat should result from upcoming measurements of the gravitational field(g¯) experienced by antihydrogen atoms in the lab.Improvements to the expected constraint on g¯ are sought through ad-vances in state-of-the-art solid-state NMR magnetometry. A broad surveyof candidate materials has been conducted in search of NMR samples thatpromise better precision at low temperature. Yttrium-bismuth is proposedas a sample with the potential to improve sensor precision by a factor of∼7 over the use of aluminium. Further gains in performance are predictedthrough the introduction of broadband (or “delay line”) probes. A brief re-view and assessment of these probes is presented.iiiLay SummaryAntimatter is not an invention of science fiction – it is a real, though ex-tremely rare and exotic substance. Its rarity is particularly troubling sinceour foremost theories of physics predict that it should be about as commonas dirt.To study antimatter, a group of scientists (calling themselves ALPHA)have built a machine to make and store antimatter atoms. This work de-scribes ALPHA’s plan to open up their antimatter container, empty thecontents, and observe the influence of gravity as the antiatoms fall out.To do this right requires a detailed understanding of the magnetic fieldsthat form the antimatter container. To this end, I implement an existingtechnique that senses magnetic fields by the microscopic “bar-magnet” onthe nucleus of many (matter) atoms. Using an in-depth understanding ofthat nuclear-bar-magnet, I propose performance improvements for this classof magnetic field sensor.ivPrefaceALPHA is an international collaboration that aims to test fundamental sym-metries in physics using experiments with antihydrogen atoms. The group iscurrently composed of about 50 scientists from 16 institutes operating an ap-paratus located at the European Organization for Nuclear Research (CERN)in Geneva, Switzerland. Any thesis regarding the goals and accomplishmentsof ALPHA necessarily relies on the work of others.As a member of ALPHA, I spent a cumulative total of approximately 2years in Geneva where I assisted with construction and operation of manyaspects of the apparatus. That apparatus and the associated gravitationalexperiment (described in Chapter 2) represents the work of the entire Col-laboration. My unique contribution to the apparatus is the design and im-plementation of arrays of NMR and Hall magnetometers.The design of the NMR probes described in Chapter 4 is primarily myown work, but was influenced by discussions with Profs. Walter Hardy andMichael Hayden. Integration of these magnetometers with the remainingapparatus was made possible by tight collaboration with Dr. Chris ØrumRasmussen, Dr. Chukman So, Prof. William Bertsche, Dr. Andrea Capra,Dr. Pierre-André Amaudruz, and Dr. Makoto Fujiwara.Dr. Chukman So performed simulations of the field profile for magnets inthe atom trap and the expected trajectories of antihydrogen during gravityexperiments. These simulations are employed in Chapter 5 to interpret NMRmagnetometer precision as a constraint on upcoming measurements of thegravitational field experienced by antihydrogen.Prof. Carl Michal and I took the data presented in Appendix A in theMichal Lab, with two exceptions. The exceptions are the data presented inFigures A.3 and A.6, which I took with the NMR spectrometer of Chapter 4.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction and Context . . . . . . . . . . . . . . . . . . . . . 11.1 Antimatter and Gravity . . . . . . . . . . . . . . . . . . . . . 21.2 Past Experiments Involving Both Antimatter and Gravity . . 41.2.1 A Thought Experiment on Anti-gravity . . . . . . . . 51.2.2 Particles that are their own Antiparticle . . . . . . . . 51.2.3 Cyclotron Frequencies of p/p¯ . . . . . . . . . . . . . . 61.2.4 Shapiro Delay from Supernova (Anti)Neutrinos . . . . 61.2.5 Eötvös Experiments . . . . . . . . . . . . . . . . . . . 71.2.6 Dropping Electrons: Gravity Experiments by Fairbankand Witteborn . . . . . . . . . . . . . . . . . . . . . . 81.3 Present Status and Current Experiments . . . . . . . . . . . 101.4 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . 11vi2 Gravitational Experiments with Antihydrogen in ALPHA-g 132.1 Historical Introduction to ALPHA . . . . . . . . . . . . . . . 132.2 The Prototypical ALPHA-g Experiment . . . . . . . . . . . . 152.3 ALPHA-g Apparatus . . . . . . . . . . . . . . . . . . . . . . 202.4 Overview of Magnetometry Techniques . . . . . . . . . . . . 282.4.1 Electron Cyclotron Resonance Magnetometry . . . . . 302.4.2 NMR Magnetometry . . . . . . . . . . . . . . . . . . 322.5 Magnetometry Strategy for Experiments in ALPHA-g . . . . 343 A Framework for Solid-State NMR Magnetometer Perfor-mance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.1 NMR and the Magnetic Field Measurement . . . . . . . . . . 383.2 Precision and the Cramér - Rao Lower Bound . . . . . . . . 423.3 The NMR linewidth . . . . . . . . . . . . . . . . . . . . . . . 453.3.1 Dipolar Broadening . . . . . . . . . . . . . . . . . . . 453.3.2 Quadrupolar Interaction . . . . . . . . . . . . . . . . 483.3.3 Knight Shift . . . . . . . . . . . . . . . . . . . . . . . 493.3.4 Motional Narrowing . . . . . . . . . . . . . . . . . . . 523.4 The NMR Signal-to-Noise Ratio . . . . . . . . . . . . . . . . 533.4.1 SNR for a Solenoid NMR Coil . . . . . . . . . . . . . 543.4.2 SNR and the Principle of Reciprocity . . . . . . . . . 553.5 NMR Probe Noise Factor . . . . . . . . . . . . . . . . . . . . 563.5.1 NMR Probe Circuits . . . . . . . . . . . . . . . . . . 573.5.2 MacLaughlin’s Method . . . . . . . . . . . . . . . . . 593.5.3 Noise Temperature for Non-Uniform Cabling . . . . . 613.5.4 Application to Various Probe Designs . . . . . . . . . 623.6 Absolute Accuracy of the NMR Magnetic Field Measurement 653.6.1 Magnetic Susceptibility of the Sample . . . . . . . . . 653.6.2 Gyromagnetic Ratio . . . . . . . . . . . . . . . . . . . 673.7 Sensor Repetition Rate and Spin-Lattice Relaxation . . . . . 693.7.1 Relaxation in Metals . . . . . . . . . . . . . . . . . . 693.7.2 Relaxation in Semiconductors . . . . . . . . . . . . . 703.7.3 Relaxation from Paramagnetic Impurities . . . . . . 703.7.4 Ernst Angle . . . . . . . . . . . . . . . . . . . . . . . 713.8 General Remarks on Low Temperature NMR . . . . . . . . 723.8.1 Metals . . . . . . . . . . . . . . . . . . . . . . . . . . 723.8.2 Phase Transitions . . . . . . . . . . . . . . . . . . . . 73vii4 Magnetometers in the ALPHA-g Apparatus . . . . . . . . . 764.1 Spectrometer Hardware . . . . . . . . . . . . . . . . . . . . . 774.2 Room-temperature NMR Probes . . . . . . . . . . . . . . . . 784.2.1 Physical Description and Performance Analysis . . . . 784.2.2 ALPHA-g Magnetic Field Environment Observed . . 844.3 Cryogenic NMR Probes . . . . . . . . . . . . . . . . . . . . . 854.3.1 Physical Description and Performance Analysis . . . . 865 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.1 NMR Magnetometry for Antihydrogen Gravity Experiments 935.1.1 Magnetic Field Homogeneity . . . . . . . . . . . . . . 945.1.2 Predicted Constraints for g¯ Measurements . . . . . . 995.2 Improving State-of-the-Art Solid-State NMR Magnetometry 1035.2.1 Solid Samples with Improved Linewidths . . . . . . . 1045.2.2 The Broadband Probe . . . . . . . . . . . . . . . . . . 1075.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118AppendixA Notes on Novel Samples for Solid-State NMR Magnetome-try . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133viiiList of Tables1.1 Summary of indirect experimental evidence concerning anti-matter and gravity. . . . . . . . . . . . . . . . . . . . . . . . . 102.1 Size and precision of ALPHA-g gravity signal in magnetic units. 183.1 Second-moment lattice factors for various lattices. . . . . . . . 473.2 Characteristics of coaxial cables for low temperature NMR. . 613.3 Demagnetizing factors for special ellipsoids. . . . . . . . . . . 663.4 Absolutely referenced gyromagnetic ratios for common NMRmagnetometer samples (aluminium, rubber, water). . . . . . . 673.5 Critical fields and temperatures are given for superconductingmaterials under consideration as NMR magnetometer samples. 754.1 List of experimental parameters describing the room temper-ature NMR magnetometers in ALPHA-g. . . . . . . . . . . . 824.2 List of experimental parameters describing low temperatureNMR magnetometers in ALPHA-g. . . . . . . . . . . . . . . . 885.1 NMR sample geometries that mitigate magnetic inhomogene-ity issues during gravity experiments in ALPHA-g. . . . . . . 965.2 Magnetic field precision from alternative NMR samples. . . . 106A.1 Linewidths for candidate low temperature NMR samples. . . 134A.2 Material properties of candidate low temperature NMR samples.135ixList of Figures2.1 The ALPHA apparatus . . . . . . . . . . . . . . . . . . . . . 142.2 Concept for the ALPHA-g experiment . . . . . . . . . . . . . 152.3 Magnetic compensation of the gravitational field vs. antiatomescape direction . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4 The ALPHA apparatus in gory detail. . . . . . . . . . . . . . 192.5 Prototypical magnetic trap that confines antihydrogen. . . . . 212.6 ALPHA-g components of the apparatus. . . . . . . . . . . . . 232.7 Wire placement in ALPHA-g magnets. . . . . . . . . . . . . . 242.8 The ALPHA-g annihilation detector. . . . . . . . . . . . . . . 252.9 Superconducting wires in ALPHA-g atom trap magnets. . . . 272.10 Overview of magnetometry techniques: accuracy and range. . 292.11 Electron cyclotron resonance magnetometry in ALPHA-g . . 312.12 Electron cyclotron resonance spectra. . . . . . . . . . . . . . . 322.13 The NMR magnetometry strategy for ALPHA-g . . . . . . . 342.14 Simulated “calibration curve” relating the NMR measurementto the magnetic field experienced by antihydrogen. . . . . . . 353.1 NMR spin magnetization. . . . . . . . . . . . . . . . . . . . . 383.2 A simple NMR experiment for magnetometry. . . . . . . . . . 403.3 Concept for magnetic field measurement with an NMR signal 413.4 CRLB vs. data acquisition parameters . . . . . . . . . . . . . 443.5 Dipolar broadening: The dipole field of one spin perturbs itsneighbours. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.6 The Knight shift . . . . . . . . . . . . . . . . . . . . . . . . . 503.7 An anisotropic Knight shift. . . . . . . . . . . . . . . . . . . . 513.8 T1 and T2 in the presence of motional narrowing. . . . . . . . 533.9 The principle of reciprocity. . . . . . . . . . . . . . . . . . . . 553.10 NMR in a cryostat. . . . . . . . . . . . . . . . . . . . . . . . . 573.11 Various NMR probe circuits. . . . . . . . . . . . . . . . . . . 583.12 RF efficiency and the NMR probe. . . . . . . . . . . . . . . . 593.13 Noise temperature for various NMR probe circuits. . . . . . . 63x3.14 Absolutely-referenced gyromagnetic ratios for common NMRmagnetometer samples (aluminium, rubber, water). . . . . . . 683.15 Spin relaxation times in rubber, near the glass transition. . . 744.1 NMR magnetometer readout system. . . . . . . . . . . . . . . 774.2 Room temperature NMR magnetometers in ALPHA-g. . . . . 794.3 Location of room temperature NMR magnetometers in theALPHA-g apparatus. . . . . . . . . . . . . . . . . . . . . . . . 804.4 A field measurement with the room temperature NMR mag-netometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.5 Effect of beamline magnets on the ALPHA-g environment. . . 844.6 Effect of overhead crane on the ALPHA-g environment. . . . 854.7 Stability of the ALPHA-g external magnetic field. . . . . . . . 854.8 Low temperature NMR magnetometers in ALPHA-g. . . . . . 874.9 A field measurement with the low temperature NMR magne-tometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.10 Precision of the low temperature NMR magnetometer. . . . . 915.1 Magnetic inhomogeneity for regions (outside mirror coil G)that accommodate low temperature NMR samples. . . . . . . 955.2 Anticipated NMR signal during gravity experiments. . . . . . 975.3 Octupole field at the NMR sample location. . . . . . . . . . . 985.4 Fields measured by NMR near analysis and mirror A/G coils. 1005.5 Improved magnetic field constraint from NMR probes. . . . . 1015.6 A constraint on g¯ from NMR magnetic field measurements. . 1025.7 Transmission line circuit model. . . . . . . . . . . . . . . . . . 1085.8 Delay line NMR probe circuit. . . . . . . . . . . . . . . . . . . 1095.9 Calculated noise temperature, quality factor and impedancefor a delay line probe. . . . . . . . . . . . . . . . . . . . . . . 1135.10 Electromagnetic mode matching for the delay line probe. . . . 115A.1 NMR signal of 115In-P at 4.5 T and 300 K . . . . . . . . . . . 136A.2 NMR signal of 23Na-Cl at 4.5 T and 300 K . . . . . . . . . . 138A.3 NMR signal of 1H in rubber at 0.3 T and 10 K . . . . . . . . 138A.4 NMR signals for two different 27Al samples at 4.5 T and 300 K140A.5 Comparison of 27Al linewidths at 4.5 T and 300 K . . . . . . 141A.6 NMR signal of 31P in copper-phosphor alloy at 1 T and 300 K 142A.7 NMR signal of 207Pb at 4.5 T and 300 K . . . . . . . . . . . . 144xiList of SymbolsBo NMR main magnetic fieldB′1 Linear RF spin excitation magnetic fieldB1 Rotating RF spin excitation magnetic field∆B Magnetic imbalance for experiments inALPHA-gBg¯ Gravitational energy difference between thetop and bottom of the ALPHA-g trap, in mag-netic unitsδB CRLB for the NMR field measurementσB Magnetic field inhomogeneityC Correction factor for the CRLB from dampedoscillationsδ Skin depthe Elementary chargee+ PositronΓ Damping rate for the NMR signalg Earth’s gravitational field, as experienced bymatterg¯ Earth’s gravitational field, as experienced byantimatterγ Nuclear gyromagnetic ratioγe Electronic gyromagnetic ratio~ Reduced Plank constant∆H/H Knight shiftI Spin numberK02 Long-lived neutral kaonK01 Short-lived neutral kaonK0 Neutral kaonxiiK0 Anti-neutral kaonk Boltzmann constantLp NMR coil inductancelcoil NMR coil loss factor, or noise factorM2 Second moment for the NMR lineMo Spin magnetization magnitude at thermalequilibriumme Electron massmg Gravitational massmI Inertial massm Spin projection quantum numberµo Permeability of free spaceN Spin densitynD Demagnetizing factorη Filling factorρ Resistivityp Protonp¯ AntiprotonRp NMR coil resistanceSNR Signal-to-noise ratioT TemperatureTNC Noise temperatureT1 Spin-lattice relaxation timetp NMR pulse widthT ∗2 Characteristic decay time for the NMR signalT2 Spin-spin relaxation timeTo NMR signal observation timeχ Magnetic susceptibilityχ2 Chi-squaredxiiiAcronymsAD Antiproton DeceleratorALPHA Antihydrogen Laser PHysics Apparatus Col-laborationALPHA-g Apparatus for ALPHA’s antihydrogen gravityexperimentsBCC Body-Centered CubicCERN European Organization for Nuclear ResearchCPT Charge-conjugation, Parity, Time-reversalsymmetryCRLB Cramér - Rao Lower BoundECR Electron Cyclotron ResonanceEEP Einstein Equivalence PrincipleELENA Extra Low ENergy Antiproton ringFCC Face-Centered CubicFWHM Full Width at Half MaximumGR General RelativityNMR Nuclear Magnetic ResonanceRF Radio-FrequencyRRR Residual Resistivity RatiorTPC radial Time-Projection-ChamberSM Standard ModelTMS TetramethylsilanexivAcknowledgementsWorking with ALPHA has been a privilege. The group is full of brilliantpeople working with an energy and an intensity that is infectious. I want tothank you all. I am particularly grateful to the leaders of the group (whoplaced a shocking amount of trust in me) and to my fellow students (whoshared in the chaos, and the grunt work, and the high times on the Rhône).I’d also like to acknowledge my colleagues at UBC. In the Supercon Lab:Walter, Pinder, James, and Doug allowed me to linger in their space duringthe first years of this work when it seemed I didn’t have anywhere else to go.Thanks also to Carl for welcoming me into Room 100, at the end.To my supervisory team, Mike Hayden, Walter Hardy, and Carl Michal:This work would not have been possible without your support and guidance.Thanks for shepherding me through all the growing pains of this process,from the beginning – when I was new, and mostly barefoot, and full ofoutlandish (sometimes bad) ideas – to the end.A special thanks to Makoto, who really believed in the ALPHA magne-tometry work before anyone else.Last, and most important, thank you to the friends and family memberswho shaped me with their love. I wouldn’t be who I am without you.xv“It’s not rocket appliances...”xviChapter 1Introduction and ContextThis is a thesis about magnetometry. The application for this work is aneffort by the ALPHA Collaboration to directly measure the force of gravityon an antimatter test particle in the Earth’s gravitational field.This introductory chapter motivates that experiment as a test of GeneralRelativity (GR) and a search for “new” physics. So far, experiments involvinggravity and antimatter have confirmed GR to a variety of precisions in diversesettings. Existing experiments in this sector are summarized in Section 1.2.Regrettably, the interpretation of these experimental results has always reliedon assumptions that are not universally accepted.To address this, antihydrogen atoms can be used by ALPHA in experi-ments that test GR more directly. The experimental procedure (and associ-ated apparatus) by which ALPHA intends to accomplish this is discussed inChapter 2. By ALPHA’s method, gravitational effects can be inferred fromthe pattern of annihilation events detected as antiatoms are carefully releasedfrom a magnetic trap. Unfortunately, the magnetic energy of a trapped an-tiatom typically dominates the gravitational energy of interest by a factorof about 104. Precise magnetic field characterization is therefore required toenable precision measurements of the gravitational field (g¯) experienced byantihydrogen atoms in the ALPHA apparatus. The magnetometry strategythat drives this characterization is described in Section 2.5. The approachuses two different techniques: the detection of the cyclotron resonance of anelectron plasma, and the detection of a nuclear spin precession frequency.Chapter 3 focuses on the spin precession technique for magnetometry.A framework for the performance of a nuclear magnetic resonance (NMR)magnetometer is compiled from first principles. Prototype NMR magne-tometers designed and installed in the ALPHA apparatus are described andcharacterized in Chapter 4. Their observed performance is well matched tothe predictions outlined in Chapter 3.Chapter 5 concludes the thesis. The NMR magnetometer performancemodels (Chapter 3), are combined with the observed magnetometer precision(Chapter 4), and characteristics of the ALPHA apparatus (Chapter 2) topredict the experimental precision with which ALPHA can constrain g¯. In1order to tighten this constraint, a general improvement in the precision ofsolid-state NMR magnetometers is sought under the guidance of theory fromChapter 3. The magnetic precision for NMR sensors using a number ofpotential samples is predicted in Section 5.2. A yttrium-bismuth sample isexpected to outperform aluminium - the current “default” sample for workat low temperature. Additional improvements are expected to result fromthe use of broadband (or “delay line”) probes. This type of probe is reviewedin Section 5.2.2.1.1 Antimatter and GravityJust after the Big Bang the Universe is believed to have converted an im-mense amount of energy into matter-antimatter particle pairs. These weresupposedly created in abundance before cooling to produce today’s observ-able universe. This model implicitly predicts a universe populated with equalparts matter and antimatter [1] – a prediction that is in stark contrast withour observations of a matter dominated universe. These observations are oftwo types:1. Antimatter content of cosmic rays.Observations of the particle content of cosmic rays show traces ofpositrons and antiprotons [2, 3], but no heavy antimatter nucleonswhich should have formed around the time of the Big Bang (e.g., an-tihelium [4–7]).2. Gamma ray radiation from matter-antimatter annihilation in space.The absence of gamma or x-ray radiation resulting from matter-antimatterannihilation in the interstellar medium, the intergalactic medium, andin collisions of galaxy clusters, constrain those regions as being domi-nated entirely by matter or antimatter. [8–10]The apparent contradiction between observation and theory, known asthe “Baryon Asymmetry Problem”, remains one of the great unsolved mys-teries in modern physics. Its solution will likely demand “new”, or beyondStandard Model (SM) physics. In spite of this, the Standard Model containswidely accepted theory that makes strong predictions about the propertiesof antimatter. It states that every particle has an antimatter counterpartwith identical mass but opposite charge and spin.Formally, this is expressed via the CPT theorem. CPT symmetry statesthat any local Lorentz-invariant quantum field theory is invariant under2the combined operations of charge-conjugation (C), parity (P), and time-reversal (T) [11]. The invariant nature of the symmetry means that thephysics of any antimatter system should be identical to that of its matteranalogue. Although the theorem provides a powerful and sweeping statementabout nearly all of physics, it does not constrain gravitational interactionsbetween matter and antimatter. This is highlighted by Nieto and Goldman[12] who argue:“Emulating Newton, we observe that if an apple falls toward theEarth in a certain way, CPT only implies that an antiapple fallstoward an antiearth in the same way. CPT says nothing abouthow an antiapple (that is to say an antiproton or a positron) fallstoward an Earth.”For descriptions of a gravitationally interacting matter-antimatter sys-tem, one might look to general relativity. General relativity (GR) consti-tutes our most advanced theory of gravity. Its guiding precept is the Ein-stein equivalence principle (EEP), which states that a uniform gravitationalfield cannot be distinguished from an accelerating reference frame [13]. TheEEP leads us to the conclusion that objects must experience acceleration ina gravitational field regardless of their mass or composition. GR thereforepredicts no gravitational asymmetry between matter and antimatter: it saysthat antimatter and matter bodies “fall the same”. Compelling reasons foraccepting this prediction can be found in the many stringent experimentalverifications of general relativity [14–17]. However, in spite of surviving thesetests, searches persist that are devoted to finding violations of the EEP orextensions to GR. These are motivated in part by the fact that general rel-ativity has not been reconciled with quantum field theory, or the StandardModel of particle physics.In addition to the Baryon Asymmetry Problem, and the non-unificationof the Standard Model with general relativity, there remains another physicsmystery relevant to both antimatter and gravitation: the cosmological con-stant problem. General relativity would describe the expansion of our uni-verse according to its mass and energy content. Observation of this expansionrate implies that we are unaware of approximately 75% of that content. Itwas once thought that this missing mass/energy might simply be that ofthe vacuum which contains virtual, particle-antiparticle pairs. However, theStandard Model energy density that should result from these virtual parti-cles is too large (by 120 orders of magnitude) to be in agreement with theobserved expansion of the universe. The wild disagreement between theory3and experiment has become known as the cosmological constant problem[18], and has led us to question the composition of the universe and supposethe existence of Dark Energy.Cosmologies involving antimatter have been discussed at times over theyears to address these mysteries [9, 19]. Meanwhile, testable models extend-ing conventional physics into the realms of antimatter, Lorentz violation,and gravity have been proposed [20].In the face of these mysteries, however, many physicists place their trustin CPT and the EEP, and expect antimatter to have gravitational propertiesthat are identical to matter. A body of indirect experimental evidence existsin support of these expectations (to be discussed in the following sections).1.2 Past Experiments Involving Both Antimatterand GravityThis section describes experimental results that have been employed to makeinferences concerning either the gravitational properties of antimatter, orviolations of the EEP. The short summaries here lean heavily on reviewsgiven by Fischler et. al. [21]; and Neito and Goldman [12], in which muchgreater detail can be found. See also Ref. [22] for a more recent review.Limits on gravitational asymmetries between matter and antimatter havebeen inferred from four different types of experimental data: gravitationallensing experiments, decay modes of the K02 meson, the cyclotron frequencyof the antiproton, and Shapiro delay of (anti)neutrinos emitted from the1987 supernova. With the exception of gravitational lensing, all of thesemeasurements employ the assumption of an absolute gravitational potential.Usually our galactic potential is used; however, this choice is a bit arbi-trary. If one instead uses the Earth’s potential, then different, typically lessstringent, constraints are obtained [21]. In addition to its arbitrary nature,the requirement for an absolute potential is theoretically unsettling, sincethroughout the rest of physics, only relative potentials are observable [12].Also in this section, I describe “Eötvös experiments” which test the equiv-alence principle in a manner similar to the proposed antihydrogen measure-ment (but at much higher precision). A brief overview of the Fairbank ex-periments is also given. Fairbank [23] was attempting to measure the gravi-tational force on a single electron. The result was to be compared to thosefrom identical experiments with positrons (which were never conducted). Inspite of the abundant confusion surrounding Fairbank’s gravitational elec-tron results, and their relative age, the experiment retains a distinguishing4uniqueness. It is the only attempt to measure the force of gravity on an el-ementary particle. The effort inspired an analogous experiment that hopedto use antiprotons, but that never came to fruition [12, 24, 25].Finally, this section begins with a thought experiment.1.2.1 A Thought Experiment on Anti-gravity“Anti-gravity” asserts that antimatter and matter experience a repulsivegravitational interaction, analogous to that of opposite charges in electro-magnetism. The matter-matter (g), and matter-antimatter (g¯) gravitationalaccelerations would be related by g = −g¯, and antimatter would “fall-up” inthe Earth’s gravitational field.In addition to the EEP, energy conservation has been invoked as a dis-proof of anti-gravity. A thought experiment put forward by Morrison [26]reasons that, under the anti-gravity paradigm, a particle-antiparticle paircould be raised from the surface of the Earth without the expenditure of anyenergy. At some greater height, the pair might annihilate and produce pho-tons. The photons could be redirected with heavy mirrors back towards theEarth’s surface, experiencing a gravitational blue shift along the way. Theblue shift thus accumulated amounts to a violation of energy conservation.For example, if the photons pair-produce at the original particle-antiparticleheight, those particles would be recreated with excess kinetic energy theydid not previously possess.This argument is widely accepted as valid, and few modern physics mod-els for anti-gravity exist [27]. However, the energy conservation argument isonly a disproof of g = −g¯. It does not necessarily rule out the possibilityg 6= g¯ [12].1.2.2 Particles that are their own AntiparticleMorrison’s thought experiment employed gravitational effects on photons;particles that are their own antiparticles. Here the gravitational lensingeffect experienced by photons is well studied, and we do not need to relyon thought experiments. Measurements of the deflection of radio wavesby our Sun [15, 28], have been interpreted as constraining the strength ofany repulsive gravitational interaction between matter and antimatter to|g¯| < 0.5%× g [29].In addition to the photon, a neutral kaon (the K02 particle) is also com-prised of both matter and antimatter. It exists as a coherent linear super-position of a particle - antiparticle pair (the K0 and K0) [30]. In 1961 Good5[31] noted that, unlike the K01 , the K02 never undergoes two-pion decay1(K02 9 pi++pi−). If, under the anti-gravity paradigm, theK0 andK0 had op-posite gravitational masses, their components in the K02 superposition wouldaccumulate differing phases. In a gravitational potential (φ) the phase valuesare Mφt/~ and −Mφt/~ respectively. The K02 would not be an eigenstate,but would instead contain some oscillating K01 component which can decayvia the pi++pi− mode. The ratio of two-pi decays to that of three-body decay,together with the lifetime of the K02 can be used to place an upper limit onthe K0 -K0 difference in relative gravitational mass of . 10−10 [31].2 A moremodern result from the CPLEAR Collaboration finds that the EquivalencePrinciple holds for this system to better than ∼ 2 × 10−9 [33]. The mostrecent reanalysis casts any K0 and K0 mass difference (< 4.8×10−19 GeV/c2[34]) as a test of CPT symmetry without wading into the gravitational ques-tion.1.2.3 Cyclotron Frequencies of p/p¯According to general relativity, clocks at different gravitational potentialstick at different rates. Therefore, if an antiproton (p¯) coupled to gravitydifferently than a proton (p), one would expect a corresponding differencein cyclotron frequencies. The experimental equality of the p/p¯ cyclotron fre-quencies to about one part in 1010 [35], combined with an assumption of anabsolute gravitational potential resulting from the super-galactic centre [32]produces a limit on |1− g¯/g| . 10−4 [21, 36]. Recently more precise measure-ments of the antiproton cyclotron frequency by the BASE collaboration havebeen interpreted as more tightly constraining deviations from conventionalgravity with antimatter [37].1.2.4 Shapiro Delay from Supernova (Anti)NeutrinosFollowing the argument of Ref. [38], LoSecco [39] used data on the arrivaltimes of neutrinos and antineutrinos from SN1987A to place an upper limiton gravitational asymmetries that might be observable in the form of ananomalous Shapiro delay. LoSecco concluded that the relative difference ingravitational delay was < 5×10−6, however this result relies on an ability todistinguish neutrinos from antineutrinos using an analysis from a statistically1This isn’t true, and the modern analysis must account for CP violation, which wasn’tdiscovered until 1964.2Another (contested [12]) limit of . 10−13 exists from Keynon [32], whose analysisincorporated more data than was available to Good.6small (Nevents = 25) number of events [40]. LoSecco frames this result aslimiting the difference between possible matter and antimatter extensions ofgeneral relativity.1.2.5 Eötvös ExperimentsA more lenient relative of the EEP is the weak equivalence principle (WEP),which states that the inertial mass and the gravitational mass of any bodymust be the same [16]. The inertial mass (mI) is that which tells us, accord-ing to Newton’s second law, how a body will accelerate under the influenceof a given force (F = mIa). The gravitational mass (mg) on the other hand,defines the gravitational force (Fg) experienced by a body in a gravitationalfield, g: Fg = mgg. The WEP is then:mI = mg, (1.1)for all bodies, regardless of their composition. However, the net gravita-tional mass of any body results from a sum over all its forms of energy. Themost prominent of these is usually the rest mass; however, binding ener-gies among particles arising from electromagnetic, weak-nuclear, and strong-nuclear forces all contribute to the mass. If any of these types of energycoupled differently to gravity, it would amount to a violation of the WEP.In this light the gravitational mass could be expressed:mg = mI +∑iηiEEi/c2, (1.2)where each contributing form of energy (Ei) is allowed its own asymme-try parameter (ηiE) describing WEP violation. For two bodies of differingmaterials we would have accelerations (due to gravity) of:a1 = (1 +∑iηiEEi1/m1c2)ga2 = (1 +∑iηiEEi2/m2c2)gThe normalized differential acceleration, ηE , has become known as theEötvös parameter:ηE =2|a1 − a2||a1 + a2| (1.3)7and is often quoted as a figure-of-merit for the upper bound on WEP viola-tion. The current best limit comes from a torsion balance type experiment[41] that measured the differential gravitational acceleration between dif-ferent materials (beryllium and titanium test masses) to be (0.6 ± 3.1) ×10−15 m/s2. The resulting Eötvös parameter is ηE = (0.3± 1.8)× 10−13.In the macroscopic realm, torsion balance experiments have, so far, pro-duced the most precise limits on ηE . However, in the quantum regime,atomic fountain experiments are now able to probe gravitational effects [42],and are sensitive to differential relative accelerations at the 10−7 level [43].The current lower limit from cold atom experiments gives an Eötvös param-eter ηE = (2.8 ± 3.0) × 10−8 by comparing two isotopes of rubidium: 85Rband 87Rb [44].It has been argued that results from these matter-only torsion balanceexperiments constrain extensions to conventional theories of gravity in thesame way as would direct antimatter experiments [45]. The argument utilizesa model with a spin-1 graviton and limits the strength of the interaction tobe less than about 10−6g.1.2.6 Dropping Electrons: Gravity Experiments byFairbank and WittebornIn the 1960s Fairbank and Witteborn sent low energy electrons through acopper “drift tube”, measuring their time of arrival at a vertically displaceddetector. They reported [23] that the electrons experienced a net force ofabout 0.09meg, and in doing so, opened a can of worms. The affair hasbeen well summarized elsewhere [12, 46]. Briefly, the difficulty with thisexperiment lies in shielding all of the electric and magnetic forces that wouldnormally dominate the gravitational force. Some theorists argued that ina metallic tube (which might perfectly shield ambient electric fields) theconduction electrons would “sink” to the bottom of the tube. There theymight collect until the electric field from these compressed electrons balancedthe gravitational force. The result is an imbalance in the density of electronsbetween the top and bottom of the shielding tube. This is known as theSchiff-Barnhill effect [47]. Any electron traversing the drift tube experiencesan electric force from these compressed metallic electrons which, Schiff andBarnhill calculated, exactly cancels the gravitational force.Witteborn and Fairbank planned to reproduce this experiment usingfreely falling positrons but had no source of low energy particles. Had thisexperiment been conducted, the expected acceleration would have been 2gsince, due to the opposite charge of the positron, the Schiff-Barnhill effect8adds to the gravitational force, rather than cancelling.Shortly afterwards Dessler, Michel, Rorschach and Trammel (DMRT)[48] pointed out that, actually, the lattice ions in the metal drift tube aremuch more massive than the conduction electrons, and are thus expected to“sag” under the force of gravity. On this reasoning, these authors predictedan effect that was about three orders of magnitude greater than that ofSchiff-Barnhill, and of opposite sign.Other factors of importance in such a measurement include: scatteringfrom residual gas, surface patch potential electric fields, thermoelectric fields,and image current distortions. Particularly problematic is the patch effect:random electric fields due either to some charged surface contaminant, orto neighbouring domain boundaries created by different crystal orientationsacross which a potential difference can appear. Electric fields from the patcheffect are expected to be as large as, or larger than, the fields associated withthe sagging lattice nuclei predicted by DMRT.In 1977 Lockhart, Witteborn, and Fairbank [49] aimed to address thedisagreement between Schiff-Barnhill and DMRT with observations of theshielding capability of their special drift tube. They reported a tempera-ture dependant shielding effect in the apparatus. Above a sharp transitiontemperature of 4.5 K the ambient force measured on free fall electrons wasfound to agree roughly with the existence of DMRT and patch effect electricfields. Below 4.5 K, however, the ambient force dropped by about 5 ordersof magnitude to a level that would have made the experiment sensitive tothe force of gravity (∼ 5×10−11 eV/m, for an electron). Despite undergoingmuch scrutiny from both theorists and experimentalists, the temperaturedependant shielding effect has never since been understood or reproduced.Fourteen years later, in 1991, the data from their 1977 publication waspartially retracted [50] when reanalysis led to the discovery of a helium leakin the cryostat that caused the ambient pressure to rise to a level muchhigher than previously thought.In spite of all the confusion surrounding these works, the authors stress intheir retraction that this helium leak did not affect their earlier experiment[23] in which the net force on a free falling electron was measured to bein agreement with the Schiff-Barnhill effect. To my knowledge, the matterremains unresolved.9SystemMeasuredQuantityRelativeConstraint Notes Ref.mattertorsionbalanceacceleration < 10−6 only for spin-1 gravi-tons[45]K0 - K0 mass < 2× 10−9 uses an absolutegravitational poten-tial[27, 31–34]SN1987Aneutrinos time < 5× 10−6 uses an absolutegravitational poten-tial[39]p - p¯ frequency < 6.9×10−7 uses an absolutegravitational poten-tial[36, 37]Table 1.1: Summary of indirect experimental evidence concerning antimatterand gravity.1.3 Present Status and Current ExperimentsA summary of results discussed in Section 1.2 is in Table 1.1. Note thatthese constraints are not easily compared. The list comprises measurementsof different physical quantities. In each case, the implied constraint is givenwith a different figure-of-merit. Sometimes a direct comparison of measuredquantities is made, but at other times limits on model dependent parametersare given (e.g., post-Newtonian parametrizations and Eötvös parameters [15,17]). Often an absolute gravitational potential must be invoked – the choiceof which can change the constraint by many orders of magnitude.ALPHA (see Chapter 2) and other collaborations (AEGIS [51], andGBAR [52, 53]) plan to measure the gravitational interaction between matterand antimatter using antihydrogen atoms. The main advantage of antihy-drogen experiments over previous work is the “direct” nature of the mea-surement. The result would be model independent, and not rely on anysuspicious assumptions.Further advantages are worth noting. First, antihydrogen is a stableatom composed entirely of antimatter (in contrast to experiments planningto use muonium [54] or positronium [55]). Additionally, antihydrogen isneutral, and the electric fields that plagued Fairbank’s efforts with elec-trons/positrons are avoided. The analogous effect in antihydrogen exper-10iments involves background magnetic forces acting on the magnetic momentof the antihydrogen atom. Addressing this issue is the primary focus of thepresent thesis.1.4 Thesis OverviewChapter 2 describes antihydrogen gravity experiments that will be conductedby ALPHA. Details of the apparatus and experimental procedure are de-scribed. The importance of magnetometry to the experiment is made clear,and an overview of magnetic sensing techniques is given.Chapter 3 develops a framework for evaluating nuclear magnetic res-onance magnetometers. “First principles” physics models are given thatdescribe the precision, accuracy, signal-to-noise ratio, and repetition ratefor this sensor type. Guidelines concerning probe construction and samplechoices are also discussed.Chapter 4 describes in detail the NMRmagnetometers that were installedand operated during a trial run of the ALPHA-g apparatus in 2018. Theframework of Chapter 3 is utilized to evaluate the performance of thesesensors. Theoretical predictions from Chapter 3 are seen to be reliable.In 2018, an attempt by ALPHA to measure g¯ was foiled by limited accessto antiproton beamtime coupled with complications encountered in buildingand operating an entirely new apparatus. During that effort, only a partial insitu characterization of ALPHA-g magnetometers (described in Chapter 4)was possible.Chapter 5 uses data from that partial characterization (together withmodels developed in Chapter 3) to predict precisions attainable for upcom-ing g¯ measurements by ALPHA. These precisions arise from uncertaintiesexpected in the determination of magnetic fields by NMR magnetometersoperating under realistic conditions in a fully operational apparatus.Improvements to the precision of magnetic field measurements made atlow temperatures using solid-state NMR magnetometry are sought under theguidance of models from Chapter 3. These improvements would not onlyadvance the state of the art in that sector, but also yield improvements toALPHA’s eventual experimental constraint for the gravitational interactionbetween matter and antimatter.At the onset of the 2020 coronavirus pandemic, an experimental char-acterization of NMR magnetometers utilizing various samples from a broadsurvey of suitable materials was underway. Performance predictions for mag-netometers based on these samples are summarized in Chapter 5. Initial11data from the aborted characterization is given in Appendix A, along withspeculative interpretation of the preliminary results.In anticipation of a wide survey of sample materials, broadband NMRprobes were constructed. A description of the “delay line” NMR probe isgiven in Chapter 5. This type of probe offers improvements in both thedynamic range and (for certain situations) the signal-to-noise ratio of theNMR magnetometer. Because of its scarcity in the literature, a review, atheoretical description, and practical construction guidelines are given.The thesis concludes that changes to both the NMR sample and probedesign should yield improvements to NMR magnetometer performance atlow temperatures. Specific recommendations for these changes are made.12Chapter 2Gravitational Experimentswith Antihydrogen inALPHA-g2.1 Historical Introduction to ALPHAHistorically, “high-energy” antihydrogen was first produced at CERN in 1996[57], and at Fermilab in 1998 [58]. A few years later the ATHENA [59] andATRAP [60] collaborations produced “low-energy” antihydrogen by mixingpositron and antiproton plasmas at cryogenic temperatures. These antiatomswere not contained and annihilated shortly after formation.However, in 2010, the ALPHA Collaboration demonstrated trapping of38 antihydrogen atoms in a magnetic potential well (see Figure 2.1) [61].A similar report by ATRAP followed in 2012 [62]. These trapped atoms(temperatures <0.5 K, containment lifetime longer than 1,000 seconds [63])brought spectroscopy experiments with antihydrogen into the realm of pos-sibility. Improvements to ALPHA’s antihydrogen synthesis scheme [64] havesince resulted in a manyfold gain in the number of antiatoms that can betrapped. In 2018, more than 1,000 antiatoms were simultaneously confinedwithin the apparatus on a regular basis.One thousand is a sufficient number of antiatoms for many experimentsthat enable precision comparisons with the matter analog system – hydrogen.Considerable progress towards this goal has already been made by ALPHA.Optical spectroscopy of the 1S-2S [65, 66] and 1S-2P [67, 68] transitionshas been performed with relative precisions of ∼ 10−12 and 10−8, respec-tively. The latter transition is a candidate for antihydrogen laser coolingexperiments [69] that are expected to further improve spectroscopic preci-sion. Microwave spectroscopy experiments have measured the ground statehyperfine splitting in antihydrogen to parts in 104 [70], and charge measure-ments [71, 72] limit deviations from neutrality of the antiatom to less thanabout 10−12 elementary charges.13Figure 2.1: Schematic illustration of the inner section of the ALPHA ap-paratus showing the Penning-Malmberg trap electrodes, the neutral trap(comprising the octupole and mirror coils) and the silicon-based annihila-tion detector. The components are not drawn to scale. Figure reproducedfrom [56]; use permitted under the Creative Commons Attribution LicenseCC-BY.A proof-of-principle gravitational experiment with antihydrogen was per-formed by ALPHA in 2013 [73]. Antiatoms were released from a magnetictrap and their annihilation patterns were scrutinized in search of gravita-tional effects. The number of annihilations in the upper / lower part of thetrap were compared with detailed simulations. The study concluded that theratio of gravitational to inertial mass F = mg/mI must satisfy the bounds−65 < F < 110. The Einstein Equivalence Principle, of course, demandsthat F = 1 exactly.The primary reason that the 2013 limits on F are so loose is that theexperiment was conducted in a horizontal apparatus. The vertical distanceover which antiatoms could “fall” was only about 4 cm.In order to place more stringent experimental bounds on F , a new, ver-tical antihydrogen machine (“ALPHA-g”) is being built. The distance overwhich antiatoms can experience free fall is much larger in the ALPHA-g ap-paratus, which has been designed to be sensitive to deviations from F = 1at the 1% level. That apparatus, and the associated gravity experiment, aredescribed in the next section.142.2 The Prototypical ALPHA-g ExperimentThe basic procedure for future antihydrogen gravity experiments is depictedin Figure 2.2. A cartoon of the vertically oriented ALPHA-g magnetic mini-mum trap (blue octupole coil and red mirror coils) is sketched at the bottomof the figure.Figure 2.2: A sketch illustrating the on-axis potential energy well experi-enced by antihydrogen atoms that are contained in a magnetic trap. Theinset at the bottom depicts the windings of the octupole (blue) and mirror(red) coils that generate the magnetic potential well. A conceptual experi-ment to measure the gravitational acceleration between matter and antimat-ter involves slowly reducing trap magnetic field barriers while maintaininga constant field difference, ∆B. This differential field controls the directionin which antiatoms escape from the trap. For ∆B > Bg¯ antihydrogen isexpected to escape preferentially upwards. For ∆B < Bg¯ atoms prefer toescape downwards.The difference in gravitational energy between the top and bottom of thetrap is∆Eg = MH¯ g¯∆z, (2.1)whereMH¯ is the antihydrogen gravitational mass, ∆z is the vertical distance15between mirror coils, and g¯ is the Earth’s gravitational field as experiencedby antimatter.At the same time, the antiatoms experience a restoring force from thetrapping magnets that provide confinement. The potential energy well inwhich the atoms reside is described according toEB = ~µH¯ · ~B(~r) (2.2)where µH¯ is the antihydrogen magnetic moment, and ~B(~r) is the magneticfield profile experienced by the antiatoms. The experiments will be con-ducted by gradually removing the magnetic barriers that provide verticalconfinement. This has to be done in such a way that the on-axis difference(∆B) between the fields at the centre of the two coils remains constant. Inthis way the energy difference between antiatoms escaping at either the topor the bottom of the trap is unchanged.In order to obtain gravitational information, the antiatom release se-quence must be repeated a number of times for various values of ∆B. Ifthe gravitational energy difference between the top and bottom coil is overcompensated (i.e., µH¯∆B > ∆Eg) then, as magnetic barriers disappear,antiatoms preferentially escape upwards, and annihilate at the top of theapparatus. On the other hand, if the gravitational energy is under compen-sated (i.e., µH¯∆B < ∆Eg), then as mirror coil fields are reduced to zero,antiatoms escape downwards and annihilate at the bottom of the apparatus.For a particular value of ∆B:∆B = Bg¯ = MH¯ g¯∆z/µH¯ , (2.3)the gravitational energy is perfectly canceled, and antiatoms will escape sym-metrically in both up and down directions. These spatial annihilation pat-terns will be detected by ALPHA using modern particle physics techniques.The expected dependence of the antiatom escape trajectory on magneticcompensation, ∆B, is shown qualitatively in Figure 2.3. The number of up-wards escaping atoms (Nup) is shown as a fraction of the total number (NT ).The gravitational interaction of matter and antimatter is measured by thethreshold magnetic imbalance (Bg¯, Equation 2.3.) that gives Nup/NT = 0.5.The Primary Systematic Uncertainty: Magnetic FieldsTo correctly infer gravitational information from this experiment, it is im-perative that the magnetic field difference (∆B) be measured precisely. Infact, the primary systematic uncertainty for ALPHA’s g¯ measurement will16Figure 2.3: The fraction of antiatoms escaping at the top of the magnetictrap as a function of magnetic compensation (∆B). The threshold value,Bg¯, provides a measure of the gravitational interaction between matter andantimatter.result from the precision with which ∆B can be known. The desired g¯ un-certainty leads us to consider precision requirements for magnetometers inthe experiment.The apparatus (discussed further in Section 2.3) has been designed toaccommodate two conceptually identical experiments that seek to achievedifferent g¯ constraints. An initial, “up-down”, experiment is intended tobe sensitive only to the sign of g¯. A follow-up, “precision”, experiment is in-tended to measure g¯ with an uncertainty of 1%. The magnetometry precisionrequirements associated with each of these experiments is given in Table 2.1.For both experiments, the precision with which ∆B is known will ul-timately limit the resulting experimental constraint on g¯. The experimentthat constrains g¯ to 1%, however, will demand precision that approaches thelimits of current magnetic sensing technology (reviewed in Section 2.4). Thedevelopment of techniques that enable this second, more difficult experimentis a central goal of this thesis. A strategy by which these techniques can beemployed in ALPHA-g is proposed in Section 2.5.Secondary Sources of Systematic UncertaintyIn addition to the primary magnetic systematic uncertainty, multiple sec-ondary sources of systematic uncertainty will also limit an experimental de-termination of g¯. Examples include the antihydrogen temperature, the time17up-down precisiongravitational signal size (Bg¯) 453 µT 708 µTdesired gravitational precision 100% 1%required magnetometer precision <400µT <7µTTable 2.1: The “size” of the gravity signal and desired precision are givenfor both an “up-down”, and a “precision” experiment (assuming g = g¯). Thesignal size is proportional to the vertical extent of the magnetic trap used ineach experiment.scale over which antiatoms are released from the trap, and instabilities inmirror coil fields. These factors are the subject of current study within theCollaboration, but it is already known that they will affect the width of thetransition (Bwidth) from Nup ∼ 0 to Nup ∼ NT in Figure 2.3.Initial simulations suggest that cold atoms, long release times, and max-imum field stability, result in the small Bwidth desired for sensitive gravityexperiments. Operating parameters typical of the existing ALPHA-2 ap-paratus (<0.5 K atom temperature, 1000 s atom release time, and tens ofmilliamps of mirror coil current noise) result in simulated values of Bwidthbetween 0.02×Bg¯ and 0.3×Bg¯. Understanding the effect of these factors onthe experiment will be important for our interpretation of the results. How-ever none of these is expected to limit the precision with which Bg¯ can bedetermined. Example results for simulated experiments are given in Chap-ter 5.18Figure 2.4: The full ALPHA apparatus. Figure courtesy of Dr. Mark Johnson, used with permission.192.3 ALPHA-g ApparatusThe ALPHA-g apparatus is much more complex than implied by the cartoonshown in Figure 2.2. A detailed schematic of the full ALPHA apparatus isin Figure 2.4. That representation depicts two separate antihydrogen traps:ALPHA-2 (in which most antihydrogen spectroscopy experiments to datehave been performed), and ALPHA-g (which is a work in progress). Manycomponents of the full apparatus are not discussed in this thesis. Here,only an overview of certain components of ALPHA-g is given insofar as theypertain to antihydrogen synthesis and trapping, as well as the upcominggravitational experiments. More elaborate descriptions of the apparatus canbe found elsewhere [56, 74, 75].Antihydrogen Synthesis and TrappingAntiprotons are produced and delivered to various experiments at a facilitycalled the Antiproton Decelerator (AD) located at CERN. There, a protonbeam, incident on an iridium target, produces antiprotons. These antipro-tons are directed into a storage and cooling ring (the AD) which reduces theirenergy to 5.3 MeV by various cooling techniques [76]. In 2021, a second stageantiproton decelerator, the “Extra Low ENergy Antiproton” (ELENA) ring,is expected to to be operational. ELENA will further reduce antiprotonenergy from 5.3 MeV to 100 keV [77].Antiprotons from the AD enter the ALPHA apparatus from the left inFigure 2.4. These particles pass through a number of thin metal foils thatabsorb antiproton energy through Coulomb interactions. Afterwards a smallfraction of particles can be contained in a dedicated Penning Trap (the “an-tiproton catching trap” of Figure 2.4) utilizing strong electric fields fromspecial high-voltage (5 kV) electrodes. In the catching trap antiprotons aremixed with electrons. In a strong magnetic field, these lighter particlesrapidly cool by emission of cyclotron radiation. The antiprotons then coolsympathetically as energy is transferred between particle species by collisions[78]. Next an electric field is quickly pulsed on and off to remove the elec-trons, which respond faster than the antiprotons due to their smaller mass.Subsequently antiprotons are cooled evaporatively [79]. These methods yieldantiproton plasmas with about 105 particles at temperatures of 40 -300 K inadvance of each antihydrogen synthesis cycle.Positrons with energies of hundreds of keV are produced via beta-decayfrom a sodium-22 source. They are directed through a film of solid neonthat acts as a moderator [80] and reduces their energy to ∼ 1 keV. Sub-20Figure 2.5: The magnetic antihydrogen trap. Currents (black arrows) in oc-tupole and mirror coils (b) produce the magnetic field profile in (a) resultingin an energy well that can confine antihydrogen atoms. Figures courtesy ofChukman So, used with permission.sequently these positrons enter a Surko type accumulator [56, 81] (on theright-hand side of Figure 2.4) that further reduces their energy through in-teractions with a dilute nitrogen gas. This method yields about 108 positronsat a temperature ∼ 1000 K in advance of each antihydrogen synthesis cycle.These particles can then be transferred to Penning traps co-located with theatom trap magnets that will eventually confine antihydrogen atoms (eitherin ALPHA-2 or ALPHA-g). In preparation for mixing with antiprotons,positron plasmas are then prepared to conditions of controlled density, tem-perature, and particle number using a technique called “strong drive regimeevaporative cooling” (SDREVC) [82]. Finally, the positrons are allowed tocool by cyclotron radiation to temperatures of ∼ 20 K.Antiproton and positron plasmas are mixed at the location where theminimum magnetic field occurs inside a set of energized atom trapping mag-nets. When these plasmas overlap, antihydrogen atoms form, primarily viathree-body collisions between two positrons and a single antiproton [83, 84].The extraneous positron carries away excess energy and momentum as theother two particles form a neutral bound state. The number of trappable21antihydrogen atoms is thought to depend mainly on the character of thepositron plasma, which should be at minimum temperature and maximumdensity.Typically 3 ×106 positrons at a temperature of 20 K and a density of108 cm−3 are mixed with 105 antiprotons at 40 K. With these parametersabout 30,000 antihydrogen atoms are created, of which about 10-30 are con-fined in the ∼ 0.5 K energy well produced by octupole and mirror magneticfields (see Figure 2.5). The plasma mixing procedure can be repeated manytimes [64] without disturbing antihydrogen trapped during previous synthe-sis cycles. The ALPHA-2 apparatus can now routinely accumulate up to1,000 antihydrogen atoms by repeating the synthesis cycle many times overthe course of a typical 8 hour antiproton availability period.MagnetsALPHA-g (Figure 2.6) contains many more magnets than are evident insimpler depictions of the apparatus (Figures 2.2, 2.5). These magnets canbe categorized according to their intended use.1. Plasma manipulations: Beamline interconnect magnets bend positronand antiproton beams vertically into the ALPHA-g structure. A back-ground mirror coil is used to assist with the transfer and prepare plas-mas in states favourable for antihydrogen synthesis. Details concerningthese components can be found in [85, 86].2. Magnetic trap: Mirror coils A-G as well as the long and short octupolescreate the magnetic minimum trap. Mirrors B-F can be controlled tooptimize trap depth and/or magnetic field homogeneity at the trapcentre.3. Gravity cancellation: Analysis coils and mirror coils A/G create fieldsthat are finely tuned to cancel the associated gravitational energy.Their field is slowly reduced to release antiatoms according to the pro-cedure outlined in Section 2.2. An “up-down” experiment using mirrorsA/G will measure Bg¯ with a precision < 100%. The result is expectedto rule out the “anti-gravity” hypothesis, and improve on the previousexperimental bounds [73] by a factor of ∼100. “Precision” experimentsperformed with the analysis coils will target a ∼ 1% constraint on g¯.The improved constraint from these experiments will result from thelarger vertical extent (and associated gravitational signal “size”), aswell as improved magnetic field control.22Figure 2.6: Schematic cross-section of the ALPHA-g apparatus (not to scale).Depiction of the cryostat surrounding the central magnet system is omitted.4. Symmetry: A mirror image of material (magnets, Penning traps, etc.)located in the lower half of the apparatus is designed symmetricallyabout the precision measurement region. Magnetic perturbations thatwould otherwise disturb the experiment thereby “cancel” – or occurequally at both the top and bottom analysis coil. Additionally, theannihilation detector is equally shielded from annihilation events oc-curring at the top and the bottom of the apparatus.5. Transfer: “Xfer”, and mirror coils A-G are used to transfer antihydrogen23Figure 2.7: A 24M pixel photograph taken during coil winding. Images ofthis type are used to record the locations of all wires on the magnet form. Adetailed numerical field model is thereby enabled. Figure from [74]: c© 2020,C. So, J. Fajans, W. Bertsche, The ALPHA-g Antihydrogen Gravity MagnetSystem, IEEE Transactions on Applied Superconductivity, June 2020.atoms from the synthesis region to the precision measurement region.Magnets on the form in Figure 2.6 were wound using CNC machinesat Brookhaven National Laboratory. During winding, each layer of everymagnet was imaged at high resolution at multiple azimuths. Analysis ofthese images provides a means of inferring the axial location of every current-carrying wire on the magnet form, and enables a detailed numerical modelof the magnetic field profile for each coil. An example image for mirror Ffrom Ref. [74] is shown in Figure 2.7.Annihilation DetectionA barrel-shaped scintillator assembly and radial time-projection-chamber(rTPC) are used to reconstruct trajectories of charged annihilation by-products(pions) [75]. Track reconstruction techniques with this rTPC will provide thetime and location of annihilations that result whenever antiatoms leave themagnetic trap and encounter matter. Simulations of the device suggest thatantihydrogen annihilations will be reconstructed with 85% efficiency, andthat the coordinates for each event will be known with resolutions of a fewmm and tens of ns in space and time respectively.This apparatus is also used to reject tracks from cosmic rays which rep-resent a dominant background signal in searches for antihydrogen annihi-lations. The addition of the barrel-scintillator is expected to improve AL-PHA’s ability to reject background events from cosmic rays by a factor of 10024Figure 2.8: Schematic of antihydrogen annihilation detection in a radial timeprojection chamber (rTPC) from (a) top and (b) side views. Figure courtesyof Dr. Lars Martin, used with permission.relative to experiments performed with ALPHA-2. Ultimately, the cosmicrejection efficiency in ALPHA-g will depend on the performance of analysistechniques that are currently under development, but the desired figure is99.9% rejection efficiency.Magnetic MaterialsTwo types of related, but crucially different, perturbations to the magneticfield profile in the apparatus affect experiments performed with ALPHA-gin a manner that cannot be accounted for with our detailed numerical modelof current-carrying wires. The first perturbation results from the magneticsusceptibility of the materials from which the apparatus is constructed. Ourimperfect knowledge of these susceptibilities prevents us from accounting fortheir effect on the simulated magnetic field profile.For example, the ALPHA-g cryostat and magnet form were painstakinglyconstructed from a non-magnetic steel alloy (316LN). These componentsof the apparatus required joints made by a welding process that modifiesthe magnetic susceptibility of the alloy at the weld seam. Using a model255150 permeability meter from Severn Engineering, the relative permeabilityof these weld joints was measured to be in the range 1.10-1.15 at roomtemperature. If the material at the joint was paramagnetic, then we wouldexpect the associated field perturbation to at least be reproducible, eventhough it is not accounted for in the numerical model. In that case thelimit with which experiments in ALPHA-g can constrain g¯ would be set bythe precision of magnetometers that are used to experimentally characterizemagnetic fields in the apparatus. Unfortunately, these seams are usuallynot paramagentic, but ferromagnetic as a result of ferrite filler materialsused in the weld. The ferromagnetic nature of the material means that theassociated magnetic perturbations will be hysteretic.Hysteretic materials in the apparatus result in a second class of mag-netic perturbation which is much worse than that from imperfect knowledgeof material susceptibility. Hysteresis will disturb the reproducibility of themagnetic field profile that is critical to experiments in ALPHA-g. A pertur-bation of this class cannot be accounted for by experimental characterization.Characterization with magnetometry can only determine the size of the hys-teretic field fluctuation that occurs as fields are periodically cycled duringexperiments with antihydrogen. The size of this magnetic field fluctuationmay limit the experimental constraint that can be placed on g¯ from experi-ments performed with ALPHA-g.In addition to ferromagnetic weld seams, superconducting material usedto construct atom trap magnets will also produce hysteretic magnetic fields.Type II superconducting material, when immersed in a magnetic field, orcarrying large currents, generally contains many “vortices” – locations wheremagnetic flux has penetrated the material. These can be pictured as micro-scopic loops of persistent current that support an associated magnetic dipolefield. When large quantities of vortices are present in such material they giverise to a macroscopic apparent magnetization referred to as “superconductormagnetization”. The effect of superconductor magnetization was a primaryconcern during the design phase of the ALPHA-g magnet system.This hysteretic perturbation to the magnetic field profile is difficult tomodel [87], but scales as the product Vs × df , where Vs is the volume ofsuperconducting material and df is the diameter of the superconductingmagnet-wire filaments. This scaling law allows order-of-magnitude estimatesof the associated field perturbation for the ALPHA-g magnets. The esti-mates will use an observed perturbation from superconductor magnetization(∼1 mT) during spectroscopy experiments in the ALPHA-2 magnetic trap(constructed from wire with 30 µm diameter filaments).The ALPHA-g octupole coils are made from a composite wire (Supercon26Figure 2.9: Cross-sectional views of composite wires used in various ALPHA-g coils. The structure of the wire involves a copper cladding on a honeycombarray of 678 NbTi filaments. The amount of superconducting material inthe wire affects the magnetic field reproducibility in the apparatus. Figurecourtesy of Chukman So, used with permission.Inc. VSF-678-0.30 mm: Figure 2.9b) with filaments 9 µm in diameter andabout 24% of the superconductor volume used in the ALPHA-2 octupole.The analysis coils are made from a composite wire (Supercon Inc. VSF-678-0.10 mm: Figure 2.9c) with filaments 3 µm in diameter and about 14%of the superconducting material used in ALPHA-2 mirror coils. Overall, a∼ 50µT field perturbation at the analysis coil saddle point is expected. Byconducting experiments that are magnetically symmetric about the analysiscoils, the perturbation is expected to be roughly the same at both the topand bottom coils resulting in an uncontrolled, hysteretic fluctuation in ∆Bof .5 µT [74] – just small enough to permit a 1% constraint on g¯.Initial “up-down” experiments seeking less stringent constraints on g¯ per-mit the inclusion of a greater amount of NbTi in nearby regions. Accord-ingly, remaining mirror coils in the apparatus (in particular A and G) havebeen wound from a wire (Supercon Inc. VSF-678-0.30 mm single wire: Fig-ure 2.9a) with more superconducting material. Mirror coils A and G haveabout 102% of the ALPHA-2 mirror coil superconductor content, and fila-ments that are 9 µm in diameter. The resulting perturbation from super-conductor magnetization is expected to be about 300 µT at the mirror coilsaddle point. The perturbation difference between mirror coils A and G will27ultimately limit the precision to which g¯ can be constrained by experimentsin this region of the apparatus – which has a degraded magnetic symmetryrelative to the “precision” region. Nevertheless, experiments with mirrors Aand G are expected to at least constrain the sign of g¯.Present StatusThe ALPHA-g apparatus is still under construction. In late 2018, at the tailend of the antiproton beamtime preceding a scheduled two-year shutdownperiod (“Long shutdown 2”, or simply LS2), the apparatus was partially con-structed, and some operational experience was obtained. Certain magnetswere commissioned and trained and antiproton annihilations were observedin the rTPC. Importantly for this thesis, prototype magnetometers were par-tially characterized during this time. Ultimately, however, a full test of themachine was not possible because of a cryostat malfunction, that preventedatom trap magnets from cooling to operational temperatures. The malfunc-tion was not rectified before the scheduled end of antiproton beamtime atthe beginning of LS2.In 2021 the full ALPHA-g apparatus will be assembled and used to per-form gravity experiments once antiprotons return to the AD.2.4 Overview of Magnetometry TechniquesSection 2.2 describes a class of gravitational experiments that will be con-ducted with antihydrogen in the ALPHA-g apparatus. That effort will re-quire precise magnetic field measurements. Many methods exist for measur-ing magnetic fields, and comprehensive overviews can be found in variousbooks and review articles [88–91]. Of the available techniques, none is a“silver bullet” that is optimal for all situations. All of them have variousstrengths and weaknesses that must be carefully considered before choosinga particular strategy for a specific application. One possible way of categoriz-ing the many different sensor types that are available is shown in Figure 2.10,where a non-exhaustive compilation of techniques is displayed according tothe measurement field range and accuracy. The ALPHA-g requirements(stated in Table 2.1) are indicated on Figure 2.10, and land squarely in theNMR region of phase space.Meanwhile, methods in which the cyclotron resonance of an electronplasma [93] is measured have been used for magnetometry in ALPHA spec-troscopy experiments since 2012 [94]. Recent improvements to electron cy-clotron resonance (ECR) magnetometry [95] have exhibited precisions that28Figure 2.10: A (non-exhaustive) ecosystem of magnetometry techniques cate-gorized by accuracy and field range. The ALPHA-g requirements are looselyindicated by the yellow star. One Tesla fields must be measured with aprecision of ∼ 1 ppm. Figure from [92]; use permitted under the CreativeCommons Attribution License CC-BY.would enable gravitational experiments with antihydrogen. This technique(described in the next subsection) is ideal for ALPHA gravity experiments.The sensing element – an electron plasma – can be readily located insidethe antihydrogen trap because an electron source and plasma diagnosticequipment already exist in the apparatus as components necessary for anti-hydrogen synthesis.Accessing the critical, on-axis location at which the field difference, ∆B,must be measured is possible with other sensors, but technically challenging.Access would require a 3 m long, movable, non-magnetic, UHV compati-ble insertion device extending from room-temperature to cryogenic locationsinside atom trap magnets.ECR is therefore expected to be the primary method that will be used fordirect measurements of ∆B in ALPHA-g. Unfortunately the ECR techniquecannot be performed simultaneously with antihydrogen gravity experiments.29NMR techniques discussed throughout the rest of this thesis are thereforeaimed at supplementing ECR with real-time monitoring of fields as part ofthe overall magnetometry strategy that will be outlined at the end of thischapter (in Section 2.5).2.4.1 Electron Cyclotron Resonance MagnetometryIn a magnetic field ( ~B), moving electrons (mass, me; charge, e) experiencea Lorentz force that induces an orbital motion (Figure 2.11b) at the angularcyclotron frequency:ωc =eBme. (2.4)These particles resonantly emit and absorb radiation at this special fre-quency, for which an experimental determination represents a magnetic fieldmeasurement. The measurement scheme is as follows.First electron plasmas are positioned and confined at the location ofinterest along the axis of the trap (e.g., the centre of a mirror coil to becharacterized, as shown in Figure 2.11a). Next they are irradiated by a mi-crowave pulse whose electric field interacts with the electron charge. Whenthe frequency of the pulse is matched to the cyclotron resonance, the plasmawill heat dramatically. By measuring the plasma temperature as a func-tion of microwave frequency, the cyclotron frequency can be inferred froma resonance in the plasma “temperature spectrum”. Unfortunately the ob-served spectra (see Figure 2.12) can be more complex than one would naivelyexpect.In addition to cyclotron motion, electrons in the plasma have other oscil-latory degrees of freedom. For example, the particles oscillate in the direc-tion (z) of microwave propagation within an approximately-harmonic electro-static containment well. These oscillations occur at the plasma “bounce” fre-quency3 and induce a lineshape that resembles a typical frequency-modulatedspectrum. Here the source of the modulation is a Doppler shift experiencedby electrons that move either into or away from propagating microwavesand accordingly perceive either higher or lower excitation frequencies. By afrequency modulation analogy, the observed plasma temperature spectrumcontains resonances corresponding to both “carrier” and “sideband” frequen-cies. The former represents the “true” cyclotron frequency that measures themagnetic field.3In ALPHA’s Penning trap, the plasma bounce frequency might typically be tens ofMHz – set by electrode geometry and voltage.30Figure 2.11: a) An electron plasma acting as the “active element” for cy-clotron resonance magnetometry. A microwave pulse at the cyclotron fre-quency will resonantly heat the electrons. b) A single electron undergoingcyclotron motion.A precise field measurement with ECR requires that the true cyclotronresonance be identified from among the many “sidebands” in the frequencymodulated spectrum. This is achieved by modifying the Penning trap elec-trode voltages that establish the shape of the electrostatic potential well, andby extension, the bounce frequency. With a modified bounce frequency, arepeated observation of the plasma temperature spectrum will reveal shifted“sideband”, but unperturbed “carrier” resonances. The procedure is exempli-fied in Figure 2.12 where multiple spectra are observed for various bouncefrequencies. The carrier, or cyclotron frequency is identified as the onlyresonance frequency that is reproduced in all these spectra.Careful characterization of the magnetometry technique described abovewas performed in the Fajans lab at UC Berkeley, where a measurementprecision of 26 ppb was achieved in a homogeneous magnetic field [95]. Theadaptation of these methods to the inhomogeneous magnetic environmentof the antihydrogen trap is expected to degrade this performance somewhat.This is currently under investigation within the Collaboration.31Figure 2.12: Spectra of plasma temperatures observed as a function ofmicrowave frequency. The Doppler shift associated with oscillatory elec-tron motion (at the “bounce-frequency”) induces a frequency modulationstructure in the spectra. A measure of the magnetic field is the cyclotronresonance, which corresponds to the modulated “carrier” frequency (here∼ 19.583 GHz). Various spectral “sideband” peaks are spaced roughly by themodulating “bounce-frequency” (see inset). Reprinted from E. D. Hunter, A.Christensen, J. Fajans, T. Friesen, E. Kur, and J. S. Wurtele. Electron cy-clotron resonance (ECR) magnetometry with a plasma reservoir. Physics ofPlasmas, 27(3), 2020 with the permission of AIP Publishing.2.4.2 NMR MagnetometryStrengths and WeaknessesNMR magnetometry is widely considered the metrological gold standard formeasuring magnetic fields. The method attains high precision and accuracy,involving few systematics. The sensor is linear, and the experimental dataare simple to interpret.A number of drawbacks should also be noted. The magnitude of thefield (| ~B|) is measured – not the vector field ( ~B). Performance is poor atlow fields where additional effort is often needed to improve the typicallysmall signal-to-noise ratio. Additionally, NMR often involves slow read-outrates (. 10 Hz), low dynamic range (. ±5%), and struggles to track rapidly32changing magnetic fields. Finally, the main disadvantage of NMRmagnetom-etry, from the perspective of the ALPHA-g experiment, is the degradation ofperformance with increasing magnetic field inhomogeneity. Implications ofthis characteristic for experiments in ALPHA-g are discussed in Chapter 5.Existing NMR MagnetometryHigh quality commercial sensors [96] presently take advantage of the strengthsof NMR while managing its disadvantages well. Unfortunately commer-cial sensors are only available for the most common experimental conditions(room temperature, ambient pressure, etc). Experimentalists with unusualrequirements are often forced to build their own NMR magnetometry sys-tems.A number of custom NMR magnetometers have already been imple-mented for applications in fundamental physics and the medical sciences [90,97–102]. Many of these were carefully characterized and achieved remarkableperformance. The prevailing wisdom [91, 103] on NMR magnetometry at lowtemperatures states that specialized samples of either helium-3 [98–101], ormetallic aluminium-27 [104, 105] are required.NMR magnetometry involving helium-3 has been demonstrated to offerfemto-Telsa precision. The advantage of this extreme precision is often out-weighed by the technical difficulty of implementing cryogenic gas handlingsystems to accommodate the (expensive) sample. Additionally, unless spe-cial care is taken, the use of helium-3 can give rise to slow sensor read-outrates that are associated with long spin relaxation times.4The first report in which an aluminium sample was used for NMR mag-netometry was published in 2001 [105], and is still frequently rememberedas the only work in which solid-state samples are used for magnetometry atlow temperatures. As a solid, the metallic aluminium sample offers a muchsimpler implementation than helium-3 NMR. This gain comes at the cost ofseverely degraded precision. To my knowledge, there has not been a carefulanalysis of magnetometer performance with aluminium samples. Chapters 3,4, 5, and Appendix A of the thesis address this issue and attempt to fill agap in the literature concerning solid-state NMR magnetometry.4Ref. [98] reports a 3He spin-lattice relaxation time of about half an hour at roomtemperature and 1.5 T.332.5 Magnetometry Strategy for Experiments inALPHA-gFigure 2.13: An isolated mirror coil used in ALPHA-g antihydrogen gravityexperiments is shown schematically in red (not to scale). An electron plasmacan be located at the center of the coil to measure the on-axis field by cy-clotron resonance methods. An NMR probe, even though it does not sit atthe critical location on the axis of the coil, can still be used to infer relevantinformation if the relationship between these two measurements is well un-derstood. Mirror coil fields are shown in dashed lines. A 1 T backgroundfield (Bext) is produced by the ALPHA-g external solenoid. Octupole fieldsare also present but are omitted from this cartoon, for the sake of clarity.The general magnetometry strategy for the ALPHA-g experiment is asfollows. The ECR technique will be used to perform a direct measurementof ∆B. Unfortunately it cannot be performed at precisely the same timeas antihydrogen gravity experiments.5 However, field measurements can beperformed at any time using NMR magnetometers located on the outer wind-ings of the mirror coil, as pictured in Figure 2.13. The implementation ofNMR sensors for this purpose is described in Chapter 4.5Microwave pulses that heat these plasmas can also induce antihydrogen “positron spinflip” state transitions. If this transition occurs the atom will be ejected from the trap.34Figure 2.14: Top panel: The simulated field profile for mirror G gives therelationship between the on-axis field (B(r = 0)) experienced by antihydro-gen, and that measured by an NMR probe (BNMR) positioned as shownin Figure 2.13. The apparent linear relationship is misleading. Deviationsfrom linearity are shown in the bottom panel. These result from the vectornature of the field and the NMR sensitivity to | ~B|. Components of the fieldtransverse to the trap axis generated by the octupole add to the total field“in quadrature” giving rise to the non-linearity.Simultaneous measurements performed using both NMR and ECR exper-iments can be used to create a “calibration” that relates the NMR measure-ment to the on-axis field. With this calibration in hand, field measurementsperformed with NMR can subsequently be used to constrain knowledge ofthe magnetic field experienced by antihydrogen atoms escaping from the trapduring gravity experiments (as described in Section 2.2). Figure 2.14 showsan example calibration curve for mirror coil G that was computed from thesimulated magnetic field profile instead of electron cyclotron resonance data.This strategy could be implemented for an “up-down” measurement im-mediately upon completion of the ALPHA-g apparatus. For the precisionmeasurement, the technique which measures the critical magnetic field (on-axis) will require refinements to the current apparatus (Figure 2.6). Possibly35a Penning trap redesign could allow ECR to be performed in the precisionregion. Or NMR probes, as described in this thesis could be used with anappropriate insertion device.36Chapter 3A Framework for Solid-StateNMR MagnetometerPerformanceThe purpose of this chapter is to provide the reader with models for theperformance of an NMR magnetometer that uses a solid-state sample. InChapter 4, these models are used to evaluate the capabilities of sensors inthe ALPHA-g apparatus. In Chapter 5 they are applied to a general searchfor improved performance utilizing different sample types and probe designs.Section 3.1 contains a description of “the basics” of NMR to orient readerswho might be new to the field. Section 3.2 gives a statistical estimate for theprecision of the NMR magnetic field measurement – the Cramér-Rao LowerBound (CRLB). The CRLB tells us that precise magnetometers result fromnarrow NMR lines with large signal-to-noise ratios (SNR). Accordingly, Sec-tions 3.3 and 3.4 contain theoretical expressions for the NMR linewidth andSNR. Section 3.5 also analyzes the noise performance of various practicallow temperature NMR probe designs. Together these tools enable quantita-tive predictions for the precision of solid-state NMR magnetometers.In addition to precision, two other sensor performance characteristics areconsidered: accuracy, and repetition rate. The accuracy of the sensor is setby the magnetic susceptibility of the sample and probe materials, as wellas our knowledge of the gyromagnetic ratio, γ. These issues are discussedin Section 3.6. The sensor repetition rate is limited by the spin-lattice re-laxation time (T1) of the NMR sample. Section 3.7 discusses expected T1sin solids and their temperature dependence. The implications for sensorsoperating at the so-called Ernst angle are discussed in case it is desired torepeat measurements faster than T1. I conclude the chapter in Section 3.8with some general remarks regarding solid-state NMR magnetometry at lowtemperatures.37Figure 3.1: A large number of spins align with an external field according toBoltzmann statistics. A common approximation treats the spin ensemble asa classical magnetic moment, | ~M | = Mo.3.1 NMR and the Magnetic Field MeasurementSpin MagnetizationA large ensemble of spins in a magnetic field ( ~Bo = Bozˆ) obey Boltzmannstatistics. The ratio of nuclei in different spin states isN−N+= e−~ωo/kT (3.1)where N+ and N− denote populations of spins states (per unit volume)separated by the energy ~ωo = ∆m~γBo. Here γ is the gyromagnetic ratioof the spin, m is the spin projection quantum number, ~ is the reducedPlanck constant, and k is the Boltzmann constant.In thermal equilibrium, a net magnetization will result from the slightexcess population of lower energy spin states (see Figure 3.1). For a totalspin density, N , the net thermal magnetization can be calculated as| ~M | = Mo = Nγ~∑m=+Im=−I m exp(γ~mBo/kT )∑m=+Im=−I exp(γ~mBo/kT )(3.2)' Nγ2~2I(I + 1)Bo3kT, (3.3)where T is the temperature of the ensemble, I is the nuclear spin, and theapproximation γ~mBo/kT 1 is used to evaluate the summations. The 1/Tdependence for Mo will benefit the signal-to-noise ratio for low temperatureNMR experiments (see Section 3.4).38Pulsed NMRFigure 3.2 gives a schematic for a very simple pulsed NMR experiment.A large number of spins are placed inside a coil. In thermal equilibrium,their net magnetization is aligned with an external magnetic field. RF cur-rent is driven through the coil resulting in a perturbing field (B′1eiωot) thatcan induce transitions between spin energy levels with ∆m = ±1. Macro-scopically, these quantum spin state transitions amount to rotations of theclassical magnetization away from its equilibrium orientation. The angle ofrotation (or “flip angle”) isθ = γB1tp (3.4)where the perturbing field6 is applied for a time, tp, called the “pulse width”.An angle θ = pi completely inverts ~M , while θ = pi/2 leaves ~M in the x− y,or transverse, plane. Following a pi/2 pulse, ~M precesses about the axis ofthe applied field, and induces a voltage, V (t), across the terminals of the coil(Figure 3.2c). The frequency of this voltage is the spin precession frequency(fo = ωo/2pi), and a magnetic field measurement is inferred from ωo = γBo.The properties of the NMR magnetic field sensor are encapsulated in theBloch equations [106], which describe the time evolution of ~M . For B1 = 0,they are:d ~Mdt= γ ~M × ~Bo − Mxxˆ+MyyˆT2− Mz −MoT1zˆ (3.5)The first term on the right hand side of Equation 3.5 describes the “spinprecession” dynamics already mentioned.The effect of spin dephasing is encompassed by the second term in Equa-tion 3.5. The transverse magnetization decays with a time constant, T2,called the spin-spin relaxation time. Long T2 times correspond to smalllinewidths that, in homogeneous magnetic fields, endow the NMR field mea-surement with improved precision. This intuition is made quantitative inSection 3.2 with the CRLB, and in Section 3.3, which examines the effectsthat set T2.The third term in Equation 3.5 describes the relaxation of spins towardsthermal equilibrium. This process occurs with characteristic time, T1, calledthe spin-lattice relaxation time. In general the maximum repetition rate for6The perturbing magnetic field that induces spin-state transitions, B1, is “rotating”,or circularly polarized. Meanwhile the field, B′1 = 2B1, from a solenoidal coil (as inFigure 3.2b) is linearly polarized.39Figure 3.2: a) Thermal equilibrium: ~M is aligned with the external field,~Bo. b) The NMR coil operating in “transmit” mode. A power amplifier (notshown) is usually used to excite the coil so that sample spins see a perturbingmagnetic field that rotates ~M away from its equilibrium orientation. c) Thecoil excitation is removed after ~M has been rotated by the flip angle (in thiscase θ = pi/2) set by the duration for which the perturbation is applied (seeEquation 3.4). Following the “pi/2 pulse”, ~M precesses about the externalfield inducing a voltage, V (t), across the coil.40Figure 3.3: A cartoon example of a noiseless NMR signal. The resonancefrequency provides a measure of the magnetic field.field measurements with an NMR magnetometer is limited to ∼ 1/T1 (seeSection 3.7), unless very small flip angles are used.The NMR signalFollowing a pi/2 pulse, the time domain NMR signal can be described byoscillations at angular frequency ωo with damping rate Γ = 1/T ∗2V (t) = Aet(iωo−Γ)+iφu(t) (3.6)where u(t) is the Heaviside step function. Equation 3.6 allows the possibilitythat the rate, Γ, may be influenced by the magnetic field inhomogeneity, σB,according to1T ∗2=1T2+ γσB. (3.7)The Fourier transform of this signal gives a Lorentzian lineshape in the fre-quency domain:S(ω) = 2piAeiφΓ− i(ω − ωo)Γ2 + (ω − ωo)2 . (3.8)Figure 3.3 shows an example of a noiseless NMR signal in both the timeand frequency domains. The resonance frequency, ωo, is easily identified andprovides a measure of the magnetic field.413.2 Precision and the Cramér - Rao Lower BoundThis section describes the Cramér-Rao Lower Bound (CRLB). In an NMRmagnetometry context the CRLB is the lower limit for the uncertainty on theNMR resonance value in the presence of noise [107]. The statistical analysisemployed is as follows. Imagine Np time-series measurements sampled fromsome oscillating signal that contains Gaussian white noise w[n] :s[n] = A exp(−Γ˜n) cos(2pif˜on+ φ) + w[n] (3.9)for datapoints n = 0, 1, 2...Np − 1, and model parameters ~Θ = [A, f˜o, φ, Γ˜].The smallest variance of any estimate for parameter Θi isVar(Θi) ≥[I−1(~Θ)]ii(3.10)with Fisher information matrix elements defined byIi,j = E[∂2 lnP (~s|~Θ)∂Θi∂Θj]. (3.11)The expected value, E, is taken with respect to the probability distributionfunction (P ) for the data (~s) given ~Θ. Broadly speaking, Ii,j describes thecurvature of P with respect to changes in model parameters. Distributionsthat are sharply peaked around the most likely parameter values generallygive tighter parameter constraints.The undamped case (Γ˜ = 0) is considered by Kay in Ref. [107]. Thelower bound on the variance of any resonance frequency estimate isVar(f˜o) ≥ 12(2pi)2(A2/2σ2)Np(N2p − 1)(3.12)for a dimensionless frequency, f˜o, that is less than the Nyquist value (f˜o <1/2), and white noise of variance σ2. Inserting explicit time dependence(f˜o → fo∆t, Np = To/∆t), and taking Np 1, we haveVar(fo) = (δf )2 ≥ 1∆t212(2pi)2(A2/2σ2)(To∆t)3 (3.13)≥ 12(2pi)2(A/σ)2∆f T 3o(3.14)≥ 24(2pi)2(SNR)2∆f T 3o, (3.15)42where To is the observation time, ∆f = 1/∆t is the measurement bandwidth,and A/σ is taken as the signal-to-noise ratio.7 An order-of-magnitude esti-mate is immediately obtained if one assumes an observation time equal tothe characteristic decay time of the NMR signal: To ∼ T ∗2 . In that case,δf & other factors× (Linewidth)3/2. (3.16)Equation 3.16 expresses the intuition that NMR samples with small linewidthsgive rise to better precision.Gemmel et al. [101] consider the damped case. These authors foundthat a signal that decays exponentially (Γ 6= 0)8 can be accounted for byintroducing a correction factorC =N3p12(1− z2)3(1− z2Np)z2(1− z2Np)2 −N2p z2Np(1− z2)2, (3.17)where z = exp(−Γ/∆f). In practice the measurement bandwidth must be∆f & Γ so that z is roughly independent of Γ and the scaling of Equation 3.16is maintained.Often data (in the form of Equation 3.9) from an NMR signal is effectivelyrecorded twice – once for each of the real and imaginary parts of the signal(as expressed by Equation 3.6). If both datasets are used, δf is reduced by afurther factor 1/√2. Finally, the minimum possible variance for a magneticfield measurement with an NMR sensor is(δB)2 ≥ 12γ2(SNR)2∆f T 3oC (3.18)where the relation δf = γδB/2pi has been used.Equation 3.18 defines δB, the minimum expected precision for an NMRfield measurement. This expression of the CRLB sets the bar for NMRsensor performance. It tells us that no matter how the data are used toextract a magnetic field value – whether by clever or simple signal processingalgorithms – the variance of the resulting field estimate cannot be less than(δB)2. The expression also enables one to form “arm chair” expectationsabout sensor precision, and to seek out improvements if experimental resultsdo not compare favourably. Equally, it allows us to view with skepticism anymeasurement with precision smaller than δB.7For the case of Γ 6= 0, Ref. [101] employs Kay’s expression using bandwidth and SNRdefinitions that are appropriate for a low frequency NMR detection scheme. Here I usethe time-domain voltage SNR for a high frequency NMR signal described in Section 3.4.The appropriate bandwidth is a factor of 2 larger than for the low frequency case.8Here Γ has dimensions s−1.43Figure 3.4: A hypothetical CRLB (δB) as a function of data acquisitionparameters. The best precision results with parameters such that Np/∆f =To T2∗.Data Acquisition ParametersThe critical parameters in Equation 3.18 are dependent on the NMR sampleand signal detection method. However, one is usually free to choose thenumber of points (Np) and bandwidth (∆f) with which data are acquired.Figure 3.4 shows the CRLB (δB) phase space as a function of these twoparameters, which are related by To = Np/∆f .When To << T ∗2 , the signal is effectively undamped. The precision fol-lows the T−3/2o power law, and the limit C → 1 is obtained. The To = T ∗2 linemarks a transition to a new regime where the damping correction factor, C,becomes important and the precision improves more slowly with observationtime. For observation times much longer than T ∗2 , C ∝ T 3/2o and δB is inde-pendent of To. This describes a situation where no more useful knowledgeabout the resonance frequency is obtained from longer observation.The theoretical best precision results in the regime where To T ∗2 forwhich there exists a complicated (but weak) dependence on ∆f .443.3 The NMR linewidthThe last section quantified our desire for narrow NMR linewidths – narrowerlinewidths result in more precise magnetometers. This section will reviewthe mechanisms that set the NMR linewidth in solids.For homogeneous magnetic fields, and in the absence of stronger, moredeleterious effects, the NMR linewidth is set by “dipolar broadening”. Thisis discussed in Section 3.3.1. In some cases, quadrupole coupling or Knightshifts can further broaden the NMR line. Cautionary discussions of theseeffects are given in Sections 3.3.2 and 3.3.3. Cubic lattices often do notsuffer from quadrupole and Knight shift broadening, and the benefits ofthose crystal types are noted. Finally a section on motional narrowing isincluded in Section 3.3.4 with relevance to the NMR linewidth in polymers.3.3.1 Dipolar BroadeningFigure 3.5: The dipolar field of one spin perturbs its neighbours.The dipolar field associated with a spin (j) perturbs the local magneticfield at nearby spin sites (k), as suggested by Figure 3.5. The associatedperturbing Hamiltonian for the spin ensemble isHˆ = 12µo4pi∑j,k~µj · ~µkr3− 3( ~µj · ~r)( ~µk · ~r)r5, (3.19)45for j 6= k, and the free space permeability, µo. From this starting point,perturbation theory will allow a calculation of a characteristic linewidth inthe form of the “second moment”M2 = 〈∆ω2〉 =ˆ ∞0(ω − ωo)2f(ω)dω, (3.20)(3.21)where f(ω) is the NMR lineshape function.The calculation of M2 was first carried out by Van Vleck [108], and isreproduced in multiple popular texts [106, 109]. In general one obtains, forneighbouring spins of the same species:M2 =34(µo4pi)2γ4~2I(I + 1)FI1nn∑j,k(1− 3 cos2(θjk))2r6jk(3.22)where rjk and θjk are the spherical coordinates for nucleus k (with zˆ ‖ ~Boand nucleus j at the origin), and where FI is the natural isotopic abundancefor the nuclear spin. If the solid is composed of multiple spin species (theheteronuclear case) then additional contributions to the linewidth follow:M2 =13(µo4pi)2γ2Iγ2S~2S(S + 1)FS1nn∑j,k(1− 3 cos2(θjk))2r6jk, (3.23)where S denotes the spin of a foreign species with natural isotopic abundanceFS . The total second moment can be determined by adding together thecontributions from various spin species. Note that the contribution fromunlike spins is smaller than that from like-spins by a factor of 4/9. For apowdered sample we can average Equation 3.22 over all orientations [106]and obtainM2 =35(µo4pi)2FIγ4~2I(I + 1)∑k1r6jk. (3.24)To proceed further, details about the arrangement of spins in the samplemust be used. These details are contained in the factor∑k1r6jkwhich hasbeen computed in terms of the lattice parameter for various crystal structures[110]; see Table 3.1.46Lattice type Like Spins Unlike SpinsSimple cubic 8.402/a6BCC 29.03/a6FCC 115.6/a6NaCl type crystal 115.6/a6 422.1/a6CsCl type crystal 8.402/a6 20.65/a6Hexagonal close packed 14.54/a6Table 3.1: Second moment summation factors (∑1/r6jk) for powdered sam-ples of some simple crystals types. These values are given in terms of thelattice constant, a. Where relevant, the contribution from unlike spins isincluded. For hexagonal close packed crystals, a refers to the hexagon sidedimension. These values are from Ref. [110].Pseudo-Dipolar BroadeningIn the case of metals with a high charge nucleus (Z & 30) [111], the couplingbetween neighbouring spins is indirectly moderated by electrons yielding the“pseudo-dipolar” expression for the second moment:M2 =34(µo4pi)2I(I + 1)FI× 1n∑j,k[B(rjk)− γ2~/r3jk]2 [1− 3 cos2(θjk)]2.(3.25)Here, B(rjk) is a complicated function describing interactions moderated viathe electron wavefunction [112].Lineshape functionIn addition to M2, the lineshape function must also be known in order toperform scrupulous comparisons of the “linewidth” when evaluating expectedmagnetometer precision via the CRLB. In particular, the statistical analysisin Section 3.2 assumed the lineshape was Lorentzian. While the full-width-at-half-maximum (FWHM) for that lineshape (the real part of Equation3.8) is Γ/pi, the second moment is infinite. Meanwhile, for a Gaussian line-shape (f(ω) ∝ exp [−(ω − ωo)2/2σ2]), we obtain M2 = σ, and a FWHM of2√2 ln 2M2 ∼ 2.4√M2.For solid NMR samples, the lineshape is often neither purely Gaussiannor Lorentzian. In Chapter 5 and Appendix A, I’ll attempt to make compar-47isons of the linewidths between various NMR samples with possibly differentlineshapes. I’ll generally use the FWHM value, regardless of lineshape.3.3.2 Quadrupolar InteractionThe interaction of the quadrupole moment of a nucleus with an electric fieldgradient will generally broaden and shift NMR lines. For NMR magnetome-try, these effects are undesirable. Several salient facts should be considered:1. Spin 1/2 nuclei have no quadrupole moment.2. The quadrupolar interaction cancels for sites of cubic symmetry.3. The m = 1/2 → m = −1/2 transition is unaffected (to first order) byquadrupolar interactions.These facts (discussed below), and our desire for small NMR linewidths willfocus attention on spin 1/2 nuclei and/or crystals of cubic symmetry.Spin 1/2 Nuclei and the Quadrupole momentThe Wigner-Eckart theorem [109] tells us that the electric quadrupole mo-ment (Q) of a nucleus is related to its spin (I) byQ ∝ I(2I − 1)/e, (3.26)which is zero if I = 1/2. Therefore, if it is possible to work with spin-halfnuclei, quadrupole shifts and broadening effects can be completely avoided.Cubic SymmetryThe quadrupole interaction gives a Hamiltonian:HˆQ = 16∑α,βVαβQαβ (3.27)=eQ6I(2I + 1)[Vzz(3I2z − I2) + Vyy(3I2y − I2) + Vxx(3I2x − I2)](3.28)which is zero when the electric field gradients satisfy Vzz = Vyy = Vxx (cubicsymmetry). Here, as with the case of spin-half nuclei, all of the undesirablequadrupole effects are avoided.48Quadrupole Effects on the m = 1/2→ −1/2 TransitionIf one can’t work with spin-1/2 nuclei, or with materials that have cubicsymmetry, then it will be necessary to consider effects related to the electricquadrupole interaction. In the presence of this interaction, NMR lines aregenerally both shifted and broadened.Transitions for which m 6= 1/2 are usually particularly broad, and shouldnot be considered for a magnetometry application. However, the centraltransition (m = 1/2 → −1/2) is only affected at second order [113]. In thecase of an axially symmetric crystal, that resonance will be shifted to lowerfrequencies by an amount:∆f =(729602I + 34I2(2I − 1)e4Q2q2h2fo), (3.29)where fo is the unperturbed transition frequency, h is Planck’s constant, eis the elementary charge, and the product eq characterizes the electric fieldgradient at the nucleus. The corresponding second moment for this line is〈∆f2〉 ' 0.93h2(9642I + 34I2(2I − 1)e4Q2q2hfo)2. (3.30)In estimating the second moment, one is still left with the problem of calcu-lating the electric field gradient from the charge distribution in the crystalstructure. As expected, the linewidth is reduced when either the quadrupolemoment or electric field gradient is small.3.3.3 Knight ShiftIn 1949 Knight found that the NMR resonance of a nucleus in a metal wasat a significantly different frequency compared to the resonance of the samenucleus in a insulating sample [114]. This difference has come to be knownas the Knight shift. The effect arises from a contact interaction betweenconduction electrons and nuclear magnetic momentsHˆe = 8pi3γeγ~2∑j∑l~Ij · ~Slδ(rlj) (3.31)where γe and S are the gyromagnetic ratio and spin of the electron, andrlj is the distance between the nucleus and electron. The shift in energylevels from first order perturbation theory accounts for Knight’s observations.For the magnetic susceptibility of conduction electrons, χe, and the Fermi49surface average of the electron wavefunction magnitude at the location ofthe nucleus, 〈|uk(0)|2〉EF , the relative Knight shift is:∆HH=8pi3〈|uk(0)|2〉EFχe. (3.32)Roughly, the local field experienced by the nucleus is raised by ∆H becauseof the magnetic polarization of the electrons; see Figure 3.6. A Knight shiftthat is not perfectly uniform across the sample can serve to broaden theNMR resonance and reduce the resolution of a magnetometer.Figure 3.6: Polarized electrons (blue) raise the magnetic field experiencedby nuclei (red).Anisotropic Knight ShiftSome lattice types give rise to anisotropic Knight shifts that depend on theorientation of the crystal axes with respect to the external magentic field.The anisotropy can be derived from the interaction of the nuclear magneticmoment with electron spins “at a distance”. If the isotropic Knight shift wasthe result of a contact interaction, ~S · ~Iδ(r), then the anisotropic part comesfromHˆanisotropic = µeγ~~I ·(~Sr3− 3~r(~S · ~r)r5)(3.33)where µe is the Bohr magneton [113]. For a sense of the effect, considerthe case of white tin, which has a tetragonal structure. Figure 3.7 showsa cartoon of a tetragonal structure in which the difference in crystal latticeconstants has been exaggerated. This results in an anisotropic distributionof electrons.50Figure 3.7: A tetragonal crystal with electron magnetic moments (blue) cen-tered on neighbouring lattice sites. The electron distribution is anisotropic.The figure exaggerates this effect to highlight the difference in field experi-enced by the nuclear spin (red) for an external magnetic field oriented parallel(a) and perpendicular (b) to the c axis. Note that the electrons are not boundas this picture might suggest, but must be viewed as relativistic travellingwaves. This is accounted for in Knight shift anisotropy calculations.A phenomenological model for the anisotropy can be presented by rea-soning that the Knight shift should be pi periodic with sample rotation. Thefrequency shift for a single crystal is∆f = ∆f‖ cos2 θ + ∆f⊥ sin2 θ, (3.34)where θ is the angle between the field and some crystal axis. Here ∆f‖is the Knight shift for θ = 0 , ∆f⊥ is the Knight shift for θ = pi/2, andthe linewidth can be estimated as ∆f‖ − ∆f⊥. A quantitative theoreticalestimate for the size of the anisotropy can be made for an axially symmetriccrystal only if one has knowledge of the electron charge distribution [115].However, experiments give typical relative linewidths (∆f‖ − ∆f⊥)/fo ofhundreds of ppm.9 Crystal structures of lower symmetry can generally beexpected to result in larger linewidths.It is important to note that, unlike the case of dipolar broadening, alinewidth that is Knight-shift broadened grows with magnetic field strength(Bo).9As an example, the anisotropy of tin is about 500 ppm [113].51Inhomogeneous Knight ShiftGenerally the largest Knight-shift-related linewidths occur for samples thatexperience inhomogeneous electron density, as is the case for some alloys[116] and heavily doped semiconductors [117]. In this case the strength ofthe shift varies between lattice sites with differing concentrations of nearbyimpurities. These lines can be 1000s of ppm wide [118], and depend both ontemperature [116] and impurity concentration [116, 117, 119].Cubic LatticesFor a cubic metal, Knight shift anisotropies occur only as second order per-turbations [120, 121], and therefore typically result in smaller linewidths. Asan example, lead has a fractional anisotropy of ∼ 20 ppm [120, 122] – morethan a factor of ten smaller than tetragonal tin.3.3.4 Motional NarrowingIn liquids, gases, and - to some extent - polymers, nuclei move in a fast,quasi-random manner. A nucleus that experiences random motion (with acorrelation time τc) perceives the usual dipolar fields of neighbouring spinsas fluctuating, magnetic noise. If the motion is isotropic, and the correlationtime is fast enough, then the dipolar fields are effectively averaged to zeroand a very small NMR linewidth results. In polymers the molecular motionis such that the averaging is present, but incomplete.Textbook treatments employing Redfield theory quantify the effect de-scribed above [109]. By making assumptions about the character of thecorrelated motion, it is possible to show that1T1= γ2[B2x +B2y] τc1 + ω2oτ2c(3.35)1T2= γ2B2zτc +12T1(3.36)where the strength of the dipolar coupling is related to the factors B2i . Thetransition from solid to liquid is shown in Figure 3.8. The relaxation timesare calculated with Bx = By = Bz = 1 × 10−3 T.10 With these values, andτc ∼ 6 × 10−8 s, Equations 3.35 and 3.36 give (correct) order-of-magnitude10These values of Bi are chosen to match the approximate linewidth of natural rubberbelow its glass transition temperature: see Figure A.352Figure 3.8: NMR relaxation as a function of correlation time, τc. Shortvalues of τc are characteristic of gases and liquids, while long values of τc arecharacteristic of solids. Polymers often have intermediate values of τc, andgenerally exhibit shorter T2 and smaller linewidths than other solids. Theminimum value of T1 occurs for τc ∼ 1/ωo. For long correlation times, themodel is inapplicable and T2 is set by dipolar broadening.estimates for the observed room temperature relaxation times in naturalrubber.For a solid, motional narrowing gives warm polymers a remarkably longT2. The linewidth is approximately a factor of 100 smaller than for a “rigid”solid – small enough that field inhomogeneity caused by the magnetic suscep-tibility of sample and probe materials may also contribute to the lineshape(see Section 3.6.1). This type of line broadening can be avoided if specialcare is taken to “susceptibility match” materials near the sample [97, 123] orif the probe is constructed with an ellipsoidal geometry [98].Unfortunately the partial motional narrowing effect found in polymersdoes not persist at low temperatures (see Section 3.8 for further details).3.4 The NMR Signal-to-Noise RatioThe CRLB tells us that the magnetic field precision of an NMR sensor isstrongly dependent on the linewidth. This section addresses the signal-to-noise ratio of the NMR signal, which influences the precision of field mea-53surements according to a slightly less dramatic scaling: δB ∝ 1/SNR. Twoclassic expressions for SNR are often employed when one is attempting toestimate the signal-to-noise ratio of an NMR experiment. These expressionsare different only in the ease with which they can be applied to different prac-tical scenarios, and the clarity with which various factors affect the results –which are equivalent. Both expressions are reproduced below.3.4.1 SNR for a Solenoid NMR CoilThe instantaneous time-domain signal-to-noise ratio for an NMR signal im-mediately following a pi/2 pulse can be derived following Abragam [106].The magnitude of an induced voltage across a coil due to an oscillatingmagnetization isε =dB∂tAη = µoωoMoAη (3.37)for magnetization Mo oscillating at angular frequency ωo, and coil area A.For a sample that only occupies the interior volume of a solenoid, the fillingfactor [124] is roughly η = Vsample/2Vcoil [125]. If the coil is placed in aseries resonant RLC circuit of quality factor Q, the build-up voltage is Qε.Meanwhile, the thermal noise isvnoise =√4kTR∆f (3.38)for coil resistance R = QLω, temperature T , measurement bandwidth ∆f ,and where k is the Boltzmann constant. The signal-to-noise ratio can thenbe writtenSNR =µoQωoMoAη√4kTR∆f. (3.39)For a coil of inductance L we can use the relationship11 A =√VcoilLµo. Finally,we have the established expression for the signal-to-noise ratio:SNR =~2√µo6k3/2Q1/2V1/2coil γ5/2I(I + 1)NB3/2o η∆f1/2T 3/2, (3.40)where we have used ωo = γBo and the expression for thermal spin magneti-zation in Equation 3.3.11This expression uses BA = ε/ω = LI, and 12LI2 = VcoilB22µo.543.4.2 SNR and the Principle of ReciprocityHoult gives an alternate expression for the NMR signal-to-noise ratio whichhas the advantage that it is applicable to all types of NMR probes (not justsolenoidal coils) [126]. This advantage results from the use of “Lorentz reci-procity” [127], or “Antenna reciprocity” [128, 129]. Hoult nicely documentedthis method from the NMR perspective [130, 131]. Here I reproduce only acursory derivation.The NMR detection scheme is sketched in Figure 3.9a, in which a nuclearspin is modeled as a small current loop with infinitesimal magnetic momentd~m = I2d ~A2.Figure 3.9: a) The NMR detection scheme. A small magnetic dipole (I2d ~A2)induces a voltage (V1) in a detection coil. b) A “reciprocal” situation thatcan be considered to ease signal-to-noise ratio calculations.We are interested in V1, the voltage across our detection coil12V1 = ωKMI2 = ωˆ~B2(x) · d ~A1. (3.41)However, this expression is difficult to evaluate directly since the magneticflux through the detection coil with cross-sectional area A1, is highly non-uniform.On the other hand, it is easy to perform the calculation for the “recipro-cal” situation (Figure 3.9b). The voltage across our small coil due a current(I1) in the detection coil isV2 = ωKMI1 = ωˆ~B1(x) · d ~A2. (3.42)The evaluation of V1 is greatly simplified by the principle of reciprocity,which tells us that the proportionality constant KM , is the same for cases12Here, KM is the mutual inductance.55(a) and (b). The signal voltage isV1 =I2I1V2 (3.43)= ωˆ ~B1(x)I1· ~MdV (3.44)where d~m has been replaced with the spin magnetization density, and theintegral is now over the NMR sample volume. For the case where B1 ishomogeneous over the sample, V1 ' ω(B1I1)MoVsample.13A second difference with respect to the previous calculation involves es-timation of the coil RF resistance (R) from its geometry and skin depth(δ):R = nRs, (3.45)where Rs = ρ/δ =√µωρ/2 is the surface resistance for a coil material withpermeability, µ, and resistivity, ρ. The ratio of coil-wire length to perimeter,n = l/p, represents the unitless surface area through which coil currentsmust flow.Combining these expressions gives Hoult’s signal-to-noise ratioSNR =B1I1ωoVsampleNγ2~2I(I + 1)Bo3kT 3/2√4k∆f lp√2µωρ. (3.46)Hoult further claims as an experimental fact: Q ∝ √ωo. In this case anoverall SNR dependence on Q, ωo, γ, and Bo is obtained that matches theexpression given by Abragam [106] (in the previous section).3.5 NMR Probe Noise FactorSome of the factors appearing in the signal-to-noise ratio expressions (Equa-tions 3.40 and 3.46) depend on sample parameters (e.g., the spin, I , for thenucleus of interest). Others depend on the details of the probe construction(e.g., the resistance, R, of the coil). Optimization of the probe can thereforeplay a significant role in obtaining a large SNR. This optimization has been13A general expression for the filling factor (η) can be quantified from expressions likeEquation 3.44 [124, 126].56discussed extensively for conventional, room temperature NMR probes [126,132–135]. This section of the thesis will focus specifically on minimizing theprobe noise factor which dominates the performance of practical low tem-perature probes. Quantitative expressions are given for the SNR penaltythat accompanies these designs.Figure 3.10: A locally-tuned NMR probe in a liquid helium cryostat. Thesample coil has inductance Lp and resistance R (not shown). At the res-onance frequency, capacitors C1 and C2 impedance match the coil to thecharacteristic impedance of the cable and spectrometer pre-amplifier (usu-ally Zo = 50 Ω). The probe is connected to an NMR spectrometer via acoaxial cable with a non-uniform temperature profile.3.5.1 NMR Probe CircuitsFigure 3.10 shows a conventional, “locally-tuned” NMR probe embedded ina liquid helium reservoir. Of the probes discussed here, this type providesthe best SNR. However it is usually impractical as a magnetic field sensor.Capacitors C1 and C2 form an impedance matching network such that powertransfer between spectrometer and coil is optimized, in both the “transmit”,and “receive” modes of operation, cf. Figure 3.2. Unfortunately this circuitis impractical for low temperature applications. The variable capacitors thatare typically used are temperature dependent to the extent that obtaining agood impedance match is challenging.Some alternative probe designs that address this challenge are shown in57Figure 3.11: NMR probe types for use at low temperature: (a) “remotely-tuned probe”, (b) “partially-matched” probe, and (c) “broadband probe”.A coaxial cable connects the low temperature components (at the right)to a room temperature space where the spectrometer and any additionalimpedance matching elements are located.Figure 3.11 – a) the remotely-tuned probe; b) the partially-tuned probe; andc) the broadband probe. These all have the property that they can be con-structed without variable elements at low temperature. This improvementcomes at the expense of SNR.The remotely-tuned and partially-tuned probes are well covered else-where [136–138]. Their performance is strongly dependent on the amountof resistive loss incurred in the coaxial cable that connects the probe to thespectrometer. MacLaughlin [139] derived the noise factor that characterizesthis effect. Here, I extend his model to the case of a cable with non-uniformtemperature. I also apply his method to the design of the broadband probe,in preparation for a deeper discussion of that design in Chapter 5.58Figure 3.12: A generic NMR probe of impedance Zp at the end of a coaxialcable of length x. Of the total power, Px, delivered to the system, only P0is transmitted past the plane x = 0, to the probe.3.5.2 MacLaughlin’s MethodConsider a generic NMR probe described by an impedance Zp = Rp+ iXp atthe end of a coaxial cable of length x (Figure 3.12). It is assumed that anyimpedance mismatch at the left-most (room temperature) end of the coaxcan be adjusted with reactive components whose loss can be neglected. Toanalyze this configuration, MacLaughlin invoked “Pierce’s rule” [140] whichstates that the total noise power (Nt) from a system of multiple components(j) can be expressed asNt =∑jgj∆fkTj (3.47)where ∆f is the bandwidth, k is the Boltzmann constant, and Tj is thephysical temperature of the component. The parameter gj = Pj/Px isthe ratio of power absorbed (Pj) by the jthcomponent of the configura-tion to the total power delivered to the entire system (Px), after accountingfor reflections at impedance discontinuities. In the context of Figure 3.12,gcoil = P0/(P0 + Px) = 1− gcoax, and the total noise power isNt = (gcoilTcoil + gcoaxTcoax)k∆f= (gcoilTcoil + [1− gcoil]Tcoax)k∆f= gcoil(Tcoil + TNC)k∆f= gcoilTcoilk∆f (1 + TNC/Tcoil) . (3.48)The noise temperature of the cable is59TNC =1− gcoilgcoilTcoax (3.49)= (lcoil − 1)Tcoax, (3.50)where lcoil = 1/gcoil is the loss factor (or noise factor), and TNC/Tcoil rep-resents the increase in noise associated with the cable. The expressions forSNR derived earlier only account for the thermal noise power from the coil:Ncoil = gcoilTcoilk∆f . By comparing the expression for the thermal noisefrom the coil with Equation 3.48 we see that the SNR of an NMR mea-surement is degraded (from that given by Equations 3.40 and 3.46) by afactor:1/√1 + TNC/Tcoil, (3.51)which can be significant unless TNC Tcoil.It remains only to calculate the loss factor, lcoil, for a particular probeimpedance, Zp. This is accomplished by considering the reflection at theplane x = 0 as a result of an impedance mismatch between the probeand the coaxial cable, which typically has a 50 Ω characteristic impedance.MacLaughlin findslcoil =e2αx − |ΓR(0)|2e−2αx1 + |ΓR(0)|2 (3.52)lcoil ' 1 + 2αx(1 + |ΓR(0)|21− |ΓR(0)|2), (3.53)where α is the cable attenuation constant in Nepers per unit length, andΓR(0) = (Zp −Zo)/(Zp +Zo) is the usual reflection coefficient, at the x = 0plane. Physically one can interpret that excess noise is caused by a standingwave on the coaxial cable leading to increased fractional power absorption(gcoax) when Zp 6= Zo.Attenuation characteristics for various coaxial cables are given in Ta-ble 3.2. It is possible to improve upon these by about an order of magni-tude if one is willing to construct “home-made” coax from large diametertubing [138, 141]. Note that the frequency dependence of α may involvelosses associated with both the resistivity of the cable metals (first termin Equation 3.54) and loss tangent of the cable dielectrics (second term inEquation 3.54). Sometimes manufactures will describe the overall cable lossaccording toα = a√f + bf, (3.54)60cableattenuationaNeper/m/√GHzbNeper/m/GHz300 K 4 K 300 K 4 KRG-223 /UPasternack 0.046 - 0.005 -Molex100067-0027 0.14 - 0.006 -SC-358/50-SCN-CN 0.077 0.033 < 0.001 0.001SC-219/50-SB-B 0.091 0.032 < 0.001 < 0.001Table 3.2: Attenuation characteristics for some coaxial cables. The con-stants a and b are obtained by fitting Equation 3.54 to manufacturer dataon cable loss (α) as a function of frequency. Data at GHz frequencies areusually employed for these descriptions so that a low frequency extrapolationis required for the present (MHz) application. Molex and RG-223 cables areconstructed from silver-plated copper, for which a low frequency, low tem-perature estimate of α can be obtained by employing an assumption aboutthe material resistivity ratio (RRR = 100 for copper at 1 T is reasonable[142]). “SC” type cables are from Coax Co. Japan, and have specified lowtemperature attenuation data. In most cases the SC-cable data is consistentwith negligible dielectric loss. Note that 1 Neper= 20/ ln 10 dB.and by giving experimental values for the constants a and b. For the fre-quency regime in which this work is centered, the ohmic loss typically dom-inates α, which scales according to a ∝ √ρ. The resistivity may change byorders of magnitude at low temperature (i.e., for a pure metal), or it may berelatively temperature independent (i.e., for an alloy).Most coaxial cables permit use of the approximation αx 1, for whichcase the loss factor becomeslcoil = 1 + αx(R2p +X2p + Z2oRpZo). (3.55)3.5.3 Noise Temperature for Non-Uniform CablingIn a cryogenic application multiple different series-connected cables are some-times used to connect the NMR coil to the spectrometer. These cables are61generally at different temperatures. The noise temperature in this situationcan be calculated by considering the cable to be a cascade of many differentelements [140].TNC = TN,1 +TN,2g1+TN,3g1g2+ . . . (3.56)TNC = TN,1 +∑i=2TN,i∏i−1j=1 gj(3.57)where each element is assigned its own noise temperature, TN,i = (li − 1)Ti,and power absorption ratio, gi = 1/li. In our case the elements are all shortsections of coaxial cable of length ∆x, each potentially having a differentphysical temperature Ti and attenuation constant αi. The loss factor lj foreach element in the cascade is obtained by rewriting MacLaughlin’s expres-sion aslj ' 1 + 2αx(1 + |ΓR(xj−1)|21− |ΓR(xj−1)|2), (3.58)and using|ΓR(xj)| = |ΓR(xj−1)|e−2αj∆x, (3.59)from transmission line theory. Here the reflection at the x = 0 plane is setby the factor ΓR(0)→ ΓR(x0) = (Zp − Zo)/(Zp + Zo).Equations 3.57, 3.59, and 3.58, allow us to calculate the noise tempera-ture for inhomogeneous cables at non-uniform temperature. The calculationis limited in principle only by the accuracy with which we know α(T ) andT (x) for any particular cabling scenario.3.5.4 Application to Various Probe DesignsThis section applies the results of the preceding section to various probedesigns. As an example, Figure 3.13 compares typical implementations ofeach design for the case of a low temperature NMR coil connected to thespectrometer using a warm cable. Note that only the noise temperature andSNR penalty associated with the cabling is accounted for. A fair comparisonof NMR probe SNR must also account for all the other influential factors(see Section 3.4) which are assumed equal in the below analysis.62Figure 3.13: SNR penalty (top) and cable noise temperature (bottom)for various probe types. For this example, αx = 0.01, Tcoax = 300 K,Tcoil =4.2 K, Qcoil = 100, Lp = 200 nH, and the design frequency isfo = 11 MHz (marked by the vertical dashed line). The local matchhas C1,local = 56 pF and C2,local = 990 pF, while the partial match hasC1,local = 390 pF, and C2,local = 890 pF. These curves assume that remotereactances have been tuned (at every frequency) so that the entire circuit isimpedance matched to Zo.Conventional ProbeIf we could use a locally-tuned probe as shown in Figure 3.10, we would haveXp = 0, Rp = 50 Ω, and loss factorlcoil = 1 + 2αx. (3.60)This is the smallest possible value for lcoil. The excess noise results only fromthe matched loss in a cable that does not support a standing wave. This typeof probe has by far the best SNR – but only at its design frequency.63A Remotely-Tuned ProbeFor the case of a purely reactive probe, the loss factor lcoil diverges as Rp → 0and the ability of the coil to dissipate power disappears. Instead of usingEquation 3.55 directly, MacLaughlin considersQeff = gcoilQcoil, (3.61)which is maximum at X2p =Z2o1+Q−2coil. In that case the loss factor becomeslcoil ' 1 + 2Qcoilαx (3.62)for Qcoil 1. The poorly matched impedance of the probe has causedreflections that create a standing wave on the cable enabling it to dissipatea large fraction of the incident power. As a result, the remotely-tuned probeis noisier than the locally-tuned probe by a factor ∼ Qcoil.In practice, if the coil is not at optimum Xp, then the factorR2p+X2p+Z2oRpZocan be uncomfortably large. A limiting case exists when the impedancematch is so poor that all power is dissipated in the cable and the noise factoris infinite.A Partially-Tuned ProbeThe partially-tuned probe can be viewed as a compromise. It exists some-where in-between the locally-tuned and the remotely-tuned probe. Clearly ifwe get the “partial” match just right the problem reduces to the conventional,locally-tuned probe. On the other hand, with C1,local →∞ and C2,local → 0,we return to the remotely-tuned case.Generally, if large SNR is needed, the design of the probe should becloser to that of a perfect (locally-tuned) probe. The improved SNR comesat the expense of “dynamic range” – the frequency range over which remotetuning reactances (C1,remote, C2,remote) are able to correct the impedancematch to Zo. This trade-off is discussed at length in Ref. [138].As an example, in comparison with the locally-tuned probe, the partialmatch in Figure 3.13 exhibits an improved tuning range, but a value ofR2p+X2p+Z2oRpZo∼ 20 that is about a factor of ten worse.Broadband ProbeAs with the conventional probe, we have Xp = 0, Rp = 50 Ω for the broad-band probe, leading to a minimum loss factor lcoil = 1 + 2αx .64This is an idealized case. Section 5.2.2 discusses realistic implementationsof this type of probe in which the impedance match is imperfect. Note thatthe broadband probe typically suffers from a small quality factor. Thismust be considered (along with the noise temperature) in order to make afair comparison between the performance of the broadband probe and otherdesigns.3.6 Absolute Accuracy of the NMR MagneticField MeasurementIt is often important to know the absolute accuracy of a sensor. Two pri-mary issues arise when estimating this accuracy for the NMR magnetometer.Firstly, the magnetic susceptibilities of the probe and sample materials canshift the field reading of the sensor. Section 3.6.1 analyses this shift. Sec-ondly, one must have accurate knowledge of the gyromagnetic ratio. Usuallythis is achieved by referring the sample value to some standard of nature.14Section 3.6.2 discusses the absolute gyromagnetic ratio of both metallic alu-minium and natural rubber with respect to the free proton value.3.6.1 Magnetic Susceptibility of the SampleThe macroscopic magnetic susceptibility, χ, of materials in the vicinity ofthe NMR sample (especially the sample itself) will both1. shift the magnetic field (Bo) seen by the sample spins from the valuethat would have otherwise existed if the probe was not present, and2. increase the magnetic field inhomogeneity (σB) at the sample location(except for very special geometries).The second issue listed can degrade the sensor precision by broaden-ing the NMR line. This usually only occurs for liquid and gaseous sampleswhere the natural linewidths are so small that the observed linewidth is setby the field inhomogeneities associated with magnetic properties of the sam-ple [98, 123]. In practice (unless a very poor choice has been made) themagnetic susceptiblities of the materials in NMR samples and probes are oforder |χ| ∼ 10−5 [143–145]. Meanwhile, solid samples usually have dipo-lar linewidths (1-100 G) that dominate the field homogeneity from weaklymagnetized samples, except at very high fields Bo & 10 T.14In metrology jargon, a “primary standard”.65nDSphere 1/3infinite cylinder || Bo 0infinite cylinder ⊥ Bo 1/2infinite plane || Bo 0infinite plane ⊥ Bo 1Table 3.3: Demagnetizing factors for special geometries.The first issue listed above pertains to the accuracy of the measurement.The “true field” one is usually trying to measure is offset from the mea-sured field. As an illustrating example, consider the situation of an externalfield ~Bo = µo ~Ho applied along some principal axis of an ellipsoidal sample.Analysis of the magnetostatic boundary value problem will yield~B′ = µo( ~H ′ + ~M) (3.63)~H ′ = ~Ho − nD ~M (3.64)~M =χ1 + nDχ~Ho, (3.65)where ~B′ and ~H ′ represent the fields inside a sample with geometry char-acterized by a demagnetizing factor nD [146]. Analytic solutions for nDyielding homogeneous B′ are known only for ellipsoids (see examples in Ta-ble 3.3). Other shapes result in a non-uniform B′ which can be approximatednumerically using finite element methods [147].The field offset experienced by the NMR sample can be obtained bycombining Equations 3.63-3.65. It works out to:~B′ − ~Bo = (1− nD)χ1 + nDχ~Bo. (3.66)The accuracy of the NMR magnetic field measurement from sample magne-tization effects is, in principle, limited by the accuracy to which χ is known.Measurements of χ are usually reported with uncertainties of a few percentso that reasonable estimates of B′ − Bo could, in principle, limit the sensoraccuracy to hundreds of ppb. However, usually this effort is unwarranted,and the accuracy of NMR sensors are given (by Equation 3.66) as roughlythe value of the susceptibility, or about ∼ 10 ppm in fractional terms.663.6.2 Gyromagnetic RatioThe gyromagnetic ratio (γ) is the factor used to transform the measuredNMR resonant frequency into a magnetic field:f =γ2piBo. (3.67)The observed frequency depends in detail on the microscopic environment inwhich the nuclei are located. In the case of a metal, the Knight shift will typ-ically modify the resonance frequency by . 1%, while in insulating samples,the chemical shift will do the same by a smaller amount (∼ 1− 100 ppm).The free proton would be an ideal primary standard for benchmarkingthese material deviations. Unfortunately it is impractical as an NMR sampleand we are forced to consider some other reference materials. For example,the gyromagnetic ratio of the proton in a spherical sample of water at roomtemperature is γ/2pi = 42.576 385 07(53) MHz/T – larger than the free pro-ton value by 26.691(11) ppm [148–150]. Even more practical as an NMRreference material is Tetramethylsilane (TMS) which has a chemical shift ofabout 4.8 ppm relative to water [151].The NMR frequencies of all sorts of materials are known with respectto TMS [152], and we can now proceed to make accurate estimates for thegyromagnetic ratio of a wide range of samples. Note that the chemical shiftof various NMR standards can depend on both temperature (. 10 ppb/K),as well as solvent type and concentration [149, 153]. These deviations willultimately limit the accuracy of gyromagnetic ratio values given here.Below example calculations are discussed for the case of two commonNMR magnetometer sample types: metallic aluminium, and natural rubber.The calculations employed are illustrated graphically in Figure 3.14.γ/2pi uncertainty uncertainty(MHz/T) (Hz/T) (ppm)AlCl3 11.094 121 21 1.9metallic Al 300K 11.112 316 24 2.1metallic Al 4K 11.112 007 30 2.7natural rubber 42.576 490 88 2.1Table 3.4: Absolutely referenced gyromagnetic ratios for typical NMR mag-netometer samples (metallic aluminium, rubber, water). These values arecalculated by employing the scaling summarized in Figure 3.14. The un-certainty on Ξ dominates the uncertainty presented here for gyromagneticratios of aluminium.67Figure 3.14: Relative gyromagnetic ratios for various samples commonlyused in magnetometry applications. The free proton value is included as aprimary standard. The solutes Al(NO3)3 and TMS are commonly used NMRreference materials. Their gyromagnetic ratios are related by the “frequencyfactor”, Ξ. Note the difference in scale to the left of the line break.AluminiumThe absolute NMR frequency of Al(NO3)3 in heavy water with respect toTMS in CDCl3 is Ξ = 26.056859% [152, 154].15 Meanwhile Knight shifts formetallic aluminium (1640 ± 1 ppm [156]) are measured using a reference ofAlCl3 in water. The aluminium line is observed by Epperlein to occur at thesame frequency in these references within about ∼ 0.5 ppm [157].For low temperature applications we need to take note of the significanttemperature dependence of the Knight shift in aluminium [158, 159]. Itsvalue is ' (1.7± 0.1)% smaller at 4.2 K than at room temperature.Utilizing this information gives gyromagnetic ratios for aluminium, ab-solutely referenced to the free proton (see Table 3.4). Note that the AlCl3reference value is frequently cited, even in documents pertaining to metallicsamples [91, 103]Natural RubberNatural rubber has three separate NMR lines corresponding to protons atdifferent locations in the molecular polymer chain. These lines occur at15Refs. [152, 154] state a value for Ξ without an uncertainty. In Table 3.4, I haveassumed 5×10−7, or about 1.9 ppm as a pessimistic estimate that represents uncertaintiesassociated with solvent type, concentration, and temperature. Also see Lindstrom [155]who gives (26.05694± 0.00010)% for aqueous AlCl3.68chemical shifts (with respect to TMS) of 5.1 ppm, 2.1 ppm, and 1.7 ppmwith amplitude ratios 1:4:3 [97]. The natural linewidth is large enough thatthe three lines are not resolved except at fields of at least a few T.The gyromagnetic ratio for natural rubber in Table 3.4 assumed a 2.1ppm uncertainty associated with TMS solvent temperature and concentra-tion, and the approximate distribution of chemical shifts in natural rubber.3.7 Sensor Repetition Rate and Spin-LatticeRelaxationFollowing a pi/2 pulse the nuclear magnetization is entirely “transverse”, i.e.,( ~M = Mxxˆ + Myyˆ). For this simple pulse-scheme, precession occurs for acharacteristic time T ∗2 during which a field measurement is made. In orderto repeat this measurement a significant fraction of the thermal equilibriummagnetization, Mo, must be present along the z-axis. The transfer of trans-verse to axial magnetization occurs as spins relax to thermal equilibriumwith characteristic time T1 – the spin-lattice relaxation time. The maximumrepetition rate with which measurements can be made is therefore of order∼ 1/T1.In general, T1 can become very long when samples are cold. The next sub-sections attempt to give a sense for the mechanisms that govern spin-latticerelaxation in solids at low temperature. This can be done quantitativelyonly for the case of metals. For semiconductors, and other solids, guidelinesfor shortening T1 are given, but an exact temperature dependence is notobtained.3.7.1 Relaxation in MetalsThe spin-lattice relaxation rate in metals is set by interactions with conduc-tion electrons – the same interactions that cause the Knight shift (∆H/H).As a result, T1 can be calculated in terms of the Knight shift. The relation-ship is expressed by the “Korringa relation” [160]:T1(∆HH)2=~4pikT(γeγ)2. (3.68)The scaling T1 ∝ 1/T leads us, as expected, to a long relaxation time atlow temperatures. However, usually the interaction is strong enough to giveT1 values such that a reasonable sensor repetition rate can be obtained.For example, T1 ' 450 ms is calculated for aluminium at 4 K with ∆H/H=1640 ppm.693.7.2 Relaxation in SemiconductorsLike metals, relaxation in semiconductors is governed by interactions withelectrons. The primary difference is that the electrons are no longer a de-generate Fermi gas – they obey Boltzmann statistics. In describing T1 forsuch a material, Abragam gives the relation [106]1T1=649pi|φE(0)|2γ2eγ2Nc(m3ekT2pi)1/2(3.69)where φE(0) is the electron wavefunction at the location of the nucleus,and me is the electron mass. Equation 3.69 gives the correct temperaturedependence only in the regime where the carrier concentration Nc is notitself a function of T . It cannot, therefore be used to make meaningfulpredictions about spin lattice relaxation for samples at low temperatures, apriori. More recently Hoch and Holcomb [161] presented a modified Korringarelation applicable to semiconductors near the metal-insulator transition.The model is useful above the Fermi temperature, which is sometimes as lowas TF ∼ 90 K. This is often somewhat high for cryogenic work.In general, semiconducting samples for NMR magnetometry will haveacceptable values of T1 only for very high dopant concentrations. In thisregime, a Korringa-like temperature dependence (T1 ∝ 1/T ) has been ob-served. See, for example, Ref. [117] where T1 was studied as a functionof dopant concentration, and the shortest low temperature relaxation timeswere T1 & 0.5 min.3.7.3 Relaxation from Paramagnetic ImpuritiesSpin-lattice relaxation for solids without conduction electrons is set by nu-clear interactions with paramagnetic impurities. The temperature depen-dence of the resulting nuclear spin-lattice relaxation rate is difficult to pre-dict, in part because paramagnetic impurities can induce nuclear spin relax-ation by a number of different interactions. In lieu of accurate predictions, Iattempt to give some intuition for the mechanisms at play.When paramagnetic impurities in the sample are dilute, they couple tomost nuclear spins via spin diffusion, for which a detailed description is givenby Abragam [106]. The spin-lattice relaxation rate for such a process is1T1= 4piNibD, (3.70)where Ni is the paramagnetic impurity density, b is an effective distance70comparable to the internuclear spacing, and D is the spin-diffusion coeffi-cient.16 The influence of paramagnetic relaxation enters through the factorb = 0.7(CiD)1/4, whereCi =25γ2eγ2~2S(S + 1)τe1 + ω2oτ2e, (3.71)is a function of the impurity spin S, and paramagnetic relaxation time τe.The paramagnetic relaxation rate itself is strongly temperature dependant.Ref. [164] gives the model1τe= c1 coth(hfe2kT)+ c2Tn +c3exp(∆/kT )− 1 , (3.72)where fe is the paramagnetic resonance frequency. Each term in Equa-tion 3.72 corresponds to the interaction between lattice phonons and para-magnetic impurity via a different process. In order to match experimentalresults the constants c1, c2, and c3, and energy ∆ are treated as free param-eters. The exponent n can also take on different values depending on theimpurity type (n = 5, 7, 9 are common).Overall, the theory is too complex to enable accurate predictions for T1at low temperature. However, some guidance is still offered – if one has asample with an uncomfortably long T1, the situation can be alleviated byincreasing Ni, the density of paramagnetic impurities. The type of impurityemployed will also affect spin-lattice relaxation time.3.7.4 Ernst AngleIf one has a sample with a prohibitively long T1, but still desires a fast rep-etition time, TR, then some degradation in signal-to-noise ratio may occur.The best possible SNR for TR . T1 occurs when spins are repeatedly pulsedat the Ernst angle [165]:θE = cos−1(e−TR/T1). (3.73)In this case, the equilibrium magnetisation is reduced toMeq = Mo1− e−TR/T11− e−TR/T1 cos θE, (3.74)16Bloembergen [162] and Kronig [163] show that spin diffusion is mediated by nucleardipolar coupling. This information is expressed in terms of the diffusion coefficient, D =Wa2, where a is the internuclear spacing, and W ∼ 1/50T2 .71relative to the thermal equilibrium magnetization (Mo). Of that reducedmagnetisation only ∼ sin θEMeq contributes to the signal-to-noise ratio,which will be less than that derived in Section 3.4.As an example, Sundfors et al. [117] measure T1 ' 3 min for a sampleof heavily doped Silicon-29 near 4 K. If that material is used as the samplefor an NMR magentometer, with magnetic field measurements desired onceper second, then one would have to accept a best-case SNR penalty ofsin θEMeq/Mo ' 0.05 with a pulse angle of θE ' 6 degrees.3.8 General Remarks on Low Temperature NMRGeneral guidelines for performing NMR at low temperatures are well summa-rized elsewhere. See, for example Refs. [141, 166]. This section is devotedto pointing out factors that are useful to keep in mind when attemptingmagnetometry with NMR at low temperature.3.8.1 MetalsMost of the solid samples considered in this thesis are metals. These mate-rials contain conduction electrons that can screen or shield the NMR signalfrom the detection coil. Typically the NMR signal from a metal sample isthe result of spins that are located within about one skin depth (δ) of thesample surface. For this reason samples with lots of surface area are usuallysought. Micropowders or thin foils are popular. The feature of interest isthat some sample dimension must be . δ =√2ρ/µω . At low temperaturesthe resistivity, ρ, of a pure metal might decrease from its room temperaturevalue by orders of magnitude (depending on the purity of the metal).For the example of an aluminium sample, a reasonable estimate for thelow temperature skin depth might go as follows. The sample might have aroom temperature resistivity ρ = 2.4 µΩ·cm with residual resistivity ratio,RRR = 100.17 At 10 MHz and 4 K, the skin depth is then δ ' 2.5µm.The signal-to-noise ratio will be degraded by a corresponding reductionin filling factor η ∝ 2δ/D. Here, D is a characteristic size of the sample –the diameter of the microparticle, or the thickness of the foil. In the case ofa microparticle, the metal is often mixed in powder form with a dielectricbinding material so that η is also reduced by the mixture ratio.1817The RRR may plausibly be between 10 and 1000 [167, 168]18I use epoxy, or paraffin, with about 60% metal by volume. Higher metal filing ratiosare possible, but give a viscosity that makes the resultant mixture difficult to work with.72Alloys and disordered materials typically have resistivities that do notdepend strongly on temperature. Therefore, if a larger low temperature skindepth is needed, a sample of imperfect purity or disordered atomic struc-ture can be considered. However note should be taken that high impurityconcentrations may result in a linewidth dominated by an inhomogeneousKnight shift. Additionally, a disordered atomic structure may degrade crys-tal structure benefits, for example, those associated with cubic symmetry.Metallic aluminium is uncommon for its tough surface oxide layer. Thislayer prevents densely packed microparticles from making electrical contact– a feature that could prove important for obtaining high SNR. Some of thealternate metal samples proposed in Chapter 5 of this thesis lack this oxidelayer. In that case the microparticles may form complex, macroscopicallyconducting structures with the potential for shielding the NMR signal andfurther reducing the SNR.3.8.2 Phase TransitionsHaving confined this work to solid-state NMR there are still two phase transi-tions to be aware of when working at low temperatures: the glass transition,and the superconducting transition.Glass TransitionThis phase transition is only relevant to polymer NMR. In a polymer, longchains of molecules are loosely bound and nuclei experience a remarkableamount of motional narrowing for a “solid”. In the polymer state this mo-tional narrowing results in a reduced linewidth. But below the glass transi-tion temperature, TG, the motion is “frozen out”, and the linewidth will beset by the mechanisms described in Section 3.3. An example of this for nat-ural rubber can be seen in Figure 3.15 where the spin-spin relaxation timeshortens by more than a factor of 100 below TG ' 220 K [169, 170]. SeeRef. [144] for a long list of polymer glass transition temperatures. The lowestvalues in that compilation are TG ' 148 K, for example, with poly(ethylene).Superconducting TransitionFrom the NMR perspective, the Meissner effect is probably the most dra-matic consequence for a sample that transitions to the superconducting state.The material becomes perfectly diamagnetic (χ = −1), and the magnetic73Figure 3.15: Temperature dependence of proton NMR relaxation times fornatural rubber. Below the glass transition temperature, TG ' 220 K, T2 be-comes short and NMR sensor precision is expected to degrade. Reproducedfrom [169] with permission from The Royal Society of Chemistry.fields that are needed to polarize spins are expelled.19 If the sample is metal-lic, both T1 and the Knight shift are expected to change as the temperaturetraverses the phase transition and conduction electrons form Cooper-pairs.A full understanding of these changes requires a discussion of the mecha-nisms underlying superconductivity, and are beyond the scope of the presentwork. For a magnetometry application, it suffices to check that a proposedsample will not be a superconductor – i.e., that the sample environment isnot below both its superconducting critical temperature and field (Tc andHc). Appendix A of this thesis includes discussion of a wide variety of NMRsamples in search of a material that outperforms the current low tempera-19Except within a small distance from the surface, known as the London penetrationdepth.74ture solid-state standard – aluminium. The critical temperatures and fieldsfor some of these candidate samples are listed in Table 3.5.Critical Field (kOe)Sample Type Tc (K) Hc Hc,1 Hc,2 Hc,3Cd I 0.52 0.03 - - -Al I 1.2 0.1 - - -Sn I 3.7 0.3 - - -Pb I 7.2 0.8 - - -V II 5 - 0.3-0.8 1.2-3.4 -Nb II 9.2 - 0.4-2.5 1.7-4.1 6-10Nb3Sn II 18 - 0.17 17-221 -Table 3.5: Critical fields and temperatures for some superconductors. Thesematerials have been ordered by Hc since NMR is usually a high field method(B & 0.1 T). The different critical field values (Hc, Hc,1, etc.) specify dif-ferent phases in the collapse of superconductivity. Nb3Sn is presented as anexample to be avoided, although Nb and V may also be problematic. Theranges of critical field that are listed reflect variations in temperature andsample preparation [144].75Chapter 4Magnetometers in theALPHA-g ApparatusIn this chapter I describe various details that are needed to characterizeNMR probes used in the ALPHA-g apparatus during the 2018 experimentalrun. Two types of probes were employed, to fundamentally different ends.The first is a room temperature probe used to monitor the environmentalmagnetic field surrounding the apparatus. A warm, polymer sample enablesa precision superior to that obtained by sensors using “rigid” solid samples.Multiple probes of this type are installed inside the external solenoid at amoderate distance from the antihydrogen magnetic trap. In Section 4.2 Idescribe the physical design of the room temperature probe. These detailsenable quantitative sensor performance predictions using models from Chap-ter 3. Using the Cramér Rao Lower Bound (CRLB) from Section 3.2, I findthat the minimum possible precision from this sensor is about 270 nT. Theprecision is sufficient to enable observation of subtle features of the ALPHA-gmagnetic environment.The second type of sensor is intended as a diagnostic for the fields ofcritical magnets in the antihydrogen atom trap. Each sensor is placed in closeproximity to a coil whose magnetic field is finely tuned to effectively cancelthe gravitational field experienced by trapped atoms. The conditions in thisregion of the apparatus are demanding. The probe needs to be physicallysmall, withstand large changes in temperature, and function for years at atime in an inaccessible location without modification or maintenance. Thisprobe is described in Section 4.3, where performance is again evaluated usingmodels described in Chapter 3. The observed experimental precision of 34µTis about the same as the minimum possible value (∼ 31µT) set by the CRLB.For both the room temperature and cryogenic sensors, NMR signals wereacquired using the spectrometer and associated electronics described in Sec-tion 4.1.764.1 Spectrometer HardwareFigure 4.1: NMR readout system. A 16-channel multiplexer (three SP6T-type switches) allows the spectrometer to address different NMR probes.Two different quarter-wave cables are included for different operating fre-quencies associated with aluminium and proton NMR. Both spectrometerand multiplexer are under computer control.NMR signals in this chapter were acquired with the electronics shownin Figure 4.1. The spectrometer is a Tecmag Apollo. The power amplifier(Spincore PA10W) can deliver 10 W at frequencies between 10-75 MHz. Itspurpose is to amplify “pulses” from the spectrometer and generate values ofB1 that can quickly rotate spin magnetization (as qualitatively illustratedin Figure 3.2b). A pre-amp is used to amplify the NMR signal from probesoperating in “receive mode” (Figure 3.2c). The radio-frequency NMR signalis then down-converted by the spectrometer to low frequencies by mixingwith a nearby reference frequency. Finally, the resulting audio frequencysignal is recorded by analog-to-digital converters inside the spectrometer.Our pre-amp is a Miteq AU-1114 with a noise temperature of 75 K, again of 36.5 dB, and a recovery time of 1µs. A multiplexer enables a singlespectrometer to interact with many NMR sensors distributed throughout theALPHA-g apparatus. It is constructed from multiple RF-switches (Paster-nack PE71S6409) with an isolation of 80 dB, and a switch time of 20 ms.77Crossed diodes are used as both a noise gate (for the power amplifier) and,in conjunction with quarter-wavelength cables, an Rx-protection circuit (forthe pre-amp). The RF-switches are actuated with a single 28 V linear powersupply delivered via a 66-channel PXI-2571 relay module (National Instru-ments) under computer control.4.2 Room-temperature NMR ProbesThis section describes NMR measurements taken with room temperatureprobes using the electronics described in the previous section. In order tocompare the results with theoretical predictions from the models of the pre-vious chapter, we must describe the sensor under consideration (shown inFigure 4.2).4.2.1 Physical Description and Performance AnalysisConstruction DetailsThe probe is of the remotely-tuned type (Figure 3.11) with a coil constructedfrom 6 turns of #34 AWG enameled copper wire tightly wound onto a bare1/16th-inch diameter natural rubber cord (McMaster-Carr item #1775T21),2.8 mm in length. The coil is bonded to the sample with a minimal amountof cyanoacrylate.20 An 8.2 m length of RG-223 coaxial cable exists betweenthe coil and the closest possible location that can accommodate remote tun-ing elements. An extra 0.85 m was added to this cable such that the netimpedance of coil and coaxial cable21 is inductive at the frequency of in-terest (∼42 MHz) and the match to Zo = 50 Ω can be accomplished withvariable capacitors (Vishay BFC280832659) rather than inductors. The coiland sample are mounted in a Kel-F fluoropolymer holder inside a phosphorbronze C510 RF shield and mounting structure. The distance between RFshield and NMR coil was of the order of the coil diameter. Kel-F is chosenas the material for coil mounting structure, since it is nominally without1H nuclei that might otherwise give rise to an unwanted background NMRsignal.20“Krazy Glue”21A transmission line circuit model for this net impedance is used to obtain an orderof magnitude estimate for the coil inductance (Lp ∼200 nH). This estimate is based onthe observed variation in resonant frequency as a function of cable length and tuningcapacitance values, for probes constructed by “trial-and-error”.78Figure 4.2: Various views of the ALPHA-g room temperature NMR probe.Several such sensors are placed inside the annhilation detector (see Figure 4.3for their exact locations). A 3-axis MV2 Hall sensor from Metrolab accom-panies each probe.Eight probes of this type were installed within the ALPHA-g rTPC in2018. Two more will be added for future experiments. The locations ofthese probes with respect to other elements of the apparatus are given inFigure 4.3.79Figure 4.3: Left: A grid of room temperature NMR and 3-axis Hall probesare located within the ALPHA-g rTPC at a radial distance r = 200 mm fromthe axis of the apparatus. Right: Top view of the rTPC volume specifyingthe azimuthal coordinates of probes with respect to the antiproton (p¯) andpositron (e+) beamline axis. A mechanical reference specifies the orientationof the rTPC (11.25 degrees from the beamline) with respect to the rest ofthe apparatus.80Figure 4.4: Top: The NMR signal from the ALPHA-g room temperatureprobe following an approximate pi/2 pulse (tp = 2.75µs) – other parame-ters as in Table 4.1. The best fit to Equation 3.6 is shown in dashed lines.Bottom: The Fourier transform of the time domain NMR signal. Only aportion of the full spectrum (bandwidth 156 kHz) is shown. The resonancefrequency (black dashed line) is fit by χ2 minimization with Equation 3.8,and yields a magnetic field measurement Bo = 1.006 693 02 (29) T. The bestfit is shown in dashed lines, and had χ2 = 512 for 508 degrees of freedom(corresponding to 512 data points and 4 fit parameters). The frequency do-main is specified as the offset from the spectrometer local oscillator frequency(fs = 42.85 MHz) which was employed for the pi/2 pulse.AccuracyThe magnetic susceptibilities of Kel-F and C510 are, respectively, χ =−11.6 × 10−6 and χ = −10.0× 10−6 [145]. An accurate value for the mag-netic susceptibility of natural rubber is not readily available. However ifwe assume a value similar to that for other polymers, we can estimate theorder of magnitude for the sensor accuracy to be ∼ 10 ppm. In this case theshift in observed field (described in Section 3.6.1) from the magnetizationof these materials dominates the sensor accuracy, which is otherwise set byimperfect knowledge of the gyromagnetic ratio for protons in the sample (seeSection 3.6.2).81Tcoil 296 KI 1/2γ/2 pi 42.57 MHz/T Table 3.4Q 10 measuredVcoil 5.5 mm3 measuredN 6.4 ×1028 m−3 Ref. [97]Bo 1.0 T measuredη 1/2 Vcoil ' Vsample∆f 156 kHz Figure 4.4Lcoax 9.0 m measuredLp 200 nH estimatedRp 0.18 Ω Equation 3.45 for ∼7 cm#34 AWG Cu wirelcoil 52 Equation 3.55α 0.01 Nepers/m Table 3.2TNC 15,000 K Equation 3.501/√1 + TNC/Tcoil 0.14 Equation 3.51SNR (experimental) 11 Figure 4.4SNR (theory) 19 Equation 3.40, corrected with 3.51T ∗2 0.44 ms Figure 4.4To 1.62 ms Figure 4.4C 5.8 Equation 3.17δB 270 nT Equation 3.18Table 4.1: Summary of parameters used to calculate the CRLB for the roomtemperature NMR probe. The CRLB, δB = 270 nT, is calculated using theexperimental SNR. Note that loss in the coaxial cable reduces Q from theintrinsic coil value of ωLp/Rp.82PrecisionAn example NMR signal from one of these probes (located at φ =0 , 288 mmbelow magnetic centre) is shown in Figure 4.4. The field is inferred fromthe best fit resonance frequency obtained by a χ2 minimization in whichthe data is modeled with Equation 3.8. The 68% confidence region22 forthe resonance centre value is an appropriate metric for comparison with thetheoretical precision calculated according to the CRLB (Equation 3.18).An example CRLB value is given in Table 4.1, which also summarizes theprobe characteristics used in the calculation. An approximate uncertainty forthe calculated CRLB is obtained from the standard deviation in δB (about12 nT) for 200 repeated field measurements. The uncertainty is a result offluctuations in the estimated linewidth and SNR used in the calculation.The SNR and T ∗2 estimated from each measurement fluctuated with a stan-dard deviation of about 0.5 and 10µs respectively. The Figure 4.4 resonancevalue fit error (290 nT) matches the calculated CRLB (δB =270 nT).Linewidth and Signal-to-Noise RatioIn Chapter 3 I show that the linewidth is an important factor influencingNMR sensor precision. The example NMR signal in Figure 4.4 is consistentwith a linewidth set by T ∗2 = 0.44 ms. Other works [169, 172] have reportedspin-spin relaxation times T2 ' 1 ms for protons in natural rubber at roomtemperature. The difference between these relaxation times suggests either:a field inhomogeneity of order σB ∼ 30µT (Equation 3.7) over the samplevolume, or a fundamental difference in natural linewidth, 1/T2.Unfortunately, T2 was not directly measured for the ALPHA-g NMRsamples, and our knowledge about the homogeneity of the external solenoid ispoor.23 However, measurements taken with this NMR sensor at a location ofimproved homogeneity – on-axis and near the magnet centre – are consistentwith T ∗2 ∼ 0.7 ms, and a corresponding σB ∼ 10µT. The implied reductionin σB suggests that the NMR lineshape from measurements taken within therTPC volume is broadened by the inhomogeneity of Bo.Additional broadening can sometimes result from field inhomogeneity dueto differences in the magnetic susceptibilities of probe construction materials.For the materials used to construct the probe, differences in χ are of order22See Ref. [171] for a review of parameter estimation using χ2 statistics.23Preliminary measurements indicate d|B|∂z'8µT/mm at r = 17.3 cm and 29 cm belowmagnetic centre. Over the sample volume, this gradient yields an inhomogeneity with theright order of magnitude: σB ∼ 10µT.83Figure 4.5: The effect of distant beamline magnets being turned on and off.The reading is offset from Bo =1.006 725 20 T.∼ 1 ppm, or about 1 µT for the Bo = 1 T ALPHA-g magnetic field – anorder of magnitude too small to account for the observed T ∗2 here.Of secondary importance to the CRLB is the signal-to-noise ratio. Whencalculating the CRLB, I use the experimental SNR as approximated fromthe data by A/σ. Here, the signal is taken from the best fit value for A(using χ2 minimization with Equation 3.6). The noise, σ, is estimated asthe standard deviation of the best-fit residuals.4.2.2 ALPHA-g Magnetic Field Environment ObservedThe room-temperature NMR sensor forms a powerful diagnostic of the envi-ronmental magnetic field within the apparatus. Various factors were ob-served to influence the measured field, including the activation of accel-erator beamline magnets (Figure 4.5) and proximity of a large, overhead,rail-mounted steel crane (Figure 4.6).The reproducibility of field measurements from this probe is set, not bythe inherent sensor precision, but by the stability of the magnetic field, Bo(see Figure 4.7). Although we cannot rule out other sources of magnetic fieldvariation, the most likely culprit causing this instability is current fluctuationin the actively powered ALPHA-g external solenoid. The nominal 191 Amagnet design current is observed by a power supply DC-current transformer[173] (DCCT) to fluctuate at a scale (∼ 2 mA) consistent with variationsobserved in NMR measurements.84Figure 4.6: A large steel crane passes over the ALPHA-g apparatus duringthe 200 - 700 s window. The reading is offset from Bo =1.006 746 61 T.Figure 4.7: Stability of the external magnetic field. These variations arespeculated to result from fluctuations in external solenoid current.The read-ing is offset from Bo =1.006 725 20 T.4.3 Cryogenic NMR ProbesIn 2018, cryogenic NMR magnetometers (Figure 4.8) were installed in theALPHA-g cryostat in close proximity to the various coils they are intendedto monitor. This section describes the probe that was installed and tested85in 2018 and compares its performance to the models outlined in Chapter 3.During construction of the ALPHA-g apparatus in 2018, a leak betweenthe helium volume and the cryostat outer vacuum chamber developed at lowtemperatures and precluded our ability to immerse atom trap magnets inliquid helium. As a result of this complication, the effect of strong magneticfield inhomogeneity associated with atom trap magnets on sensor precisionhas not yet been investigated experimentally. The best experimental evalu-ation of cryogenic NMR sensor performance occurred during an attemptedcooldown, when the cryostat reached temperatures of about 20 K.4.3.1 Physical Description and Performance AnalysisConstruction DetailsThe sample is a 60-40% mixture (by volume) of aluminium powder withparaffin wax. The aluminium was from Hunan Jinhao Aluminum IndustrialCo. Ltd. It was an atomized powder of approximately 3µm spherical par-ticles. The stated purity of the aluminium is 99.99%. X-ray diffraction wasperformed to confirm its crystalline structure24, and both scanning electronmicroscope (SEM) and transmission electron microscope (TEM) images weretaken to confirm the approximate particle size. The probe is of the remotely-tuned type (Figure 3.11) with a coil constructed from 7 turns of #34 AWGenameled copper wire wound tightly onto a bare 2.5 mm diameter cylindricalsample, 2.6 mm in length. The coil is bonded to the sample with a minimalamount of cyanoacrylate. The coil and sample are mounted in a phenolicholder inside a phosphor bronze C510 RF shield and mounting structure.The distance between RF shield and NMR coil was of the order of the coildiameter.The cabling that connects the coil to remote impedance matching el-ements25 is arranged as follows. A 6.2 m length of cold Molex 100067-0027 coaxial cable connects the sample coil to a cryostat feedthrough re-gion. A length of 1.16 m SC-219/50-SB-B coaxial cable26 (Coax Co. JapanLtd.) connects the cold coax to a room temperature, hermetically sealed,feedthrough. The feedthrough is connected to impedance matching elementswith a 2.5 m length of room temperature RG-223. This information, together24Actually, X-ray diffraction is not very sensitive to amorphous structures. Thesemeasurements only confirm that some fraction of the aluminium was crystalline. It remainspossible that a significant fraction was amorphous.25Variable capacitors GME50301 and GME50601 from Sprauge Goodman.26To be replaced in future experiments with SC-358/50-SCN-CN (Coax Co. JapanLtd.), to avoid use of toxic beryllium-copper.86Figure 4.8: The ALPHA-g cryogenic NMR magnetometer. The bottominset shows an SEM image of the sample, which is formed from aluminiummicrospheres mixed with a paraffin binder. A Hall sensor (HGCA-3020) andthermometer (CX-1050-AA) from Lakeshore Cryotronics are also installed.A cylindrical sample is pictured here. Future realizations of this probe willinvolve rectangular sheet-like samples. In the notation of Chapter 5, thegreen and magenta markers denote the placement of B-type and C-typesamples, respectively (see Table 5.1)87Tcoil 25 KI 5/2γ/2 pi 11.11 MHz/T Table 3.4Q 10 measuredVcoil 12.8 mm3 measuredN 6.0 ×1028 m−3 calculatedBo 1.0 T measuredη 0.3 Vcoil ' Vsample, 60% 27Al∆f 555 kHz Figure 4.9Lcoax 9.9 m measuredLp 200 nH estimatedRp 0.013 Ω Equation 3.45 for 9.5 cm#34 Cu wire, RRR=100TNC 21,700 K Section 3.5.31/√1 + TNC/Tcoil 0.034 Equation 3.51SNR (experimental) 10 Figure 4.9SNR (theory) 34 Equation 3.40, corrected with 3.51T ∗2 28 µs Figure 4.9To 108 µs Figure 4.9C 5.3 Equation 3.17δB 31 µT Equation 3.18Table 4.2: Summary of parameters used to calculate the CRLB for the cryo-genic NMR probe. The CRLB, δB = 31µT, is calculated using the experi-mental SNR, which is lower than expected.88with a cable spatial temperature profile, specifications in Table 3.2, and anassumption about the RRR of copper, allows for a noise temperature cal-culation with the method described in Section 3.5.3. The resulting noisetemperature is given in Table 4.2 under the assumption of a linear temper-ature gradient (from Tcoil to room temperature) over the full length of the“SC”-type cable – which is connected to a “cold” Molex cable at uniformtemperature, Tcoil.Figure 4.9: Top: The NMR signal from the ALPHA-g cryogenic probe fol-lowing an approximate pi/2 pulse (tp = 6.5µs) – other parameters as inTable 4.2. The best fit to Equation 3.6 is shown in dashed lines. Bottom:The Fourier transform of the time domain NMR signal. Only a portion ofthe full spectrum (bandwidth 555 kHz) is shown. The resonance value (blackdashed line) is determined from a χ2 minimization with Equation 3.8 as thefit function and yields the field measurement Bo = 1.005 784 (31) T. The fitis shown in the lower panel with dashed lines, and had χ2 = 120 for 116degrees of freedom (corresponding to 120 data points and 4 fit parameters).The frequency domain is specified as the offset from the spectrometer localoscillator frequency (fs = 11.1 MHz) which was employed for the pi/2 pulse.AccuracyAs with the room temperature sensor, the accuracy of the cryogenic NMRsensor is set by the shift in observed field due to the magnetization ofprobe construction materials. The magnetic susceptibilities for phosphorbronze C510 and aluminium are, respectively, χ = −10.0× 10−6, and χ =89+20.9× 10−6 [145]. Accurate values for the magnetic susceptibilities of phe-nolic and paraffin wax are not readily available. However if we assume27, forphenolic, the value of a similar composite (e.g., FR4 with χ = −3× 10−6),and for wax, the value of paraffin oil (χ = −9× 10−6) we can make an orderof magnitude estimate for the sensor accuracy: ∼10 ppm.PrecisionAn example signal from this probe is shown in Figure 4.9 for the experimentalparameters listed in Table 4.2. In the case of the cryogenic sensor, a precisionmetric can be evaluated from the results of repeated experiments. Figure 4.10gives the outcome of 200 field measurements, repeated during a time periodwithout any known magnetic field fluctuations. An estimate for the sensorprecision (34µT) is obtained from the standard deviation of these readings.That value should be compared to both the calculated CRLB (δB = 31µT,Table 4.2), and the 68% confidence bound (31µT) on the best fit resonancevalue in Figure 4.9. The minimum possible sensor precision is attained.Linewidth and Signal-to-Noise RatioThe linewidth is expected to be set by the dipolar broadening mechanismdescribed in Section 3.3.1. For an aluminium FCC type lattice, with latticeconstant of 404 pm, the calculated second moment isM2 = 7.5 Gauss2, fromwhich T2 ' 40µs can be inferred – in rough agreement with previous obser-vations [104, 110, 174]. Under the assumption that T ∗2 = T2, the data fromour sample (Figure 4.9) implies a significantly shorter spin-spin relaxationtime, T2 = 28 ± 2µs. To account for the discrepancy by a magnetic fieldinhomogeneity, and allow for T ∗2 6= T2, requires a suspiciously large inho-mogeneity: σB ∼ 1 mT. For comparison, the on-axis field gradient in thisregion of the apparatus was measured in 2018 to be about 4µT/mm. Followup experiments with two different microparticulate aluminium samples in ahomogeneous field yield room temperature linewidths that are compatiblewith the low temperature data presented here – see Appendix A.The calculated SNR for this probe is in Table 4.2. There, the fillingfactor η = Vsample/2Vcoil × 60% accounts for a sample that fills the coilwith a 60-40% mixture of aluminium-paraffin. The spin density is calculatedaccording to the molar mass and mass density for aluminium. The exper-imental SNR stated in Table 4.2 is estimated from the data by the same27These susceptibility values are taken from Ref. [143].90Figure 4.10: Two hundred repeated field measurements were taken nearBo = 1.005 826 T with the probe shown in Figure 4.8. No environmentalmagnetic perturbations were known to be present during this time. a) givesa histogram of the measurement results – shown as a time series in (b). Foreach point in (b), the CRLB is calculated and the fit uncertainty is obtainedfrom χ2 methods. These values are given in the histogram of (c) where theycan be compared to the standard deviation (σ) from (a). The range of CRLBvalues results from fluctuating SNR and T ∗2 , which are estimated from thedata. The dataset fulfills the expectation: CRLB ' fit uncertainty ' σ. Anestimate for sensor precision is 34µT – the CRLB is attained.method as for the room temperature probe. The experimental SNR variedwith a standard deviation of 1.0.It is possible that the discrepancies between expected and observed val-ues of T ∗2 and SNR result from the material properties of our sample. Forexample, a disordered aluminium phase would both broaden the line andreduce the SNR. However, other possible explanations cannot be ruled out– the lineshape could be broadened by an unexpectedly large field inhomo-geneity, and the signal-to-noise ratio may be weaker than expected becauseof environmental noise.An Incomplete AnalysisThe performance analysis described above relies on measurements taken ina malfunctioning cryostat, under conditions that are different from those to91be encountered during gravity experiments. In Chapter 5 I will extrapo-late the cryogenic magnetometer precision reported in this chapter to reflectthe expected performance under realistic operating conditions in ALPHA-g.The factors which limit sensor precision will be identified, enabling specificsuggestions for improvement. In particular, the magnetic field homogeneityfrom antihydrogen atom trap magnets will degrade the precision that can beachieved with these cryogenic NMR probes. In the next chapter, the simu-lated profile of atom trap magnetic fields is used to estimate and alleviatethe anticipated degradation in sensor precision. The result is interpreted asa systematic uncertainty for measurements of g¯ in upcoming experiments.92Chapter 5OutlookChapter 4 described a cryogenic NMR magnetometer with a precision ofabout ±34µT under conditions that were slightly different than those ex-pected during antihydrogen gravity experiments. This chapter attempts toidentify and account for these various differences in operating conditions,and thereby estimate the implied constraint on g¯ from future experimentsconducted with the ALPHA-g apparatus.The expected magnetic precision of the cryogenic ALPHA-g NMR sen-sors are calculated in Section 5.1. Differences between the expected preci-sion and that observed in the last chapter result mainly from changes insample temperature and field inhomogeneity. The calculated precision is in-terpreted as a constraint on g¯ for experiments with ALPHA-g. Section 5.2then considers general improvements that might advance the state of theart in cryogenic solid-state NMR magnetometry. Various novel samples areconsidered, including some which are expected to outperform the aluminiumsample that was used in the magnetometers described in Chapter 4. Animproved potential limit on g¯ is predicted for the case of a magnetometerwith a yttrium-bismuth sample.The impact of adopting a broadband probe for magnetometry applica-tions is also discussed. The SNR from this type of probe is expected tooutperform remotely-tuned probes in certain situations. Due to the scarcityof broadband probes in the literature, a brief review is included. Practicalnotes on building such probes, drawn from my own experiences, are alsopresented.5.1 NMR Magnetometry for AntihydrogenGravity ExperimentsMultiple factors need to be considered in order to extrapolate magnetometerprecision from the observed value in Section 4.3, to that expected duringtests of gravitational interactions in an operational ALPHA-g apparatus.Firstly, the sample temperature will be different from that during the93magnetic field measurements described in Section 4.3. The ALPHA-g cryo-stat will be completely immersed in liquid helium resulting in an NMR op-erating temperature of Tcoil = 4 K. The magnetic field (Bo) experienced bythe NMR sample will also change significantly as mirror coil currents arereduced to release antihydrogen from the trap. Both these factors influencethe expected signal-to-noise ratio and resulting CRLB (δB, Equation 3.18).They can be accounted for by rescaling the observed SNR according toEquation 3.40 after accounting for the effect of cable noise temperature.Secondly, the NMR line is broadened by the inhomogeneity of the mag-netic field produced by atom trap magnets, which were not used during theexperiments reported in Chapter 4. The anticipated effect of this broaden-ing is to reduce the sensor precision according to the 3/2 power law impliedby the CRLB (Equation 3.16). The amount of line broadening that is in-duced depends on the mirror coil current and NMR sample geometry (seeFigure 5.2). Sample geometries tailored to minimize the NMR linewidth areconsidered.Finally, and most importantly, magnetometers that will be used to mon-itor mirror coil fields during gravity experiments are necessarily in the wronglocations. The magnetic field that is relevant for a gravity experiment isthe on-axis field experienced by antihydrogen atoms as they escape the trap.The NMR sensors, by contrast, must be located off-axis, outside the volumeoccupied by the octupole and mirror coils. To account for this fact, simu-lated magnetic field profiles are used to relate the field measured by the NMRsensor to the on-axis field, as described in Section 2.5. In the future, the sim-ulated relationship between the field measured with NMR, BNMR, and theon-axis magnetic field, B(r = 0), can be validated by direct measurementsusing electron cyclotron resonance magnetometry (Section 2.4.1).5.1.1 Magnetic Field HomogeneityThis section discusses the effect of the antihydrogen trap magnets on theperformance of the cryogenic NMR sensors. Antihydrogen is trapped bythe application of inhomogeneous magnetic fields. Meanwhile NMR mag-netometers work best in very homogeneous fields – in particular when thebroadening caused by field inhomogeneity is much smaller than the naturallinewidth of the sample. Here, the expected field inhomogeneity is shown tobe manageable if the sample geometry is carefully chosen.Figure 5.1 gives a sense for the field homogeneity at NMR-accessiblelocations outside a mirror coil that is critical to the experimental protocoldescribed in Section 2.2. The figure shows the approximate inhomogeneity at94Figure 5.1: The magnetic field inhomogeneity (calculated as the standarddeviation of the simulated field distribution within a 1 mm3 volume) as afunction of spatial coordinates in the r− z plane outside mirror coil G. Thetop panel shows the homogeneity for a fully energized mirror coil (at thestart of an antihydrogen gravity experiment). The bottom panel shows theconditions in the same region when there is no current in the mirror coil.In the latter case, the field inhomogeneity is set by the radial decay of theoctupole fields.both the start and end of the antihydrogen release sequence. In general, theend of the experimental sequence is the most important time for antihydrogengravity measurements since it is at this point that the coldest antiatomsescape the magnetic trap. The low mirror coil currents at these times resultin a magnetic field gradient that is set primarily by octupole fields and islargest in the radial direction.Sample GeometryTo mitigate precision degradation from the radial field gradient, I considerNMR sample geometries that are radially small. Table 5.1 lists proposedlocations and geometries of four sample types to be implemented in futureversions of the apparatus. Sample A corresponds to the sample shown in95Sample Experiment Coil Position(r, φ)mm, deg.GeometryA - Mirrors A/G 53.0 , 0B up-down Mirrors A/G 54.7 , 0C precision Analysis 54.9 , 0D precision Analysis 54.9 , 0Table 5.1: Proposed sample geometries and locations for ALPHA-g cryogenicNMR magnetometers. Operational experience was obtained using sample A(see Section 4.3). Remaining samples are for a future version of the ALPHA-g apparatus. Their geometry and position has been chosen to complimentthe field profile for a specific coil as discussed in the text. The z coordinate issuch that all samples are centred at a mirror coil midplane. The azimuthalcoordinate, φ = 0, corresponds to the centre of a bundle of longitudinaloctupole wires.96Figure 5.2: During an antihydrogen gravity experiment mirror coil currentsare reduced to zero. The mean (|B|) and standard deviation (inhomogene-ity) of the field over the NMR sample volume evolve as indicated by thetrajectories shown. Anticipated results are plotted for four sample types.The arrows indicate the parametric variable – increasing time, or decreasingmirror coil current. Whenever the field inhomogeneity across the samplevolume exceeds the NMR natural linewidth (FWHM), sensor precision isexpected to degrade.Figure 4.8 for which magnetometer precision was analyzed in Chapter 4.Samples B, C, and D have geometries specially tailored to optimize magnetichomogeneity at their respective locations.Figure 5.2 shows the mean field and approximate inhomogeneity experi-enced by each sample during a simulated antihydrogen gravity experiment.This homogeneity is set by the total field gradient from a superposition ofoctupole, mirror coil, and external solenoid fields. In the NMR sensor region,the mirror coil field (which opposes the external field) decreases with increas-ing radius and the axial field (Bz) grows with r. Meanwhile the transverseoctupole field decays rapidly with r. At a particular mirror coil current,a field inhomogeneity minimum can occur when the transverse and axial97field gradients roughly cancel. For most locations accessible to NMR mag-netometers, this field homogeneity optimum occurs in-between maximumand minimum mirror coil energization – i.e., somewhere in the middle ofthe antihydrogen release sequence when the mirror coil is only part-way off.As before, this inhomogeneity is calculated as the standard deviation of thesimulated magnetic field profile over the sample volume.The field profile in the vicinity of the analysis coil is significantly moreuniform than that near coils A and G. This degree of homogeneity allowsus to consider improvements to the magnetometry situation from the use ofNMR samples with smaller natural linewidths. The advantage of a smallerlinewidth can be realized with sample D, as discussed in Section 5.2.1. Therange of fields experienced by samples C and D is reduced (in comparisonwith that for samples A and B) because of the smaller magnetic fields pro-duced by analysis coils for experiments in that region of the apparatus.Figure 5.3: The Top panel: Transverse components of the field producedby the “short octupole”. Middle panel: Transverse components of the fieldproduced by the “long octupole”. Bottom panel: Magnitude of the magneticfield experienced by NMR sample B (blue, left axis) and sample C (black,right axis) for magnetic field conditions near the end of the antihydrogenrelease sequence (1 T external field, fully energized octupole, no mirror coilfields).Octupole FieldsIn addition to their (radial) field gradient, some other features of the octupolefields are considered for their potential impact on the NMR measurement.98Figure 5.3 shows the calculated octupole fields as a function of the azimuthalcoordinate at which the NMR sample is placed. The transverse octupole fieldis independent of φ, and any φ-dependent field inhomogeneity is mainly aresult of imperfections in the octupole nature of the coil caused by the nearby“end turns”. For the short octupole√B2x +B2y = 0.28 T at the sample Blocation. For the case of the “long octupole” at the sample C location,√B2x +B2y = 0.061 T. In both cases, the φ dependence of the inhomogeneityis negligible. For the long octupole, the end-turns are so far away that ournumerical model cannot constrain their effect on the field experienced bysample C.Since these NMR samples are intended to be placed directly over top ofthe longitudinal windings of the octupole, the vector ~Bo field will not be quiteperpendicular to the coil axis. This will reduce the NMR SNR since somecomponent of the oscillating spin magnetization no longer contributes to theNMR signal. In the reciprocity picture (Figure 3.9), this SNR reduction isdescribed by a magnetic moment with effective area (d ~A2) that is not parallelto the coil axis ( ~A1). The SNR penalty associated with the misalignment isBz/|B| – about 0.96 for sample B and 0.99 for sample C.5.1.2 Predicted Constraints for g¯ MeasurementsFigures 5.4a and 5.4c show the time evolution of the expected difference infield (∆BNMR = BbotNMR−BtopNMR) measured by NMR sensors on mirror coilsat the bottom (BbotNMR) and top (BtopNMR) of the antihydrogen magnetic trap.The time evolution in the figure is set by the particular protocol by whichmirror coil currents are reduced to zero. The timing details of this protocoldo not affect NMR precision.Red bands indicating the estimated 68% confidence region are obtainedby scaling the ±34µT precision established for the type-A sample in Sec-tion 4.3 according to expected changes in NMR linewidth and SNR, while ac-counting for a slight reduction in cable noise temperature (TNC = 21, 000 K,for Tcoil =4 K). In this way, differences in field, temperature, sample volume,and linewidth can be accounted for. Additionally, whatever factors have de-graded the experimental SNR from the theoretical value in Table 4.2 areassumed to be present. The linewidth in the presence of the simulated fieldinhomogeneity is calculated according toΓ = Γo + 2.4piγ√Var(B) (5.1)where Γo = 1/T2 is the natural linewidth, and a numerical factor (2.4pi) is in-99Figure 5.4: a) The anticipated difference in field (∆BNMR) measured bysample B type magnetometers located near coils A and G. b) The corre-sponding difference between on-axis mirror A and G fields (∆B) as a ratioof Bg¯ (Equation 2.3). c) The anticipated difference in field (∆BNMR) mea-sured by sample C type magnetometers located near the two analysis coils.d) The corresponding difference between on-axis top and bottom analysis coilfields (∆B) as a ratio of Bg¯. Red bands indicate 68% confidence intervals,as discussed in the text.cluded in order to equate the FWHM of Lorentzian and Gaussian lineshapes(discussed in Section 3.3.1).Sensor precisions estimated by the method outlined above are ±45µTand ±5.5µT for samples B and C, respectively, under end-of-sequence ho-mogeneity conditions.Figures 5.4b and 5.4d account for the fact that NMR sensors are in thewrong spot. Because these probes are necessarily displaced from the axis of100Figure 5.5: Expected constraint for the magnetic imbalance, ∆B, from mag-netometers based on a C-type Al sample and a D-type YBi sample. Fromthe perspective of the antihydrogen gravity experiment, g¯ is constrained toabout ±4.7%, and ±1.5%, respectively, in these two cases.the apparatus, they experience a different magnetic field than antihydrogenatoms due to the spatial profile of the mirror coil field. The expected fieldmeasurements (BbotNMR and BtopNMR) are extrapolated to on-axis equivalents,after which the difference is taken to obtain a measure of ∆B – the magneticimbalance that is critical for experiments in ALPHA-g (Section 2.2).The extrapolation to the axis degrades the field constraint by a “sensi-tivity” factor – approximately 2.2 and 4.3 for samples B and C, respectively.This factor28 describes a coil with a stronger field on-axis than at the NMRlocation due to its spatial profile. For example, a 1 µT shift in field at theNMR probe is equivalent to 2.2µT and 4.3µT shifts in the on-axis fieldsfor samples B and C, respectively. The red bands on Figures 5.4b and 5.4dcan be interpreted as the ±1σ uncertainty to which g¯ is constrained. Therelative constraints expected for the two cases are ±30% and ±4.7%, respec-tively. Potential improvements proposed in Section 5.2 are foreshadowed inFigure 5.5, in which the same bounds are shown for a magnetometer basedon a D-type YBi sample.28Set by the slope of a “calibration” curve, such as the example shown in Figure 2.14.101Figure 5.6: Antihydrogen release experiments are simulated for various mag-netic imbalances (∆B). The top panel shows results anticipated for experi-ments conducted with mirror coils A and G which are seperated by a verticaldistance ∆z = 256.1 mm. Error bars reflect expectations from NMR sampleB. The bottom panel shows anticipated results for experiments conductedwith analysis coils seperated by a vertical distance ∆z = 400 mm. Errorbars reflect expectations from a C-type aluminium sample and a D-typeYBi sample. No uncertainties are given for Nup/NT , since these will dependon factors beyond the scope of this work such as the annihilation detectorperformance, and number of antiatoms that are trapped and released.Figures 5.4 and 5.5 illustrate expectations for a single magnetic imbal-ance (∆B = Bg¯), which is maintained as antihydrogen is released from thetrap. In order to show that the antiatom escape direction is controlled bythis magnetic imbalance, the experiment must be repeated for various valuesof ∆B. Figure 5.6 shows the anticipated results of a number of such simu-lated experiments. Each point represents the outcome of an experiment (cf.Figure 5.4) in which the numbers of antiatoms escaping upwards (Nup) anddownwards (Ndown) are recorded by the annihilation detector. For a totalatom number NT , a transition from Nup ∼ 0 to Nup ∼ NT is expected near∆B = Bg¯. The 68% confidence bands in Figures 5.4b/d and 5.5 appear inFigure 5.6 as horizontal error bars bounding the magnetic imbalance.Systematic LimitationsThe NMR sensors described in Chapters 4 and 5 are precise enough thatsystematic effects may dominate the magnetic field constraint and thereby102degrade the precision of the g¯ measurement. For example, a fluctuatingrelationship between the on-axis field and that measured by the NMR probeswould weaken the g¯ constraint. The scale of magnetic field fluctuationsexpected to result from hysteretic superconductor magnetization effects arediscussed in Section 2.3.In the vicinity of mirror coils A and G, the magnetic field is expectedto be reproducible to . 300µT – significantly larger than the precision ofthe sample B NMR probe. Meanwhile at locations near the analysis coils,the field reproducibility is expected to be about 5µT – the same order ofmagnitude as the precisions expected from NMR probes with C and D typesamples.In the case of experiments using mirror coils A and G the resulting con-straint on g¯ is likely to be limited by systematic effects that cannot be char-acterized and accounted for by the magnetometry strategy of Section 2.5.For precision-g¯ experiments performed using the analysis coils, the factorthat will limit the g¯ constraint remains unclear.Ultimately, the magnetic field reproducibility in the apparatus remains tobe studied experimentally as a part of a Collaboration-wide magnetometryeffort. Whatever the field reproducibility turns out to be, the experiment willbe better off relying on a more precise magnetometer that enables improvedstudy of the magnetic systematics at play.5.2 Improving State-of-the-Art Solid-State NMRMagnetometryThere are two immediately obvious ways to improve the precision of magneticfield measurements discussed up to this point. The first would be to designa probe with a superior signal-to-noise ratio. The second would be to choosea sample material with a smaller linewidth. In this section I try for both.Section 5.2.2 describes a type of broadband NMR probe that shouldgive larger signal-to-noise ratio than the remotely-tuned probes described inChapter 4. Section 5.2.1 discusses various samples with linewidths that aresmaller than or similar to the aluminium linewidth. Quantitative estimatesare made for the CRLB expected when using these samples in an NMR mag-netometer. From these, an improved potential constraint on g¯ is estimated.1035.2.1 Solid Samples with Improved LinewidthsTable A.1 in Appendix A lists a number of materials with linewidths thatare (or are expected to be) competitive with that of aluminium. Theoreticalexpectations for these linewidths are compared to experimental results whenpossible. The NMR signal-to-noise ratio is (in part) dependant on sampleparameters. These are summarized in Table A.2. Together the informationin these tables allows for the calculation of the CRLB for a given probeconstruction. This metric for the precision of an NMR field measurement isgiven (for a perfectly homogeneous field) and theoretical SNR in Table 5.2for the probe described in Section 4.3 at a temperature Tcoil = 4 K, witha sample C volume of 50 mm3, and a cable noise temperature of TNC =21, 000 K.29The smallest CRLB in Table 5.2 is that for a metallic compound ofyttrium-bismuth (YBi), for which the theoretically expected field precisionis calculated to be ±100 nT. This represents an improvement over aluminiumby a factor of about 7. For use in ALPHA-g, the YBi linewidth is sufficientlysmall that yet another sample geometry is required in order to take advantageof the potential improvement. Sample D (Table 5.1, Figure 5.2) addressesthis. It is shaped in such a way that the field inhomogeneity over the samplevolume is smaller than the FWHM of the natural lineshape (120 µT).The analysis outlined in Section 5.1.2 is applied to a type-D YBi NMRsample in Figure 5.5 where it is compared with an earlier result for a type-C aluminium sample. The anticipated precision of magnetic field mea-surements is calculated by scaling the experimentally-determined precision(±34µT) obtained in Section 4.3 according to expected changes in the NMRlinewidth and SNR (Equations 3.40 and 3.18), and accounting for a slightreduction in cable noise temperature (TNC = 21, 000 K). Differences in nat-ural linewidth, spin density, gyromagnetic ratio, spin, field, temperature,sample volume, and field inhomogeneity are accounted for. As before, scal-ing the experimentally-observed precision is preferred to direct calculation ofthe CRLB. This is intended to account for an experimental performance thatmay be worse than theoretically expected (as observed in Section 4.3). Theprecision predicted by the analysis is ±1.8µT – larger than the Table 5.2value as a result partly of line broadening from field inhomogeneity. Thecorresponding constraint on g¯ would be ±1.5% – illustrated in Figures 5.5and 5.6.29The dependence of TNC on operating frequency is ignored.104Sample CharacterizationAt the onset of the 2020 coronavirus pandemic, an experimental characteri-zation of the NMR samples listed in Table 5.2 had just begun. That work isfar from complete, but initial data on some of these samples are presentedin Appendix A, along with a few reflections on the suitability of each samplefor magnetometry applications.For the characterization of many different low temperature NMR magne-tometer samples, a tuned probe would require a unique circuit to be designedand built for operation with every nucleus at every magnetic field of interest.A campaign involving about 20 sample types would have required design andfabrication of a large number of probes. Instead, I intended to use a broad-band probe. With that approach, all of the samples listed in Table 5.2 couldbe characterized over a wide range of magnetic fields and temperatures usinga single probe.Only preliminary designs and realizations of a broadband probe for sam-ple characterization were completed before the effort was interrupted by thecoronavirus pandemic.105Sample Nucleus SNR T2 (µs) δB (µT)YBi 209Bi 800 380 0.10InP 115In 1000 100 0.23YSb 121Sb 350 210 0.30MoP 31P 300 120 0.35NaCl 23Na 340 140 0.40YP 31P 200 140 0.45YSb 123Sb 150 480 0.56Rubber 1H 1300 7 0.52PEEK 1H 800 11 0.59V 51V 1700 18 0.67Al 27Al 950 28 0.74 ← current solutionLiCl 7Li 530 33 0.77CuP 31P 90 100 1.3InP 31P 120 58 1.9Pt 195Pt 50 220 2.2TSP 31P 40 30 11Pb 207Pb 10 160 17Cd 113Cd 8 70 42Cd 111Cd 8 73 43Table 5.2: The CRLB (δB, Equation 3.18) is calculated for a variety ofpotential NMR samples. Values for T2 are from the NMR lineshape FWHMslisted in Table A.1. The signal-to-noise ratio is calculated from Equation3.40 using a noise temperature of TNC = 21, 000 K (associated with cablingin a remotely-tuned probe). The parameters in Table A.2 were used alongwith the sample C volume (50 mm3, Table 5.1), a probe quality factor Q =10, filling factor η = 0.3, field Bo = 1 T, temperature Tcoil = 4 K, anda repetition time TR T1. The CRLB is calculated for a measurementbandwidth of 10× γ · FWHM/2pi, and an observation time 10× T2.1065.2.2 The Broadband ProbeIn this section I suggest that the broadband probe is likely a better optionthan the remotely-tuned probe for NMR magnetometry. A brief descriptionof the probe type, its behaviour, and suggested avenues for improvement aregiven.A basic description is helpful because broadband probes are fairly uncom-mon. Only a handful have been reported in the literature [102, 175–186].Furthermore, despite a recent review article [187], few practical guidelinesexist for building and optimizing broadband probes.Most realizations of broadband probes described in published reportswere developed by following the simple design rules of a delay line circuittheory [175]. Only a few reports actually present RF characterizations ofthe device [102, 176, 179, 183, 185, 186], and almost none quantify the typesof errors likely to be encountered in probe construction, or their impact onNMR experiments.30In the absence of better methods, I evaluate the broadband probe withMacLaughlin’s method (discussed in Chapter 3). I also document somelessons-learned attempting to construct homemade broadband probes, aswell as suggested avenues to be explored in search of future improvements.There exist two broadband probe designs not discussed here. They arebased on fundamentally different circuit designs to which the discussion be-low does not directly apply. Webb built (possibly the first) broadband probefrom a traveling wave helix [182]. Fratila et al. realized a broadband probefrom a low impedance (Z ∼ 5 Ω) planar spiral inductor [180].DescriptionA broadband probe – often called a “delay line probe” or a “transmissionline probe” – is constructed from a network of distributed inductance andcapacitance. This situation (sketched in Figure 5.7) leads to the telegrapher’sequations that describe genuine transmission lines such as coaxial cablesand striplines [188]. That description can be used to show that a losslesstransmission line of infinite length has an impedance (the “characteristic”impedance) that is the same at all frequencies:Zo =√L/C (5.2)30Exceptions include Ref. [176] which discusses the issue of RF field homogeneity andRef. [102] in which speculative suggestions for probe optimization are given.107Figure 5.7: A circuit model for a transmission line. Distributed inductance(L) and capacitance (C) result in a frequency-independent characteristicimpedance Zo. Imperfect, or “lossy”, transmission lines can be modeled bydistributed resistance (R) and conductance (G).where L and C are the inductance and capacitance per unit length. Theseparameters depend on the transmission line geometry, which is designed suchthat Zo = 50 Ω for most coaxial cables.The goal of a delay line probe is to emulate the transmission line in sucha way that an NMR sample can be included. Usually this is done by windinga physical coil and attempting to provide a well-distributed capacitance toground from all points on the coil. A circuit that resembles this physicalprobe design more closely is shown in Figure 5.8, where the distributed Land C are replaced by multiple (Ni) small units Li and Ci.The electrical behaviour of a delay line probe has been examined in de-tail by various authors utilizing circuit theory [175, 176, 179]. If the unitinductance and capacitance are chosen such that√Li/Ci = Zo, then a ter-mination with resistor RT = Zo will result in an impedance match from DCup to an angular cutoff frequency,31ωc =√2LiCi. (5.3)If the speed of electromagnetic wave propagation in the delay line is less thanin the cabling that connects to the spectrometer, a (modest) build up of RFenergy will occur and improve the SNR of the NMR measurement. Thecircuit model presented in Ref. [175] provides us with an effective qualityfactor for the probeQeff = 2Niωωc1√1− (ω/ωc)2), (5.4)which, in the low frequency limit, reduces to Qeff = ωtNi , for a “delay time”,31The factor√2 in Equation 5.3 results from a subtlety of the circuit model used inRef. [175]. Slightly different cutoff frequencies can be obtained from different delay linecircuits, and according to the exact definition of cutoff. See Ref. [189] for an example.108Figure 5.8: A circuit model resembling the physical design of most delayline probes. The distributed inductance and capacitance of the transmissionline are approximated with multiple (Ni) small units with inductance andcapacitance Li and Ci. If this ladder sequence is terminated with a resistanceRT =√Li/Ci = Zo, the spectrometer-cabling and probe will be impedancematched over a wide range of frequencies. In this example sketch, Ni = 5.tNi . I note here that the low frequency Q can be derived more simply fromthe quality factor definitionQeff = 2piEnergy storedEnergy dissipated per cycle.The “stored” energy is that propagating in the structure at the reduced speed,v. The “dissipated” energy is the amount that leaves the probe in a singleperiod (2pi/ω). For a wave of energy density Ud that transits a probe oflength b and cross-sectional area A,Qeff =2pibAUd(2pi/ω)vAUd=ωb/v (5.5)=ωtNi .Here the delay time, tNi , is the time required for a wave to propagate a dis-tance b at speed v. Equation 5.5 does not reflect high frequency dispersiveproperties of the probe, but has the advantage of a simple physical interpre-tation. Furthermore, we can connect this result back to the case of an idealtransmission line that supports waves travelling at speedv =1√LC(5.6)109or, for the delay linev =dx√LiCi=b√LtCt(5.7)where dx is the length of the electrical unit i, and Lt = NiLi and Ct =NiCi are the total probe inductance and capacitance. The effective lowfrequency Q can then be transparently approximated from readily accessiblecoil properties. For v = bωc/Ni,32 Equation 5.5 becomesQeff =ωb/v (5.8)=ω√LtCt (5.9)=ωLt/Zo (5.10)=ωLt/RT , (5.11)which is the usual expression for a coil. Unlike a typical NMR coil, however,the probe resistance, RT , is much higher than normal so that the resultingeffective quality factor is usually small.The signal-to-noise ratio for the NMR signal in a delay line probe canbe computed by considering the power delivered to the receiver by spinsprecessing in the coil [175]P =µoM2ω2VcoiltNiκ28(5.12)where Vcoil is the coil volume, M is the spin magnetization density, and κis a factor that accounts for the unequal coupling of spins to each unit ofthe delay line. For a thermal noise voltage,√4kT∆fZo , and low frequencyQeff, one obtains exactly Equation 3.40, if κ/2 is interpreted as the fillingfactor (η = Vsample/2Vcoil).Construction MethodsDelay line probes have been realized in practice by a few different methods.One common construction strategy is to make an NMR coil from enameledwire, and then apply a conductive-paint ground-plane to the enamel [177,178, 184]. Usually paint is added while monitoring the probe impedance(for example, with a network analyzer). The capacitance (Ci) to groundestablished by this paint can be approximated from the thickness of theenamel layer and the surface area of the coil wire.32I use ωc = 2/√LiCi, which is a slightly different definition than given by Equation 5.3.110This method is “finicky”. If the impedance is observed to be suboptimal,it is not clear how much paint to add or remove, or which locations on the coilneed changes. Furthermore, if the capacitance is suspected to be too highit is difficult to remove paint, and usually easier to start again from scratchwith a new coil. Finally, the capacitance that is added has a tendency todrift as the paint dries.A variant of the above method is to press a grounded copper foil onto theouter layer of the NMR coil [185]. In practice I find that the foil is difficultto set reliably and the resulting capacitance is extremely sensitive to thepressure with which the foil is pressed into place.A different approach is to use discrete circuit components to constructthe probe [175, 176, 179, 183, 186]. In this method the coil is made withbare or stripped wire so that short taps can be added to each turn. Eachtap can then be grounded via an appropriate capacitor.The methods described above are difficult to realize, and have forced me(and others [179, 183]) to resort to construction by “trial and error” – wheremultiple probes are made until one is found that performs suitably.Two existing designs did not use physical coils as a part of a delay line,but instead incorporated the sample more directly into a transmission linestructure [102, 181]. They are discussed below in the context of suggestionsfor improved probe designs.A Design ProcedureAccording to established guidelines from circuit theory, a design procedurefor a delay line probe might go as follows:1. Require Li and Ci such that√Li/Ci = Zo is constant (usually at50 Ω).2. Maximize both Li and Ci to obtain the slowest possible wave speed,v. This will yield higher SNR as a result of both the improved Q andlarger sample volume associated with a physically bigger coil.3. Minimize both Li and Ci to obtain a high cut-off frequency.Points (2) and (3) are in conflict and require a compromise. Generally thelargest Li and Ci should be chosen such that satisfactory working frequenciescan be obtained. Once approximate design requirements are fulfilled, it isbest to infer a target value for Ci from the measured inductance of thecoil – which can be significantly different from values that are calculated orspecified by a manufacturer.111Performance AssessmentTo compare the expected signal-to-noise ratio of the broadband probe to thatof other probes it is necessary to calculate both the effective quality factor,and the noise temperature associated with the cabling connecting probe andspectrometer.The noise temperature can be computed for the broadband probe fromits frequency dependant impedance. In what follows I’ve chosen to numer-ically calculate this impedance from a quasi-transmission line circuit model(as in Figure 5.7) with Ni = 6 “units” terminated with a 50 Ω resistor. Theresult of this noise temperature calculation is shown in Figure 5.9 for unitcircuit values Li = Ldx = 33 nH, Ri = ωLi/Qi, Qi = 3, and Ci = Li/Z2o .Also shown is the result for another probe having an imperfect unit capaci-tance as a possible consequence of construction error. The “mistuned” probehas a similar noise temperature and Qeff, but significantly different cut offfrequency. These differences quantify this type of error as being relativelyminor. Both probes have a noise temperature that is degraded from theidealized case discussed in Section 3.5.Generally one can expect the NMR signal-to-noise ratio for the broad-band probe to be quite weak. All other things being equal, the SNR isexpected to scale as√Q/(1 + TNC/T ). To provide a rule of thumb to distin-guish situations where the broadband probe outperforms the remotely-tunedprobe, let’s approximate the former by assuming Qeff = 1 and letting thenoise temperature equal the physical temperature: TNC, broadband ∼ T . Fora remotely-tuned probe (Q ∼ 10), the signal-to-noise ratio is preferable onlywhen TNC, remote/T . 20.To apply this strategy to the design of NMR probes for ALPHA-g wemust explicitly compare the parameters listed in Table 4.2 with the char-acteristics of broadband probes illustrated in Figure 5.9. If an aluminiumsample is employed at 1 T (giving rise to an 11 MHz working frequency)the SNR of a broadband probe (Qeff ∼ 0.25, TNC, broadband ∼ 10 K) will beapproximately a factor of 6 higher than the narrow-band probe described inSection 4.3 (Q = 10, TNC, remote ∼ 21, 000 K). In this case, the degradationin broadband probe SNR from the lower Qeff is outweighed by the vastlyimproved noise temperature. To my knowledge, improvements in SNR fromthe lower noise temperature of a broadband probe remain an untapped ad-vantage for NMR in challenging environments.112Figure 5.9: Top: The calculated real and imaginary impedance of two hypo-thetical broadband probes. The intended design (solid lines) has Li = 33 nH,Ri = ωLi/Q, Q = 3, and Ci = Li/Z2o . The result for a second, imperfectprobe, is shown for Ci = Li/2Z2o (dashed lines). Bottom: The effective qual-ity factor for the two probes (purple) and the expected noise temperature forthe cabling situation described in Section 4.3. Differences in TNC and Qeffquantify performance degradation resulting from a probe constructed withnon-ideal capacitance. The quality factor was calculated using equations 5.4and 5.3.Outlook and Potential ImprovementsIn attempting to make homemade probes by the “painted-ground” method, Ifound (in agreement with observations reported in Ref.[176]) that the shapeof the ground plane could significantly influence the performance of theprobe. A possible explanation is that an imperfect impedance match resultsfrom an electromagnetic field pattern in the probe that is too dissimilar tothe field patterns of the coaxial cable to which it is coupled. Modifying theground plane geometry may change the return path for currents that partlydetermine the electromagnetic mode propagating in the probe structure.To illustrate how a poor “mode match” can effect the impedance matchbetween the probe and the coaxial cable, consider the case of two “crossedwaveguides” shown in Figure 5.10a. Naively, two identical waveguides haveexactly the same impedance. However, when the crossed-guide assemblymistake is made in the lab, reflections of ∼ 20 dB or more can be observed.The reason is that the electromagnetic waves that are able to propagate113in the first waveguide are largely orthogonal to those of the second. It ispossible to couple from one orientation to the other if a gradual transitionis employed, as in the case of a waveguide twist (Figure 5.10b).What is needed for the broadband NMR probe is a reliable transition(analogous to the waveguide twist) that can gradually change the field pat-terns from those of a coaxial cable (Figure 5.10c) to those of the probe(Figure 5.10d). A concept for such a transition is sketched in 5.10e.An alternative method to improve the mode match is to choose a probestructure that is more similar in geometry to the coaxial cable. Two pre-viously reported designs incorporate the NMR sample into modified trans-mission lines. These transmission-line-like probes [102, 181] generally havefield patterns that are more similar to those of the coaxial cable than thoseof the delay line solenoid. Whether or not the performance of these probesbenefited from this effect remains a matter of speculation.114Figure 5.10: a) Two “crossed” waveguides. The incorrect orientation causes a large reflection when the guidesare joined in this way. b) A twist can couple the two orthogonal waveguides by gradually transitioning the guidegeometry from one orientation to the other. An analogous mismatch exists between the electromagnetic fieldpatterns of the coaxial cable (c) and delay line (d). e) A transition region that gradually blends one field patterninto the other may improve the match. Note the delay line ground and dielectric spacer are shown in cross sectionto expose the coil.1155.3 ConclusionThis has been a thesis about NMR magnetometry. I began by telling you(in Chapter 1) that antimatter and gravity are mysterious sectors of modernphysics. The outstanding mysteries were intended to motivate an experi-ment (Chapter 2), planned by ALPHA, to measure the gravitational field(g¯) experienced by antihydrogen atoms in the lab. The dominant source ofsystematic uncertainty in the experiment is expected to result from mag-netic fields that mask the effects of gravity unless very precise magneticcharacterization is achieved. The strategy for that magnetic characteriza-tion (discussed in Section 2.5), necessitated the development of solid-stateNMR magnetometers compatible with the harsh conditions expected in theapparatus – in particular for low temperatures. These conditions precludedthe use of many existing implementations of high-performance NMR mag-netometers (described in Section 2.4).In Chapter 3 I developed a theoretical framework for the behaviour ofsolid-state NMR magnetometers. It summarized the ways in which, overthe course of my graduate work, I have come to think about the NMRmagnetometry challenge presented by the ALPHA gravity experiment. InChapter 4 I described NMR magnetometers that were built and installed inthe ALPHA-g apparatus, and applied the framework developed in Chapter 3to compare their theoretical and observed performance.The precision of a cryogenic magnetometer using an aluminium sample(Chapter 4) was experimentally evaluated in situ using small amounts of dataacquired during the commissioning of a newly built, and largely malfunction-ing apparatus. In particular, the failure of the ALPHA-g cryostat preventedenergization of atom trap magnets that would have enabled the Collabora-tion to directly test the Einstein equivalence principle with antihydrogen.That same failure also prevented a careful experimental characterization ofthe magnetometer under the operating conditions expected during gravityexperiments.In lieu of direct experimental characterization, simulations of the mag-netic fields expected during gravity experiments, together with the modelsin Chapter 3, were used (in Chapter 5) to estimate the expected precisionof NMR magnetometers in the functioning ALPHA-g apparatus. 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Metal samples with cubiclattices also usually avoid the effects of Knight-shift broadening. A largenumber of metals are considered here, in part for their naturally short T1s.Table A.1 lists the expected linewidth for each material. Sample param-eters that contribute to the NMR SNR are given in Table A.2.Also presented in this appendix are preliminary data from the begin-ning of a campaign to experimentally characterize these samples. With twoexceptions,33 data presented here were aquired using a broadband probe con-structed from an air core inductor (Coilcraft 2929SQ-501) 14 mm long withan inner diameter of about 6 mm and 14 tightly spaced turns. A conductivecopper paint (843AR-340G MG Chemicals) was used to apply a ground planeto the outer windings of the coil. The manufacturer specified coil inductanceand quality factor are 500 nH and 180, respectively.Unfortunately, efforts to characterize both the samples and the probewere interrupted by the coronavirus pandemic. Therefore, the analysis andinterpretation of NMR data that follows remains mostly incomplete. Ad-ditionally, a detailed RF description of the broadband probe used in theexperiments is missing. The probe had a reflection coefficient, S11 < −15 dBfor frequencies from DC - 404 MHz, as measured with a network analyzer.33NMR experiments with rubber (0.3 T), and copper-phosphor alloy (1 T) were per-formed using remotely-tuned probes.133Sample Nucleus Ref. T Expected Linewidth FWHMKnight 2.4√M2 obs.shift(K) (ppm) (µT) (µT)YBi 209Bi [190] - 180 120YSb 123Sb [190] - 190 120MoP 31P this work expected 150 -YP 31P [190] - 110 130YSb 121Sb [190] - 160 150CuP15 31P this work 300 ∼170 - see textNaCl 23Na this work 300 - 160 200LiCl 7Li [162] 2.1-14 - 3101 590Pb 207Pb [122] 1.2 20 2002,3 230InP 31P [191] 4.2 - 4802 320InP 115In this work 300 - 2202 340Pt 195Pt [192] 78 - 962,3 160Cd 111Cd [193] 480 601,2 -Cd 113Cd [193] 480 601,2 -TSP 31P [194] 300 - - 610PEEK 1H [195] 300 - - 680Al 27Al see Section 4.3 25 - 650 1000Rubber 1H this work 10 - - 1000V 51V [196] 25 - 1000 1 16001 Calculated in this work.2 Possibly important “pseudo-dipolar” contributions to the linewidth are ne-glected.3 A linewidth dependence on temperature is expected due to contributionsfrom spin-lattice relaxation.Table A.1: Various samples with linewidths comparable to that of alu-minium. The temperature corresponds to that at which the linewidth wasobserved. Both the expected and observed linewdith are from the tabu-lated reference, unless otherwise stated. When not found in the literature,I compute the second moment (M2) – except for “difficult” structures (TSP,polymers). For yttrium compounds, the temperature at which the linewidthmeasurement was made is not specified, but the study encompassed the range1.5-600 K.134Sample Nucleus γ/2pi spin density I T1(MHz/T) (×1028 m−3) (ms)YBi 209Bi 6.9 1.6 9/2 340YSb 123Sb 5.5 0.7 7/2 970MoP 31P 17.3 3.5 1/2 -YP 31P 17.3 2.2 1/2 1600YSb 121Sb 10.2 1.0 5/2 290CuP15 31P 17.3 1.2 1/2 see textNaCl 23Na 11.3 2.2 3/2 -LiCl 7Li 16.5 2.7 3/2 -Pb 207Pb 8.9 0.5 1/2 6.3InP 31P 17.3 2.0 1/2 -InP 115In 9.4 1.9 9/2 see textPt 195Pt 9.2 2.2 1/2 7.5Cd 111Cd 9.0 0.6 1/2 150Cd 113Cd 9.5 0.56 1/2 150TSP 31P 17.3 2.8 1/2 -PEEK 1H 42.6 3.3 1/2 -Al 27Al 11.1 6.0 5/2 450Rubber 1H 42.6 6.7 1/2 see textV 51V 11.2 7.2 7/2 200Table A.2: Sample parameters needed to compute NMR signal-to-noise ra-tios. The spin lattice relaxation time, T1, corresponds to the value expectedat 4 K. For the yttrium compound metals, T1 is calculated using Equa-tion 3.68 and the Knight shifts reported in [190]. For all other metals, T1is calculated from Korringa constants given in [197]. Values of T1 are notgiven for MoP (where no data is available), or for non-metals (where it de-pends on paramagnetic impurities). Spin densities are calculated from thematerial density, molar mass, and natural isotopic abundance. In the case ofPEEK and rubber, the molar mass was that of the repeating polymer chain:C19H12O3, and C5H8, respectively.Yttrium CompoundsThe compounds YBi, YSb, and YP are expected to have promising param-eters for low temperature solid-state magnetometry. The anticipated signal-to-noise ratio is acceptable, and T1 is short. As cubic (NaCl type) crystals,the linewidths of these compounds are expected to be set by dipolar in-teractions. This expectation is fulfilled by noting the close match between135Figure A.1: An example spectrum for a 115In-P sample at room tempera-ture and 4.5 T. Dashed lines represent the best fit with Equation 3.8. Theobserved linewidth for 115In-P in Table A.1 is obtained from this fit. Thefrequency offset is fs = 42.571 MHz.the theoretical second moment and observed linewidths, both of which aresmaller than the corresponding values for aluminium by almost an order ofmagnitude. One drawback is that these compounds are not always readilyavailable from commercial sources, and must sometimes be synthesized inthe lab by a chemist or crystal grower.115In-PThe cubic symmetry of the indium-phosphide structure suggests that 115Inmay have both a narrow line, and high SNR (as a result of its large spin,9/2). As a semiconductor, the low temperature T1 for 115In may be longunless dopant concentration is sufficient to yield Korringa-like spin-latticerelaxation.Figure A.1 shows an example 115In-P spectrum that I acquired at 4.5 T.The sample was made from single crystal In-P, grown by InPACT, witha sulphur dopant concentration between (3-8)×1018 cm−3. Many rod-likeelements were cleaved from a wafer to dimensions 0.35 mm × 2 mm ×15 mm and then stacked to form the sample. The linewidth observed here(3.2 kHz) is slightly larger than observed by Han (2.42 kHz) [198]. Han’sresult is closer to the theoretical value given by a second moment calculated136(in this work) by considering coupling to the 8 nearest indium neighbours(Table A.1). Contributions to the linewidth from “pseudo-dipolar” couplinghave been shown to be significant for the 31P nuclei in this material [191], butthis mechanism has not been considered here for the 115In nuclei. The roomtemperature spin-lattice relaxation time for 115In was of order T1 ∼ 100 ms.If the doping is high enough that this semiconductor is metallic, the Korringarelation would give a spin-lattice relaxation time of 8 s at 4 K.Mo-31PPhosphorous (31P) is a generally attractive nucleus for NMR experiments.A high signal-to-noise ratio usually results from its large gyromagnetic ratioand 100% naturally abundant spin 1/2 isotope. Its spin number implies thatquadrupole broadening of the NMR lineshape will not occur. Finally, in theform of a metallic compound (e.g., Mo-P), we can hope that T1 will be short.Unfortunately the crystal structure of this conductor is not cubic. It hasa tungsten-carbide (WC) structure. Therefore, an anisotropic Knight shift isexpected to broaden the line to some extent. A lower limit for the linewidthis given by the second moment. The value calculated in this work (given inTable A.1) includes contributions from the 2 nearest, and 6 second-nearest31P neighbours, as well as the 6 nearest Mo neighbours. Crystal structuredetails that enabled the calculation can be found in Ref. [199].23NaClSalt is a commonly available material with the cubic symmetry that oftenimplies a narrow NMR lineshape. An acceptable SNR results from reason-able values of spin density, gyromagnetic ratio, and spin number (3/2). Ofthe readily available, cheap materials listed in Table 2.1, salt is anticipatedto give the best magnetometer precision. Unfortunately, obtaining short T1at low temperatures will likely require doping with paramagnetic impurities.An 23Na NMR signal at 4.5 T is shown in Figure A.2. The observed linewidthagrees reasonably well with that calculated from the second moment.RubberThe proton NMR line is notorious for its large SNR. It is in part for thisreason that our room temperature sensors (as well as those used by others[96, 172]) rely on rubber samples. Below its glass transition temperature(∼220 K), however, the linewidth is known to increase by orders of magni-137Figure A.2: An example NMR signal (and corresponding Fourier transform)from powdered 23NaCl at room temperature and 4.5 T. Black dashed linesrepresent the best fit of Equation 3.8 to the data. The lineshape is notparticularly Lorentzian. The observed linewidth for 23NaCl in Table A.1 isobtained from this fit. The frequency offset is fs = 51.346 MHz.Figure A.3: An example 1H NMR signal (and corresponding Fourier trans-form) from a rubber sample at T = 10 K and 0.3 T. Dashed lines representthe best fit of Equation 3.8 to the data. The observed linewidth for rubber inTable A.1 is obtained from this fit. The frequency offset is fs = 13.803 MHz.138tude. Also, the low temperature spin-lattice relaxation time is expected tobe long and to depend on the density of paramagnetic impurities.To my knowledge no NMR data on rubber has been published for tem-peratures below 77 K [169, 170]. Figure A.3 shows an example 1H spectrumacquired from rubber at 10 K and 0.3 T. The sample was Adafruit Industries519 rubber cord impregnated with carbon black. The best fit of Equation3.8 to the data implies a FWHM of 1 mT. Unfortunately, T1 was not mea-sured, but experiments repeated 100 s apart did not show significant SNRdegradation.PEEKPEEK is included in this NMR sample survey as one of the few sample typesthat would be UHV compatible. The FWHM of the 1H NMR line has beenpreviously observed to be 29 kHz or about 700 µT [195]. As with the caseof the rubber sample, T1 is likely to be long, and dependant on the amountof paramagnetic impurities present.VanadiumVanadium (51V) is a metal with cubic symmetry. As such it is expected tohave a short T1 and a (small) linewidth set by dipolar interactions. Addition-ally, its high spin density and spin (7/2) may yield a healthy SNR. On theother hand, vanadium is a Pauli paramagnetic material with susceptibilityχ ∼ 450 × 10−6 (measured at 113 K [196]). This is larger than the corre-sponding value for many other (usually diamagnetic) NMR samples. Abovefields of about 2 T, the 51V natural linewidth (1600 µT [115, 196, 197]) isexpected to be broadened by field inhomogeneity from the magnetization ofthe sample. Vanadium is also a superconductor and becomes problematicfor NMR below its transition temperature Tc ' 5 K (see also Table 3.5 for arange of critical magnetic fields above which operation is desirable when atlow temperature).AluminiumIn Section 4.3 the spin-spin relaxation time of 27Al microparticulate alu-minium was observed to be T2 = 28± 2µs using the cryogenic NMR probesin the ALPHA-g apparatus. This value is significantly shorter than expected(T2 ' 40µs) by comparison with both previous observations and the theo-retical value implied from a calculation of the second moment.139Figure A.4: Example 27Al NMR signals (and corresponding Fourier trans-forms) from two different aluminium samples at T = 300 K and 4.5 T. Blackdashed lines represent the best fit of Equation 3.8 to the data. a) and b)show data taken with a -325 mesh aluminium sample. c) and d) show datataken with an atomized aluminium sample. In each case, a total of 40,000scans were acquired and averaged in order to obtain a large SNR. Thefrequency offset is fs = 50.662 MHz.To investigate the possibility that the lineshape was broadened by eitherthe environmental ALPHA-g field inhomogeneity, or by anomalous sampleproperties, spectra from two different aluminium samples were recorded at4.5 T and room temperature in a homogeneous magnetic field. The firstsample uses an identical material to that in NMR probes described in Sec-tion 4.3: an atomized powder of ∼ 3µm spherical particles with 99.99%purity from Hunan Jinhao Aluminum Industrial Co. Ltd. in a paraffin waxbinder. The second sample used -325 mesh aluminium powder of 99.5%purity from Fisher Scientific (CAS No. 7429-90-5) mixed with Slygard 184silicone rubber (Dow Chemical) as a binder.The complex spectra were subjected to a χ2 minimization with Equa-tion 3.8 as fit function. The best-fit linewidths for each sample implyT2 = 29µs, for atomized aluminium, and T2 = 30µs for -325 mesh alu-minium powder. The time series data and associated spectra are shown inFigure A.4. The NMR lineshapes as compared in Figure A.5 are nearly iden-tical. Neither is very Lorentzian. The discrepancy between the observed andexpected linewidth for aluminium samples has not be resolved by this work.140Figure A.5: Two 27Al NMR spectra from different samples of microparticu-late aluminium at room temperature and 4.5 T. The magnitude of the com-plex spectrum (normalized to its peak value) is shown so that the linewidthscan more easily be compared. Both linewidths are consistent with thoseobserved in Section 4.3. The frequency offset is fs = 50.662 MHz.LiClLithium-Chloride is a salt with the cubic symmetry that usually yeilds anarrow NMR lineshape. A healthy SNR results from reasonable values ofspin density, spin (3/2), and large gyromagnetic ratio. An example for thelow temperature T1 of doped LiCl has been given by Bloembergen, whofound T1 = 90 s at 2.1 K [162].Copper-Phosphor AlloyCopper-phosphor is a metallic alloy with the NMR-attractive nuclei 31P. Asa metallic sample, we can hope that T1 will be short. Unfortunately, as adisordered alloy, we might also expect that the lineshape will be Knight-shiftbroadened.I acquired NMR spectra for 31P in a CuP alloy at 1 T and 4.5 T. Thesedata were obtained using different probes, magnets and spectrometers. Fig-ure A.6 shows the spectrum acquired at Bo ∼ 1 T and room temperature.The sample was a powdered copper-phosphor alloy from Pometon Powder,nominally containing 13-16% phosphorus. The powder (particles sized be-tween 45-106 µm) was mixed to a 60% volume fraction with Slygard 184silicone rubber (Dow Chemical). The sample volume is 0.3 cm3. A total of141Figure A.6: An example 31P NMR signal from powdered copper-phosphoralloy (and corresponding Fourier transform) at room temperature and 1 T.Dashed lines represent the best fit of Equation 3.8 to the data. The frequencyoffset is fs = 17.31 MHz. The Knight-shift broadened linewidth listed inTable A.1 is discussed in the text.500 scans were acquired and averaged in order to obtain a reasonable SNR.The observed FWHM of the NMR line at 1 T was about 3.1 kHz – or about180 ppm. At 4.5 T and room temperature the FWHM was about 12 kHzimplying a fractional width of 150 ppm. These data alone are far from con-clusive, but the rough proportionality of linewidth to field suggests that thissample, like other alloys, has a Knight-shift broadened line.The spin-lattice relaxation time was measured to be 8 ms at 1 T androom temperature. At 4 K this would yield T1 ∼ 600 ms if a Korringa-typetemperature dependence is assumed.In-31PIndium-phosphide is a semiconductor with cubic symmetry and with a highdensity of the desirable 31P nuclear spin. The linewidth is less than thecalculated second moment by a factor of ∼2. This is explained [191] witha “pseudo-dipolar” coupling that partially cancels the conventional dipolarbroadening mechanism – which would otherwise be dominated by couplingto the 115In spins. As a semiconductor, T1 depends strongly on the dopantconcentration.142PlatinumAs a spin-1/2 nucleus in an FCC type metallic lattice, platinum was investi-gated for possible magnetometry advantages from a small linewidth and shortT1. A number of sources report various (small) values for the linewidth of195Pt, including 2.4 kHz at room temperature [115]; 250 µT at room temper-ature and 140µT at 78 K [192]; as well as a range of similar values that maydepend on field and temperature in the presence of paramagnetic impurities[197, 200]. The value for T2 implied by the small second moment is longerthan T1 (listed in Table A.2), so that spin-lattice relaxation contributes tothe linewidth. The temperature dependence of T1 may therefore explain thelinewidth temperature dependence [192]. A lower limit for the Knight shiftanisotropy is given as 5 ppm [197]. The 4 K spin lattice relaxation timeT1=7.5 ms is calculated from the Korringa relation: T1T = 0.03 K·s fromRef. [197].Trisodium Phosphate (TSP)TSP is considered in this sample survey out of a naive hope that, as a read-ily available non-metal containing 31P, the sample might have the desirednarrow lineshape and large SNR. The spin-spin relaxation time is reportedto be about 30µs [194], implying a FWHM linewidth of 600µT. The widthis expected to depend on the water content of the sample. The locations ofwater molecules in the compound are unknown and prevent a reliable calcu-lation of the second moment from dipolar broadening. A lower limit for thelinewidth might be considered for the case of anhydrous TSP. UnfortunatelyI have not been able to find a reliable description of the crystal structurethat would enable a second moment calculation.LeadLead is a metal sample with cubic symmetry. It is considered here for itsshort T1 and (hopefully) small linewidith. In actuality the linewidth may beset in part by the Knight shift anisotropy, which has been observed by Schrat-ter and Williams [122] to be of order 200 Hz / 9 MHz ∼ 20 ppm. That workalso found that the linewidth had an angular dependence spanning a range ofabout 1 kHz. Meanwhile Refs. [116, 201] assert that the dominant linewidthmechanism for lead is “pseudo-dipolar” broadening, with spin-lattice relax-ation contributing to a lesser extent. Luders [111] reports a slightly largerlinewidth (3.3-4.4 kHz) at T = 4.2 K than found in Table A.1 (which gives143Figure A.7: An example NMR signal (and corresponding Fourier transform)from powdered lead at room temperature and 4.5 T. The observed linewidthis larger than expected. Black dashed lines represent the best fit of Equa-tion 3.8 to the data. The linewidth presented in Table A.1 is not obtainedfrom these data. The frequency offset is fs = 40.581 MHz.the value measured by Schratter and Williams). The 4 K spin lattice relax-ation time for 207Pb, T1 = 6.3 ms, is calculated from the Korringa relation:T1T = 0.025 K·s [197].An example 207Pb spectrum taken at 4.5 T and room temperature isshown in Figure A.7. The sample was -200 mesh lead powder (99.9% purity;Alfa Aesar 10223) mixed to a 60% volume fraction with Slygard 184 siliconerubber (Dow Chemical). The sample volume was 0.3 cm3. A total of 40,000scans were acquired and averaged in order to obtain a reasonable SNR. Theobserved FWHM of the NMR line was about 22 kHz - significantly widerthan expected (∼ 7 kHz) from theoretical considerations, even after contri-butions from spin-lattice relaxation and anisotropic Knight shift broadeningare included.CadmiumCadmium was originally included in this material survey for the desirableNMR magnetometry properties usually associated with a spin-1/2 metallicsample. However, the small signal-to-noise ratio and Knight-shift broadenedlinewidth suggest that this is a sample to be avoided for magnetometryapplications.144
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Physics
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Solid-state nuclear magnetic resonance magnetometry at low temperature with application to antimatter gravity experiments by ALPHA
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