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A linear Paul trap for barium tagging of neutrinoless double beta decay in nEXO Lan, Yang 2020

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A linear Paul trap for barium taggingof neutrinoless double beta decay innEXObyYang LanM.Sc., The University of British Columbia, 2014B.Sc., Huazhong University of Science and Technology, 2012A 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)November 2020© Yang Lan 2020The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, thedissertation entitled:A linear Paul trap for barium tagging of neutrinoless double beta decay in nEXOsubmitted by Yang Lan in partial fulfillment of the requirements forthe degree of Doctor of Philosophyin PhysicsExamining Committee:Jens Dilling, Physics and Astronomy, UBCSupervisorChris Waltham, Physics and Astronomy, UBCSupervisory Committee MemberKirk Madison, Physics and Astronomy, UBCUniversity ExaminerDon Douglas, Chemistry, UBCUniversity ExaminerAdditional Supervisory Committee Members:Reiner Kruecken, Physics and Astronomy, UBCCo-supervisorJason Holt, TRIUMFSupervisory Committee MemberiiAbstractnEXO is the next-generation Enriched Xenon Observatory searching forneutrinoless double beta decay (0νββ) in 136Xe. If observed, 0νββ willvalidate the neutrino to be its own anti-particle and determine the absolutemass scale of the neutrinos. nEXO’s sensitivity is limited by the backgroundlevel. Barium tagging is the ultimate background rejection method usingthe coincidence detection of 136Ba as the daughter nucleus.A linear Paul trap (LPT) is needed for the barium tagging concept innEXO or a future gaseous experiment. The theory of an ideal LPT wasstudied from first principles to obtain analytical solutions of the trappedions and to validate a simulation method. Then simulations were doneto optimize the design of a realistic final LPT. The final LPT has beenmanufactured and is being set up. Meanwhile, prototypes of key componentsof the LPT were built for the experimental developments.A prototype of the LPT’s quadrupole mass filter (QMF) achieved massresolving power m/∆m around 140 and exceeded its requirement. A 3Dprinted prototype of the novel ion cooler demonstrated successful ion cooling,trapping and ejection.Based on the progress with the prototypes, improvements were made tothe design of the final setup. The final LPT will be installed between an RFfunnel and a high precision mass spectrometer for barium tagging of nEXO.iiiLay SummaryThe nEXO collaboration is studying a rare nuclear reaction (neutrinolessdouble beta decay) which is much slower than the current age of the universe(1.38 billion years) and in fact has never been observed. The reaction is im-portant because its discovery can reveal some hidden secrets of the neutrino– currently one of the most mysterious fundamental subatomic particles.In order to help discover this rare nuclear reaction, I studied and builtan ion trap which can capture and identify an ion from the reaction. Ialso developed experiments to test the ion trap and proved it is capable ofseparating different ions and capturing the ions we need. The ion trap willbe combined with other ratus to further test its functionalities and help toreveal the nature of neutrinos.ivPrefaceThe work presented in this dissertation contains contributions from manyindividuals and groups in the three collaborations I have been a member of:EXO-200 (Enriched Xenon Observatory), nEXO (next-generation EnrichedXenon Observatory) and TITAN (TRIUMF’s Ion Trap for Atomic andNuclear science).The EXO-200 collaboration is formed by researchers from 26 institutes in7 countries. The development of the experiment started in the early 2000s;the data-taking was done between 2010 to 2018. I participated in shifts torun the EXO-200 and took data from June 20 to 25 of 2016, August 23 toSeptember 16 of 2017, December 14 to December 23 of 2017 and August27 to September 11 of 2018. In addition, I was the data quality analyzerfrom January 2017 to December 2018. I performed routine inspections ofevents measured by EXO-200 and provided weekly data quality reports.The data quality reports helped to steer the operations of the EXO-200 ata low background level and determine the cut of data with a high level ofbackground or abnormal events.My shift work and data quality analysis of the EXO-200 contributed tothe two publications below. The manuscripts were prepared by C. Licciardiin 2018 and G.S. Li in 2019.J. B. Albert, et al. (EXO-200 collaboration). Search for neutrinoless double-beta decay with the upgraded EXO-200 detector. Physical Review Letters120, 072701, 2018.G Anton, et al. (EXO-200 collaboration). Search for neutrinoless double-βdecay with the complete EXO-200 dataset. Physical Review Letters 123,161802, 2019.The key result from the above papers was used in Section 1.3.1 andFigure 1.4 of this dissertation.The nEXO collaboration was formed around 2014 by most of the re-searchers from EXO-200 and a few other institutes. I participated in thedevelopment of a barium tagging technique for nEXO.vThe barium tagging technique was based on the Monte Carlo simulationsof an RF funnel by V. Varentsov at FAIR (Facility for Antiproton and IonResearch) and the experimental study of an RF funnel prototype at StanfordUniversity by T. Brunner (now at McGill University) and D. Fudenberg.Figure 4.2 of this dissertation is plotted using simulated results of the RFfunnel by V. Varentso and D. Fudenberg (unpublished).I mainly focused on developing a linear Paul trap (LPT) downstreamfrom the RF funnel for the barium tagging. The development of the LPTwas done at the TITAN group at TRIUMF (TRI-University Meson Facility,Canada’s particle accelerator centre). Contributions to the development ofthe LPT from different individuals and groups are listed below:• In Chapter 2, the theory and analytical solutions of an ideal LPT werederived by me.• In Chapter 3, the simulations were done by me. I reported the sim-ulations results in regular meetings with the TITAN group and thebarium tagging group of nEXO; the feedback from these meetings con-tributed to the progress of the simulations.• In Chapter 4, the mechanical design of the LPT was done by me withconsultations provided by J. Langrish at TRIUMF’s design office andM. Good of the TITAN collaboration.• In Chapter 5 and Appendix A, the prototypes of the LPT were de-signed and machined by me. The experimental development of theelectronics, control and DAQ (data acquisition) systems was done byme with discussion and advice from members of the TITAN group andthe nEXO barium tagging group. The data of all the experimentalmeasurements were taken by me. The machining of the parts for thefinal LPT setup was done by the Physics department machine shop ofthe Universite´ de Montre´al. The partial assembling of the final LPTwas done by X. Shang and H. S. Rasiwala at McGill University.• In Appendix B, the mechanical drawings of the LPT were made byX. Shang based on the 3D Solidworks models I designed. Many im-provements to the design have been contributed by X. Shang.Based on the material of this dissertation, the following publications arein preparation. The manuscripts were written by me and will be revisedbased on communications with the co-authors.vi• Y Lan and J Dilling. Analytical solutions for the performance ofquadrupole mass spectrometers and ion guides. Adapted from Chap-ter 2.• Y Lan, T Brunner, A Kwiatkowski and J Dilling. A novel ion coolerand buncher with tapered electrodes. Adapted from Section 4.4.2 andSection 5.3.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Abbrevations . . . . . . . . . . . . . . . . . . . . . . . . . . xxAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Neutrino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Neutrinoless double beta decay . . . . . . . . . . . . . . . . . 31.2.1 0νββ experiments . . . . . . . . . . . . . . . . . . . . 41.3 EXO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.1 EXO-200 . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.2 nEXO . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Barium tagging . . . . . . . . . . . . . . . . . . . . . . . . . 112 Theory of the linear Paul trap . . . . . . . . . . . . . . . . . 162.1 Radio frequency quadrupole . . . . . . . . . . . . . . . . . . 162.1.1 Ion dynamics in an RFQ . . . . . . . . . . . . . . . . 162.1.2 Pseudopotential well model . . . . . . . . . . . . . . . 182.1.3 Full solution of Mathieu equation . . . . . . . . . . . 202.1.4 Ion motion . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Ion acceptance and emittance . . . . . . . . . . . . . . . . . 292.3 RFQ ion guide . . . . . . . . . . . . . . . . . . . . . . . . . . 38viii2.4 Quadrupole mass filter . . . . . . . . . . . . . . . . . . . . . 392.4.1 Mass scan of QMS . . . . . . . . . . . . . . . . . . . . 392.4.2 QMS mass resolving power and transmission efficiency 422.5 Linear Paul trap . . . . . . . . . . . . . . . . . . . . . . . . . 442.5.1 Ion cooling with buffer gas . . . . . . . . . . . . . . . 462.5.2 Ion cooling in LPT . . . . . . . . . . . . . . . . . . . 472.5.3 Optimum gas pressure for ion cooling . . . . . . . . . 502.5.4 Equilibrium ion temperature . . . . . . . . . . . . . . 533 Simulations of the linear Paul trap . . . . . . . . . . . . . . . 543.1 Electric potential in an LPT . . . . . . . . . . . . . . . . . . 543.1.1 Quadrupole electrode geometries . . . . . . . . . . . . 553.2 Ion transmission simulations in an RFQ . . . . . . . . . . . . 593.2.1 Ion acceptance simulations in a pure quadrupole po-tential . . . . . . . . . . . . . . . . . . . . . . . . . . 593.2.2 Ion acceptance in non-perfect quadrupole potentials . 643.3 RFQ ion guide simulation and optimization . . . . . . . . . . 653.3.1 Influence of higher-order spatial harmonics in electricpotential . . . . . . . . . . . . . . . . . . . . . . . . . 653.3.2 Electrode geometries for RFQ ion guide . . . . . . . . 703.4 QMS simulation and optimization . . . . . . . . . . . . . . . 733.4.1 Influence of higher-order spatial harmonics . . . . . . 733.4.2 Electrode geometries for QMS . . . . . . . . . . . . . 753.5 RFQ ion cooler . . . . . . . . . . . . . . . . . . . . . . . . . . 833.5.1 Ion drift velocity and mobility . . . . . . . . . . . . . 833.5.2 Ion cooling rate . . . . . . . . . . . . . . . . . . . . . 853.5.3 Ion temperature . . . . . . . . . . . . . . . . . . . . . 883.5.4 Ion trapping in the longitudinal direction . . . . . . . 923.6 RFQ ion buncher . . . . . . . . . . . . . . . . . . . . . . . . 984 A linear Paul trap system for barium tagging . . . . . . . . 1054.1 LPT system requirements . . . . . . . . . . . . . . . . . . . . 1054.1.1 Ion acceptance requirement . . . . . . . . . . . . . . . 1074.1.2 Vacuum requirements . . . . . . . . . . . . . . . . . . 1104.1.3 Mechanical tolerance requirements . . . . . . . . . . . 1114.2 Mechanical design . . . . . . . . . . . . . . . . . . . . . . . . 1124.3 QMF design . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.4 Ion cooler design . . . . . . . . . . . . . . . . . . . . . . . . . 1144.4.1 Pre-cooler as the differential pumping channels . . . . 1154.4.2 Ion cooler . . . . . . . . . . . . . . . . . . . . . . . . 117ix4.5 Laser spectroscopy ion trap (LSIT) design . . . . . . . . . . 1214.6 Ion buncher design . . . . . . . . . . . . . . . . . . . . . . . . 1214.7 Vacuum system of the LPT . . . . . . . . . . . . . . . . . . . 1254.8 Manufacturing of the LPT . . . . . . . . . . . . . . . . . . . 1275 Experiments and results . . . . . . . . . . . . . . . . . . . . . 1285.1 Test stand setup . . . . . . . . . . . . . . . . . . . . . . . . . 1285.1.1 Ion source . . . . . . . . . . . . . . . . . . . . . . . . 1305.1.2 Ion detector . . . . . . . . . . . . . . . . . . . . . . . 1315.1.3 Tests with ion source and detectors . . . . . . . . . . 1335.2 Quadrupole mass filter prototypes . . . . . . . . . . . . . . . 1365.2.1 QMF2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 1375.3 RFQ cooler prototype . . . . . . . . . . . . . . . . . . . . . . 1545.3.1 Installation of the cooler prototype in test stand . . . 1575.3.2 Ion transmission tests with the cooler . . . . . . . . . 1575.3.3 Experiments with ion trapping . . . . . . . . . . . . . 1625.4 Ion temperature in the LPT . . . . . . . . . . . . . . . . . . 1765.5 The final LPT system . . . . . . . . . . . . . . . . . . . . . . 1776 Conclusion and future work . . . . . . . . . . . . . . . . . . . 179Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182AppendicesA QMF prototypes . . . . . . . . . . . . . . . . . . . . . . . . . . 194A.1 Prototype QMF1 . . . . . . . . . . . . . . . . . . . . . . . . 194A.1.1 QMF1 machining . . . . . . . . . . . . . . . . . . . . 194A.1.2 QMF1 mechanical precision measurement . . . . . . . 196A.1.3 Installation of QMF1 in test stand . . . . . . . . . . . 199A.1.4 Ion transmission test . . . . . . . . . . . . . . . . . . 199A.1.5 Summary for QMF1 . . . . . . . . . . . . . . . . . . . 202A.2 QMF V2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203A.2.1 QMF2.1 design and machining . . . . . . . . . . . . . 203A.2.2 QMF2.1 mechanical precision measurement . . . . . . 204A.2.3 Installation of QMF2.1 in test stand . . . . . . . . . . 206A.2.4 Ion transmission test . . . . . . . . . . . . . . . . . . 206A.2.5 QMF2.1 for mass measurement as a QMS . . . . . . 209A.2.6 Mass measurement with square wave RF signal . . . 212xA.2.7 Mass measurement at higher RF amplitude . . . . . . 213B Mechanical drawings of the LPT . . . . . . . . . . . . . . . . 218xiList of Tables2.1 Calculated C2n for a few typical values of q when a = 0. . . . 282.2 Parameters of 136Ba+ ion cooling in helium buffer gas at dif-ferent pressures. . . . . . . . . . . . . . . . . . . . . . . . . . . 484.1 The acceptance  and phase independent acceptance PI fora few sizes of RFQ operating at 1 MHz and (q = 0.45, a = 0).The RF voltage needed is shown in the last row of the table. 110A.1 Stability parameter q for different ions at a few RF frequen-cies. Values between 0 and 0.908 are emphasized in bold font. 201xiiList of Figures1.1 Electron energy spectrum for ββ and 0νββ . . . . . . . . . . 51.2 Engineering design rendering of the cleanroom containing theEXO-200 detector. . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Illustration of the EXO-200 TPC . . . . . . . . . . . . . . . . 81.4 Best fit to the energy spectrum for Phase I (top) and PhaseII (bottom) data of EXO-200. . . . . . . . . . . . . . . . . . . 91.5 Pre-conceptual design of nEXO. . . . . . . . . . . . . . . . . . 101.6 Sensitivity of nEXO with and without barium tagging . . . . 122.1 Illustration of a linear Paul trap . . . . . . . . . . . . . . . . 172.2 Calculated Mathieu stability diagram . . . . . . . . . . . . . 242.3 Mathieu stability diagram in the region of 0 < β < 1 . . . . . 252.4 Stability diagram for both y and z axis . . . . . . . . . . . . . 262.5 Solutions of the Mathieu equation for a few q values annotatedin each plot when a = 0 . . . . . . . . . . . . . . . . . . . . . 302.6 Stable (top) and unstable (bottom) solutions of the Mathieuequation obtained by numerical integrations. . . . . . . . . . 312.7 Ellipses in phase space . . . . . . . . . . . . . . . . . . . . . . 332.8 Ellipse expressed in Twiss parameters . . . . . . . . . . . . . 362.9 Analytically calculated acceptance . . . . . . . . . . . . . . . 382.10 Combined acceptance in both axes . . . . . . . . . . . . . . . 392.11 Acceptance of RF ion guide . . . . . . . . . . . . . . . . . . . 402.12 QMS mass scan with voltage sweep . . . . . . . . . . . . . . . 412.13 QMS mass scan with frequency sweep . . . . . . . . . . . . . 432.14 Mass resolving power and transmission efficiency of QMS . . 442.15 Linear Paul trap with segmented quadrupole electrodes . . . 452.16 Analytical and numerical calculation of ion cooling with buffergas using the ion mobility at different gas pressures. Thestability parameters are (q = 0.1, a = 0) . . . . . . . . . . . . 492.17 Ion velocity of analytical and numerical solutions shown inFigure 2.16. . . . . . . . . . . . . . . . . . . . . . . . . . . . 50xiii2.18 Analytical and numerical calculation of ion cooling using theion mobility at different gas pressures. The stability param-eters are (q = 0.5, a = 0). . . . . . . . . . . . . . . . . . . . . 523.1 Electric potential of hyperbolic electrodes . . . . . . . . . . . 573.2 Electric potential of round electrodes . . . . . . . . . . . . . . 583.3 Ion transmission simulation through an RFQ in SIMION. . . 603.4 Distribution of initial and transmitted ions in the RFQ sim-ulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.5 Simulated ion acceptance compared to theoretical values . . . 633.6 Simulated ion transmission and acceptance at q = 0.64 anda = 0 with different RF cycles. . . . . . . . . . . . . . . . . . 643.7 Ion transmission simulation for an RF ion guide with added6th spatial harmonics . . . . . . . . . . . . . . . . . . . . . . 663.8 Ion transmission simulation for an RF ion guide with added10th spatial harmonics . . . . . . . . . . . . . . . . . . . . . . 673.9 Ion transmission simulation for an RF ion guide with added14th spatial harmonics . . . . . . . . . . . . . . . . . . . . . . 683.10 Ion transmission simulation for an RF ion guide with added18th spatial harmonics . . . . . . . . . . . . . . . . . . . . . . 693.11 Simulated ion acceptance for the electric potential with thepresence of the 6th spatial harmonic component. . . . . . . . 703.12 Simulated ion acceptance for the electric potential with thepresence of the 10th and above spatial harmonic component. 713.13 Simulated acceptance of an RF ion guide with hyperbolicelectrodes of different truncation. See text for details. . . . . 723.14 Simulated acceptance of an RF ion guide with round elec-trodes of different radius re = ηr0. See text for details. . . . . 733.15 (Top) Ion acceptance of an ideal QMS with pure quadrupolepotential. (Bottom) The mass resolving power is derived fromthe ion acceptance and shown as a function of a. . . . . . . . 743.16 Ion transmission simulation in a QMS with added 6th spatialharmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.17 Peak shape of a QMS with pure quadrupole potential (theory)and added 6th order spatial harmonic (simulations) . . . . . . 773.18 Peak shape of a QMS with pure quadrupole potential (theory)and the added 10th (top), 14th (middle) and 18th (bottom)higher-order spatial harmonic (simulations). . . . . . . . . . . 783.19 Peak shapes of QMS with hyperbolic electrodes of differenttruncation at a = 0.236. . . . . . . . . . . . . . . . . . . . . . 79xiv3.20 Simulation of a QMS with hyperbolic electrodes. . . . . . . . 803.21 QMS with round electrodes with radius re = 1.14511r0. . . . 813.22 QMS with round electrodes with radius re = 1.13r0. . . . . . 823.23 Simulation of ion drift velocity vd in 0.1 mbar helium gas withdifferent electric field strength. . . . . . . . . . . . . . . . . . 843.24 Ion cooling simulation in an LPT with stability parameter(q = 0.1, a = 0) at different helium gas pressures. . . . . . . . 863.25 Ion cooling simulation in an LPT with stability parameter(q = 0.5, a = 0) at different helium gas pressures. . . . . . . . 873.26 Velocity as a function of time for an ion cooled in an LPTwith 1 mbar of helium gas. . . . . . . . . . . . . . . . . . . . 883.27 Motion of ions trapped in a simple harmonic potential wellshown in the position-momentum (u− p) phase space. . . . . 893.28 Temperature of ions in the LPT with 0.1 mbar helium buffergas at different stability parameter q. The ions were confinedonly in the radial direction. See text for details. . . . . . . . . 913.29 A simplified LPT for the simulation of ion trapping in thelongitudinal direction. . . . . . . . . . . . . . . . . . . . . . . 933.30 Secular frequency ω¯ and reduced secular frequency ω¯′ of ionmotion in an LPT. . . . . . . . . . . . . . . . . . . . . . . . . 953.31 Temperature of ions in an LPT with different longitudinaltrapping potential depth U2. . . . . . . . . . . . . . . . . . . . 963.32 Longitudinal emittance x and transverse y,z of cooled ionsin the LPT at different longitudinal trapping potential U2 andRF frequency fRF . . . . . . . . . . . . . . . . . . . . . . . . 983.33 An RFQ ion buncher for the simulation of ion ejection. . . . . 993.34 Kinetic energy K¯E (top plot) and energy spread σKE (bottomplot) of ions ejected from the buncher at different electric fieldstrength Ex and helium buffer gas pressure. . . . . . . . . . . 1003.35 Simulated time of flight (ToF) of the ejected ions to reachthe exit of the ion buncher as a function of the longitudinalelectric field strength Ex . . . . . . . . . . . . . . . . . . . . . 1023.36 Ion energy KE and time-of-flight tToF of bunched ions as afunction of helium gas pressure in the ion buncher. See textfor the details of the different labeled pressure regions. . . . . 1034.1 A conceptual design of the LPT system as of 2017. . . . . . . 1064.2 Distrubution of ions extracted from the RF funnel . . . . . . 1074.3 The phase independent acceptance obtained as the overlap ofacceptances at 36 RF phases. . . . . . . . . . . . . . . . . . . 109xv4.4 The phase independent acceptance PI at different q value. . 1104.5 Rendered drawings of the mechanical design of the LPT sys-tem. See text for details. . . . . . . . . . . . . . . . . . . . . . 1134.6 Rendered drawings of the finalized design of QMF. . . . . . . 1144.7 Rendered drawings of the design of the pre-cooler. . . . . . . 1164.8 Electric field penetration for quadrupole electrodes of differ-ent width. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.9 Multipole expansion of potential inside the cooler. (a) and(b) show the simulated electric potential distribution for theelectrode width 2 mm and 4 mm. (c) shows the coefficient ofthe spatial harmonics as a function of the electrode width w. 1204.10 Mechanical design of the cooler . . . . . . . . . . . . . . . . . 1224.11 Comparison between the previous design (left) of the ion trapfor barium tagging and the new design (right) in this study. . 1234.12 Mechanical design of the laser spectroscopy ion trap. . . . . . 1244.13 Rendered drawings of the laser spectroscopy ion trap as anion buncher with a pulse drift tube (indicated). . . . . . . . . 1254.14 Schematics of the vacuum system of the LPT. . . . . . . . . . 1265.1 The test stand for the experimental development of the LPTsystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.2 TIS assembly and ion sources . . . . . . . . . . . . . . . . . . 1305.3 A custom made Faraday cup (FC) . . . . . . . . . . . . . . . 1325.4 CEM ion detector . . . . . . . . . . . . . . . . . . . . . . . . 1335.5 A configuration of the test stand for testing ion source anddetectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.6 Measurement of ion detectors reading at different ion sourcefloating voltages. . . . . . . . . . . . . . . . . . . . . . . . . . 1355.7 Photos of QMF 2.2. See text for details. . . . . . . . . . . . . 1385.8 Photo of a mechanical precision problem in the QMF2.2 as-sembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.9 Photos of QMF2.2 installed in the vacuum chamber of teststand. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405.10 Frequency response of the function generator and the gain ofthe RF amplifier. . . . . . . . . . . . . . . . . . . . . . . . . . 1425.11 Amplitudes of function generator (top) and amplified RFvoltages with RF balancing (middle). The amplitude of CH1in the top plot has a similar shape as the gain difference(CH2-CH1)/CH1 in the bottom plot as a result of the RFbalancing. See text for details. . . . . . . . . . . . . . . . . . 144xvi5.12 Ion mass measurements with QMF2.2 at different U/V values.The cutoffs of the ion stability patterns near the lower rightcorners are explained in the text. . . . . . . . . . . . . . . . . 1455.13 Ion mass measurements with QMF2.2 at different U/V val-ues. The cutoff in the ion stability diagram as described inFigure 5.12 has been fixed as explained in the text. . . . . . . 1475.14 QMS V2.2 mass spectrometry measurement . . . . . . . . . . 1485.15 Measured and simulated ion transmission at different RF toDC voltages U/V . Ion energy is around 50 eV. See text fordetails. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1505.16 Measured and simulated ion transmission at different RF toDC voltages U/V . Ion energy is around 5 eV. See text fordetails. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1525.17 Measured and simulated ion transmission at different RF toDC voltages U/V . Ion energy is around 1 eV. See text fordetails. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1535.18 Photos of the RFQ ion cooler prototype. See text for details. 1555.19 Photos of the RFQ ion cooler prototype during and after as-sembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1565.20 Installation of the cooler to the test stand’s vacuum chamber.See text for details. . . . . . . . . . . . . . . . . . . . . . . . . 1585.21 Ion transmission measurement with the cooler at different DCvoltage U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1595.22 Ion transmission measurements with the cooler using ionsfrom an almost pure cesium ion source (#2). . . . . . . . . . 1605.23 Ion transmission measurements of the cooler with helium gaspressure of 5.5× 10−3 mbar. . . . . . . . . . . . . . . . . . . . 1625.24 Ion transmission measurements of the cooler with helium gaspressure of 1.9× 10−2 mbar. . . . . . . . . . . . . . . . . . . . 1635.25 Simulated DC potential along the RFQ cooler’s central axisfor ion trapping and ejection when the metal tube voltage wasset at Utube = −150 V. The inset is a zoomed-in plot. . . . . . 1655.26 An example showing ion signal detected by the CEM ion de-tector from the ejected ions as a function of time of flight(ToF). The ions were trapped with the aperture voltage of240 V. After the aperture voltage was switched to 0 V, theions started to fly out and hit the CEM detector later. TheToF of the ions is obtained from the ion signals. . . . . . . . 1665.27 Ion count of cooled and ejected ions as a function of injectionenergy at 5.2× 10−3 mbar helium buffer gas pressure. . . . . 167xvii5.28 Cooling and trapping of ions with different injection energyat 1.9× 10−2 mbar helium buffer gas pressure. . . . . . . . . 1685.29 Measured ion time of flight (ToF) mean tToF and ToF spreadσtToF as a function of the ejection voltage UE at trappingvoltage UT from 50 V to 110 V. . . . . . . . . . . . . . . . . . 1695.30 Measured ion time of flight (ToF) mean tToF (top plot) andToF spread σtToF (bottom plot) as a function of the ejectionvoltage UE at larger trapping voltage UT from 120 V to 180 V. 1705.31 Simulation of cooler’s ion ejection ToF as a function of theejection voltage UE at different trapping voltage UT . Thetrapping voltage UT is from 50 V to 180 V with a 10 V incre-ment as represented by the size of the marker and the colorbar.1715.32 Charge buildup problem near the exit of the cooler. Thecharge buildup occurred after lots of ions deposit on the innersurface of the electrode holder (highlighted in blue). . . . . . 1725.33 Ion cloud split by the charge buildup, resulting in two ionbunches. The ejection voltage used in this measurement wasUE = −20 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1735.34 Measurement of ion numbers in the cooler as a function ofstorage time. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1755.35 Temperature of 136Ba+ ions trapped in the cooler with differ-ent RF voltage obtained from simulation. . . . . . . . . . . . 177A.1 Machining of the QMF1’s monolithic holder. . . . . . . . . . 195A.2 The first prototype QMF1 before and after assembly. . . . . . 196A.3 QMF electrode measurement . . . . . . . . . . . . . . . . . . 197A.4 Photos of the QMF1 installed in the vacuum chamber of thetest stand. See text for details. . . . . . . . . . . . . . . . . . 200A.5 Ion transmission test of QMSV1 with RF frequency scan. . . 202A.6 Maching of the holders for QMFV2.1. . . . . . . . . . . . . . 204A.7 The QMF2.1 prototype before and after assembly. . . . . . . 205A.8 QMF2.1 installed in the vacuum chamber of the test stand. . 207A.9 Ion transmission test of QMF2.1 with frequency scan. . . . . 208A.10 Photo of the RF&DC mixing boxes with the electronics an-notated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209A.11 Mass measurement using QMF2.1 as a quadrupole mass spec-trometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211A.12 Mass measurement using QMF2.1 as a QMS with square waveRF signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214A.13 Photos of the RF amplifier Aigtek ATG2022H. . . . . . . . . 215xviiiA.14 RF amplitude (and gain) of the RF amplifier at 1 V inputfrom 0.01 to 5 MHz. . . . . . . . . . . . . . . . . . . . . . . . 215A.15 Mass measurement with QMF2.1 at higher RF amplitude. . . 217xixList of Abbrevations0νββ Neutrinoless double beta decay.CEM Channel Electron Multiplier.CLB Cooler, Laser spectroscopy and Buncher.DAQ Data ACquisition.EXO Enriched Xenon Observatory.FC Faraday Cup.LPT Linear Paul Trap.LSIT Laser Spectroscopy Ion Trap.MR-TOF Multi-Reflection Time-Of-Flight.nEXO next-generation Enriched Xenon Observatory.QMF Quadrupole Mass Filter.QMS Quadrupole Mass Spectrometer.RFQ Radio Frequency Quadrupole.TMP Turbo Molecular Pump.xxAcknowledgmentsI would like to thank my academic and research supervisor Dr. Jens Dillingfor introducing me to the exciting research at EXO and TITAN and for thecontinued guidance over the years. I am especially grateful for the knowledgeand experience he shared with me on how to build an ion trap. I am alsograteful for not being laughed at with my usually too optimistic timelineand for not being pressured when I was lagging behind.I would like to thank my co-supervisor Dr. Reiner Krucken for introduc-ing me to the interdisciplinary field of isotopes and for the financial supportthrough the IsoSiM (Isotope for Science and Medicine) fellowship program.During my adventure of developing and building an ion trap, I got themost direct and valuable advice from Dr. Thomas Brunner who leads theneutrino physics group at McGill and Dr. Ania Kwiatkowski who leads theTITAN group at TRIUMF. I am also grateful for their efforts in keepingthe teams and labs safe, healthy and friendly.I appreciate the opportunity provided by Dr. Giorgio Gratta for my re-search stay at Stanford University when I first joined EXO, and I appreciatethe lab experience shared with me by Thomas and Dan.I also appreciate the opportunity provided by Dr. Michael Block for myresearch stay at GSI in Germany. I am grateful for the hospitality of theSHIPTRAP members and for the trust of Dr. Sebastian Raeder in givingme full control of a quadrupole mass filter setup. The experience and skills Igained there were invaluable later when I developed the experimental setupof the ion trap.I would like to thank my collaborators Brian and Jon for introducing andaccompanying me to the unique underground facility of EXO-200 during myshifts. It was very warm at -2150 ft on those winter days.I thank all the other 60+ TITAN, EXO-200 and nEXO colleagues andcollaborators I worked and chatted with. I cherish those productive andhappy experiences with each of them.xxiThe experimental development of the ion trap benefited from the ex-pertise of many people at TRIUMF. Especially, I thank John Langrish forbeing the ideal consultant when I was doing the mechanical design of theion trap and I thank Mel Good for keeping the vacuum pumps running.The simulations presented in this work were made possible by ComputeCanada and WestGrid, especially Orcinus. Without these high-performancecomputing clusters, these simulations would take more than 20 years to runon my quad-core desktop computer.Finally and most importantly, I would like to thank my wife Meiling fortaking care of the family during the months when I was spending all mytime with experiments and writing.xxiiChapter 1IntroductionThe Enriched Xenon Observatory (EXO) is searching for the extremely rareneutrinoless double beta decay (0νββ) in 136Xe. Experimental detection ofthis lepton number violating process would prove the neutrino to be its ownantiparticle.Assuming the neutrino to be a light Majorana neutrino is currently themost favored explanation to 0νββ [DPR19, Car18]. Under this theoreticalmechanism, the measured half-life time of 0νββ would provide critical in-formation for determining the absolute mass scale of the neutrino.1.1 NeutrinoThe neutrino was postulated by Wolfgang Pauli [Pau30] in 1930 in orderto explain energy and momentum conservation in beta decay. The elusivenature of the neutrinos, due to lack of electromagnetic and strong interactionwith matter, makes them very difficult to detect with current experimentalmeasurements.The first direct detection of neutrinos (electron antineutrino) was in 1956by Cowan and Reines [RC56] using two large tanks filled with 200 L of waterto react with neutrinos emitted from a nearby nuclear reactor. The reactionused for the neutrino detection was via the inverse beta decay ν¯e+p→ n+e+.The positron e+ generated in the reaction would annihilate with a nearbyelectron and emit a pair of gamma rays. The water was also filled with40 kg of CdCl2 to absorb the neutron generated in the reaction and emitanother gamma ray. The gamma rays of expected sequences were detectedby photomultiplier tubes inside three tanks filled with liquid scintillatorwhich were arranged beside and between the water tanks. Even though theneutrino flux from the reactor was 5× 1013 s−1cm−2, the detected event ratewas only 2.88± 0.22 counts/hr in the entire volume.In subsequent experiments, neutrinos were found to have three types(flavors) associated with each of the three leptons: electron neutrino νe,1muon neutrino νµ [DGG+62] and tau neutrino ντ [KUA+01].The standard model of particle physics constructed in the 1970s assumedthe neutrino to have zero mass. However, breakthroughs in neutrino oscilla-tion experiments in the last few decades proved that neutrinos have non-zeromass and they can change flavor when travelling through space [FHI+98a,AAA+02]. Neutrinos are in states of a definite flavor when being producedor when interacting with matter; otherwise they are in states of definitemass when propagating in space. The neutrino flavor states[νe νµ ντ]Tandmass states[ν1 ν2 ν3]Tare related via a matrix transformationνeνµντ =Ue1 Ue2 Ue3Uµ1 Uµ2 Uµ3Uτ1 Uτ2 Uτ3ν1ν2ν3 , (1.1)where Uαi are elements of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS)matrix [Ber14]. Neutrino oscillation experiments such as the Super-Kamiokande[FHI+98b], Sudbury Neutrino Observatory (SNO) [CDDJ+98], KamLAND[EEF+03], MINOS [MAA+06] and T2K [AAA+12] determined most of theparameters in the PMNS matrix and the differences of squared massesamong the three mass eigenstates: ∆m221 ≈ 7.5 × 10−5 eV2 and |∆m232| ≈2.4 × 10−3 eV2 [CFL+14], where ∆m2ij = m2i −m2j . Since |∆m232|  ∆m221and the sign of ∆m32 is unknown, there are two possible mass hierarchies:∆m232 > 0 =⇒ m1 < m2 < m3 “Normal hierarchy”, (1.2)∆m232 < 0 =⇒ m3 < m1 < m2 “Inverted hierachy”. (1.3)In order to identify the correct neutrino mass hierarchy and obtain theabsolute mass scale of the neutrinos (their masses’ offset from zero), thereare a few possible approaches [BK18]:• Beta decay spectrum end pointA classical and direct way to measure the electron neutrino mass isto measure the beta decay spectrum endpoint of normal beta decays.The Mainz and Troitsk experiments measuring the beta decay of tri-tium obtained mve ≤ 2.3 eV [KBB+05] and mve ≤ 2.2 eV [ABB+11].A combined collaboration from Mainz, Troitsk and other experimentsare setting up a large experiment, KATRIN (Karlsruhe Tritium Neu-trino Experiment), aiming at lowering the limit by a factor of 10 to20.2 eV [Cho17]. As of the end of 2019, KATRIN has been operatingand obtained the experimental result of me < 1.1 eV [AAA+19].• Cosmological observationThe neutrino is the most abundant matter particle in the universe, sothe neutrino mass could have an observable effect on the density ofthe universe if it are heavy enough. Via measurement of the cosmicwave background, the Planck experiment reported the sum of neutrinomass∑imi < 0.12 eV [AAA+18a].• Neutrinoless double beta decay (0νββ)0νββ experiment is an indirect way of measuring the neutrino mass.However, it potentially has the best sensitivity by probing the neu-trino mass around 10 meV [DMVV16] or lower; a few next-generation0νββ experiments aiming at such sensitivity are already under devel-opment [DPR19].1.2 Neutrinoless double beta decayDouble beta decay (ββ) is a process where a nucleus with proton numberZ and mass number A (Z, A) undergoes two beta decays simultaneouslyand becomes (Z + 2, A). For an ordinary double beta decay, two neutronsinside the nucleus turn into two protons; two electrons and two-electronantineutrino are emitted. For 136Xe as an example, the ββ process is136Xe→ 136Ba++ + 2e− + 2ν¯e. (1.4)This ordinary double beta decay is also called a two neutrino double betadecay (2νββ) to be distinguished from the neutrinoless double beta decay(0νββ) which does not emit any neutrinos:136Xe→ 136Ba++ + 2e− + 0ν¯e, (1.5)which would be possible assuming that the two neutrinos annihilated eachother and have violated the conservation of lepton number.Double beta decay only happens to even-even isotopes [GP12] wherethe process is kinematically forbidden from the ordinary beta decay due toexcessively large binding energy from the nucleon pairs while emitting two3electrons is kinematically allowed. The ββ is extremely rare with a half-lifein the order of 1020 years. 0νββ is even rarer and in fact has never beenexperimentally observed.The theoretical 0νββ half-life time is related to the effective neutrinomass [AIEE08] via1T 0νββ1/2= G|M |2〈mββ〉2 ≈ 1028(0.01 eV〈mββ〉)2, (1.6)where G is the phase space factor for emission of the two electrons, M is thetheoretically derived nuclear matrix element for this second order processand 〈mββ〉 is the effective Majorana mass of the electron neutrino.The calculated value of G ≈ 10−25 yr−1eV −2 for the 0νββ of 136Xe isavailable from literature [KI12, SM13]. The currently available value of thenuclear matrix element M for the 0νββ of 136Xe varies in the range of 1.89to 4.2 depending on the different nuclear models [ME13]. Inserting thesevalues into Eq. (1.6), the half-life of the 0νββ of 136Xe can be approximatelyexpressed as [DPR19]T 0νββ1/2 ≈ 1028(0.01 eV〈mββ〉)2year. (1.7)The effective Majorana mass 〈mββ〉 of the electron neutrino in Eq. (1.6)and Eq. (1.7)〈mββ〉 =3∑i=1miU2ei, (1.8)where Uei is the PMNS matrix element with the electron flavor state and miis the mass of the neutrinos in their mass eigenstate used in the definitionof the neutrino mass hierarchy.1.2.1 0νββ experimentsIn double beta decay, the two neutrinos are practically impossible to bedetected due to their small rate of interaction (or interaction cross-section)with any detector material. So the only practical way to identify the ββor 0νββ event is via the energy spectrum of the two electrons from the decay.4For 2νββ, the Qββ is the sum of the kinetic energy of all the decayproducts. The two electrons’ summed energy spectrum would be contin-uous, similar to that of the normal beta decay because the two neutrinoscarry away some kinetic energy. The ion always shares 0.003% or less ofthe kinetic energy as a result of the conservation of momentum. For thespecial case of 0νββ when there is no neutrino, the two electrons carry atleast 99.997% of the kinetic energy. So the 0νββ spectrum would appear asa narrow peak exactly at the Qββ value (see Figure 1.1).Figure 1.1: Electron energy spectrum for ββ (blue) and 0νββ (red). The0νββ spectrum is based on a detector resolution of 2%Qββ and has beenscaled up by at least 104 to make it visible in the plot. Figure credit:Walton, 2016 [Wal16].There are 35 known naturally existing isotopes that could undergo ββdecay. For 12 of them, 2νββ has been observed and a few among them canbe candidates for a 0νββ experiment. 136Xe appears to be a good choice fora 0νββ experiment because of the following arguments:• Xenon is a noble gas element, so it is chemically inert and can becleaned to high purity in order to remove radioactive contaminants.• Xenon has ionization and scintillation response to the ββ decay prod-ucts, so the same source material can also be used as a detector tominimize radioactive contaminants.• With either liquid or gaseous xenon, a small test detector can be po-tentially upscaled to a ton-level experiment.5• Last and most importantly, xenon is practically the only 0νββ candi-date which can have the daughter isotope barium identified (bariumtagging) to ultimately eliminate background events.1.3 EXOThe EXO collaboration has completed the experiment EXO-200 which madeuse of 175 kg (initially planned to be 200 kg) of liquid xenon enriched to80.6% of 136Xe. The liquid xenon is in a recirculation system for periodic pu-rification. The next generation of the experiment next-EXO (nEXO) [KAA+18]is in the R&D process and will use 5 tonnes of liquid xenon.1.3.1 EXO-200The EXO-200 experiment [AAB+12] was located at the Waste IsolationPilot Plant (WIPP) in New Mexico. The facility is 655 meters underground,providing ∼ 1620 meters of water equivalent shielding against cosmic rayswhich can cause background for the experiment. The experiment setup andsupporting equipment were housed in class 100 clean rooms. An engineeringdesign rendering of the cleanroom containing the EXO-200 detector is shownin Figure 1.2. The central part of the experiment was a Time ProjectionChamber (TPC) containing 110 kg of liquid xenon. The TPC was placed in acryostat filled with cryogenic fluid 1 to keep the xenon at a stable liquid phaseunder atmospheric pressure as well as to shield the TPC from radioactivebackgrounds. The cryostat was further shielded with 25 cm of lead on allsides. Despite being underground, a small number of cosmic muons couldstill reach the EXO-200 detector. In order to veto events induced by thesecosmic muons, scintillator panels (veto panels) were used to cover more than95% of the cleanroom.The TPC was a cylinder of ∼ 40 cm in diameter and ∼ 44 cm in length.An illustration of the TPC is shown in Figure 1.3. When a ββ event oc-curred inside the TPC, the two electrons from the decay dissipated theirkinetic energy to the liquid xenon in the form of ionization electrons andscintillation light. The scintillation light was collected by avalanche pho-todiodes (APDs); the ionization electrons drift to the charge collectionwires in the electric field between the cathode and the wires. The eventenergy was obtained via the total detected amount of scintillation lightand ionization electrons. When the electrons reached the charge collection13M Novec 7000 Engineered Fluid6Figure 1.2: Engineering design rendering of the cleanroom containing theEXO-200 detector. The detector is placed in a cryostat shielded by lead onall sides. The detector is covered by veto panels. See text for details. Figurecredit: EXO collaboration (2012) [AAB+12].7wires, their transverse positions were recorded. The longitudinal position ofthese ionization electrons can be calculated by their drift speed in the TPCand the time difference between the scintillation light signal and the chargecollection signal. Thus, the 3D topologies of the events were reconstructed.The topology information was used to distinguish the true ββ events fromthe background; it could also locate the barium ion to facilitate bariumtagging in-situ or for extraction of the volume containing the barium ionout of the TPC for barium tagging at a later stage.Figure 1.3: Illustration of the EXO-200 TPC. The TPC consisted of twosymmetric halves with the cathode in the central plane, charge collectionwires and avalanche photodiodes (APDs) on each side. See text for details.Figure credit: EXO-200 collaboration.EXO-200 started data-taking in May 2011. In August 2011, the first evermeasurement of 2νββ in 136Xe was made to be T 2νββ1/2 = 2.11± 0.04(stat)±0.21(syst) × 1021 years [AAA+11]. In 2014, EXO-200 updated the lowerlimit of 0νββ in 136Xe to be T 0νββ1/2 > 1.1 × 1025 years at 90% confidencelevel [c+14].EXO-200 underwent an upgrade of electronics and cathode high-voltageduring a shut-down from 2014 to 2016. Data-taking was resumed after theupgrade until the completion of EXO-200 at the end of 2018. The data ofthe phase I (September 2011 to February 2014) and phase II (May 2016 toDecember 2018) were analyzed. The energy spectrum of the data with thebest fit is shown in Figure 1.4. The combined data with a total live timeof 1181.3 days resulted in no statistically significant evidence for 0νββ and8the lower limit of 0νββ in 136Xe was set to be T 0νββ1/2 > 3.5 × 1025 years at90% confidence level.Figure 1.4: Best fit to the energy spectrum for Phase I (top) and Phase II(bottom) data of EXO-200. The insets shows the zoomed in plots for theregion-of-interest near Qββ = 2458 keV. Figure from [ABB+19].1.3.2 nEXOnEXO is being designed with the goal to be as close as possible to theEXO-200 concept but scaled up to 5 tonnes of liquid xenon as illustratedin Figure 1.5. The experiment is planned to be set up in the cryopit of theSNOLAB, where a deeper 2070 m (6010 m water equivalent) undergroundshielding will provide further reduction of the rate of cosmic ray comparedto EXO-200.In the later stage of the operation of nEXO, a barium tagging setup maybe added to the experiment to reject all the backgrounds except 2νββ events9(a)(b)Figure 1.5: Pre-conceptual design of nEXO. (a) Engineer design renderingof nEXO in the SNOLAB cryopic. (b) Cut view of the nEXO TPC. See textfor details. Figures from [AAA+18b].10at the tail of their energy spectrum near Qββ .1.4 Barium taggingDue to the very long half-life, the expected event rate of 0νββ is extremelylow even after considering a large amount of source material. For the currentEXO-200 experiment which has 110 kg of enriched liquid xenon in thedetector (3.9× 1026 136Xe atoms), there is only one 2νββ event every 250seconds. The 0νββ event rate is 1/10,000 of that or lower, depending onthe effective Majorana electron neutrino mass 〈mββ〉. Meanwhile, there arebackground events from radioactive contaminants and cosmic rays makingthe 0νββ signal more difficult to be distinguished from the backgroundsignals. In such cases, it is critical to reduce these background events andideally eliminate all of them. The EXO experiments have been considered inevery possible aspect to reduce background events. The ultimate approachto reject the backgrounds for EXO is via barium tagging.The idea of barium tagging, i.e. identification of the barium ion asthe 0νββ daughter of 136Xe, was first proposed by Moe [Moe91]. Moe’sproposed barium tagging technique is based on the classic experiment withthe successful observation of single Ba+ ions in ion trap by Neuhauser etal. [NHTD80].The technique works by illuminating the Ba+ ions with a blue (493 nm)and red (650 nm) laser; the Ba+ ions would emit fluorescent light at bothwavelengths. The presence of barium ions is then determined by the de-tection of the signature fluorescent light. In this way, only an event whichalso produces a barium ion at the event location (reconstructed by theTPC) will be verified as a 2νββ or 0νββ event; all the other events withoutcoincidence production of a barium ion will be rejected.Figure 1.6 shows the expected improvement of nEXO’s sensitivity af-ter including barium tagging. Assuming 100% barium tagging efficiency(identify all 136Ba ions generated by 0νββ), running nEXO for 1.1 yearscan reach the same sensitivity as running it for 10 years without bariumtagging (improvement by a factor of 9). Even at 50% of barium taggingefficiency, the sensitivity can be greatly improved such that nEXO couldprobe the effective Majorana neutrino mass down to 3 meV in the normalmass hierarchy region after running for 10 years. The minimum bariumtagging efficiency required to bring improvement to nEXO is 11%.11Figure 1.6: Sensitivity of nEXO with and without barium tagging. The lefty-axis represents the sensitivity to 0νββ half-life time, the right y-axis isthe sensitivity of the effective Majorana mass of the electron neutrino. Thetwo color bands are ranges of the normal and inverted mass hierarchies; theranges are set by the neutrino’s absolute mass scale and the uncertaintyin the calculation of the nuclear matrix element. Figure credit: nEXOsimulation group, 2015.The research and development of barium tagging started at the earlystage of the EXO collaboration. Because of the importance of bariumtagging and the challenges in achieving it, multiple techniques have beenunder development.12Barium tagging techniquesThe most direct approach to barium tagging is to shine lasers to where abarium ion is generated during the decay inside the TPC and subsequentlydetect the fluorescent light [DDD+00]. The mobility of barium ions insideliquid xenon has been measured and found to be small; the small ion mo-bility leads the barium ions to move slowly and makes a tagging laser easyto follow the ion while producing fluorescent light [Jen04]. However, furtherdevelopment of this technique found that the fluorescent light cannot bereliably identified within the whole detector volume [Hal12].An alternative approach to barium tagging is to insert a probe into theTPC and attract the barium ion to the tip of the probe with electrostaticforce. Then the barium ion identification can possibly be done in-situ usinglaser spectroscopy through optical fiber to send the lasers and collect thefluorescent light. The probe can also possibly be removed from the TPCalong with the barium ion (or atom), then transfer the barium to a dedicatedsetup for identification.A cold probe is being developed for barium tagging in xenon ice [Wam07a].Barium identification in solid xenon ice at an external setup obtained a flu-orescence image sensitive to fewer or equal than 104 atoms [MCW+15]. Fur-ther progress on this has led to imaging of individual barium atoms [CWF+19].An RSI (laser resonance ionization) probe is also being developed [Die12,TKD+14, Wam07b]. The tip of the RSI probe is a clean semiconductingsubstrate such as silicon or silicon carbide. The barium ions in liquid xenonwould neutralize on the surface of the substrate, then the probe is trans-ferred to an external setup to resonantly ionize the barium atom with lasers.The barium ions were then identified via time-of-flight mass spectrometry.Following the classic experiment of Neuhauser et al. [NHTD80], an iontrap has been developed for EXO and achieved single barium ion identifica-tion [Wal05, FGW+07]. Subsequently, a second generation of the ion trapwas developed for the additional purpose of testing ion ejection (loading)to (from) a cold probe [Gre10]. The attempt to recover ions from the coldprobe to the ion trap has not been successful yet.The second generation of the ion trap was transferred to Carleton Uni-versity and the research on laser spectroscopy continues there. Throughimprovements of the laser scheme, in particular the inter-modulation of theblue and red lasers, the background photo count of the fluorescent light hasbeen significantly reduced [Kil15].13This work focuses on developing an ion trap for barium tagging in gaseousxenon. At the early R&D stage of EXO, a large (40 m3) gaseous TPC withbarium tagging has been proposed [DDD+00]. Later, both EXO-200 andnEXO were designed with a smaller liquid TPC to reduce the engineeringchallenges. In the future, an experiment may be built with a gaseous TPCto study the electron correlations in the double beta decay and requiresbarium tagging in gaseous xenon.The concept of barium tagging in gaseous xenon will follow these steps:• Extract the Ba++ ions from the high pressure xenon gas.• Convert the Ba++ ions into Ba+ [Rol11] if needed.• Capture and confine the Ba+ ions in an ion trap.• Unambiguously identify the Ba+ ions via laser spectroscopy or massspectrometry.Extraction of ions from high pressure xenon gas of up to 10 bar to10−6 mbar vacuum environment has been demonstrated [BFV+15] using anRF funnel. The RF funnel is made of 301 thin concentric electrodes withdecreasing inner diameters to form the shape of a funnel. The electrodesare connected to an RF potential to confine the ions along the axis, whilethe neutral xenon gas is vented through gaps between the electrodes.A linear Paul trap (sometimes also called a linear quadrupole ion trapor a linear radio frequency ion trap) will be used at the downstream endof the RF funnel to capture, cool and store the extracted ions to allow ionidentification. This work particularly focuses on the development of such alinear Paul trap.This linear Paul trap may be used for the barium tagging of the upcomingnEXO if future studies of transferring barium ions from liquid xenon to thetrap are successful.The linear Paul trap and the RF funnel are also planned to be usedwith a new ion extraction approach under development at Carleton Univer-sity [Wat19]. The new ion extraction approach aims at extracting bariumions from liquid xenon using a specially designed capillary to vaporize theliquid xenon along with the ions. If successful, the ions extracted by thecapillary will be sent to the RF funnel and the linear Paul trap as described14above.The key features and goals of the linear Paul trap are the following:• Transmit and trap close to 100% of the ions extracted from gaseousxenon through the RF funnel, or from liquid xenon through an alter-native ion extraction method.• Cool the ions to a temperature close to that of the buffer gas and trapthe ions a few seconds for identification via laser spectroscopy.• Eject the ions as fine bunches to a multi-reflection time-of-flight (MR-TOF) mass spectrometer for additional ion identification via high pre-cision mass spectrometry. The ion bunch needs to have a small energyspread (typically within 2%) and a small time spread (typically tensof nanoseconds).In order to achieve the above goals, the theory of the linear Paul trapwas studied from the first principles to obtain analytical solutions for ionstrapped in an ideal linear Paul trap as described in Chapter 2. The ana-lytical solutions were used to validate simulations of the same ideal linearPaul trap, then simulations were done for the design and optimization of arealistic linear Paul trap as described Chapter 3. The mechanical design ofthe optimized linear Paul trap is described in Chapter 4. Finally, the exper-imental development and tests with prototypes are described in Chapter 5.15Chapter 2Theory of the linear PaultrapThe linear Paul trap (LPT) was developed based on the concept leadingto winning the Nobel prize by Wolfgang Paul [PS53]. A LPT consists ofone or more sets of radio frequency quadrupoles (RFQ) to transmit or trapions. The theory and principles of the RFQ are studied and presented inSection 2.1 followed by the characterization of its performance in Section 2.2.Such RFQs can also be configured to work as a simple ion guide to transmitall ions without filtering as described in Section 2.3 or a quadrupole massfilter in Section 2.4. The details of ion cooling and trapping in a linear Paultrap are described in Section 2.5.2.1 Radio frequency quadrupoleA radio frequency quadrupole (RFQ) consists of four electrodes as illustratedin an example in Figure 2.1(a). The two diagonal pairs of the electrodes aresupplied with opposite polarities of a DC potential U and an oscillatingRF potential V cos Ωt, where V is the RF amplitude and Ω is the angularfrequency. The electrodes create a quadrupole potentialφRF = (U − V cos Ωt)(y2 − z2)r20, (2.1)in the center of the y-z plane as shown in Figure 2.1(b), where r0 is thedistance from the inner surface of the electrodes to the x-axis.2.1.1 Ion dynamics in an RFQThe ion dynamics, or in other words, how the ion trajectories are determinedin an RFQ, are governed by the time-dependent electric field (the effect ofgravity is typically much smaller and can be ignored). At a radial position16(a) (b)Figure 2.1: (a) Illustration of an RFQ and the voltage configuration on itselectrodes. (b) Quadrupole potential inside the trap for ion confinement inthe transverse directions; adapted from [KP15].(y, z) as shown in Figure 2.1, the electric field at time t is:E(y, t) = (U − V cos Ωt)2yr20(2.2)E(z, t) = (U − V cos Ωt)−2zr20, (2.3)so the ion motion can be described independently in the y and z directions.For a positive singly charged ion of mass m, the equations of motion are:md2ydt2=−2eyr20(U − V cos Ωt), (2.4)md2zdt2=2ezr20(U − V cos Ωt), (2.5)where e = 1.6 × 10−19 C is the elementary charge. The equations can berewritten in form of the Mathieu equation:d2udξ2+ (a− 2q cos 2ξ)u = 0, (2.6)ξ =Ωt2. (2.7)The variable u denotes a coordinate of either y or z, while a and q are the17so-called stability parameters:a = ay = −az = 8eUmΩ2r20, q = qy = −qz = 4eVmΩ2r20. (2.8)When a and q are small, the ion dynamics can be simplified and approxi-mated by the so-called pseudopotential well model developed by Wuerker [WSL59]and extended by Dehmelt [Deh68].2.1.2 Pseudopotential well modelIn this model the ion motion is described by a combination of a micromo-tion δ with the same frequency as the RF potential, and a lower frequencymacromotion u¯:u = u¯+ δ. (2.9)Plugging Eq. (2.9) into the Mathieu equation Eq. (2.6) results in theequationd2u¯dξ2+d2δdξ2= −(a− 2q cos(2ξ))(u¯+ δ). (2.10)When the following conditions are met:δ  u¯, d2u¯dξ2 d2δdξ2, a q; (2.11)Eq. (2.10) can be simplified asd2δdξ2= 2q cos(2ξ)u¯. (2.12)The micromotion is then solved asδ = −qu¯2cos(2ξ). (2.13)To solve for the macromotion, plug Eq. (2.13) back into Eq. (2.10):d2u¯dξ2+d2δdξ2= −au¯+ aqu¯2cos 2ξ + 2qu¯ cos 2ξ − q2u¯ cos2(2ξ). (2.14)18Then average Eq. (2.14) over one RF cycle. The micromotion term d2δdξ2onthe left and the two terms on the right containing cos 2ξ vanish:∫ pi0d2δdξ2dξ =∫ pi02qu¯ cos 2ξdξ = 0, (2.15)∫ pi0(aqu¯2cos 2ξ + 2qu¯ cos 2ξ)dξ = 0. (2.16)After integrating the remaining terms, Eq. (2.14) becomes〈d2u¯dξ2〉=∫ pi0d2u¯dξ2dξ = −(a+ q22)u¯. (2.17)When the change of u¯ (du¯) is small over one RF cycle,d2u¯dξ2= −(a+ q22)u¯. (2.18)Then replace the variable ξ with t using the relationship dξ = Ω2 dt,d2u¯dt2= −Ω24(a+q22)u¯. (2.19)This Eq. (2.19) describes u¯ in a simple harmonic motiond2u¯dt2+ ω¯2u¯ = 0, (2.20)where the secular frequency isω¯ =Ω2√a+q22. (2.21)For a positive singly charged ion with mass m, the potential well can bedescribed asD¯(u¯) =mΩ28e(a+q22)u¯2. (2.22)The depth of the pseudopotential well is D¯(u¯) when u¯ = r0:D¯ =mΩ2r208e(a+q22). (2.23)19Substituting a and q from their definition in Eq. (2.8) revealsD¯ = U +qV4. (2.24)Specifically, for the y and z axesD¯y = U +qyV4(2.25)D¯z = −U + |qz|V4. (2.26)The pseudopotential well is similar to a simple harmonic well when theconditions in Eq. (2.11) are met. The depth D¯ determines the maximumnumber of ions that can be trapped and the kinetic energies of the ions.2.1.3 Full solution of Mathieu equationThe full solution of the Mathieu equation (2.6) is known to be an infiniteseries expansion [MT05]:u(ξ) = Γeµξ∞∑n=−∞C2n,u exp(2niξ) + Γ′e−µξ∞∑n=−∞C2n,u exp(−2niξ), (2.27)where Γ and Γ′ are constants dependent on the initial values of the positionu0, velocity u˙0 and the RF phase ξ0; C2n,u are coefficients of these termsand their values only depend on a and q; the value of µ leads to differentcategories of the solution:1. µ is a non-zero real number; the solution u(ξ) will be unstable due toexponential growth with ξ in either eµξ or e−µξ.2. µ is a complex number; the solution u(xi) will still be unstable due tothe real part of µ.3. µ = im, where m is an integer number; the solution will be periodicbut unstable.4. µ = iβ, where β is not an integer number; this is the only situationfor the solution u(ξ) to be periodic and stable.20Only solutions described in the last category are stable and suitable for ionconfinement within a finite volume. The stable solution can be rewritten asu(ξ) = A∞∑n=−∞C2n cos[(β + 2n)ξ] +B∞∑n=−∞C2n sin[(β + 2n)ξ], (2.28)where A = Γ + Γ′, B = i(Γ− Γ′). The solution describes the ion motion asa Fourier series with frequenciesωn =Ω2(β + 2n). (2.29)Analytical solution for βThe value of β can be obtained by inserting Eq. (2.28) back to the Mathieuequation (2.6). Keep only the cosine terms for now:−A∞∑n=−∞C2n,u(β + 2n)2 cos[(β + 2n)ξ]+(a− 2q cos 2ξ)A∞∑n=−∞C2n,u cos[(β + 2n)ξ] = 0.(2.30)Applying the trigonometric relationshipcos(2ξ) cos(β + 2nξ) =12(cos(β + 2n+ 2)ξ + cos(β + 2n− 2)) (2.31)to Eq. (2.30) and re-arranging the cosine terms:A∞∑n=−∞[a− (β + 2n)2qC2n − C2n+2 − C2n−2] cos(β + 2n)ξ = 0. (2.32)Working on the sine terms would yield the same result as Eq. (2.32).Eq. (2.32) implies that for every term of nD2nC2n − C2n+2 − C2n−2 = 0, (2.33)whereD2n =a− (β + 2n)2q. (2.34)21For n = 0,D0 =a− β2q=C2C0+C−2C0. (2.35)Re-arranging Eq. (2.33) also leads to two recursive relationshipsC2n+2C2n= D2n − C2n−2C2n= D2n − 1D2n−2 − 1D2n−4−···(2.36)C2n−2C2n= D2n − C2n+2C2n= D2n − 1D2n+2 − 1D2n+4−···(2.37)which can be inserted into Eq. (2.35) to solve for β:a− β2q=1D−2 − 1D−4−···+1D2 − 1D4−···. (2.38)Finally, using the definition of D2n from Eq. (2.34) and re-arranging it towrite β in a recursive form:β =√√√√a− q2a− (β − 2)2 − q2a−(β−4)2−···− q2a− (β + 2)2 − q2a−(β+4)2−···.(2.39)An equivalent expression of β is given by Eq. (2.94) of [MT05].For small a, q and hence small β values (β  4), Eq. (2.39) can beapproximated asβ ≈√(a+12q2). (2.40)Then the fundamental frequency of the ion motion described by Eq. (2.28)isω0 =Ω2β =Ω2√a+q22, (2.41)which is exactly the secular frequency of the macromotion derived in thepseudopotential well model in Section 2.1.2.Stability diagramFor larger q and a values, the approximation in Eq. (2.40) is no longer validbecause β  4 is no longer true. As an original method proposed in thisstudy, β can be calculated by numerical iteration of Eq. (2.39) using some22initial guesses. Such a numerical calculation was carried out with n up to 20in D2n and the recursive formula of Eq. (2.39). The initial guess for β wasempirically set to be β0 = 1 for every q and a coordinate.1 After 1000 iter-ations, the results in the range of |q| < 10 and |a| < 5 converged to changessmaller than 1 × 10−15 (which is the limit of the double-precision floating-point number) except around and outside the stable-unstable boundaries.2In this way, the value of β is effectively solved analytically. The precise valueof β around the stable-unstable boundaries is obtained after more iterations.The values of β at the end of the iterative calculation are plotted inFigure 2.2, showing a stability diagram in the (q, a) parameter space. Thevalue of β in the color-coded stable region is a real number and non-integral,making the Mathieu equation’s solution Eq. (2.28) stable. The stabilitydiagram is found to be symmetric along the line q = 0 as predicted byEq. (2.39).Most RF quadrupoles work in the region of 0 < β < 1. A detailedstability diagram of this region is shown in Figure 2.3.For the ion described at the beginning of this section, it would havestable motion in the y coordinate for qy = q and ay = a within thosestability diagrams. For the z coordinate qz = −qy and az = −ay. Therefore,the stability diagram for the z coordinate only needs to be flipped alongthe line a = 0 due to the symmetry in the q axis, see Figure 2.4(a). In theoverlapped stable region shown in Figure 2.4(b), the ion would be confinedin both y and z axis.There are two noticeable characteristics of the stability diagram in thisregion.• For a = 0, the maximum q is found to beqm = 0.90804633. (2.42)• The upper tip of the stable region is found to beqt = 0.7059961, at = 0.2369940. (2.43)1I found complex numbers needed for these numerical iterations so in fact β0 = 1 + 0iwas used; at the end of the calculation, only the real part of β was kept.2This numerical iterative method doesn’t converge in some regions of large q and avalues. But the range of |q| < 10 and |a| < 5 already fully cover their values used for ionconfinement in RFQs and Paul traps.23Figure 2.2: Calculated stability diagram of the Mathieu equation. The valueof β is calculated by numerical iterations. The solution of the Mathieuequation is stable in the colored region.24Figure 2.3: Calculated stability diagram for 0 < β < 1. Equal-β lines from0.1 to 0.9 are plotted on top.25(a)(b)Figure 2.4: (a) Stability diagram for the y and z axes of an RFQ. (b)combined stable region where the solution is stable in both axes. See textfor details.26These values are obtained after 100,000 iterative calculations. The precisevalues are useful for validating and benchmarking alternative approachessuch as numerical integration in Section 2.1.4 and simulations in Chapter 3.2.1.4 Ion motionNow, the exact ion motion described by the Mathieu equation’s solution canbe obtained both analytically and numerically.Analytical solutionsFor an ion with initial position u0 and velocity u˙0 at ξ0 = 0, the followingtwo relationships can be obtained from Eq. (2.28):u0 = u(0) = A∞∑n=−∞C2n, (2.44)u˙0 =dudξ∣∣∣ξ=0= B∞∑n=−∞C2n. (2.45)For the convenience of calculation, set the amplitude of the fundamentalfrequency term C0 = 1, then the other C2n values can be obtained via therecursive relations of Eq. (2.36) and Eq. (2.37).As an example of the calculation, Table 2.1 shows values of the C2n(−10 ≤ n ≤ 10) terms for q = 0.1, 0.5 and 0.908 with a = 0.1 The sumof C2n shown in the last row of the table can be used to relate the initialconditions u0 and u˙0 to A and B in Eq. (2.44) and Eq. (2.45). Then for anygiven value of (q, a) within the stable region, and with initial condition (u0,u˙0), the full solution can be explicitly expressed in the form of Eq. (2.28).In this way, for the (q , a) values listed in Table 2.1 and initial conditionu0 = 1, u˙0 = 0; (2.46)the evaluations of u over ξ are shown in Figure 2.5 as the curves labeled“Analytical”. The horizontal axes are also labeled as time t which is relatedto ξ ast =2ξΩ, (2.47)1For each corresponding negative q value, their C2n has the same absolute values butare all positive.27Table 2.1: Calculated C2n for a few typical values of q when a = 0.q 0.1 0.5 0.908β 0.07085 0.37374 0.99362C−20 8.9319e-30 2.2625e-22 9.5528e-19C−18 -3.5475e-26 -1.743e-19 -3.8005e-16C−16 1.1404e-22 1.083e-16 1.2105e-13C−14 -2.8935e-19 -5.289e-14 -3.0022e-11C−12 5.614e-16 1.9641e-11 5.5931e-09C−10 -7.989e-13 -5.3096e-09 -7.4617e-07C−8 7.8762e-10 9.8401e-07 6.6652e-05C−6 -4.9519e-07 -0.00011445 -0.0036027C−4 0.00017408 0.0072451 0.09938C−2 -0.026875 -0.19043 -0.98563C0 1.0 1.0 1.0C2 -0.023322 -0.088943 -0.10169C4 0.00014073 0.0023255 0.0037055C6 -3.8186e-07 -2.8624e-05 -6.8805e-05C8 5.8623e-10 2.0412e-07 7.7245e-07C10 -5.7801e-13 -9.4838e-10 -5.8035e-09C12 3.967e-16 3.0971e-12 3.1212e-11C14 -2.0036e-19 -7.4953e-15 -1.2607e-13C16 7.7578e-23 1.3979e-17 3.9639e-16C18 -2.3757e-26 -2.0703e-20 -9.9769e-19C20 5.8973e-30 2.493e-23 2.0554e-21∑C2n 0.95012 0.73006 0.01215728with the unit of 2TRF where TRF is the RF periodTRF =2piΩ. (2.48)Numerical integrationFor each set of given (q, a) and (u0, u˙0), the solution u(ξ) can also beobtained by applying numerical integration to the Mathieu equation (2.6).The Runge-Kutta method to the 4th order is used to calculate u(ξ) forthe same parameters and also shown in Figure 2.5 with label “RK4”. Theanalytical and RK4 results agree well.For q = 0.909 which is outside of the stable region, the value of β is nota real number and cannot be obtained in the method described above. Soonly the result of the numerical integration is shown. The envelope formedby the maxima of u increases exponentially along with ξ and therefore thesolution u(ξ) is indeed unstable.In order to validate the numerical integration approach, more calcula-tions were run with a = 0 and q around 0.908 to obtain the boundary be-tween stable and unstable solutions. When the integration step width h isset to beh =TRF200(2.49)or smaller, the maximum q allowing a stable solution was found to be0.90804633 < qm < 0.90804634 (2.50)(see Figure 2.6). The value of qm agrees with the result obtained in theprevious analytical approach in Eq. (2.42).For (q, a) far away from the stability boundaries, a larger integrationstep width up to h = TRF20 can be good enough to correctly obtain thenumerical solutions.2.2 Ion acceptance and emittanceThe ion motions in an RFQ can be better understood and characterized inthe position-velocity phase space (u, u˙) with the concept of acceptance andemittance.290 5 10 15 20 25 30 35 40 (2 ), Time (2TRF)101Position (mm)q=0.1AnalyticalRK40 5 10 15 20 25 30 35 40 (2 ), Time (2TRF)101Position (mm)q=0.5AnalyticalRK40 20 40 60 80 100 (2 ), Time (2TRF)2001000100200Position (mm)q=0.908AnalyticalRK40.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 (2 ), Time (2TRF)5000500Position (mm)q=0.909RK4Figure 2.5: Solutions of the Mathieu equation for a few q values annotatedin each plot when a = 0. The solutions are stable for |q| < 0.90804633; forq = 0.909 shown in the last plot, the solution is unstable.30Figure 2.6: Stable and unstable solutions of Mathieu equation obtained bynumerical integrations. The filled regions are actually dense curves with anoscillation period TRF as shown in the zoomed-in plots in the insets.31The acceptance of an RFQ a is the region of (u, u˙) that allows ions tobe stably confined or transmitted, while the emittance e is the phase spacearea occupied by the ions existing in the RFQ. Usuallya = e (2.51)when there is a sufficient amount of ions in the system.Acceptance ellipsesAs an example, for an ion with (q = 0.1, a = 0) and initial condition(u0 = 1, u˙0 = 0), Figure 2.7(a) shows its position and velocity in the phasespace corresponding to the y axis. An interesting fact is, if plotting onlythese specific (u, u˙) exactly one RF cycle apart, then they land perfectly onellipses [Daw75] as shown in Figure 2.7(b). The fact is true for any otherinitial condition (u0, u˙0) and any other (q, a) values within the stable region.The RF phase labeled for each ellipse in Figure 2.7 is the initial phaseϕ0 in the RF potentialφRF = (U − V cos(Ωt+ ϕ0))(y2 − z2)r20. (2.52)The requirement for an ion to be accepted in the RFQ is that its maxi-mum displacement from the x axis must be within r0, which is the physicalboundary defined by the electrodes as shown in Figure 2.1. In Figure 2.7,the maximum displacement of the ion is the semi-major of the ellipse withinitial RF phase ϕ0 = pi. Thus the ions which can be accepted are the oneswith initial condition (u0, u˙0) within this ellipse. For any other initial RFphase ϕ0, the acceptance is the corresponding ellipse.It is noteworthy that, in the z axis, an ion’s maximum displacementis the semi-major of the ellipse with initial RF phase ϕ0 = 0. The differ-ence comes from the sign of q and hence the resulting C2n coefficients inEq. (2.28). C2n are all positive when a − 2q > 0. For a − 2q < 0, C2n isnegative when n is an odd number.The equation of these ellipses can be derived analytically as below.321.00.50.00.51.0Position (mm)RF phase: 0RF phase: /2RF phase: RF phase: 3 /20 2 4 6 8 10Time (2Trf)0.100.050.000.050.10Velocity (mm/TRF)(a)1.0 0.5 0.0 0.5 1.0Position (mm)0.100.050.000.050.10Velocity (mm/TRF)RF phase: 0RF phase: /2RF phase: RF phase: 3 /2(b)Figure 2.7: (a) Position and velocity of an ion from the solution of theMathieu equation. (b) The same position and velocity plotted in phasespace (u, u˙), showing ellipses corresponding to each initial phase.33Derivation of ellipses from Mathieu equation solutionsThe non-zero initial phase ϕ0 in the RF potential adds a variable in theMathieu equationd2udξ2+ [a− 2q cos(2ξ + ϕ0)]u = 0. (2.53)Then the solution of Eq. (2.53) isuϕ0(ξ) = A∞∑n=−∞C2n cos[(β+2n)(ξ+ϕ02)]+B∞∑n=−∞C2n sin[(β+2n)(ξ+ϕ02)].(2.54)The above expression can be expanded using the trigonometric relationsto separate the terms for ξ and ϕ0uϕ0(ξ) =A∞∑n=−∞C2n cos[(β + 2n)ξ] cos[(β + 2n)ϕ02]−A∞∑n=−∞C2n sin[(β + 2n)ξ] sin[(β + 2n)ϕ02]+B∞∑n=−∞C2n sin[(β + 2n)ξ] cos[(β + 2n)ϕ02]+B∞∑n=−∞C2n cos[(β + 2n)ξ] sin[(β + 2n)ϕ02].(2.55)For ξk = kpi (k = 0, 1, 2, . . . ) corresponding to the successive points(uϕ0(ξk), u˙ϕ0(ξk)) on each ellipse,cos(β + 2n)ξk = cosβξk, sin(β + 2n)ξk = sinβξk (2.56)Then the solution Eq. (2.55) can be simplified asuϕ0(ξk) = uϕ0(0) cosβξk + uϕ0(ξK) sinβξk, (2.57)whereuϕ0(0) = A∞∑n=−∞C2n cosϕ02(β + 2n) +B∞∑n=−∞C2n sinϕ02(β + 2n). (2.58)34anduϕ0(ξK) = −A∞∑n=−∞C2n sinϕ02(β + 2n) +B∞∑n=−∞C2n cosϕ02(β + 2n);(2.59)ξK = Kpi =2mpi + pi2β, (2.60)where m is an integer.In a similar way,u˙ϕ0(ξk) = u˙ϕ0(0) cosβξk + u˙ϕ0(ξK) sinβξk, (2.61)whereu˙ϕ0(0) = −A∞∑n=−∞(β+2n)C2n sinϕ02(β+2n)+B∞∑n=−∞(β+2n)C2n cosϕ02(β+2n)(2.62)andu˙ϕ0(ξK) = −A∞∑n=−∞(β+2n)C2n cosϕ02(β+2n)−B∞∑n=−∞(β+2n)C2n sinϕ02(β+2n).(2.63)Now uϕ0(ξ) and u˙ϕ0(ξ) can be written in the form of an ellipse’s standardparametric equationsuφ0(ξ) = uM cos(βξk + θ1) (2.64)u˙φ0(ξ) = u˙M sin(βξk + θ2), (2.65)whereuM =√uϕ0(0)2 + uϕ0(ξK)2 (2.66)u˙M =√u˙ϕ0(0)2 + u˙ϕ0(ξK)2 (2.67)andθ1 = arctan2(−uϕ0(ξK), uϕ0(0)) (2.68)θ2 = arctan2(u˙ϕ0(0), u˙ϕ0(ξK)). (2.69)The area of the ellipse Ae isAe = piuM U˙M cos(θ2 − θ1) = pi[−u(0)u˙(ξK) + u(ξK)u˙(0)]. (2.70)35Twiss parametersThe ellipses can also be expressed in the so-called Twiss parameters whichare often used in accelerator physics to describe the transverse beam dy-namics of charged particles:CTu2 + 2ATuu˙+BT u˙2 = , (2.71)with the constraint ofCTBT −A2T = 1. (2.72)Then =Aepi, (2.73)where Ae is the area of the ellipse. The angle θ between the ellipse’s major-axis to the horizontal coordinate isθ =12arctan2ATCT −BT . (2.74)A few characteristic points on the ellipse are calculated and shown inFigure 2.8. The intercepts of the ellipse with the horizontal or vertical axisare obtained by setting u˙ = 0 or u = 0. The coordinates of the maximum uor u˙ on the ellipse are obtained by finding dudu˙ = 0 ordu˙du = 0.√²CT√²BT√²BT−AT√²BT√²CT−AT√²CT) θuu˙Figure 2.8: An ellipse corresponding to the Twiss parameters.36The Twiss parameters are then related to the parameters in Eq. (2.66)– Eq. (2.70) in the following way:uM =√BT (2.75)u˙M =√CT (2.76)tan(θ2 − θ1) = − 1AT(2.77)− u(0)u˙(ξK) + u(ξK)u˙(0) = . (2.78)Analytical calculation of acceptanceThe acceptance of the RFQ in the y axis can be calculated with the specialcase of initial phase φ0 = 0 and B = 0 in Eq. (2.54). Then = −u(0)u˙(ξK) (2.79)andu(0) = A∞∑n=−∞C2n, (2.80)u˙(ξK) = −A∞∑n=−∞(β + 2n)C2n. (2.81)In order to account for the physical dimensions of the RFQ, the accep-tance  needs be normalized to the maximum ion position r0 limited by thequadrupole electrodesr0 = A∞∑n=−∞|C2n|. (2.82)Acceptance of the RFQ in the y axis analytically calculated in this way isshown in Figure 2.9.For the z axis, the acceptance is Figure 2.9 flipped vertically along a = 0.The combined acceptance yz of an RFQ for transmitting ions is the productof its acceptances in the y and z axesyz = yz. (2.83)The calculated values of yz are shown in Figure 2.10.37Figure 2.9: Acceptance of the RFQ in one of the transverse directions ycalculated analytically from the Mathieu equation’s solution.2.3 RFQ ion guideAn RFQ ion guide is a straightforward application of the RFQ. Usually theDC voltage U of the ion guide is set to 0 to obtain larger acceptance (seeFigure 2.10 along a = 0) and consequently higher ion transmission efficiency.Acceptance of the ion guide yz as a function of q (proportional to the RFvoltage V ) when U = 0 is shown in Figure 2.11. The results are from bothanalytical calculation and the area of a fitted ellipse using the relationshipyz =Ae,ypiAe,zpi. (2.84)Values of the two sets of results agree well as shown in the figure.The acceptance is maximum when q = 0.577. To have better than halfof the maximum acceptance, q needs to be between 0.271 to 0.817.38Figure 2.10: Acceptance of the RFQ in both the y and z axes.2.4 Quadrupole mass filterA quadrupole mass filter (QMF) uses the stability parameters near theupper tip of the stability diagram in Figure 2.4 with q ≈ 0.706 and a ≈0.237. In this case, only the ions within a narrow range of mass can passthrough the QMF.The QMF can also work as a quadrupole mass spectrometer (QMS) byscanning the operating parameters while the ions are continuously injectedinto the system. The ions of different mass will subsequently pass throughthe QMS and produce a measured mass spectrum.2.4.1 Mass scan of QMSThe mass scan of a QMS can be done with two different approaches as below.390.0 0.2 0.4 0.6 0.8 1.0q0.000.010.020.030.04Acceptance ² yz(pi2r4 0T2 RF) AnalyticalAe, yAe, zpi2Figure 2.11: Acceptance of the RF ion guide when a = 0 from two sets ofcalculations.Approach 1: QMS with voltage sweepThe approach to operating the RFQ as a QMS with voltage sweep is oftenused when the RF voltage of the QMS is provided by a resonant circuitthat performs best at a fixed frequency. The mass scan is done by sweepingboth the DC and RF voltages to let the ions of different mass pass throughthe tip of their stable region one after another, see Figure 2.12.The voltages V and U are normalized with units shown in the labels ofFigure 2.12 qm1 and am1 which are the stability parameters of the ion withmass m1. According to Eq. (2.43), the mass scan for this ion reaches the tipof its stable region whenqm1 = qt = 0.7059961, (2.85)am1 = at = 0.2369940. (2.86)The mass m is related to the RF voltage V at the tip of its stable regionviam = V · 4eΩ2r20qt, (2.87)as labeled on the top axis of Figure 2.12.40Figure 2.12: Mass scan of the QMS by sweeping both RF and DC voltagesV and U . The overlapping shaded regions are stable regions of ions withdifferent masses m1 – m5. The slope of the scan lines is s =UV . Thevalues on horizontal and vertical axes are dimensionless for the stabilityparameters qm1 and am1 corresponding to an ion of mass m1. The mass ofions corresponding to each tip of the stable region is given as the top axis.41The DC voltage is set to beU = sV, (2.88)where the slope s = UV as the mass scan line is related to the mass resolvingpower R of the QMS:R =m∆m. (2.89)Details of the mass resolving power of the QMS will be discussed in Sec-tion 2.4.2.Approach 2: QMS with frequency sweepThis approach is possible when a wide frequency band of the RF voltagecan be provided by the hardware. Then both the RF and DC voltages canbe fixed and only sweep the RF frequency Ω (see Figure 2.13). The mass ofions m is related to Ω asm =1Ω2· 4eVr20qt. (2.90)For different mass resolving power R, the DC voltage U is set at differentvalues of s or aU = sV =a2qtV. (2.91)2.4.2 QMS mass resolving power and transmissionefficiencyThe number of ions which can be transmitted by a QMS is related to theacceptance yz. For the special case when there are a large number of ionsrandomly distributed uniformly in the phase space (uy, u˙y) and (uz, u˙z), theion transmission efficiency T is proportional to the acceptance yzT ∝ yz. (2.92)Then ∆mFWHM , defined as the full-width at half-maximum (FWHM) ofthe acceptance yz, can be used to calculate the mass resolving power R viaR = m/∆mFWHM . Analytically calculated acceptance of an RFQ shown inFigure 2.10 is used to derive the mass resolving power (see Figure 2.14).The mass resolving power R as a function of the stability parameter a (orthe DC to RF ratio s) shown in Figure 2.14 is only true for a theoretically42Figure 2.13: Mass scan of the QMS by sweeping the RF frequency. For moredetails see Figure 2.12 and text.idealized QMF with infinite length to allow the ions to experience an infinitenumber of RF cycles. The mass resolving power R of a realistic QMF is alsolimited by the length of the quadrupole electrodes.Limitation of mass resolving power from RF cyclesAn ion of mass mi in the unstable region of a QMF will have an exponen-tially increasing amplitude of its oscillatory motion (similar to the bottomplot shown in Figure 2.5). Depending on the stability parameters (qi, ai)and the ion’s initial position and velocity, the ion takes a certain numberof RF cycles nRF,i to have its amplitude reach r0 and hit the electrodes tobe filtered. If the number of RF cycles experienced by the ion in the QMFis smaller than nRF,i, the ion can fly through the QMF even though itsstability parameters (qi, ai) are outside of the theoretical stable region. Insuch cases, the transmission of the unwanted ions reduces the mass resolving430.00 0.05 0.10 0.15 0.20 0.25a, DC to RF ratio s× 2qt100101102Mass resolving power R10-310-2Acceptance ² yz(pi2r4 0T2 RF)Figure 2.14: Mass resolving power of a QMS and its acceptancecorresponding to ion transmission efficiency.power of a QMF.The limitation of mass resolving power by the number of RF cycles canbe empirically expressed as [Dou09]Rn =n2RFh, (2.93)where h ≈ 12.25 was obtained by Paul et al. [PRVZ58]. Later, h ≈ 20 wasfound by Austin et al. based on more experimental results [AHL76]. Theexact value of h is also dependent on the QMF such as the detailed electrodegeometry.Note that these experimentally obtained values of h are based on massscans with RF amplitude (shown in Figure 2.12). The value of h would besmaller for mass scans with RF frequency as shown in Figure 2.13.2.5 Linear Paul trapA linear Paul trap (LPT) uses RFQs to confine ions in the transverse direc-tions y and z. The DC potential U of an LPT is usually set to 0 to have44larger ion acceptance, similar to an RF ion guide described in Section 2.3.To confine ions in the longitudinal direction, a DC trapping potentialalong the x axis is needed. One way of forming the longitudinal potentialis to use a segmented RFQ with configurable DC voltage for each segment(see Figure 2.15(a)). The DC voltage Un is the same for the two diagonalpairs of electrodes, as shown in Figure 2.15(b). Note that in this case, theDC voltage Un doesn’t contribute to the quadrupole potential, hence thestability parameter a = 0.(a) (b)0 50 100 150 200 250 300 350 400Longitudinal position (mm)200DC Potential (V)Trapping potentialEjection potential(c)Figure 2.15: (a) Illustration of a linear Paul trap consisting of segmentedquadrupole electrodes. (b) Electric connection for each set of quadrupoleelectrodes. (c) DC potential of the LPT along the x axis. See text fordetails.The LPT for this study also needs to provide a cooling effect to trapand store the ions. The ion cooling can be achieved by filling the LPT withbuffer gas to damp the ions via ion-gas collisions [Kim97]. The collisions45were found to be cooling down the ions when the gas molecules are lighterthan the ions [MD68]. Helium is a good choice as the buffer gas due to itslight atomic mass. In addition, helium is a noble gas so it is chemicallyinert and can reach high purity to reduce ion loss due to charge exchangefrom other contaminate gas molecules.A typical configuration of the DC potential for the LPT along the x axisis shown in Figure 2.15(c). The majority of the segments’ purpose is to forman electric field (drag field) to guide the cooled ions to the location of thelowest potential near the exit. After enough ion cooling and accumulation,the DC voltages of the last three segments are switched to form an ejectionpotential, and the ions will be ejected as an ion bunch out of the LPT withsmall emittance, time spread and energy spread.2.5.1 Ion cooling with buffer gasIon mobilityThe motion of ions in a buffer gas can be quantitatively described using theconcept of ion mobility K, which describes a constant drift velocity of ionsvd in the buffer gas with the presence of an electric field Evd = KE. (2.94)For a singly charged positive ion, the electric force experienced by the ionis balanced by a frictional force from the buffer gas from the averaged effectof ion-gas collisions:Ff = −eE = −evdK. (2.95)For an ion of mass m with initial velocity vi, the ion’s equation of motionismdvdt= −evK+ eE = −e(v − vd)K, (2.96)then the ion’s velocity as a function of time isv(t) = vd + (vi − vd)e−tτv , (2.97)whereτv =mKe(2.98)46is the time constant of the ion slowing down from vi to vd.1.For 136Ba+ ions in helium gas, when the ion drift velocity vd < 0.268 mm/µs(kinetic energy Ke < 0.05 eV) the experimentally measured ion standardmobility is found to be [VM95].K0 =NN0K = 1.66× 10−3 m2V−1s−1, (2.99)where N0 = 2.687 × 1025 m−3 is the standard number density of ideal gasat 0°C and 1 bar. The ion mobility is smaller at larger ion velocities. Attemperature 313 K (the temperature used in the literature [VM95]) andbuffer gas pressure of 0.01 mbar, the ions’ slowing down time constant forsmall initial and final drift velocity is calculated to be τv = 272 µs, which isinversely proportional to the buffer gas pressure.2.5.2 Ion cooling in LPTThe ion cooling in an LPT can be approximately described by including adamping term in the ion’s equation of motionmd2udt2=−2eur20(U − V cos Ωt)− eKdudt. (2.100)Using the same parameters q, a and ξ in Section 2.1, the equation becomesd2udξ2+ 2kdudξ+ (a− 2q cos 2ξ)u = 0, (2.101)where the damping coefficient k = emKΩ comes from the ion cooling effectof the buffer gas.Following McLachlan [McL51], Whetten [Whe74] and Kim [Kim97],Eq. (2.101) can be solved by definingu = u1e−kξ, (2.102)then Eq. (2.101) will come to a similar form as the Mathieu equation:d2u1dξ2+ (a− k2 − 2q cos 2ξ)u1 = 0. (2.103)1Note that Eq. (2.97) and Eq. (2.98) are no longer valid when K is dependent on theion velocity.47After further using the substitution of an adjusted stability parametera¯u = a− k2, (2.104)u1 will have the same solutions as described in Section 2.1.1 in the stableregion of (a¯u, q).In this way, for any initial position u and velocity u˙, the evolution ofu as a function of ξ or t with any initial condition (u0, u˙0) is obtained asEq. (2.102). The e−kξ term in the solution of u leads to the exponentialdecay of the u with a time constantτu =2mKe. (2.105)Such calculations were done using the standard mobility in Eq. (2.99)for 136Ba+ in helium buffer gas. The derived parameters for helium pressurefrom 0.01 mbar to 10 mbar are shown in Table 2.2.Table 2.2: Parameters of 136Ba+ ion cooling in helium buffer gas at differentpressures.pressure (mbar) 0.01 0.1 1 10Ion mobility K (m2V−1s−1) 192.7 19.27 1.927 0.1927Damping coefficient k 5.86× 10−4 5.86× 10−3 5.86× 10−2 0.586Adjusted stabi. param. a¯u −3.4× 10−7 −3.4× 10−5 −3.4× 10−3 -0.34Cooling time const. τu (µs) 544 54.4 5.44 0.544Using the values of k and a¯u from Table 2.2, solutions of u were obtainedfor (q = 0.1, a = 0) in the form of Eq. (2.102). The coordinate z was used asu so that the initial condition of (u0 = 1, u˙0 = 0) at t = 0 has the maximumposition, this is equivalent to having initial RF phase ϕ0 = pi in Figure 2.7.The results are plotted in Figure 2.16 with label Analytical.The analytical result was not obtained at 10 mbar pressure because(q = 0.1, a¯u = −0.34) leads to an imaginary number of β and hence thesolution u1 is outside of the stable region of the Mathieu equation’s solution.Due to the e−kξ term in Eq. (2.102), the solution u = u1e−kξ will still bestable but needs to be solved differently than Section 2.1.1. In fact u willhave stable solution in an extended stable region up to <(µ) < k [Whe74],where µ = iβ is introduced in the previous Eq. (2.27).480 250 500 750 1000 1250 1500 1750 2000Time ( s)101Position (mm)1/e544pressure: 0.01 mBarAnalyticalRK40 25 50 75 100 125 150 175 200Time ( s)01Position (mm)1/e54.4pressure: 0.1 mBarAnalyticalRK40 20 40 60 80 100Time ( s)0.00.51.0Position (mm)1/e5.44pressure: 1 mBarAnalyticalRK40 25 50 75 100 125 150 175 200Time ( s)0.51.0Position (mm)1/e0.544pressure: 10 mBarRK4Figure 2.16: Analytical and numerical calculation of ion cooling with buffergas using the ion mobility at different gas pressures as annotated in eachplot. The stability parameters are (q = 0.1, a = 0). See text for details.49Numerical solutionsThe damped Mathieu Eq. (2.101) can also be solved numerically using theRunge-Kutta method similarly as described in Section 2.1.4. The resultsare plotted in Figure 2.16 with label RK4 and overlaid with the analyticalsolutions for buffer gas pressures of 0.01 mbar to 1 mbar.The analytical and numerical solutions are almost the same for the gaspressure of 0.01 and 0.1 mbar when the damping coefficient k is small. Forgas pressure of 1 mbar, there is a noticeable discrepancy as shown in Fig-ure 2.16. In this case, the analytical solution is found to be incorrect becausethe e−kξ term leads to a large decrease of the amplitude from the beginning,when the velocity u˙ is still small. In these cases, the RK4 numerical solutionobtained the correct evolution of u(t).0 20 40 60 80 100Time ( s)0.40.30.20.10.00.10.20.3Velocity (mm/s)pressure: 1 mBarAnalyticalRK4Figure 2.17: Ion velocity of analytical and numerical solutions shown inFigure 2.16.2.5.3 Optimum gas pressure for ion coolingThe ion cooling time constant τu is inversely proportional to the buffer gaspressure in Eq. (2.105) and Table 2.2. The values of τu annotated in theplots of Figure 2.16 correctly indicate when the envelope of the waveformdecreases to 1/e (e = 2.71828 . . . ) for the gas pressure of 0.01 and 0.1 mbar.50However, at a higher pressure of 1 mbar and 10 mbar, the actual ion coolingtime constant is significantly larger. In fact, the ions are cooled more slowlyat 10 mbar than at 0.1 mbar.Macromotion as damped oscillationThe unexpected slower ion cooling rate at higher gas pressure was explainedby Kim in Section 4.2 of his dissertation [Kim97], where the macromotionof the ions is described as a damped oscillation with the presence of buffergas:d2u¯dt2+ 2k¯du¯dt+ ω¯2 = 0, (2.106)wherek¯ =e2mK(2.107)and the other variables u¯ and ω¯ are the same as in Eq. (2.9) to Eq. (2.21).The fastest ion cooling occurs at the critical damping when k¯ = ω¯. Forq = 0.1 and RF frequency fRF = 1 MHz, this corresponds to a helium gaspressure of 1.2 mbar. The slower ion cooling rate at 10 mbar is caused byover-damping.For a larger value of q = 0.5, around which is used commonly in RFQ ioncoolers, the critical damping occurs at a higher helium pressure gas pressureof 6.0 mbar. Calculations of ion cooling were done for this q value andshown in Figure 2.18. As expected, the actual ion cooling rate at 10 mbaris shorter than when q = 0.1 in Figure 2.16.For the lower helium gas pressure, the macromotion is under-dampedand the time constantτu¯ =1k¯=2mKe(2.108)is the same as τu obtained earlier in Eq. (2.105).The buffer gas pressure in RFQ coolers is usually limited to be lowerthan 0.1 mbar by the vacuum system. At this range, higher gas pressure ismore optimum in achieving a faster ion cooling rate.Ion cooling in the axial directionIn the axial direction x, the ions experience only a DC trapping potentialas shown in Figure 2.15. At the final stage of ion cooling, the axial DC510 200 400 600 800 1000Time ( s)101Position (mm)1/e544pressure: 0.01 mBarAnalyticalRK40 25 50 75 100 125 150 175 200Time ( s)101Position (mm)1/e54.4pressure: 0.1 mBarAnalyticalRK40 10 20 30 40 50Time ( s)01Position (mm)1/e5.44pressure: 1 mBarAnalyticalRK40 10 20 30 40 50Time ( s)0.00.51.0Position (mm)1/e0.544pressure: 10 mBarRK4Figure 2.18: Analytical and numerical calculation of ion cooling using the ionmobility at different gas pressures as annotated in each plot. The stabilityparameters are (q = 0.5, a = 0). See text for details.52potential is close to a quadrupole potential, hence the ion motion can bedescribed asd2xdt2+ 2k¯dx¯dt+ ω2x = 0, (2.109)where ωx is the axial oscillation frequency corresponding to the shape ofthe axial DC potential. Usually the axial DC potential is flatter than thepseudopotential in the radial direction, hence the axial oscillation frequencyωx will be smaller, resulting in a lower gas pressure at critical damping.The optimum gas pressure for ion cooling in the axial direction needs tobe considered according to the actual configuration of DC trapping potentialand will be discussed in more details in Chapter 3.2.5.4 Equilibrium ion temperatureThe equilibrium temperature is an important characteristic of the ion cool-ing and needs to be considered in addition to the analytical and numericalcalculations in Section 2.5.2. Because the ions cannot be cooled to a tem-perature lower than that of the buffer gas, the ions will be cooled to anion cloud of a size corresponding to the equilibrium temperature insteadof infinitely approaching the zero position as shown in Figure 2.16 andFigure 2.18. A lower ion temperature corresponds to a smaller ion cloudsize and smaller emittance for the extracted ions.The equilibrium temperature of ion cooling has been extensively studiedin a 3D Paul trap [Lun92, LBM92, Gha96] and LPTs [Kim97, Smi05]. Theion temperature was found to be higher than the buffer gas temperatureand depends on the stability parameter q of the ion trap. The more detailedstudy of the ion temperature via simulation will be discussed in Section 3.5.3.53Chapter 3Simulations of the linearPaul trapContemporary computational hardware and software have enabled simula-tion of many scientific apparatus before they were built. Such simulationscan usually significantly cut down the time and resources needed for exper-iments and improve their design and performance.For the linear Paul trap (LPT), electrostatic simulations are needed todetermine the electrodes’ geometry and voltage settings as described in Sec-tion 3.1. The characteristic of ion transmission in an RFQ was simulatedin terms of ion acceptance as described in Section 3.2. Then the specificsimulations and optimizations were done for an RFQ ion guide (Section 3.2)and a QMF (Section 3.4). Finally, simulations of ion cooling with buffer gasin an LPT is described in Section 3.5.3.1 Electric potential in an LPTElectric potential in any charge-free space, such as the center of an ion trap,follows the simple form of Laplace equation∇2φ = 0. (3.1)In polar coordinates, the general solution isφ(r, θ) = A0 ln r+B0 +∞∑n=1(An cos(nθ)+Bn sin(nθ))(Cnrn+Dnr−n). (3.2)For ideal quadrupole electrodes, the solution can be reduced to keep onlythe spatial harmonic terms φn[DGKS99]φ(r, θ) =∞∑n=0Anφn, (3.3)54φn = cos(nθ)(rr0)n, (3.4)where r0 is the distance from an electrode’s inner surface to the central axisof the electrodes as illustrated in Figure 2.1. In 2D Cartesian coordinatesof the y − z plane,φn = <[(y + izr0)n]. (3.5)For four ideally positioned quadrupole electrodes with rotational symmetry,the first non-zero terms areφ0 = A0 (3.6)φ2 = A2y2 − z2r0(3.7)φ6 = A6x6 − 15x4y2 + 15x2y4 − y6r60(3.8)φ10 = A10x10 − 45x8y2 + 210x6y4 − 210x4y6 + 45x2y8 − 710r100. (3.9)3.1.1 Quadrupole electrode geometriesThe coefficients An in Eq. (3.3) depend on the geometry of electrodes asthe boundary condition, and the pure electric potential of each term can becreated by having electrodes follow the equipotential lines φn(y, z) = V .For example, to produce an ideal quadrupole potentialφ(y, z) = Vy2 − z2r20, (3.10)the electrodes needs to have hyperbolic shape defined by the equipotentiallines of φ(y, z) = V :y2 − z2 = r20. (3.11)These equipotential lines are infinitely long, while in the real world, theelectrodes need to be at least truncated.Electric potential in real-world quadrupole electrodes are simulated usingcommercially available software SIMION [D10]. In a SIMION simulation,electric potential in a 3D or 2D space is defined by a potential array (PA).55The PA corresponding to electrodes is given fixed voltage values, then thepotential in the space between electrodes is solved by the Finite BoundaryMethod (FBM). Effects of the electrodes’ shape are described below.Hyperbolic electrodesA set of hyperbolic electrodes placed inside a grounded metal tube of radius4r0 is simulated. The two pairs of electrodes are set at +1 V and -1 V.Results within |y| < r0 and |z| < r0 are used for least-square fitting todetermine the coefficients of the spatial harmonics terms in Eq. (3.3). Sim-ulations were done for electrodes of different truncation. Results of differenttruncations in units of r0 are shown in Figure 3.1. The higher-order termsA6, A10, A14 and A18 are found to be smaller than 1× 10−5 for truncationT > 1.6r0.These results also validate the reliability of using SIMION for electro-static simulations of the quadrupole electrodes.Round electrodesHyperbolic electrodes are difficult to manufacture and assemble to highprecision. As an alternative, round electrodes have been explored as thequadrupole electrodes and are currently used in most cases.The configuration of round electrodes has only one independent variableη = re/r0, (3.12)where re is radius of the electrode. To best approximate a pure quadrupolepotential, a “magic” value ofη = re/r0 ≈ 1.14511 (3.13)is known to make the first higher-order term A6 = 0 [LWY71, RSM+96,DGKS99].This “magic” value is double checked in this study by running simula-tions of η from 0.5 to 1.5 and in fine steps from 1.144 to 1.146. The resultsare shown in Figure 3.2. The zero-crossing of A6 is found to be between1.1451 and 1.1452. The fine agreement was made possible with the surfaceenhancement feature for electrode geometry introduced in the 8.1 version ofSIMION.561.0 1.2 1.4 1.6 1.8 2.0 2.2 2.40.951.00Quadrupole termA21.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4Truncation (r0)0.020.000.020.040.060.080.10Higher order termsA6A10A1410×A181.4 1.6 1.8 2.0 2.2 2.41012 1e 4(a)20 10 0 10 20y (mm)201001020z (mm)0.90.60.30.00.30.60.9Voltage (V)(b) Truncation = 1.1r020 10 0 10 20y (mm)201001020z (mm)0.90.60.30.00.30.60.9Voltage (V)(c) Truncation =1.8r0Figure 3.1: (a) Spatial harmonic terms of electric potential in hyperbolicelectrodes with different truncations. (b) Electrodes truncated close tor0, causing deviations of the potential from a pure quadrupole. Thedeviations are more obvious at large radial positions (close to r0). (c)Electrodes truncated at a larger distance (1.8r0), forming almost a purequadrupole potential. The white dashed lines are equipotential lines of apure quadrupole potential at the given values in the color bar.570.6 0.8 1.0 1.2 1.40.951.00Quadrupole termA20.6 0.8 1.0 1.2 1.4re/r00.040.020.000.020.040.06Higher order terms1.145A610×A10100×A141000×A18(a)1.1440 1.1445 1.1450 1.1455 1.14601.0001.002Quadrupole termA21.1440 1.1445 1.1450 1.1455 1.1460re/r00.00020.00010.0000Higher order terms1.14511A60.1×A10A1410×A18(b)(c) re/r0 = 0.5 (d) re/r0 = 1.14511Figure 3.2: (a) Spatial harmonic terms of round electrodes with differentradius re. (b) Detailed simulation around the “magic” value of re/r0 =1.14511. (c) Non-ideal potential formed by round electrodes with smallradius re = 0.5r0. (d) Electrodes with radius re = 1.14511r0 to bestapproximate a pure quadrupole potential in the inner area.583.2 Ion transmission simulations in an RFQSIMION also allows simulation of ion trajectories in a time-dependent elec-tric potential such as the case of a Radio Frequency Quadrupole (RFQ).Such simulations were done to characterize the ion transmission perfor-mance of an RFQ.In a SIMION simulation, the electrodes were represented by a potentialarray file such as electrode.PA#; each set of electrodes with the same volt-age is solved independently and saved as a potential array (electrode.PAn).Then the electrical potential inside the RFQ is the superposition of thepotential from each electrode set at each time step of the simulation, andthe effect of alternating voltages on electrodes at radio-frequency is created.The simulations in this section were done with an RFQ wth the followingparameters:• r0 = 5 mm• RF frequency fRF = 1 MHz• RFQ length L = 350 mm• Initial RF phase ϕ0 = 0◦.An example of the simulation is shown in Figure 3.3. The round elec-trodes are for demonstration only and do not represent the exact electrodegeometries used in the simulations. The four electrodes are shielded by agrounded metal tube to ensure a well defined electric boundary condition.Ions with given initial position and velocities start near the entranceof the RFQ in the left of Figure 3.3, then SIMIION calculates the ions’trajectories for each time step using a modified Runge-Kutta method to the4th order. Ions with initial conditions that meet the ion acceptance aretransmitted to the exit of RFQ while the other ions get lost by hitting theelectrodes or the shielding tube.3.2.1 Ion acceptance simulations in a pure quadrupolepotentialTo validate the reliability of SIMION, ion transmission simulations weredone for an ideal RFQ with pure quadrupole potential so that the results59Figure 3.3: Ion transmission simulation through an RFQ in SIMION. Thetrajectories of the ions are shown as the blue curves.can be compared to the theory in the previous chapter.The virtual ions were initiated at x = 10 mm. The ion’s velocity com-ponent along the x axis is vx = 5 mm/µs, hence the number of RF cyclesthe ions take to travel through the RFQ isnRF =Lvx= 70. (3.14)In the transverse directions y and z, the ions’ position and velocityare represented in the phase space (y, vy) and (z, vz). The voltages on theelectrodes determine the ion transmission characteristics of the RFQ as theion acceptance discussed in Section 2.2.Figure 3.4(a) shows Ntotal = 10, 000 randomly generated ions with uni-form distribution in (y, vy) and (z, vz). For the DC voltage U = 0 V (a = 0)and RF voltage V = 34.8 V (q = 0.1), the transmitted ions are shown inFigure 3.4(b).The ion acceptance in units of pir20/µs is shown in the figure’s legend. Thetheoretical acceptance was obtained using the analytical method describedin Section 2.2. For N transmitted ions, assuming the ion acceptance is anelliptical area, then the simulated acceptance is calculated as = 4σuσv√1− r2uv, (3.15)601 0 1y (r0)0.750.500.250.000.250.500.75v y (r 0/s)1 0 1z (r0)0.750.500.250.000.250.500.75v z (r 0/s)(a) 10,000 ions uniformly distributed in (y, vy) and (z, vz)1 0 1y (r0)0.750.500.250.000.250.500.75v y (pir 0/µs)Theory:²y =0.064Simulation: ²y =0.0611 0 1z (r0)0.750.500.250.000.250.500.75v z (pir 0/µs)Theory:²z =0.064Simulation:²z =0.064(b) q = 0.1, a = 0: 117 ions transmitted.1 0 1y (r0)0.750.500.250.000.250.500.75v y (pir 0/µs)Theory:²y =0.205Simulation: ²y =0.2071 0 1z (r0)0.750.500.250.000.250.500.75v z (pir 0/µs)Theory:²z =0.205Simulation:²z =0.207(c) q = 0.577, a = 0: 1198 ions transmitted.Figure 3.4: Distribution of initial (a) and transmitted (b, c) ions in the RFQsimulation.61where u represents the ion positions y or z, and v represents the ion velocitiesvy or vz; σu and σv are the standard distribution of the ions’ position andvelocity; ruv is the Pearson correlation coefficientruv =σuvσuσv=∑Ni=1(ui − u¯)(vi − v¯)√∑Ni=1(ui − u¯)2√∑Ni=1(vi − v¯)2. (3.16)Due to the elliptical shape of the simulated acceptance, the uncertainty ofthe acceptance  isσ = √pi√N. (3.17)Figure 3.4(c) shows the maximum ion acceptance for the given r0 andfRF when a = 0 and q = 0.577 (RF voltage V = 200 V). Correspondingly,more ions were able to be transmitted compared to the smaller ion accep-tance of Figure 3.4(b).Ion acceptance and transmission efficiencyThe ion transmission efficiency T as defined previously in Eq. (2.92) is re-lated to the acceptance yz. For ions uniformly distributed in both (y, vy)and (z, vz), the ion transmission efficiencyT =NNtotal(3.18)is proportional to the combined ion acceptance yzyz = y × z. (3.19)The uncertainty of yz isσyz =√(yσz)2 + (zσy)2 = yz√2pi√N. (3.20)Results of the combined acceptance yz from the simulation of a = 0 andq from 0 to 1 are compared to theoretical results as shown in Figure 3.5.The ion counts which represent the transmission efficiency are plotted inred dots and appear proportional to the acceptances as expected.620.0 0.2 0.4 0.6 0.8 1.0q0.000.020.040.06Acceptance ² yz(pi2r4 0T2 RF) TheorySimulation0500100015002000Ion countIon countFigure 3.5: Simulated ion acceptance compared to theoretical values. Theerror bars are the uncertainties calculated using Eq. (3.20). The abnormalityaround q = 0.64 is explained in text.Ion acceptance abnormality around q = 0.64The ion acceptances obtained from simulations agree well with theory ex-cept for an abnormality around q = 0.64 in Figure 3.5. Figure 3.6(a) showsthe transmitted ions at q = 0.64. The unexpected large ion acceptanceresults from the transmitted ions outside of the elliptical acceptance area.To rule out possible issues of SIMION or the simulation settings, the sim-ulations were also done through a custom-written Python script and thesame abnormality was observed.Further detailed study and simulations were done to understand theabnormal acceptance around q = 0.64. The first thing to check is the num-ber of RF cycles in the ion transmission simulation because the theoreticalacceptance assumes the ions travel down an RF ion guide of infinite lengthand experience infinite RF cycles. When the longitudinal velocity of theions is set to vx = 0.5 mm/µs so that the ions experience 700 RF cycles, theabnormality disappears as shown in Figure 3.6(b).The abnormally large ion acceptance around a = 0 and q = 0.64 maybe used as a benefit in RF ion guides to improve ion transmission efficiency.For ions experiencing 70 RF cycles or less, operating the ion guide at thissetting should increase its ion transmission efficiency by 50%.631 0 1y (r0)0.750.500.250.000.250.500.75v y (pir 0/µs)²y =0.202²y =0.2431 0 1z (r0)0.750.500.250.000.250.500.75v z (pir 0/µs)Theory:²z =0.202Simulation:²z =0.246(a) vx = 5 mm/µs: 70 RF cycles1 0 1y (r0)0.750.500.250.000.250.500.75v y (pir 0/µs)Theory:²y =0.202Simulation: ²y =0.1931 0 1z (r0)0.750.500.250.000.250.500.75v z (pir 0/µs)Theory:²z =0.202Simulation:²z =0.205(b) vx = 0.5 mm/µs: 700 RF cyclesFigure 3.6: Simulated ion transmission and acceptance at q = 0.64 anda = 0 with different RF cycles.3.2.2 Ion acceptance in non-perfect quadrupole potentialsA realistic RFQ with non-hyperbolically shaped electrodes have higher-order spatial harmonics in the electric potential as described in Section 3.1.1.In this case, there is no known analytical solution to ion motion, hencenumerical simulations are the only approaches.Simulations similar to those shown in Figure 3.4 were done for quadrupolepotentials with added higher-order spatial harmonics terms. For a = 0 andq = 0.577, ion transmission simulations with the presence of the higher-order spatial harmonics in the electric potential are shown in Figure 3.7 toFigure 3.10. The effect of these spatial harmonic terms is negligible when64they are smaller than 0.001. With larger higher-order spatial harmonics,the boundary of the ellipse becomes blurred and the amount of transmittedions decreases.With the presence of higher-order spatial harmonics in the electric po-tential, the ions transmitting through an RFQ no longer follow an ellipticalacceptance. So, the acceptance can not be calculated from Eq. (3.38) andEq. (3.20). Instead, the acceptances shown in the legends of Figure 3.7 toFigure 3.10 are calculated from the count of transmitted ionsyz =NNtotaltotal, (3.21)where total is the acceptance that gets all the initial ions transmitted. Forthe ions used for these simulations in phase space range shown in Fig-ure 3.4(a), total = 0.365pi2 r40T 2RF.3.3 RFQ ion guide simulation and optimizationThe theory of the RFQ ion guide is introduced in Section 2.3. For anideal RFQ ion guide with pure quadrupole potential, the theoretical andsimulated ion acceptance is shown in Figure 3.5.For practical applications of RFQ, the influence of higher-order spatialharmonics in the electric potential needs to be considered as discussed inSection 3.2.2, and in more detail in Section 3.3.1.3.3.1 Influence of higher-order spatial harmonics in electricpotentialSimulations were done for an RFQ ion guide with the presence of the 6thspatial harmonic term for q values between 0 and 1. Acceptances calculatedusing Eq. (3.21) from the simulations are shown in Figure 3.11. The ab-normally large ion acceptance around q = 0.64 as discussed near the end ofSection 3.2.1 is still visible for |A6| < 0.01.Similar simulations were done for an RFQ ion guide with the 10th andabove higher-order spatial harmonics in the electric potential. Acceptancesderived from the simulations are shown in Figure 3.12.651 0 1y (r0)0.750.500.250.000.250.500.75v y (pir 0/µs)Theory:²y =0.205Simulation: ²y =0.1981 0 1z (r0)0.750.500.250.000.250.500.75v z (pir 0/µs)Theory:²z =0.205Simulation:²z =0.198(a) A6 = 0.001: 1121 ions transmitted1 0 1y (r0)0.750.500.250.000.250.500.75v y (pir 0/µs)Theory:²y =0.205Simulation: ²y =0.181 0 1z (r0)0.750.500.250.000.250.500.75v z (pir 0/µs)Theory:²z =0.205Simulation:²z =0.182(b) A6 = 0.01: 936 ions transmitted1 0 1y (r0)0.750.500.250.000.250.500.75v y (pir 0/µs)Theory:²y =0.205Simulation: ²y =0.0991 0 1z (r0)0.750.500.250.000.250.500.75v z (pir 0/µs)Theory:²z =0.205Simulation:²z =0.112(c) A6 = 0.1: 273 ions transmittedFigure 3.7: Ion transmission simulation for an RF ion guide with added 6thspatial harmonics in electric potential for a = 0 and q = 0.577.661 0 1y (r0)0.750.500.250.000.250.500.75v y (pir 0/µs)Theory:²y =0.205Simulation: ²y =0.1971 0 1z (r0)0.750.500.250.000.250.500.75v z (pir 0/µs)Theory:²z =0.205Simulation:²z =0.203(a) A10 = −0.001: 1048 ions transmitted1 0 1y (r0)0.750.500.250.000.250.500.75v y (pir 0/µs)Theory:²y =0.205Simulation: ²y =0.1691 0 1z (r0)0.750.500.250.000.250.500.75v z (pir 0/µs)Theory:²z =0.205Simulation:²z =0.17(b) A10 = −0.01: 791 ions transmitted1 0 1y (r0)0.750.500.250.000.250.500.75v y (pir 0/µs)Theory:²y =0.205Simulation: ²y =0.1321 0 1z (r0)0.750.500.250.000.250.500.75v z (pir 0/µs)Theory:²z =0.205Simulation:²z =0.123(c) A10 = −0.1: 437 ions transmittedFigure 3.8: Ion transmission simulation for an RF ion guide with added 10thspatial harmonics in electric potential for a = 0 and q = 0.577.671 0 1y (r0)0.750.500.250.000.250.500.75v y (pir 0/µs)Theory:²y =0.205Simulation: ²y =0.1971 0 1z (r0)0.750.500.250.000.250.500.75v z (pir 0/µs)Theory:²z =0.205Simulation:²z =0.199(a) A14 = 0.001: 1048 ions transmitted1 0 1y (r0)0.750.500.250.000.250.500.75v y (pir 0/µs)Theory:²y =0.205Simulation: ²y =0.1761 0 1z (r0)0.750.500.250.000.250.500.75v z (pir 0/µs)Theory:²z =0.205Simulation:²z =0.18(b) A14 = 0.01: 871 ions transmitted1 0 1y (r0)0.750.500.250.000.250.500.75v y (pir 0/µs)Theory:²y =0.205Simulation: ²y =0.1371 0 1z (r0)0.750.500.250.000.250.500.75v z (pir 0/µs)Theory:²z =0.205Simulation:²z =0.133(c) A14 = 0.1: 518 ions transmittedFigure 3.9: Ion transmission simulation for an RF ion guide with added 14thspatial harmonics in electric potential for a = 0 and q = 0.577.681 0 1y (r0)0.750.500.250.000.250.500.75v y (pir 0/µs)Theory:²y =0.205Simulation: ²y =0.21 0 1z (r0)0.750.500.250.000.250.500.75v z (pir 0/µs)Theory:²z =0.205Simulation:²z =0.201(a) A18 = 0.001: 961 ions transmitted1 0 1y (r0)0.750.500.250.000.250.500.75v y (pir 0/µs)Theory:²y =0.205Simulation: ²y =0.1831 0 1z (r0)0.750.500.250.000.250.500.75v z (pir 0/µs)Theory:²z =0.205Simulation:²z =0.176(b) A18 = 0.01: 784 ions transmitted1 0 1y (r0)0.750.500.250.000.250.500.75v y (pir 0/µs)Theory:²y =0.205Simulation: ²y =0.1441 0 1z (r0)0.750.500.250.000.250.500.75v z (pir 0/µs)Theory:²z =0.205Simulation:²z =0.141(c) A18 = 0.1: 496 ions transmittedFigure 3.10: Ion transmission simulation for an RF ion guide with added18th spatial harmonics in electric potential for a = 0 and q = 0.577.690.0 0.2 0.4 0.6 0.8 1.0q0.000.010.020.030.040.05Acceptance ² yz(pi2r4 0T2 RF) TheoryA6 = − 0.001A6 =0.001A6 = − 0.01A6 =0.01A6 = − 0.1A6 =0.1Figure 3.11: Simulated ion acceptance for the electric potential with thepresence of the 6th spatial harmonic component. The abnormality aroundq = 0.64 is discussed in text.From these simulation results, an RFQ ion guide needs to have higher-order spatial harmonics close to 0.001 or smaller to avoid any noticeableloss of ion transmission efficiency due to decreased ion acceptance. Suchrequirements can be met by an ion guide with either truncated hyperbolicelectrodes or round electrodes as discussed in Section 3.1.1, and in moredetail in Section 3.3.2.3.3.2 Electrode geometries for RFQ ion guideHyperbolic electrodeThe ideal choice of the electrode geometry to ensure minimum higher-orderspatial harmonic terms is the hyperbolic shape as shown in Figure 3.1.Ion transmission simulations were done for an RFQ ion guide with suchtruncated hyperbolic electrodes to determine the influence of the truncationon ion acceptance. Acceptances derived from the simulations are shown inFigure 3.13.For electrodes truncated closer to r0, less of the hyperbolic profile isleft hence larger spatial harmonic terms exist in the electric potential. Fig-ure 3.13 shows the RFQ ion guide with electrode truncated at 1.04r0 and1.1r0 has much lower ion acceptance than the ideal case.Electrodes truncated at 1.4r0 or larger were found to have optimum ion700.0 0.2 0.4 0.6 0.8 1.0q0.000.020.040.06Acceptance ² yz(pi2r4 0T2 RF) TheoryA10 = − 0.001A10 =0.001A10 = − 0.01A10 =0.01A10 = − 0.1A10 =0.10.0 0.2 0.4 0.6 0.8 1.0q0.000.010.020.030.040.05Acceptance ² yz(pi2r4 0T2 RF) TheoryA14 = − 0.001A14 =0.001A14 = − 0.01A14 =0.01A14 = − 0.1A14 =0.10.0 0.2 0.4 0.6 0.8 1.0q0.000.010.020.030.040.05Acceptance ² yz(pi2r4 0T2 RF) TheoryA18 = − 0.001A18 =0.001A18 = − 0.01A18 =0.01A18 = − 0.1A18 =0.1Figure 3.12: Simulated ion acceptance for the electric potential with thepresence of the 10th (top) 14th (middle) and the 18th (bottom) spatialharmonic component.710.0 0.2 0.4 0.6 0.8 1.0q0.000.020.040.06Acceptance ² yz(pi2r4 0T2 RF) TheoryTrunc: 1.04 r0Trunc: 1.1 r0Trunc: 1.2 r0Trunc: 1.3 r0Trunc: 1.4 r0Trunc: 1.8 r0Figure 3.13: Simulated acceptance of an RF ion guide with hyperbolicelectrodes of different truncation. See text for details.acceptance, because the higher-order spatial harmonics terms are all smallerthan 2 × 10−4 as shown in Figure 3.1. The optimum ion acceptances arelarger than the theory in some q values because the larger hyperbolic shapedsurface of electrodes was used as the boundary for ions instead of a squareof r0 used in the theory.Round electrodeRound rods are used as quadrupole electrodes in many RFQ ion guidesbecause of the easier manufacturing and assembly processes. Simulationswere done for an RFQ with round electrodes of different radius. Acceptancesderived from the simulations are shown in Figure 3.14.For electrodes with radius re = 1.0r0 and re = 1.2r0, the higher-orderspatial harmonic terms are in the order of 0.01 and caused the RFQ to havenoticeably lower acceptance for some regions of the q value. However, itis noteworthy that the “magic” value of re = 1.14511r0 which enables theRFQ to have A6 = 0 didn’t produce the optimum ion acceptances. Instead,re between 1.1r0 and 1.3r0 leads to the optimum ion acceptances of theRFQ ion guide.720.0 0.2 0.4 0.6 0.8 1.0q0.000.010.020.030.040.05Acceptance ² yz(pi2r4 0T2 RF) Theoryη=1.0η=1.13η=1.145η=1.16η=1.2Figure 3.14: Simulated acceptance of an RF ion guide with round electrodesof different radius re = ηr0. See text for details.3.4 QMS simulation and optimizationThe theory of QMS is introduced in Section 2.4. An ideal QMS with purequadrupole potential has the optimum performance in mass spectrometry.Ion acceptance of such an ideal QMS is shown in Figure 3.15 for the uppertip of the stable region used for mass spectrometry.Figure 3.15 also shows the performance of the QMS in terms of massresolving power R and the acceptance yz as a function of the a value. Asa approaches at = 0.2369940 when q = qt = 0.7059961, the mass resolvingpower R becomes infinitely large while the ion acceptance becomes infinitelysmall. So a trade-off needs to be made between these two. Usually, a massresolving power of a few hundred is achievable in a realistic QMS.For the real world applications of QMS, the influence of higher-orderspatial harmonics in the electric potential needs to be considered as discussedin Section 3.4.1.3.4.1 Influence of higher-order spatial harmonicsSimulations similar to Section 3.2.2 were done for q = 0.706 and a = 0.23 toinvestigate the influence of higher-order spatial harmonics on ion acceptanceof the QMS. Due to the smaller ion acceptance at a larger a value, 100,000ions were simulated for the QMS. The ions were randomly generated withnormal distribution in the same phase space range as Figure 3.4(a).73Figure 3.15: (Top) Ion acceptance of an ideal QMS with pure quadrupolepotential. (Bottom) The mass resolving power is derived from the ionacceptance and shown as a function of a.74The simulation results of ion transmission and acceptance are shown inFigure 3.16. The ion acceptance is affected more obviously compared to thecase of an RF ion guide in Figure 3.7 when q = 0.577 and a = 0.The affected ion acceptances change the peak shape of the mass spec-trometry measurement of the QMS, and are related to the performance ofthe QMS such as mass resolving power R and ion transmission efficiencyT . For a = 0.23, the theoretical peak shape of an ideal QMS with purequadrupole potential is shown in Figure 3.17 as the blue solid line. Sim-ulated peak shape for a QMS with and without the presence of the 6thspatial harmonic is also shown in the figure.For a = 0.23, the ideal QMS has theoretical mass resolving powerRa=0.23 =qt∆q= 50.5, (3.22)where qt ≈ 0.706 and ∆q is the peak’s full-width at half-maximum (FWHM).The simulation of the ideal QMS (A6 = 0) agrees well with the theory.With the presence of the 6th spatial harmonic in the electric potential,the peak shape changes. When A6 = ±0.001, the mass resolving power isclose to the ideal QMS; the ion transmission efficiency at the peak decreasesabout 10% and the peak is deformed. For larger A6, both the mass resolvingpower and the ion transmission efficiency is significantly affected. Also, theposition of the peak is shifted and would lead to a mass shift in the massspectrometry.The influence of the 10th and higher spatial harmonics on the peakshape is shown in Figure 3.18.The presence of these higher-order spatial harmonic terms is dependenton the geometry of the electrodes as discussed in Section 3.1.1.3.4.2 Electrode geometries for QMSHyperbolic electrode for QMSThe ideal choice of electrode geometry for the QMS is the hyperbolic shapeto minimize the higher-order spatial harmonic terms. The influence of75(a) A6 = 0.001: 5904 ions transmitted(b) A6 = 0.01: 4418 ions transmitted(c) A6 = 0.1: 2076 ions transmittedFigure 3.16: Ion transmission simulation in a QMS with added 6th spatialharmonics in electric potential for a = 0.23 and q = 0.706. The three setsof plot show the effect of the 6th spatial harmonics of different amplitude.760.66 0.68 0.70 0.72 0.74q0.00000.00050.00100.00150.0020Acceptance ² yz(pi2r4 0T2 RF) TheoryA6 =0A6 = − 0.001A6 =0.001A6 = − 0.01A6 =0.01A6 = − 0.1A6 =0.1Figure 3.17: Peak shape of a QMS with pure quadrupole potential (theory)and added 6th order spatial harmonic (simulations).truncation of electrodes on a QMS’s performance was studied in the simu-lation when a = 0.236; corresponding to a theoretical mass resolving powerR = 345.1.To have the simulations correctly represent this large mass resolvingpower, the ions need to go through enough RF cycles to filter out the ionswith unstable trajectories as discussed in Section 2.4.2.The longitudinal ion velocity of vx = 5 mm/µs used in the previoussimulations corresponds to nRF = 70 and Rn ≈ 245 < 345.1. So, a slowerlongitudinal velocity vx = 2 mm/µs was also used in simulations to havemore RF cycles ( nRF = 175). The simulation results shown as acceptancesand peak shapes are shown in Figure 3.19.The simulations show that when the electrodes are truncated close to r0and hence have only a small surface area, the peak shape is broad due to theinfluence of the higher-order spatial harmonics. For electrodes truncated at1.4r0 or larger, the peak shape is the same as the theory.More detailed simulations were done for a QMS with hyperbolic elec-trodes truncated at 2r0 so that the higher-order spatial harmonic termsare all smaller than 1× 10−5 and negligible. Results of the simulations areshown in Figure 3.20.770.66 0.68 0.70 0.72 0.74q0.00000.00050.00100.00150.0020Acceptance ² yz(pi2r4 0T2 RF) TheoryA10 = − 0.001A10 =0.001A10 = − 0.01A10 =0.01A10 = − 0.1A10 =0.10.66 0.68 0.70 0.72 0.74q0.00000.00050.00100.00150.0020Acceptance ² yz(pi2r4 0T2 RF) TheoryA14 = − 0.001A14 =0.001A14 = − 0.01A14 =0.01A14 = − 0.1A14 =0.10.66 0.68 0.70 0.72 0.74q0.00000.00050.00100.00150.0020Acceptance ² yz(pi2r4 0T2 RF) TheoryA18 = − 0.001A18 =0.001A18 = − 0.01A18 =0.01A18 = − 0.1A18 =0.1Figure 3.18: Peak shape of a QMS with pure quadrupole potential (theory)and the added 10th (top), 14th (middle) and 18th (bottom) higher-orderspatial harmonic (simulations).780.67 0.68 0.69 0.70 0.71 0.72 0.73q10-510-410-3Acceptance ² yz(pi2r4 0T2 RF) TheoryTruncation: 1.15Truncation: 1.2Truncation: 1.25Truncation: 1.3Truncation: 1.4Truncation: 2Figure 3.19: Peak shapes of QMS with hyperbolic electrodes of differenttruncation at a = 0.236.For ions with longitudinal velocity vx = 2 mm/µs, Figure 3.20 shows theQMS’s performance of both mass resolving power R and ion acceptance yzare close to theory. For ions with vx = 5 mm/µs, R and yz starts to deviatefrom theory from a = 0.236. In this case, the QMS’s mass resolving powerR is limited up to 1000 by the RF cycle nRF = 70.Round electrode for QMSIn practice, hyperbolic shaped electrodes are difficult to machine and as-semble to high precision in a QMS, so round electrodes are often used. A“magic” value of η = re/r0 = 1.14511 is known to have the first higher-orderspatial harmonic Am6 = 0 [LWY71, RSM+96, DGKS99]. This configurationwas believed to enable the best mass spectrometry performance in the earlydays of QMS research. However, at this configuration the next higher-orderspatial harmonic terms are not all negligible:Am10 = −2.44× 10−3 (3.23)Am14 = −2.73× 10−4 (3.24)Am18 = −1.57× 10−5. (3.25)A QMS with round electrodes of re = 1.14511r0 were simulated and theresults shown in Figure 3.21 reveal some issues:• ion transmissions still occur outside of the QMS’s theoretical stableregion,790.690 0.695 0.700 0.705 0.710 0.715q0.2300.2310.2320.2330.2340.2350.2360.2370.238aSimulation vx =2mm/µs10-510-410-30.230 0.231 0.232 0.233 0.234 0.235 0.236 0.237 0.238a, DC to RF ratio s× 2qt102103104Mass resolving power RSimulation vx =2mm/µsSimulation vx =5mm/µsTheory10-510-410-3Acceptance ² yz(pi2r4 0T2 RF)0.690 0.695 0.700 0.705 0.710 0.715q0.2300.2310.2320.2330.2340.2350.2360.2370.238aSimulation vx =5mm/µs10-510-410-3Acceptance ² yz(pi2r4 0T2 RF)Figure 3.20: Simulation of a QMS with hyperbolic electrodes.80• the boundary of the original stable region becomes unclear,• ion transmissions inside the original stable region decrease.0.70 0.71q0.2310.2320.2330.2340.2350.2360.2370.238aSimulation vx =2mm/µs10-510-410-30.230 0.232 0.234 0.236 0.238a, DC to RF ratio s× 2qt102103Mass resolving power RSimulation vx =2mm/µsSimulation vx =5mm/µsTheory10-510-410-3Acceptance ² yz(pi2r4 0T2 RF)0.70 0.71q0.2310.2320.2330.2340.2350.2360.2370.238aSimulation vx =5mm/µs10-510-410-3Acceptance ² yz(pi2r4 0T2 RF)Figure 3.21: QMS with round electrodes with radius re = 1.14511r0.These issues lead to a decrease in the performance of the QMS. The ionacceptance is about 60% of the theoretical value for most of the a value. Themass resolving power R starts to deviate from the theory around a = 0.2362when R ≈ 400. The maximum mass resolving power achievable is R ≈ 700around a = 0.2372.In later QMS research, it was found that a slightly smaller ratio ofη = re/r0 leads to better QMS performance [GT01, DK02]. Following thesefindings, simulations were also done for a QMS with round electrodes of81re = 1.13r0 and the results are shown in Figure 3.22.0.70 0.71q0.2310.2320.2330.2340.2350.2360.2370.238aSimulation vx =2mm/µs10-510-410-30.230 0.232 0.234 0.236 0.238a, DC to RF ratio s× 2qt102103Mass resolving power RSimulation vx =2mm/µsSimulation vx =5mm/µsTheory10-510-410-3Acceptance ² yz(pi2r4 0T2 RF)0.70 0.71q0.2310.2320.2330.2340.2350.2360.2370.238aSimulation vx =5mm/µs10-510-410-3Acceptance ² yz(pi2r4 0T2 RF)Figure 3.22: QMS with round electrodes with radius re = 1.13r0.In comparison to Figure 3.21, there is less ion transmission outside of thetheoretical stable region. The simulation with vx = 2 mm/µs only starts todeviate from theory from a = 0.2367 when R ≈ 1000. Thus the QMS withround electrodes of re = 1.13r0 indeed has better performance than the onewith electrodes of re = 1.14511r0. The QMF for this study will be designedwith round electrodes with re = 1.13r0.823.5 RFQ ion coolerThe physics of ion cooling with buffer gas has been described with theconcept of ion mobility in Section 2.5. Simulation of the ion cooling is doneby including randomized ion-gas collision in the numerical calculation of theion trajectories. In this way, the simulations can obtain some more detailedion cooling characteristics, such as the coolng time for the ions to reachthermal equilibrium and the equilibrium ion temperature.Simulation of the ion cooling process is done using SIMION with a user-customizable hard-sphere model HS1 [Man07]. In the simulations, the he-lium buffer gas is treated as an ideal gas at room temperature with ran-domized Maxwell-Boltzmann distribution. The occurrence of the collisionsis related to the mean-free pathl =vionv¯σn(3.26)of the ions in the gas. In Eq. (3.26) v¯ is the mean relative velocity betweenthe ion and gas, n is the number density of the helium gas, andσ = pi(rion + rgas)2 (3.27)is the combined cross-section of the collisions.3.5.1 Ion drift velocity and mobilityTo get started with the ion-neutral simulations and to benchmark the SIMIONHS1 model for 136Ba+ ions in the helium buffer gas, simulations were donefor 1000 randomly generated ions in a constant electric field E with thepresence of 0.1 mbar helium gas. The velocities of the ions stabilize afterflying 1000 µs and were obtained after the drift velocity vd.At first, the Van Der Waals radius of helium rhelium = 143 pm [Web20a]and the Pauling ionic radius of Ba+ rBa+ = 153 pm [Web20b] were usedfor the cross-section calculation in Eq. (3.27). But the simulated ion driftvelocities were found to be significantly larger than the previously experi-mentally measured values [VM95]. The drift velocity was also found to beinversely proportional to the collision cross-section σ, so later an empiricalvalue of Ba+ reBa+ = 220 pm was used. The results of the simulations are83shown in Figure 3.23. The ion mobility is derived asK =vdE, (3.28)and the standard ion mobility isK0 =NN0K =pgaspatmT273.15K, (3.29)where pgas = 0.1 mbar is the helium gas pressure and patm = 1013.25 mbaris the standard atmospheric pressure.010002000300040005000v d (m/s)ExperimentSIMION HS10 50 100 150 200 250 300E/N (Td)0.000500.000750.001000.001250.001500.00175K 0 (m2 V1 s1 )ExperimentSIMION HS1Figure 3.23: Simulation of ion drift velocity vd in 0.1 mbar helium gaswith different electric field strength. The unit of the horizontal axis isTownsend (1 Td = 10−21 Vm2). The standard ion mobility K0 is derivedfrom Eq. (3.29). Previously published experimental results [VM95] are alsoplotted for comparison.The results from simulation agree well with experimental data whenthe ion velocity is small, e.g. vd < 1000 m/s. At larger ion velocity, thediscrepancy is likely a result of the simplification of the hard sphere collisionmodel HS1. An alternative model using the realistic potential [Kim97,Smi05, Sch06] between the ion and gas molecules to calculate the collisionalparameters ha better agreement with experimental data.843.5.2 Ion cooling rateIon cooling simulations in SIMION were done for ions in an LPT with RFfrequency fRF = 1 MHz. The initial RF phase is ϕ0 = 0; the initial ionposition is (x = y = 0, z = 10) and the initial velocity is (vx = 0.1 mm/µs,vy = vz = 0). For the stability parameter (q = 0.1, a = 0), the results areplotted in Figure 3.24. The results are compared to numerical solutionsobtained using ion mobility as described in Section 2.5.2.For helium gas pressure of 0.01 mbar and 0.1 mbar, the ion trajectoriesshown in the top and middle plot of Figure 3.24 are found slowing downfaster than the calculated time constant from the ion mobility. This is be-cause at larger ion velocity the ion mobility is smaller, as shown in Fig-ure 3.23. The smaller ion mobility corresponds to a larger damping effectfor the ion.For a larger helium gas pressure of 1 mbar, the ion cooling is found tobe taking longer than the RK4 solution. This is also expected because theion is being over-damped as discussed in Section 2.5.3.Simulations were also done for the stability parameter (q = 0.5, a = 0)with the same RF frequency and initial RF phase. The initial ion position is(x = y = 0, z = 1) and the initial velocity is (vx = 0.1 mm/µs, vy = vz = 0).The results are shown in Figure 3.25.The ion velocity of an ion being cooled at 1 mbar is shown in Figure 3.26and compared with the RK4 solution from ion mobility. The ion-neutralcollisions can be seen as the abrupt changes in ion velocity, in contrast tothe smooth decrease in the RK4 solution.Figure 3.24 to Figure 3.26 also reveal that the ion position and velocitydoesn’t keep decreasing to infinitesimal. Instead, both the position andvelocity reach thermal equilibrium with the helium gas.850 200 400 600 800 1000Time ( s)10010Position (mm)10/e544pressure: 0.01 mBarIon mobility RK4SIMION HS10 25 50 75 100 125 150 175 200Time ( s)010Position (mm)10/e54.4pressure: 0.1 mBarIon mobility RK4SIMION HS10 25 50 75 100 125 150 175 200Time ( s)0510Position (mm)10/e5.44pressure: 1 mBarIon mobility RK4SIMION HS1Figure 3.24: Ion cooling simulation in an LPT with stability parameter(q = 0.1, a = 0) at helium gas pressure of 0.01 mbar (top), 0.1 mbar (middle)and 1 mbar (bottom).860 200 400 600 800 1000Time ( s)101Position (mm)1/e544pressure: 0.01 mBarIon mobility RK4SIMION HS10 25 50 75 100 125 150 175 200Time ( s)101Position (mm)1/e54.4pressure: 0.1 mBarIon mobility RK4SIMION HS10 25 50 75 100 125 150 175 200Time ( s)01Position (mm)1/e5.44pressure: 1 mBarIon mobility RK4SIMION HS1Figure 3.25: Ion cooling simulation in an LPT with stability parameter(q = 0.5, a = 0) at helium gas pressure of 0.01 mbar (top), 0.1 mbar and1 mbar (bottom).870.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0Time ( s)1.51.00.50.00.51.0Velocity (mm/s)pressure: 1 mBarIon mobility RK4SIMION HS1Figure 3.26: Velocity as a function of time for an ion cooled in an LPT with1 mbar of helium gas.3.5.3 Ion temperatureThe temperature of ions in a linear Paul trap has been extensively studiedby Kim [Kim97]. Some notations and derivations below are borrowed fromthere.The distribution of ions after reaching thermal equilibrium with thebuffer gas can be generically described using the Boltzmann distributiondNdE=N0kBTe− EkBT , (3.30)where dN is the number of ions in the energy interval dE, N0 is the totalion number, kB is the Boltzmann constant and T is the ion temperature.Temperature of ions in a simple harmonic well potentialFor ions trapped in a simple harmonic well, the phase space distribution ofthe ions follows the Gibbs distribution∂6N∂S= Ae− EkBT , (3.31)88where A is a normalization constant. When the ion motion is independentin the coordinates, the Gibbs distribution in one of the coordinates u is∂2N∂u∂pu= A exp(− EkBT). (3.32)For a singly charged positive ion of mass m, the ions experience a restor-ing force F from the electric field Eu. The electric field and the resultingrestoring force are proportional to the displacement u in the harmonic wellF = −eEu = −ku. (3.33)The energy of the ion in the harmonic well isE =p2u2m+ku22=p2u2m+mω2u22, (3.34)where ω is the harmonic oscillation frequency of the ion in the well. In thephase space, the oscillation of the ions is represented by ellipses as shownin Figure 3.27. The area of the ellipses is proportional to the ion energy.u2p2u3p3u1p1upuFigure 3.27: Motion of ions trapped in a simple harmonic potential wellshown in the position-momentum (u− p) phase space.Inserting Eq. (3.34) back to Eq. (3.32) reveals that the ions have Gaus-sian distribution of both their position u and momentum pu. The standarddeviation of u and pu isσu =√kBTmω2, (3.35)σpu =√mkBT . (3.36)89The ion temperature can be derived from either σu or σpu . Note that theion temperature can be different in the three coordinates.Temperature of ions in an LPTThe ion trapping in the longitudinal direction x of an LPT is similar to thecase of a harmonic potential well. According to Eq. (3.36) and pu = mvu,the ion temperature Tx is derived from σvu :Tx =mσ2vu2kB. (3.37)In the transverse directions y and z of an LPT, the ions are confined bya pseudopotential formed by the RF potential. The ions undergo a macro-motion and a micromotion as discussed in Section 2.1.2 and Section 2.1.3.In the phase space, the ion motions evolved along ellipses with the secularfrequency ω¯ as the macromotion. In each RF cycle, the ellipse tilts to dif-ferent angles depending on the RF phase as a result of the micromotion asshown in Figure 2.7(b). As a result, both σu and σpu varies along the RFphase. However, the ion emittance is independent of the RF phase = 4σuσv√1− r2uv, (3.38)where ruv is the Pearson correlation coefficient defined in Eq. (3.16). Ac-cording to Eq. (3.35), Eq. (3.36) and pu = mvu, the ion temperature isderived asTu =mω¯2σuσv√1− r2uvkB, (3.39)where ω¯ = ω0 =βΩ2 , and β is obtained from Eq. (2.39).Simulations were done for 4000 136Ba ions in an LPT filled with 0.1 mbarof helium buffer gas. To first study the effect of the RF confinement, no DCpotential was applied in the longitudinal direction. The ions were set tohave a small initial longitudinal velocity vx = 0.1 mm/µs to make sure theydo not fly out of the simulation volume.The RF frequency of the simulation was set at 1 MHz, the RF ampli-tude was set within 24.7 V< V < 221.9 V corresponding to the stabilityparameter 0.1 < q < 0.9 for an LPT with r0 = 4.21 mm. For each time step,the ions’ standard deviation of the position σu and velocity σvu for each90coordinate was recorded, the Pearson correlation coefficient ruv betweenthe ion positions and the velocities was recorded for each coordinates aswell. The ions reached thermal equilibrium with the buffer gas after a fewhundred microseconds. The value of σu, σvu and ruv recorded during 800 to1000 µs was used to calculate the ion temperatures according to Eq. (3.37)and Eq. (3.49) and shown in Figure 3.28.0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8q30040050060070080090010001100Ion temperature (K)TbuffergasTxTyTzFigure 3.28: Temperature of ions in the LPT with 0.1 mbar helium buffergas at different stability parameter q. The ions were confined only in theradial direction. See text for details.The ion temperature in the radial directions y and z were found to bedependent on the q value. The ion temperature is close to the buffer gastemperature of 300 K for a small q value. For larger q values, the ions aredriven by the larger micromotion and have more intensive collisions with thebuffer gas. The excess micromotions and collisions heat up the ions knownas the RF heating [BKQW89, PWM+91, RZS05]. The ion temperature Tyand Tz in the radial directions were found to be below 400 K when q < 0.6.The ion temperature in the longitudinal direction x is mostly the sameas the buffer gas temperature at 300 K. The slight effect of RF heating isonly noticeable for large q values.913.5.4 Ion trapping in the longitudinal directionThe trapping of ions in the longitudinal direction in an LPT was studiedin a simplified LPT with three sets of short quadrupole electrodes as il-lustrated in Figure 3.29(a). In the longitudinal direction x, the length ofeach set of quadrupole electrodes is 4.5 mm; the gap between each set ofquadrupole electrodes is 0.5 mm. The DC potential of the first and thirdset of quadrupole electrodes was set to be U1 = U3 = 0 V; U2 was set to anegative voltage to form a longitudinal trapping potential.Electric potential along the LPT’s central axis obtained from SIMIONsimulation is shown in Figure 3.29(b). The potential is proportional to theapplied voltage of U2. For every -1 V of U2, the minimum potential at thelongitudinal center xc = 7.25 mm is -0.59 V. The simulation result fromU2 = -10 V within the range of the second (central) set of the quadrupoleelectrode 5 < x < 9.5 was used for a least-square fitting with the formulaUA = B0 +B2 (x− x0)2. (3.40)The fitted result areB0 = −5.897± 0.002(V), (3.41)B2 = 0.253± 0.001(V/mm2), (3.42)x0 = 7.248± 0.002(mm). (3.43)(3.44)The Chi-square of the fitting is χ2 =∑(Usimulation − UA) = 6.8× 10−3.The small value of χ2 indicates axial potential in the region of the central setof quadrupole electrodes is close to a pure quadratic potential well. There-fore, the ions will do damped harmonic oscillations in the longitudinal di-rectionmd2x′dt2+eKdx′dt+ 2eB2x′ = 0, (3.45)where x′ = x− 7.25, K is the ion mobility and e is the elementary charge.The ion’s equation of motion in Eq. (3.45) can be rewritten asd2x′dt2+ 2ζωx0dx′dt+ ω2x0 = 0, (3.46)where ωx0 =√2eB2m is the natural resonance frequency of the ions in the xdirection when there is no damping, and ζ = emKωx0 is the damping ratio.92xyz4.5mm0.5mm2r0=8.42mmU1U2U3(a) Illustration of a simplified LPT0.00 7.25 14.50Longitudinal position x (mm)0.000.592.955.90Axial DC Potential UA (V)U2=1VU2=5VU2=10VU2=10V fitting(b) Electric potential along the central axisFigure 3.29: A simplified LPT for the simulation of ion trapping in thelongitudinal direction.93When U2 =-1 V, B2 = 2.53×104V/m2. The natural resonance frequencyof the barium ions in the longitudinal direction is ω0 = 3.02 × 104 · 2piHz.In this case, ζ = 0.23 < 1 hence the ions are under-damped with actualoscillation frequency ωx1 = ωx0√1− ζ2 = 0.95ωx0.For a deeper trapping potential U2, the natural oscillation frequency ω0of the ions will be larger, hence the damping ratio ζ will be smaller and theactual oscillation frequency of the ions will be closer to ωx0.Expelling potential in the radial directionAccording to the solution of the Laplace equation ∇2φ = 0, the trappingpotential along the longitudinal direction x expressed by Eq. (3.40) wouldcause an expelling potential in the radial direction that expels the ions:Ur = B0 −B2r2, (3.47)where r =√y2 + z2.The expelling electric potential causes a decrease in the effective pseu-dopotential for ion trapping in the radial direction. When the pseudopoten-tial is weaker than the expelling electric potential, the LPT can no longerconfine ions.The reduced pseudopotential corresponds to a reduced secular frequencyω¯′ of the ion motion which can be theoretically expressed asω¯′ =√ω¯2 − ω2x0/2. (3.48)For comparison, SIMION simulations were done for 10 ions in the simplifiedLPT with different trapping voltage U2. The ion trajectories were recordedand the secular frequency of the ions’ motion was obtained from the FFT(fast fourier transform) of the ions’ motion and velocity. The results areshown in Figure 3.30.The good agreement between the theoretical and simulated reducedsecular frequency ω¯′ of the ion motion validates the effect of the expellingpotential in the radial direction. Note that for the trapping depth U2 =−10 V, the pseudopotential of q < 0.2 is smaller than the expelling potentialhence the ions cannot be trapped in the LPT and the theoretical reducedsecular frequency is invalid.940.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9q0.00.10.20.30.4Reduced secular frequency ′ (2MHz)Theory Theory ′: U2= 1VTheory ′ U2= 10VSimulation Simulation ′: U2= 1VSimulation ′ U2= 10VFigure 3.30: Secular frequency ω¯ and reduced secular frequency ω¯′ of ionmotion in an LPT. Most of the error bars are too small to be seen.Effect of longitudinal trapping depth on ion temperatureThe ion temperature in an LPT with longitudinal trapping was studied viaSIMION simulations of the simplified LPT cooler. The simulations weredone with the same RF frequency and voltages as in Figure 3.28. .For each simulated longitudinal trapping voltage, the ion temperaturein the longitudinal direction was obtained using Eq. (3.37). In the radialdirections y and z, the ion temperature was calculated fromTu =mω¯′2σuσv√1− r2uvkB. (3.49)The results are shown in Figure 3.31.For both longitudinal trapping depths U2 = −1 V and U2 = −10 V,the ion temperature is similar to Figure 3.28 when there is no longitudinal950.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8q300400500600700Ion temperature (K)TbuffergasTxTyTz(a) Longitudinal trapping depth U2 = −1 V.0.3 0.4 0.5 0.6 0.7 0.8q300400500600Ion temperature (K)TbuffergasTxTyTz(b) Longitudinal trapping depth U2 = −10 VFigure 3.31: Temperature of ions in an LPT with different longitudinaltrapping potential depth U2.96trapping potential. In the longitudinal direction, the ion temperature isalways close to the buffer gas temperature. In the longitudinal direction,the RF heating is more obvious at a larger q value. At the smallest q valuethat still allows the LPT to trap ions, the ion temperature appeared to beunexpectedly higher but with a larger uncertainty.Emittance of cooled ions in LPTSimulations were done to study the emittance of cooled ions in the simplifiedLPT. In the simulation, 1000 barium ions were cooled for 1000 µs in theLPT with 0.1 mbar of helium buffer gas. The standard deviation of the ionposition σu and velocity σvu were recorded for each time step for each coor-dinate. The Pearson correlation coefficient ruv (defined in Eq. (3.16)) wasalso recorded. The ions were cooled to thermal equilibrium after around 300microseconds. The values of σu, σvu and ruv from 800 to 1000 microsecondswere used to calculate the ion emittance for each coordinate uu = 4σuσv√1− r2uv. (3.50)At this point, the LPT was studied with a fixed q value of q = 0.5.Simulations were done for the longitudinal trap depth U2 from -1 V to -30 Vat the RF frequencies fRF of 0.5 MHz, 1 MHz and 2 MHz. The longitudi-nal emittance x and the transverse emittance y,z were calculated usingEq. (3.50) and are shown in Figure 3.32.For all RF frequencies, the longitudinal ion emittance x was found tobe smaller with deeper longitudinal trapping potential.In the transverse direction, a larger RF frequency fRF corresponds to ahigher RF amplitude hence a deeper pseudopotential. The transverse ionemittance y,z was also found to be smaller at a deeper pseudopotential.For the lowest RF frequency fRF = 0.5 MHz, the transverse emittancewas found to be larger at a deeper U2. This is a result of the decreasedeffective pseudopotential depth as described above. When |U2| > 12 V, theLPT can no longer trap ions at 0.5 MHz.At fRF = 1 MHz and above, the LPT’s pseudopotential is strong enoughto trap ions in the whole range of simulated range of U2 from -1 V to -30 V.The transverse ion emittance y,z is slightly larger at a deeper U2 as expected.970 5 10 15 20 25 300.20.4² x (mm·mm/µs)fRF =0.5 MHzfRF =1 MHzfRF =2 MHz0 5 10 15 20 25 30Longitudinal trapping depth |U2| (V)0.00.20.40.60.81.0² y,z (mm·mm/µs)fRF =0.5 MHzfRF =1 MHzfRF =2 MHzFigure 3.32: Longitudinal emittance x and transverse y,z of cooled ions inthe LPT at different longitudinal trapping potential U2 and RF frequencyfRF .3.6 RFQ ion buncherThe RFQ ion buncher is needed to eject the cooled ion cloud in the LPT asan ion bunch to an ion detector or downstream experiments. An ion bunchershown in Figure 3.33(a) is studied in this section.The geometry of the ion buncher is the same as the simplified LPTin Figure 3.29(a) except two additional aperture plates are needed. Theelectric potential gradient (electric field strength Ex) used to eject the ionsis formed by the different DC potentials applied on the three sets of thequadrupole electrode and the aperture plates. The potential along thecentral axis of the buncher is obtained via electrostatic simulation usingSIMION and shown in Figure 3.33(b).98xyz4.5mm0.5mmU1 U2U3(a) Illustration of an RFQ ion buncher0.0 2.5 5.0 7.5 10.0 12.5 15.0Longitudinal position x (mm)050010001500200025003000Axial DC Potential UA (V)Ex=10V/mmEx=100V/mmEx=200V/mmTrapped ions(b) Electric potential along the central axisFigure 3.33: An RFQ ion buncher for the simulation of ion ejection.99The purpose of the ion buncher in this study is to form fine ion bunchesfor a downstream MR-TOF mass spectrometer. The MR-TOF requires theion bunch to have a small energy spread (typically around 2%) and a smalltime spread (typically tens of nanoseconds).Energy spread of bunched ionsThe kinetic energy KE of the ejected ions is determined by the DC po-tential UT at the position of the trapped ion cloud. Consequently, Keis proportional to Ex as the simulated results shown in Figure 3.34. Theresults come from ion ejection simulation of 1000 ions which has been cooledfor 1000 µs and trapped as an ion cloud in the center of the ion buncher.The key parameters of the simulation are: q = 0.5, fRF = 1 MHz andU2 = −10 V. The ions stop at the exit of the ion buncher in the simulation,the ions’ kinetic energy KE and time-of-flight (ToF) is recorded for analysis.050010001500Average energy KE (eV)Pressure 0mBarPressure 0.001mBarPressure 0.1mBar25 50 75 100 125 150 175 200EX (V/mm)050100Energy spread K E (eV)eEx xPressure 0mBarPressure 0.001mBarPressure 0.1mBarFigure 3.34: Kinetic energy K¯E (top plot) and energy spread σKE (bottomplot) of ions ejected from the buncher at different electric field strength Exand helium buffer gas pressure.100The kinetic energy spread σKE originates from the positional spread σxof the ion cloud before the ion ejection (the initial kinetic energy spreadfrom velocity is around 0.01 eV and can be ignored). In the ideal case,σKE = eExσx, (3.51)where σx = 0.215 mm when U2 = −10 V. The ion energy spread calculatedin this way is also plotted in Figure 3.34 and agrees well with the simulationwhen the ion buncher has ideal vacuum with 0 mbar pressure.Similar simulations were then done for an ion buncher with realistichelium buffer gas pressure. The results are also shown in Figure 3.34. Atthe pressure 0.1 mbar, the ejected ions have noticeably decreased kineticenergy and increase energy spread as a result of excessive collisions with thehelium gas. At 0.001 mbar, the collisional effect of the gas on the ion energyis almost negligible.Time spread of bunched ionsThe time spread of bunched ions comes from both the velocity spread σvxand positional spread σx of the ion cloud before the bunching.When the ions are initially accelerated in the electric field, the initialtime spread isσt0 = σvx/eExm, (3.52)where σvx is the initial velocity spread of the trapped ion cloud before theion ejection.When the ions reach the exit of the ion buncher, there is an additionaltime spread as a result of the ions’ initial positionσt1 =σxvx, (3.53)where vx is the longitudinal velocity of the ions after reaching the exit ofthe ion buncher.Finally, the spread of the ions’ time-of-flight (ToF) isσtToF =√σ2t0 + σ2t1. (3.54)101The average and spread of the ions’ time-of-flight tToF were obtainedfrom the same simulations described above and shown in Figure 3.35. Thetime spread calculated from theory using Eq. (3.54) is also plotted andagrees reasonably well with the simulation when the helium gas pressure is0.001 mbar or lower.0.51.01.5Average ToF t ToF (s) Pressure 0mBarPressure 0.001mBarPressure 0.1mBar25 50 75 100 125 150 175 200EX (V/mm)0.010.020.030.04ToF spread t ToF (s) TheoryPressure 0mBarPressure 0.001mBarPressure 0.1mBarFigure 3.35: Simulated time of flight (ToF) of the ejected ions to reach theexit of the ion buncher as a function of the longitudinal electric field strengthEx. The simulations were done at ideal vacuum (0 mbar), 0.001 mbar and0.1 mbar. Theoretical values of the ToF spread were calculated according toEq. (3.54).For a higher helium gas pressure of 0.1 mbar, the simulated ion timespread is noticeably larger than the theory as a result of excessive collisionswith the helium gas during the ion ejection.102Effect of helium gas pressure on ion bunchingThe effect of helium gas on the bunched ions’ energy spread and time spreadwas studied in more detail for the electric field strength Ex = 100 V/mm.Ion ejection simulations were done for the ion buncher with helium gaspressure from 10−5 mbar to 0.6 mbar. The ions’ kinetic energy KE andtime-of-flight tToF obtained from the simulations are shown in Figure 3.36.The energy spread σKE and time spread σtToF are shown as the error bars.200400600800Energy KE (eV)Good Acceptable Bad10 5 10 4 10 3 10 2 10 1 100Pressure (mBar)0.500.550.600.65ToF t ToF (s)Figure 3.36: Ion energy KE and time-of-flight tToF of bunched ions as afunction of helium gas pressure in the ion buncher. See text for the detailsof the different labeled pressure regions.The effect of the helium gas on the ion energy and ToF was found tobe negligible when the gas pressure is lower than 10−3 mbar (labeled as103“Good”). At higher gas pressure, both the ion energy spread σKe and timespread σtToF increase significantly.For the simulated ion cloud cooled in the longitudinal trap depth U2 =−10 V, the relative energy spread σKE/KE = 2.9% is already higher thanthe requirement of the MR-TOF. In theory, the ion energy spread scaleslinearly with 1/√|U2|. Even though a deeper longitudinal trapping depthcan reduce the ion energy spread, it is still critical to limit the increaseof σKE due to helium gas during ion ejection. For the ion buncher of thisstudy, the helium gas pressure in the buncher is required to be lower than7 × 10−3 mbar in order to limit the increase of ion energy spread to within50% (labeled as “Acceptable”). When the helium gas pressure is higherthan 7 × 10−3 mbar (labeled as “Bad”), the increase of the bunched ion’stime spread and energy spread are more significant.104Chapter 4A linear Paul trap system forbarium taggingIn the following the design of a linear Paul trap (LPT) system based on thetheory and simulations in the previous chapters is described. The system isto be installed downstream from an RF funnel to capture the extracted ions.Together they will be useful to carry out detailed studies of ion trappingand identification from xenon gas.4.1 LPT system requirementsThe LPT system is required to capture ideally 100% of the ions from theRF funnel. The ions need to be first cooled by the helium buffer gas,then trapped sufficiently long to allow barium ion identification via laserspectroscopy.In previous experiments of ion extraction from high pressure xenon gaswith a prototype of the RF funnel, significant amounts of ion contaminantwere found [Fud18]. Therefore, a multi-reflection time-of-flight (MR-TOF)mass spectrometer is being designed and will be installed downstream fromthe LPT to systematically study the extracted ion species using high preci-sion mass spectrometry.For the MR-TOF to work at high precision, the ions from the LPT needto be ejected as fine ion bunches with small energy spread (typically within2% of the kinetic energy of the ions) and small time spread (typically a fewtens of nanoseconds).The rate of ions that can be cooled in the RFQ cooler is limited. Thetotal number of ions that can be stored in the laser spectroscopy ion trap andion buncher is limited as well. Therefore, too many contaminant ions mayreduce the trapping efficiency of the barium ions. Too many contaminant105ions can also overwhelm the downstream MR-TOF mass spectrometer.In order to ensure the required conditions, a pre-filter system is needed.This can be realized by including a quadrupole mass filter (QMF) to removemany of the contaminant ions. The mass resolving power required for theQMF is R = m/∆m > 80 as this allows us to filter out most of the possiblecontaminant ions other than the isobar 136Xe+.According to these requirements, a conceptual design of the LPT systemwas completed in 2017 as shown in Figure 4.1 [Lan18b]. The design sepa-rates the major components of the LPT system into two vacuum chambers:a QMF chamber contains the quadrupole mass filter; a CLB chamber con-tains the cooler, laser spectroscopy ion trap and the buncher. A pulse drifttube is designed to be placed at the exit of the CLB chamber for adjustingthe energy of the ions transferred into the MR-TOF mass spectrometer.Differential pumping channelsGMacor / PEEKElectrode / detectorCCD Screw / pinQuadrupole mass filter CoolerTMPScrool pumpG G GQMF chamberCLB chamberG Vacuum GaugeBuncherLaser spectroscopyXYZValveTurbo molecular pump (TMP)Bunched ions toMR-TOFIons from RF funnelorion sourcePulsed drift tubeFigure 4.1: A conceptual design of the LPT system as of 2017. The majorcomponents are annotated in bold font.The conceptual design was used as a guideline for the ion optics, me-chanical and vacuum design of the LPT system. Some changes and im-provements have been made. Notably, a pre-cooler was added between theQMF and the cooler; the laser spectroscopy ion trap and the buncher arecombined to share the same trapping region and electrodes.1064.1.1 Ion acceptance requirementThe characteristics of ions coming from the RF funnel are shown in Fig-ure 4.2 in one simulated example of 910 ions [Fud15]. The ion bunch hastransverse emittance ofyrms ≈ zrms = 0.16mm ·mm/µs (4.1)calculated fromrms = σuσv√1− r2uv, (4.2)where u is y or z and v is vy or vz; ruv is the Pearson correlation coefficientdefined in Eq. (3.16).2 0 2y (mm)1012v y (mm/µs)²3rms2 0 2z (mm)101v z (mm/µs)²3rms0 10000 20000Ion time of flight (µs)012345Ion energy (eV)(a) (b) (c)Figure 4.2: Phase space distribution in the transverse directions y (a) z (b)and the longitudinal direction x (c) of ions extracted from the RF funnel.The ion acceptance 3rms is defined in Eq. (4.4).For ions with 2D Gaussian distribution of position u and velocity vu ineach coordinate, the probability of ions being captured with an acceptancecorresponding to nrms · σu and nrms · σvu isPn rms = 1− exp(−n2rms/2). (4.3)In order to capture 98% or more of the ions in both y and z axis, nrms ≥ 3is needed. The corresponding ion acceptance required is3rms = 9rms = 1.44 mm ·mm/µs, (4.4)107which is represented by the ellipses also shown in Figure 4.2.The ion acceptance ellipse of an RFQ depends on the RF phase (seeSection 2.2). The continuous ions coming from the RF funnel can enterthe LPT system at any RF phase. To meet the required close to 100% iontransmission efficiency, the ion emittance needs to be within the ion accep-tance ellipse of any RF phase [Daw75]. So the overlap of the ion acceptanceat different RF phases referred to as phase independent acceptance needsto be considered.Phase independent acceptancePractically, 36 equally spaced phases were used to obtain the phase inde-pendent acceptance as illustrated in Figure 4.3 for the stability parameters(q = 0.5, a = 0).The phase independent acceptance for other q values when a = 0 wasobtained in the same way as shown in Figure 4.3 and shown in Figure 4.4.The maximum phase independent acceptance is obtained at q = 0.45:PI,max = 0.063 r20Ω/2. (4.5)At fRF =1 MHz and the stability parameters (q = 0.45, a = 0), theacceptance  and the phase independent acceptance PI are calculated foran RFQ with realistic mechanical properties. A few choices of commerciallyavailable high precision stainless steel rod as the quadrupole electrodes wereconsidered. The calculated values are shown in Table 4.1. The configurationof r0 = re/1.13 is used for each size of rod.The phase independence acceptance for all three sizes of QMS shown inTable 4.1 meets the requirement of 3rms in Eq. (4.4). The larger acceptanceenables more than 99% of ions from the RF funnel to be captured andtransmitted.1081.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00Position (r0)0.40.20.00.20.4Velocity (r 0/TRF)0.000.250.500.751.001.251.501.752.00RF phase ()Figure 4.3: The phase independent acceptance PI obtained as the overlap(plotted in black) of acceptance ellipses at 36 RF phases. The ion motionin the phase space is plotted in the thin black line. At each time stepcorresponding to each RF phase, the ion is represented by a different coloras shown in the colorbar. An ellipse is obtained for each phase by least-square fitting to these points and shown in the same color.1090.0 0.2 0.4 0.6 0.8q0.000.020.040.06Acceptance ² PI (r2 0Ω/2)q=0.45²PI=0.063Figure 4.4: The phase independent acceptance PI at different q value.4.1.2 Vacuum requirementsThe LPT system requires different vacuum levels for the different com-ponents. The ion cooler requires ∼0.1 mbar of helium as a buffer gas toeffectively trap and cool the 136Ba+ ions as discussed in Section 3.5. The ionbuncher required a pressure lower than 7× 10−3 mbar to avoid heating thebunched ions as discussed in Section 3.6. The laser spectroscopy ion traprequires a gas pressure smaller than 1×10−3 mbar to avoid the scattering oflaser lights. The QMF requires the gas pressure to be around 1×10−5 mbarTable 4.1: The acceptance  and phase independent acceptance PI for afew sizes of RFQ operating at 1 MHz and (q = 0.45, a = 0). The RF voltageneeded is shown in the last row of the table.re (inch) 1/8 5/32 3/16re (mm) 3.18 3.97 4.76r0 (mm) 2.81 3.51 4.21 (mm mm/µs) 4.81 7.51 10.80PI (mm mm/µs) 1.56 2.44 3.51V (V) 49.4 77.1 111.0110or lower so that the ion-neutral collisions will be negligible when the ionstraverse through the QMF.4.1.3 Mechanical tolerance requirementsIn all the previous analyses and simulations of the RFQ, the quadrupoleelectrodes are assumed to be perfectly machined and positioned. However,in a realistic RFQ, the electrodes could have displacement from their de-signed positions due to mechanical tolerance.The manufacturing imperfection and displacement of the quadrupoleelectrodes cause changes to the electric potential in the RFQ. Noticeably,the electrodes will no longer have a four-fold rotational symmetry along thex-axis, hence any order of spatial harmonics described by Eq. (3.5) can exist.The QMS requires the higher-order spatial harmonics to be small in orderto maintain a high mass resolving power R and a large ion acceptance. Asdiscussed in Section 3.4.1, in order for a QMS to have mass resolving powerR = 50.5 without significantly reduced acceptance, the higher-order spatialharmonic terms A6, A10, A14 and A18 need to be 0.01 or smaller. Similarsimulations can be done to study the effect of additional higher-order spatialharmonics such as A3, A4, A5, A7 and so on.Such simulations were not systematically performed during this studydue to the limited time and computing power. Instead, preliminary sim-ulations were done by displacing one of the quadrupole electrodes in themodel [Lan18a]. The simulation reveals that in order for the QMS to havemass resolving power R ≥ 60, the displacement of the electrode needs tobe within 0.014r0. For reference, past experiments revealed that a relativemechanical tolerance of around 0.01 is necessary to achieve mass resolvingpower R ≥ 100 [AHL76].For QMS with round electrodes, stainless steel rods are commerciallyavailable (such as from McMaster-Carr) with a relative precision of the ra-dius better than 0.001. So the mechanical tolerance from the manufacturingimperfection of the quadrupole electrodes can be ignored in comparison tothe possible displacement of the electrodes in the assembly.The cooler, the laser spectroscopy ion trap and the ion buncher werefound to require a less strict mechanical tolerance at around 0.05r0.1114.2 Mechanical designThe mechanical design of the LPT system was done following the conceptualdesign as shown in Figure 4.1. The main outer mechanical structures consistof two ConFlat (CF) 6-way crosses, one CF 6-way cube and various flangesfor electrical and gas feedthrough as shown in Figure 4.5 (a). On the bottomof each 6-way cross, a turbo molecular pump (TMP) will be installed.The electrodes and holders are installed on the custom machined flangesas shown in Figure 4.5 (b) and (c). The mechanical and ion optics design ofthe QMF, ion cooler and the laser spectroscopy ion trap (and ion buncher)are described in the following sections.4.3 QMF designThe design of the QMF is shown in Figure 4.6. The QMF has three sets ofquadrupole electrodes. The two sets of short quadrupole electrodes at theentrance and exit are commonly called the Brubaker filters [Bru68] whichhave only RF voltage applied to mitigate the fringing field at the end of thelong quadrupole electrodes during mass filtering.The mechanical precision of the quadrupole electrodes is the most criti-cal consideration for the design of the QMF. There are high precision stain-less steel rods with a diameter tolerance of 0.0002” (5 µm) from McMaster-Carr which are suitable for the quadrupole electrodes of the QMF. Forthe QMF of this design, the rod diameter is chosen to be 5/16” (9.74 mm)using the #1255T15 stainless steel rod from McMaster-Carr. The ratio ofre/r0 = 1.13 was used as discussed in Section 3.4.2. The ion emittance and phase independent emittance PI at this dimension and fRF = 1 MHzas shown in Table 4.1 allow an ion transmission efficiency larger than 99%when the RFQ is operated as an ion guide with maximum ion acceptanceat q = 0.45.The positional precision of the rods is also critically important to achiev-ing an overall tight mechanical tolerance for the QMF. A monolithic elec-trode holder was specially designed so that the positioning of all the elec-trodes is fully defined by the holder and human error during the assemblycan be minimized. The holder can be machined in a computer numericalcontrolled (CNC) milling machine.112QMF chamberCLB chamberFigure 4.5: Rendered drawings of the mechanical design of the LPT system.See text for details.113Figure 4.6: Rendered drawings of the finalized design of QMF. (a) sideview. (b) cut view showing the electrodes of the QMF and structure of theholder. (c) cut view of the QMF showing the positioning and mounting ofthe quadrupole electrodes.The QMF is designed to have two aperture lenses at the entrance andone at the exit. The aperture lenses are used for both creating a well definedelectric potential boundary and for adjusting the ion energy.Additional details and dimensions of the QMF are shown as mechanicaldrawings in Appendix B (drawing number LPT2Q).4.4 Ion cooler designThe ion cooler downstream from the QMF needs to be filled with approx-imately 0.1 mbar of helium gas to effectively capture and cool the bariumions. Between the QMF and the cooler, the gas pressure is different by afactor of around 104. The different gas pressures required in the vacuumsystem can be achieved via differential pumping. However, a single-stagedifferential pumping is not practical to achieve the 104 pressure differencein this case. Therefore, a pre-cooler is needed as a second stage for thedifferential pumping and for transmitting the ions.1144.4.1 Pre-cooler as the differential pumping channelsAn RFQ ion guide is designed to function as the pre-cooler. The pre-cooler transmits the ions while allowing the excess helium gas from thecooler to escape from the space between the quadrupole electrodes. Thegas conductance CC2Q from the cooler to the QMF through the pre-coolerneeds to be as small as possible to enable effective differential pumping.Considering the vacuum pumps to be used for the vacuum system, CC2Q <5 L/s is required.A small gas conductance is commonly achieved through a small aperture.However, the size of the aperture has to be large enough for the flight pathof the ions to fall within. Depending on the RF phase, the ions comingfrom the upstream QMF can have positional spread as large as rQMF0 =3.51 mm. From Eq. (3.25) of [MDCG09], the conductance of an aperture atthe molecular flow region isCa = 3.7√TMAL/s, (4.6)where T is the gas temperature in units of Kelvin, M is the mass of thegas molecule in units of Dalton and A is the area of the aperture in cm2(conventionally centimeter is the unit used more commonly in the calculationof the conductance). For an aperture with radius 3.51 mm, the calculatedconductance Ca = 12.4 L/s is larger than the required 5 L/s.An alternative approach to achieving a small conductance is using along tube. From Eq. (3.20) of [MDCG09], the conductance of a tube withdiameter D and length L in cm in the molecular flow region isCt = 2.6× 10−4v¯D3LL/s, (4.7)where v¯ is the average molecular velocity of the gas in cm/s.The idea of using an RFQ ion guide with partially filled cross-sectionalarea between the electrodes as a long differential pumping channel orig-inates from the BECOLA (BEam COoler and LAser spectroscopy) ioncooler[Bar14]. The channel is formed along the center of the ion guide afterfilling the outer space between the quadrupole electrodes with an insulatingmaterial like MACOR (Machinable Glass Ceramic) or PEEK (Polyetherether ketone). The design of the original differential pumping channel hasbeen modified for this study to simplify the mechanical design as shown in115Figure 4.7. The concept of simplification stems from the author’s experiencewith designing a flute as a musical instrument made of two halves allowingthe inner geometry of the flute to be custom machined [LW16].Figure 4.7: Rendered drawings of the design of the pre-cooler. (a) side view.(b) cut view showing the quadrupole electrodes of the pre-cooler. (c) frontview of the precooler showing the holder as two halves assembled together;the cross-section of the differential pumping channel is highlighted as bluesketches.In this design, the electrode holder of the pre-cooler is made of two halves.The channel for differential pumping is formed between the four quadrupoleelectrodes and the straight inner walls of the electrode holders. The heliumgas from the cooler is mainly pumped out through 6 venting holes on the sideof the pre-cooler because the venting holes are much larger compared to thecross-section of the channel. The cross-section of the channel is highlightedin Figure 4.7 (c). According to its geometry, the cross-sectional area of thechannel is calculated asAC = (2re + 2r0))2 − 2pir2e − (2r2e) = 5.02r20 (4.8)for the fixed ratio of re = 1.13r0. In this case, the cross-sectional areaof the differential pumping channel is equivalent to a tube of diameterDC = 2.53r0.116The pre-cooler was designed to have re = 3/16” (4.76 mm) and r0 =4.21 mm using commercially available stainless steel rods. The larger stain-less steel rod is used for the quadrupole electrodes of the pre-cooler than theQMF for the following reasons:1. When the RF voltage is shared with the QMF, a larger trap size ofr0=4.21 mm leads to a smaller stability parameter q =0.55 and enablesthe RFQ to work at a more preferable region for ion cooling and lowerion temperature as discussed in Section 3.5.3.2. The ion acceptance is larger because of the larger r0 and the value ofq as discussed in Section 2.3 and Section 4.1.1.3. A larger differential pumping channel can be formed to allow ionsfrom the RFQ with larger initial positional spread to be accepted intothe pre-cooler and the cooler.The area of the differential pumping channel of the designed pre-cooleris equivalent to a tube with a diameter DC = 10.7 mm. The conductance ofthe channel for the side connecting to the QMF with length LQ = 80 mm isCQ = 3.1L/s. (4.9)The conductance of the channel for the side connecting to the cooler withlength LC = 158 mm isCC = 1.6L/s. (4.10)The other details and dimensions of the pre-cooler are shown as me-chanical drawings in Appendix B (drawing number LPT2CP).4.4.2 Ion coolerThe ion cooler is filled with around 0.1 mbar of helium buffer gas to cap-ture and cool the ions as studied in theory in Section 2.5.2 and simulationSection 3.5. The ion cooler needs to have a drag field (electric potential gra-dient) to guide the ions and accumulate them in the expected longitudinalposition with the lowest DC potential as illustrated in Figure 2.15.The pre-cooler is designed to operate without such a drag field becausethe ions will have sufficient kinetic energy to be only slowed down to still inthe ion cooler region.117DC drag fieldThe original approach of creating the drag field was to use segmented quadrupoleelectrodes with different DC voltages. Such a segmented RFQ design hasbeen used as ion coolers of ISOLTRAP [HDK+01], JYFLTRAP (EXO-TRAPS) [NHJ+01] and TITAN [BSB+12]. This approach requires tens ofsegments of the RFQ to create a smooth enough drag field.To reduce the complexity of electrical configurations, alternative ap-proaches to forming the drag field have been devised and used such as in thesecond generation of the ion cooler for ISOLTRAP [AFG+04], and the ioncooler in LEBIT [SBR+16]. A new way of creating a drag field using onlythe variation of the quadrupole electrode geometry is proposed in this study.Flat quadrupole electrodesQuadrupole electrodes with a flat inner surface and different widths werestudied via electrostatic simulation and the results are shown in Figure 4.8.The electrodes were placed inside a square- shaped metal tube with a biasedvoltage Vtube = 1 V while the electrodes were held at 0 V. The electrostaticpotential penetration at the central axis of the quadrupole electrodes wasfound to be dependent on the width of the electrodes.In this way, a set of flat quadrupole electrodes with gradual width changecan create a DC drag field. For the electrodes with r0 = 4.21 mm and widthsof 2 mm and 4 mm from one end to the other, every 100 V of biasing voltageon the metal tube creates a DC potential difference of 3.6 V.The electric potential from the RF voltage for ion confinement in thetransverse direction at the center of the flat quadrupole electrodes wassimulated and fitted as described in Section 3.1. The coefficient of thequadrupole potential A2 and the higher-order spatial harmonics are shownin Figure 4.9.The higher-order spatial harmonic terms A6 and above are larger thanthe case of an RFQ with hyperbolic or round electrodes. As a result, thecooler will have reduced acceptance as discussed in Section 3.2.2. However,it can be assumed that the ions have already been cooled to some extentafter traveling through the pre-cooler. The reduced acceptance of the coolercan be considered to be enough to trap all the incoming ions using the11810 0 10y (mm)10010z (mm)0.000.150.300.450.600.750.90Voltage (V)(a) Electrode width w = 2 mm10 0 10y (mm)10010z (mm)0.000.150.300.450.600.750.90Voltage (V)(b) Electrode width w = 4 mm1 2 3 4 5w (mm)0.020.040.060.080.100.12Potential at trap center (V)0.0160.052(c) Electric potential along the central axisFigure 4.8: Electric field penetration for quadrupole electrodes of differentwidth. (a) and (b) show the simulated electric potential distribution for theelectrode width 2 mm and 4 mm. (c) shows the potential at the center ofthe quadrupole electrodes as a function of the electrode width w.11910 0 10y (mm)10010z (mm)0.90.60.30.00.30.60.9Voltage (V)(a) Electrode width w = 2 mm10 0 10y (mm)10010z (mm)0.90.60.30.00.30.60.9Voltage (V)(b) Electrode width w = 4 mm0.2 0.4 0.6 0.8 1.0 1.20.81.0Quadrupole termA20.2 0.4 0.6 0.8 1.0 1.2w/r00.050.000.050.100.15Higher order terms0.48 0.96A6A10A1410×A18(c) Electric potential along the central axisFigure 4.9: Multipole expansion of potential inside the cooler. (a) and (b)show the simulated electric potential distribution for the electrode width2 mm and 4 mm. (c) shows the coefficient of the spatial harmonics as afunction of the electrode width w.120optimum settings for the helium buffer gas pressure.The cooler is designed with flat electrodes as described above. The widthof the flat electrodes tapers from 4 mm where the ions enter to 2 mm wherethe ions exit. The mechanical design of the cooler is shown in Figure 4.10.An aperture plate is located at the exit of the cooler. The 2 mm diameterhole is both relevant for the ions to exit the cooler and for the differentialpumping of the helium buffer gas to be achieved.Additional mechanical details and dimensions of the cooler are shown asmechanical drawings in Appendix B (drawing number LPT2CC).4.5 Laser spectroscopy ion trap (LSIT) designThe laser spectroscopy identification of the trapped barium ion requires ablue (493 nm) and a red (650 nm) laser to shine precisely on the trappedions. The fluorescent light emitted by the ions is collected by a CCD (chargecoupled device) or PMT (photomultiplier tube) detector through the gapbetween the quadrupole electrodes. The previous generation of ion trapdeveloped for the barium tagging of EXO [Gre10] suffered background noisefrom the laser light scattered by the quadrupole electrodes. The amountof fluorescent light that can be collected by the detector is limited by theopening angle θfluo = 25.5◦ between the quadrupole electrode as illustratedin Figure 4.11.In this study, the quadrupole electrodes for the laser spectroscopy iontrap were designed using a blade-shape following the ion trap at Innsbruckused for quantum computing [SKHG+03]. The blade-shaped electrodes havea much larger angle between the electrode θfluo = 75◦ for the fluorescentlight detection; the smaller surface area of the electrode around the laserbeam also reduces the scattering of the laser light. The electrode holderwill be manufactured from black insulating material to reduce the reflectionof scattered laser light. The mechanical design of the laser spectroscopy iontrap is shown in Figure 4.12.4.6 Ion buncher designDuring the study of the laser spectroscopy ion trap (LSIT), it was foundthat the same trapping region may be employed to work as an ion buncher.121Figure 4.10: Rendered drawings of the mechanical design of the cooler. (a)side view. (b) cut view showing the tapered flat quadrupole electrodes. (c)cross-sectional view of the cooler near the ion exit.122CCD/PMTCCD/PMTfluorescent lightPrevious design This designRed laser Blue laserfluorescent lightFigure 4.11: Comparison between the previous design (left) of the ion trapfor barium tagging and the new design (right) in this study.At the current design, the only difference between the LSIT and the ionbuncher described in Section 3.6 is the shape of the quadrupole electrodes.Simulations were done to the LSIT with the voltage configuration forion ejection. The axial potential along the center of the LSIT was found tobe the same as for the ion buncher as shown in Figure 3.33(b). Therefore,we decided to also use the LSIT as an ion buncher.A pulse drift tube (PDT) is added to the exit position of the ion buncher(LSIT) as shown in Figure 4.13. The voltage of the PDT can be quicklyswitched during the time the ion bunch flies inside the tube, thus allowingthe ions to be ejected at different ion energies.Additional mechanical details and dimensions of the laser spectroscopyion trap and the ion buncher are shown as mechanical drawings in Ap-123Figure 4.12: Rendered drawings of the mechanical design of the laserspectroscopy ion trap. (a) Side view facing the through hole for theentrance of lasers to the trap center. (b) Cut view showing the blade-shaped quadrupole electrodes and an aperture plate where the ions exit.(c) cross-sectional view from the ion entrance direction; the bottom showsa spherical mirror for reflecting the fluorescent light to the light detectorabove the ion trap. The electrode holder will be manufactured from blackinsulating material to reduce the scattering of laser light.124Figure 4.13: Rendered drawings of the laser spectroscopy ion trap as an ionbuncher with a pulse drift tube (indicated).pendix B (drawing number LPT2CL).4.7 Vacuum system of the LPTThe vacuum system of the LPT is designed to consists of two vacuum cham-bers with differential pumping through the pre-cooler. The QMF chamberhouses the quadrupole mass filter; while the CLB chamber houses the coolerand laser spectroscopy ion trap which also works as an ion buncher. Eachof the chambers is pumped by two 500 L/s turbo molecular pumps (TMP).The mechanical design of each component of the LPT had been donewith considerations for the vacuum system as well. The gas pressure and125flow through the LPT were calculated and the results are shown in Fig-ure 4.14.QMF chamber CLB chamberCoolerPc=0.1 mbarC₁=3.1 L/sC₂=1.6L/sC₃=1.0L/sHelium gas inPCLB=5.1×10⁻⁴ mbarPQMF=3.2×10⁻⁶ mbarSp1=500 L/s Sp2=500 L/sS1=490 L/sS2=312 L/s S3=195 L/sTMP TMPBacking pump Figure 4.14: Schematics of the vacuum system of the LPT. The gas pressureP and flow S are calculated as described in the text.The calculation of the gas pressure starts from the required pressurePC = 0.1 mbar inside the cooler. The conductance of the differential pump-ing channel is C1 = 3.1 L/s and C2 = 1.6 L/s as calculated in Eq. (4.9) andEq. (4.10). The conductance of the 2 mm diameter aperture lens at the exitof the cooler is C3 = 1.0 L/s calculated using Eq. (4.6).The total gas flow from the cooler to the CLB chamber isQC = (PC − PCLB)C2 + (PC − PCLB)C3. (4.11)The total gas flow exiting the CLB is the sameQC = PQMF SP1 + PCLB SP2, (4.12)where SP1 = SP2 = 500 L/s is the pump speed of the two TMPs. Thepressure between the QMF chamber and the CLB chamber is determinedthrough the differential pumping of the pre-cooler:PQMFSP1 = (PCLB − PQMF )C1. (4.13)126By solving the equations from Eq. (4.11) to Eq. (4.13), the pressures areobtained as PCLB = 5.1× 10−4 mbar and PQMF = 3.2× 10−6 mbar. Thesepressures meets the vacuum requirement of the LPT in Section 4.1.2.Note that the xenon gas coming from the RF funnel upstream of theLPT is ignored in the above calculations because the estimated xenon par-tial pressure in the QMF chamber is smaller than 1 × 10−6 mbar when theRF funnel is coupled to the LPT with a sextupole ion guide [BFV+15].4.8 Manufacturing of the LPTThe mechanical design of the LPT system and its components passed adesign review at TRIUMF to make sure the requirements in Section 4.1were satisfied. The mechanical drawings of the LPT as shown in Appendix Bwere sent for manufacturing at the Physics department machine shop ofthe Universite´ de Montre´al. The commercially available vacuum chambers,flanges and feedthrough were ordered.127Chapter 5Experiments and resultsThe machining of the parts for the LPT took a few months. In the mean-time, prototypes of the QMF and the RFQ cooler were built to validatethe feasibility of their mechanical design and to facilitate the experimentaldevelopment of the LPT’s electronics, control and DAQ (data acquisition)systems. Some measurements and studies were carried out along with theseexperimental developments. The remaining components of the LPT, includ-ing the laser spectroscopy ion trap (which also functions as a buncher) andthe pulse drift tube were not experimented in the duration of this study. Thefinal LPT will be set up later based on the prototypes and the experimentaldevelopments.5.1 Test stand setupA test stand shown in Figure 5.1 was set up in the ISAC-1 experiment areaof TRIUMF, Canada’s particle accelerator center.The main vacuum chamber is a ConFlat (CF) 6-way cross with 8 inchflanges. A Varian TV 551 turbomolecular pump (TMP) was installed onthe bottom of the 6-way cross. The TMP is backed by a scroll pump. Thevacuum level inside the chamber is measured by a MKS Convectron Piranigauge and an Agilent IMG-100 inverted magnetron gauge. When nothingwas put into the vacuum chamber, the pressure reached a stable value of2.4×10−7 mbar after pumping for a few hours. The pressure level is limitedby the Viton O-rings used for sealing CF flanges. In the final setup ofthe LPT system, copper gaskets will be used and a better vacuum level isexpected.The other apparatus was later added for the development of the LPTsystem. The key components are annotated in Figure 5.1. The backgroundbelonging to other experiments is set to black and white in the photo.128Figure 5.1: The test stand for the experimental development of the LPTsystem.1295.1.1 Ion sourceA test ion source (TIS) assembly on loan from the TITAN group as shownin Figure 5.2(a) was used throughout this work. The TIS was designedto use the HeatWave Model 101139 aluminosilicate ion source as shown inFigure 5.2(b) and 5.2(c). The aluminosilicate as the ion emitting material isfused into a porous tungsten disk (some aluminosilicate is left on the surfaceof the disk as shown in Figure 5.2(b) and (c)). The tungsten disk as theion emitter is isolated from the metallic ion source body by a heater cavity,which is filled with non-conductive alumina. A molybdenum wire coil passesthrough the alumina and heats up the ion source when an electrical currentis applied. When the ion source is heated to 950 °C or higher, ions areproduced through thermionic emission.(a) TIS assembly (b) Ion source #1 (c) IS#2 (SN358)Figure 5.2: (a) Photo of the test ion source (TIS) assembly installed ona 4.62 inch ConFlat (CF) flange. (b) and (c) are two HeatWave Model101139 ion sources, compatible to be installed in the TIS assembly. Theouter diameter of the ion sources is 1/4 inch (6.35 mm).The ions are emitted from the surface of the ion emitter more effectivelywhen there is a positive electric field gradient. The TIS can have the bodyof the ion source floated to a positive voltage, and an aperture plate is infront of the ion emitting surface as an anode. The anode can be biased to a130negative voltage. Both voltages can be used to control the number of ionscoming out of the 4 mm diameter aperture.Initially, an alkali ion source (#1, shown in Figure 5.2(b)) which comeswith the TIS with unknown properties was used. Later on, an almost purelyCs+ ion source (#2, shown in Figure 5.2(c)) was used.5.1.2 Ion detectorTwo types of ion detectors were used throughout this work as below.Faraday cup (FC)A Faraday cup (FC) is a metal container used to directly measure thecurrent of ions or any other charged particles by depositing them in theinner wall of the container.An FC has been designed and machined by the author for this work asshown in Figure 5.3. The FC is made from aluminum for ease of machining.The inner diameter of the cup is 1 inch (25.4 mm), except the opening forion entrance is tapered to 3/4 inch. The inward tapering is designed tomitigate the leakage of secondary electrons generated when the ions hit thecup’s inner surface. An electron suppressor lid is added at the ion entrancewith the same 3/4 inch opening.The bottom of the FC is made to be interchangeable so that adapterswith various apertures can be installed to allow a small percentage of ionsto pass through and be detected by a second ion detector as describedbelow. The purpose of these adapters is for ion detector calibration, and forabsolute ion transmission efficiency measurement by placing the FC withthe aperture in front of a QMS or an RFQ ion cooler.The FC is connected to an electrometer (Keithley 6514) via a BNCcable. Since the BNC cable doesn’t have electromagnetic shielding as goodas a tri-axial cable, the electrometer is placed close to the vacuum chamberto allow a 6 inch (152 mm) short cable to be used. Compared to a 2 meterBNC cable used previously, the short cable led to a reduced noise level bya factor of 10 measured by the electrometer. Ion current from the FC canbe measured to the 10−14 A level.131(a) FC with ion entrance on top (b) FC with different adapters for the bottomFigure 5.3: Photo of a custom made Faraday cup (FC) used in theexperiment. In (b), the three aperture adapters have diameters from leftto right: 1.6 mm, 1.1 mm and 0.6 mm.Channel electron multiplier (CEM)A channel electron multiplier (CEM) can amplify a signal by typically afactor of 107 with a few stages of amplification via secondary electron emis-sion along a channel-shaped dynode with high voltage applied across. Inthis way, each individual ion hitting the entrance of the CEM can generatea large enough pulse shaped signal to be detected. In this work, an Adaptas(previously DeTech) Model 2403 CEM is used. Figure 5.4(a) shows theCEM installed on a 4.5 inch CF flange. The CEM without holder (availableseparately as Model 2125) is shown in Figure 5.4(b))).The output signal from the CEM for single ions is typically in the milli-volt range. In order for the signal to be detected by a counter (SainSmartMHS-5200A 1) which requires input signal at TTL (transistor-transistorlogic) level, the signal is further amplified by an Ortec VT120 fast timingpreamplifier.1This a low-cost signal generator which also comes with the counter function tomeasure count rate up to 60 MHz.132(a) (b)Figure 5.4: (a) The CEM assembly installed on a 4.5 inch CF flange. (b)The CEM without holder; the larger side of the cone has a diameter of10 mm.5.1.3 Tests with ion source and detectorsA simple configuration of the test stand was set up as shown in Figure 5.5to check the working condition of both the ion source and the ion detectors.The ion source’s 4.62” flange was installed to one of the 6-way cross’sflange via an 8” to 4.62” CF reducer. The ion detectors were installed about400 mm away on the opposite side of the 6-way cross via an 8” to 4.5” CFreducer. The Faraday cup was used with the adapter of a Da=1.6 mmdiameter aperture as shown in Figure 5.5(b). If the ions coming into theFaraday cup were uniformly distributed, then the aperture allows a ratio ofηa =AaAFC=pi(Da/2)2pi(DFC/2)2= 0.7% (5.1)133(a)(b) Faraday cup (c) CEM ion detectorFigure 5.5: A configuration of the test stand for testing ion source anddetectors. (a) Configuration of the vacuum chambers. The location of theion source and ion detectors are annotated. (b) The Faraday cup witha 1.6 mm diameter aperture on the bottom. (c) The CEM ion detectorpositioned behind the Faraday cup. 134of ions to pass through, where DFC = 19.1 mm is the diameter of the FC’sopening.The Faraday cup was installed inside the tube of the 8” to 4.5” CFreducer and the aperture on the bottom was positioned about 10 mm infront of the CEM ion detector as shown in Figure 5.5.The ion source was heated to 1.35 A (4.1 W heating power) and given afew hours for it to stabilize. The ion source was floated to a voltage between0 to 80 V, while the anode voltage was fixed to 0 V. For each ion sourcefloating voltage, a 50 seconds measurement was done to record 100 samplesof ion current in the FC. The control of the power supply voltage and thedata recording are automated using Labview. The results are plotted inFigure 5.6.0 10 20 30 40 50 60 70 80Ion source floating voltage (V)0.00.10.20.30.40.5Current on Faraday cup (pA)05001000150020002500Ion count rate on CEM (Hz)Figure 5.6: Measurement of ion detectors reading at different ion sourcefloating voltages.The ion current increased along with the increase of the floating voltage,as a result of a stronger electric field gradient around the ion emittingsurface [PS78, KSP+10].135The CEM was operated by applying -2.3 kV 1 to the cone where incomingions are collected. The ion count rate was recorded and also plotted inFigure 5.6. The ion count rate appears to be proportional to the ion currentin FC, demonstrating that both ion detectors detect the number of ions in astable condition. For every 1 pA of uniformly distributed ion current in theFaraday cup, the rate of ions coming out of the aperture is calculated to beRcem =1 pAeηa = 4.37× 104Hz. (5.2)The measured ion count rate on the CEM corresponds to an ion detectionefficiency of around 12%.The ion source and detector tests demonstrated their reliability to beused in experiments with the prototypes in the following sections.5.2 Quadrupole mass filter prototypesA few prototypes of the QMF have been built and tested for differentpurposes.The first QMF prototype, QMF1, was built to test the practicality ofthe mechanical design described in Section 4.3 and the mechanical rigidityof the design. The details of the QMF1’s manufacturing, installation andpreliminary ion transmission tests are described in Appendix A.1.The tests found that the mechanical design is viable. Low amplitude(10 V) RF voltages were used to operate the QMF1 as an ion guide. Theion transmission properties of the QMF1 were as expected even with its0.2 mm positional precision of the quadrupole electrodes.The second QMF prototype, QMF2.1, was designed and built with theaim of achieving better mechanical precision using the available machineshop resources. The details of the QMF2.1’s design, manufacturing, installa-tion, ion transmission tests and mass measurements as a QMS are describedin Appendix A.2.The QMF2.1’s quadrupole electrodes were found to have 40 µm posi-tional precision, limiting its maximum mass resolving power to beRFWHM = m/∆mFWHM ≈ 51.1This voltage needed was relatively large, probably because the CEM has been heavilyused previously and was near the end of its lifetime.136The RF, electronics and automated control and DAQ systems weredeveloped along with the QMF prototypes.The third QMF prototype, QMF2.2, was built using the same design asQMF2.1. The parts were more carefully machined to aim at higher mechan-ical precision. The details of QMF2.2 are described in Section 5.2.1.5.2.1 QMF2.2The mechanical errors in QMF2.1 mainly come from the mismatch betweenthe electrodes and slightly larger holes in the holder. A new 5/16 inch endmill cutter was purchased and used on the same vertical milling machine tomake the holders for QMF2.2. Due to the lack of PEEK material at thetime of machining, acetal (a different type of engineering plastic) was used.The finished holders for QMF2.2 are shown in Figure 5.7(a). The quadrupoleelectrodes and aperture plates from QMF2.1 were repurposed for QMF2.2.To define a better electric potential at the aperture plate for adjustingion energy, each of the aperture plates was overlaid with stainless steel wovenmesh as shown in Figure 5.7(b). The mesh has 400 openings per inch andthe open area that can allow ions through isαmesh = 31%. (5.3)The assembled QMF2.2 is shown in Figure 5.7(c).Mechanical precision of QMF2.2The fitting between the quadrupole electrode rods and the holes of the hold-ers was extremely tight. Based on the machining precision of the holder’sfour holes, the positioning of the quadrupole electrodes was within 10 µm.This precision was double checked with measurement of the spacings be-tween the quadrupole electrodes with a digital caliper using the same pro-cedure as described for QMF2.1. The measured spacings were all within20 µm compared to the expected value except one measured spacing was0.15 mm too small, which was caused by an electrode’s positioning problemas shown in Figure 5.8.The 0.15 mm gap shown in the figure leads to an inward displacementof one end of that electrode. The electrode’s displacement would adversely137(a) Holders for QMF2.2 (b) Apertures with mesh(c) Assembled QMF2.2 with wires and electrical connectorsFigure 5.7: Photos of QMF 2.2. See text for details.affect the performance of the QMF2.2. The gap is hard to notice andwas only found during troubleshooting after the measurements had beencompleted. The measurements revealed a possible mechanical precisionproblem of the QMF2.2Installation of QMF2.2 in test standThe QMF2.2 was installed into the vacuum chamber of the test standas shown in Figure 5.9. The Faraday cup was added to measure the ioncurrent in front of the QMF2.2. The bottom of the Faraday cup has a 4 mmdiameter aperture to allow a small percent of ions to pass through as shownin Figure 5.9(d).After sealing the vacuum chamber and starting pumping, the pressurereached 1 × 10−6 mbar in 12 hours. The ultimate pressure after two daysof pumping was 4.2× 10−7 mbar, which is 27% higher than the pressure of138Figure 5.8: Photo of a mechanical precision problem in the QMF2.2assembly. There was an unexpected 0.15 mm gap between the holder andone of the electrodes.QMS2.1 and 75% higher than when there is nothing in the vacuum chamber.The Faraday cup was installed with an adapter also made of acetal asshown in Figure 5.9(d), so there was more acetal material in the vacuumchamber for QMS2.2 than the amount of PEEK for QMS2.1. The acetalis not used as commonly as PEEK in vacuum due to its higher outgassingrate. But it was found that the outgassing rate of acetal is no more than27% higher than PEEK in the tested setups of this study.Test of RF amplitude stability and balanceIn order for a QMF to operate at a mass resolving power R = m/∆m,the RF amplitude V needs to be stabilized and balanced to V/∆V = R orbetter.In order to systematically check the stability of the RF amplitude andthe balance between the two channels, the function generator (FG) wasconfigured to have fixed RF amplitude VFG = 3 V at both output channels.The RF signals were amplified by the RF amplifier with a fixed gain of139(a) Position of major components in vacuum chamber(b) Side view of quadrupole electrodes in vacuum chamber(c) View from ion entrance (d) View of Faraday cup from ion entranceFigure 5.9: Photos of QMF2.2 installed in the vacuum chamber of test stand.14030 times for both channels. The amplified signal was then mixed with DCvoltages and sent to the quadrupole electrodes; “T” shaped connectors wereused to allow for measurements of the voltages using an oscilloscope duringthe operation of the RFQ.The Tektronix TDS2024C oscilloscope has an 8-bit ADC and thereforelimits the measurement precision to be 0.4%. The measurement precision ofthe amplified RF amplitudes over the scanned frequency range was around1% due to the range setting of the voltage measurement; this limited themass resolving power to R ≈ 100.To obtain better mass resolving power with QMF2.2, a PicoScope 5442Dwith a 16-bit ADC was used. The PicoScope was configured to work at14-bit with 125 MS/s sampling rate, corresponding to an amplitude mea-surement precision better than 0.01%.Measurements were done from 0.1 MHz to 3 MHz. For each frequency,the amplitude of the input signal from the function generator (FG) and theamplified RF voltage was measured and recorded 10 times by the PicoScope.The frequency response of the FG and the gain of the RF amplifier areshown by the results in Figure 5.10.The stability of the amplified RF voltage is characterized by their stan-dard deviations σV , which are plotted as the error bars in the central plot ofFigure 5.10. The error bars are too small to be seen because σV /V <0.1%for almost all data points.The difference between the two amplified RF voltages is plotted in theright axis of the central plot in Figure 5.10. The differences at some frequen-cies are as large as 2%. These differences originate from the independentfrequency response of the RF amplifier’s dual channels, which are plottedas the gains in the bottom plot of Figure 5.10.The gain difference between the RF amplifier’s dual channels was com-pensated by RF amplitude balancing. This was done as the following:1. During the operation of QMF2.2, both channels of the function gen-erator were first set with an initial amplitude, e.g. VFG =3 V.2. After amplifying both channels and mixing them with the DC voltages,the voltage was measured with an oscilloscope.1412.902.953.00FG amplitude (V)20406080RF amplitude (V)CH1CH20.0 0.5 1.0 1.5 2.0 2.5 3.0RF frequency (MHz)1015202530RF amplifier gainCH1CH20.040.020.00(CH2-CH1)/CH1(CH2-CH1)/CH10.060.040.020.00(CH2-CH1)/CH1(CH2-CH1)/CH1Figure 5.10: Frequency response of the function generator (FG, top plot)and the gain of the RF amplifier (bottom plot). The middle plot shows theamplitude of the amplified RF voltages. The error bars for all lines are toosmall to be seen.1423. At each scanned RF frequency, the secondary channel of the functiongenerator was fixed with an RF amplitude of VFGCH2 =3 V while theamplitude of the main channel was adjusted.The RF frequency adjustment only needs to be done to the main channeland was synchronized to the secondary channel.An example of the balanced RF amplitude is shown in Figure 5.11. Thetop plot shows that channel 1 (CH1) of the function generator’s amplitudewas adjusted at each frequency to compensate for the gain difference of theRF amplifier’s two channels. The middle plot shows the two amplified RFvoltages are balanced to within 0.2%. In the middle plot of Figure 5.11,the amplitude of the function generator’s CH1 can be seen adjusted at thebeginning and each frequency later to compensate the RF amplifier’s gaindifference. The RF amplifier gain shown in the bottom plot in Figure 5.11is mostly the same as the one in Figure 5.10, this is expected because of thehardware properties of the RF amplifier.The gain of the RF amplifier’s frequency response was found to onlychange if the load (capacitance between the RFQ’s electrodes) has beenchanged. If a faster mass scan is needed, the measured RF amplifier’s gaincan be used as calibration data for RF balancing of each scan.Ion transmission tests with QMF2.2The ion source was heated to 1.45 A to produce sufficient ions for the testswith QMF2.2 as the two layers of mesh on the aperture plate at the entranceof QMF2.2 allow 9.6% of ion transmission. The ion source was floated at10 V; an optimum anode voltage of -120 V was set to produce 18.0 pA ofion current in the Faraday cup.Mass measurements were done at different DC voltage U correspondingto different mass resolving power. The RF amplitude and the range offrequency scans were set the same as for the measurement of QMF2.1 withthe RF amplifier. The RF amplitudes of the dual channels were balancedas described above, but using an 8-bit oscilloscope. The results are shownin Figure 5.12.The maximum ion count rate was Imax = 85 000 Hz when the DC voltageU = 0. Assuming the ion current entering the Faraday cup was uniformly1432.852.902.953.00FG amplitude (V)CH1CH220406080RF amplitude (V)CH1CH20.0 0.5 1.0 1.5 2.0 2.5 3.0RF frequency (MHz)1015202530RF amplifier gainCH1CH20.0100.0050.0000.0050.010(CH2-CH1)/CH1(CH2-CH1)/CH10.040.020.00(CH2-CH1)/CH1(CH2-CH1)/CH1Figure 5.11: Amplitudes of function generator (top) and amplified RFvoltages with RF balancing (middle). The amplitude of CH1 in the top plothas a similar shape as the gain difference (CH2-CH1)/CH1 in the bottomplot as a result of the RF balancing. See text for details.144Figure 5.12: Ion mass measurements with QMF2.2 at different U/V values.The cutoffs of the ion stability patterns near the lower right corners areexplained in the text.distributed, the expected ions entering the QMF2.2 isIexpected =18 pAeηaα2mesh = 119 000 Hz, (5.4)where ηa = (2mm19.05 mm)2 as a result of the second aperture plate’s 2 mmdiameter opening.After considering the CEM detector efficiency of ηCEM ≈ 80% at thisrange of ion energy, the absolute ion transmission efficiency of QMF2.2 atthe maximum ion count rate is preliminarily calculated asηI =ImaxηcemIexpected≈ 89%. (5.5)145The stable region of ions shown in Figure 5.12 has unexpected cutoffs atthe lower right corners as compared to Figure 2.13. The cause of the cutoffwas found to be an incorrect electrical connection to the entrance and exitBrubaker filters. Two 1 nF capacitors were used to remove the DC voltageand supply the RF only voltages to the Brubaker filters. In this situation,ion deposition on the quadrupole electrodes can create charge buildup andcreate a parasitic DC potential on these Brubaker filters. This parasitic DCpotential also existed with the QMF2.1 but its effect was not noticed. Theeffect of the parasitic DC potential was found to be more obvious when theions were at lower kinetic energy.The parasitic DC potential problem was fixed by adding a DC connec-tion to the Brubaker filters via two 1 MΩ resistors. This also allows thequadrupole electrodes to be floated to any given DC potential. Later theQMF2.2 was always floated to DC potential of Uf = 15.5 V. Hence theDC quadrupole potential U applied to the electrodes can be either positiveor negative relative to Uf without re-configuring the electrical connectionto the power supply. Results of such measurements are shown in Figure 5.13.Now the stability diagram corresponding to each ion appears symmetricalong the U = 0 axis and there is no longer cutoff around U = 0. Duringthe measurement, the ion source was floated at 16 V and the anode was setat -200 V. The QMF2.2 was floated at Uf = 15.5 V, hence the incomingions have energy around 1 eV.The smaller ion energy caused fewer ions to enter the QMF2.2. In orderto get more ion counts, the Faraday cup was also biased to -200 V. The ioncurrent couldn’t be measured in such configuration, hence the absolute iontransmission efficiency was not obtained.The data used for Figure 5.13 was also used to plot 1D mass spectra.Such a mass spectrum measured at U/V = 0.16 is shown in Figure 5.14.The lower two figures show the zoomed-in plots of the mass spectrum for thepotassium and rubidium isotope pairs. Least square fittings with empiricallychosen Gaussian distribution were done to the measured data and also shownin the plots.146Figure 5.13: Ion mass measurements with QMF2.2 at different U/V values.The cutoff in the ion stability diagram as described in Figure 5.12 has beenfixed as explained in the text.Maximum achievable mass resolving power of QMF2.2The maximum mass resolving power Rmax of QMF2.2 was studied by de-tailed mass measurements focused on the 39K and 41K isotopes after con-sidering the RF amplitude stabilization and balancing. The Rmax is limitedby the RF cycles of the ions flying through the RMF as discussed in Sec-tion 2.4.2. For a realistic QMF with a finite and fixed length, the RF cycleis determined by the ion energy.A few sets of measurements were done with the ion source floating atdifferent voltages; the anode voltage was set to -145 V.1470 25 50 75 100 125 150Mass (u)100101102103Ion count rate (Hz)23Na 39K 85Rb 133Cs30 40 50Mass (u)10 1100101102103104Ion count rate (Hz)39K41KFitData80 85 90 95Mass (u)10 1100101Ion count rate (Hz)85Rb87Rb FitDataFigure 5.14: Mass spectrometry measurement of QMS V2.2 at U/V = 0.16.See text for details.148At first, the ion source was floating at 50 V. The ions entering the QMFhave energy around 50 eV and energy spread around 1 eV. The ion velocityin the longitudinal direction is vx ≈ 15.5 mm/µs. The corresponding RFcycle is nRF ≈ 17 when the ions fly through the QMF. The measurementresults are shown in Figure 5.15 along with results from the simulation.The SIMION simulations were done for 39K ions flying through an elec-tric potential array model of the same geometry and voltages as the QMF2.2.The displacement of one of the electrodes as shown in Figure 5.8 was in-cluded modeled in the simulation. The simulation also modeled the 2 mmdiameter aperture at the entrance of the QMF2.2.For every DC to RF voltage U/V in the measured and simulated rangeof 0.154 to 0.170, the mass spectrum for 39K was fitted as demonstratedin Figure 5.14. The ∆mFWHM of the mass spectrum’s full-width at half-maximum from the fitting is used to calculate the mass resolving power R =m39K/∆mFWHM . The R was obtained in this way for both measurementand simulation and shown in the middle plot of Figure 5.15.The maximum mass resolving power of QMF2.2 for ions of 50 eV wasfound to be Rmax ≈ 35 from measurements. In this case, Rmax was mainlylimited by the number of the RF cycles. The obtained Rmax = 35 corre-sponds to h ≈ 8 in Rn = n2RFh as discussed in Section 2.4.2. The value of hhere for the mass scans with frequency sweep is smaller than the literaturevalue of h = 20 which was obtained for mass scans with voltage sweep. Thefrequency sweep mode of mass scan resulted in a larger Rmax by a factor of2.5.The bottom plot of Figure 5.15 shows the peak of the fitted mass spec-trum as a function of U/V for the measurements and simulations. Thevalue of the peaks is proportional to the ion transmission efficiency of theQMF at the different DC to RF voltages U/V and the corresponding massresolving power R.For the second set of measurements, the ion source was floated at 5 V.The ions entering the QMF have energy around 5 eV and energy spreadaround 1 eV. The ion velocity in the longitudinal direction is vx ≈ 5.0mm/µs.The corresponding RF cycle is nRF ≈ 53 when the ions fly through theQMF. The measurement results are shown in Figure 5.16 along with resultsfrom the simulation.14937.5 40.0Mass (u)0.15500.15750.16000.16250.16500.1675DC to RF ratio U/VMeasurement0.65 0.70q0.1450.1500.1550.1600.1650.170DC to RF ratio U/VSimulation0.154 0.156 0.158 0.160 0.162 0.164 0.166 0.168 0.170101Mass resol. power RRmax ≈ 35MeasurementSimulation0.154 0.156 0.158 0.160 0.162 0.164 0.166 0.168 0.170DC to RF ratio U/V100101102103104Peak count rate (Hz)Measurement101102103104Count rate (Hz)10-510-410-3Acceptance ² yz(pi2r4 0T2 RF)10-610-510-410-3Peak acceptances ² yz(pi2r4 0T2 RF)SimulationFigure 5.15: Measured and simulated ion transmission at different RF toDC voltages U/V . Ion energy is around 50 eV. See text for details.150The calculated maximum mass resolving power using h ≈ 8 as obtainedabove is Rmax = 346. However, the experimentally obtained Rmax frommeasurement is Rmax ≈ 80. In this case, the Rmax is most likely limitedby the mechanical error of one of the quadrupole electrodes as shown inFigure 5.8.A final set of measurements was done with the ion source floated at 1 V.The ions entering the QMF have energy around 1 eV and energy spreadaround 1 eV. The ion velocity in the longitudinal direction is vx ≈ 2.2mm/µs.The corresponding RF cycle is nRF ≈ 120 when the ions fly through theQMF. The measurement results are shown in Figure 5.17 along with resultsfrom the simulation.The maximum mass resolving power was found to be Rmax ≈ 140 forthe ions with 1 eV energy. In this case, the Rmax is still limited by themechanical error of the QMF2.2 as shown in Figure 5.8. However, Rmax ≈140 already meets the requirement of R > 80 for the purpose of eliminatingsufficient contaminant ions from the RF funnel as described in Section 4.1.15137.5 40.0Mass (u)0.15500.15750.16000.16250.16500.1675DC to RF ratio U/VMeasurement0.65 0.70q0.1450.1500.1550.1600.1650.170DC to RF ratio U/VSimulation0.154 0.156 0.158 0.160 0.162 0.164 0.166 0.168 0.170100101102Mass resol. power RRmax ≈ 80MeasurementSimulation0.154 0.156 0.158 0.160 0.162 0.164 0.166 0.168DC to RF ratio U/V100101102103104Peak count rate (Hz)Measurement101102103Count rate (Hz)10-510-410-3Acceptance ² yz(pi2r4 0T2 RF)10-610-510-410-3Peak acceptances ² yz(pi2r4 0T2 RF)SimulationFigure 5.16: Measured and simulated ion transmission at different RF toDC voltages U/V . Ion energy is around 5 eV. See text for details.15238 40Mass (u)0.15500.15750.16000.16250.16500.1675DC to RF ratio U/VMeasurement0.65 0.70q0.1450.1500.1550.1600.1650.170DC to RF ratio U/VSimulation0.154 0.156 0.158 0.160 0.162 0.164 0.166 0.168 0.170100101102Mass resol. power RRmax ≈ 140MeasurementSimulation0.154 0.156 0.158 0.160 0.162 0.164 0.166 0.168DC to RF ratio U/V10-1100101102103Peak count rate (Hz)Measurement101102103Count rate (Hz)10-510-410-3Acceptance ² yz(pi2r4 0T2 RF)10-710-610-510-410-3Peak acceptances ² yz(pi2r4 0T2 RF)SimulationFigure 5.17: Measured and simulated ion transmission at different RF toDC voltages U/V . Ion energy is around 1 eV. See text for details.1535.3 RFQ cooler prototypeA prototype of the RFQ cooler has been built as shown in Figure 5.18(hereafter referred to as the cooler). The metal tube and the electrodeswere machined by the author in the TRIUMF ISAC-II machine shop. Theelectrode holders have a more complicated structure, so these holders were3D printed from nylon powder using SLS (selective laser sintering) tech-nique. The 3D printing was done by a commercial company Shapeways 1.The components for the cooler are shown in Figure 5.18(a). The assembledcooler is shown in Figure 5.18(b), the inset shows the positioning of thequadrupole electrodes.The machined quadrupole electrodes have a mechanical precision ofaround 0.2 mm. The 3D printed metal electrodes were also tested. Thetop quadrupole electrode shown in Figure 5.18(b) was 3D printed usingsteel fused with bronze material. However, the post-processing required forthis material led to a 2 mm shrinkage of the electrode in the longitudinaldirection and eventually it was not used.The lowest quadrupole electrodes shown in Figure 5.18(a) were 3D printedfrom aluminum powder using the SLM (selective laser melting) technique,also by Shapeways. The 3D printed aluminum electrode has a rough surfaceand was manually polished with sandpapers. Afterward, the mechanicalprecision of the electrode is measured and estimated to be around 0.1 mm.The 3D printed electrode was used with the 3 conventionally machinedaluminum electrodes in the cooler.The tapered quadrupole electrodes with width converging from 4 mm atthe ion entrance side to 2 mm at the ion exit side are shown in Figure 5.19(a)and (b) before the aperture plates were installed. The aperture plates eachhave a 2 mm diameter hole in the center for both the ion entrance/exitand the differential pumping of helium gas. The cooler with these apertureplates installed is shown in Figure 5.19(c) and (d). The four extended metalstructures were improvised to position that side of the cooler inside the tubeof the 8” to 4.5” reducer.1https://www.shapeways.com/154(a) Parts for RFQ cooler prototype(b) Assembled RFQ cooler prototypeFigure 5.18: Photos of the RFQ ion cooler prototype. See text for details.155(a) View from ion entrance (b) View from ion exit(c) Assembled (d) Ion exit with aperture plateFigure 5.19: Photos of the RFQ ion cooler during (top figures) and after(bottom figures) assembly.1565.3.1 Installation of the cooler prototype in test standThe cooler was installed in the vacuum chamber of the test stand as shownin Figure 5.20. The paths for helium gas to enter the cooler and for thepressure to be measured by a vacuum gauge are shown in Figure 5.20(b).The position of the cooler relative to the ion source and the ion detector isshown in Figure 5.20(c).After closing the vacuum chamber, the vacuum pressure reached 1 ×10−6 mbar after pumping for 8 hours. The ultimate pressure after pumpingfor two days was 4.3 × 10−7 mbar, which is 79% higher than when thevacuum chamber was empty. It demonstrated the outgassing rate of thenylon used for 3D printing for these holders was not significantly higherthan acetal and PEEK.5.3.2 Ion transmission tests with the coolerThe cooler was first tested as a poorly built QMF with non-ideal electrodegeometry and bad mechanical precision (0.2 mm) in order to fully validateits electrical connections and its ion transmission properties. These testswould also reveal the cooler’s performance as an ion guide in some of thetests when only RF voltage was applied.The tests were done both with and without helium buffer gas in thecooler to study the vacuum requirement of a QMS and the effect of buffergas in an RFQ ion guide.Cooler tests without helium gasThe cooler was first tested without helium buffer gas to check the perfor-mance of the flat and tapered quadrupole electrodes.The ion source was heated to 1.35 A and floated at 50 V, the RFQ coolerwas floated at 45.5 V. The energy of incoming ions was around 5 eV.Ion transmission measurements were done with the cooler at differentDC voltage U . The RF voltage of the function generator and the RFamplifier was set to be the same as during the measurements with the QMFprototypes. The RF frequency was scanned from 4.5 MHz to 0.6 MHz foreach of the DC voltage U . The results are shown in Figure 5.21. The157(a) (b)(c) Position of major components in vacuum chamberFigure 5.20: Installation of the cooler to the test stand’s vacuum chamber.See text for details.158patterns of stability diagram can still be recognized for the 23Na, 39,41K and85Rb ions.Figure 5.21: Ion transmission measurement with the cooler at different DCvoltage U .Later a different ion source (#2) was used as shown in Figure 5.2(c).This ion source produced > 99.9% of 133Cs and < 0.1% of 39K. Similarmeasurements were done as above and the results are shown in Figure 5.22.As expected, this time there is only a single pattern of the stability diagramfor the 133Cs ion. This ion source was used for all later measurementsdiscussed below.Both measurements shown in Figure 5.21 and Figure 5.22(a) have maxi-mum mass resolving power R ≈ 20. In this case, the mass resolving power ofthe cooler was limited by the higher-order spatial harmonics in the electricpotential, also by the varying magnitude of the quadrupole term A2 as aresult of the changing width of the electrodes.Figure 5.22(b) shows the ion transmission of the cooler working as anRFQ ion guide when the DC voltage U = 0. The horizontal axis isq = qtmpkmCs, (5.6)where qt ≈ 0.706 , mpk is the value on the horizontal axis of Figure 5.22(a)159(a) Ion count rate at different U0.0 0.2 0.4 0.6 0.8q050100150200Ion count rate (Hz)(b) Ion count rate at U = 0Figure 5.22: Ion transmission measurements with the cooler using ions froman almost pure cesium ion source (#2).160and mCs is the mass of133Cs. The 133Cs ions were transmitted between0 < q < 0.908 as expected in an ion guide. However, there are obviousfluctuations in the ion count rate as a function of q. The fluctuations areprobably caused by the non-ideal electric potential inside the cooler. Thefluctuations were found to be smaller when the ions flew through the coolerat a smaller velocity.Ion transmission tests with helium buffer gasThe cooler was filled with some helium gas after slightly opening a needlevalve which is connected to a helium gas bottle. The pressure in the vacuumchamber (outside of the cooler) also increased due to helium gas flowingout of the 2 mm diameter apertures on the entrance and exit of the cooler.After a few hours, the pressure in the vacuum chamber stabilized at 4.0 ×10−6 mbar, while the pressure inside the cooler was 5.5 × 10−3 mbar dueto the differential pumping as measured by the vacuum gauge shown inFigure 5.20(b).An ion transmission test was done with the same configuration (ionsource voltages and the RF&DC voltages) and the results are shown inFigure 5.23.The smoother variation of the ion count rate at different RF and DCvoltage indicated the ions traveled at a lower velocity through the cooler.The effect of ion velocity on the count rate can be understood by comparingFigure A.15(a) and Figure 5.12. The measurements were done at the sameconfiguration of the ion source voltages, so the only possibility is that theions were cooled and slowed down by the helium buffer gas.To further validate the ion cooling effect of the helium buffer gas, a setof measurements was done at higher helium gas pressure of 1.9×10−2 mbar;the pressure in the vacuum chamber was 1.3 × 10−5 mbar. The results areshown in Figure 5.24.The even smoother variation of the ion count rate in Figure 5.24 corre-sponds to an even smaller ion velocity inside the cooler and validates thehelium gas’s cooling effect. The ion transmission as indicated by the countrate also increased because of ion cooling. The maximum mass resolvingpower of the cooler is R ≈ 20 at both gas pressures.161Figure 5.23: Ion transmission measurements of the cooler with helium gaspressure of 5.5× 10−3 mbar.The turbo molecular pump and the CEM ion detector are both ideal towork at low gas pressure to ensure a long operation lifetime. So no testswere done at helium gas pressure higher than 1.3×10−5 mbar in the vacuumchamber (1.9× 10−2 mbar in the cooler).5.3.3 Experiments with ion trappingThe RFQ cooler was tested as a standalone ion trap to validate ion coolingand trapping with the novel tapered quadrupole electrodes. Note that inthe final LPT, the ions are intended to be trapped in the laser spectroscopyion trap following the ion cooler.The RF voltage applied to the ion cooler’s quadrupole electrodes wasset to be V = 71.0 V and the RF frequency was fRF=0.8 MHz. The stabilityparameter for the 133CS+ ions in the cooler is q = 0.46. The metal tube wasbiased to a negative voltage Utube to form the drag field. The quadrupoleelectrodes in the RFQ cooler were floated at Uf =45.5 V. The cooler wasenclosed by an aperture plate with a 2 mm diameter hole where the ionsexit. When the aperture plate voltage Uap was set to be higher than Uf , atrapping potential can be formed as shown in solid lines in Figure 5.25. The162(a) Ion count rate at different DC voltage U0.0 0.2 0.4 0.6 0.8q0100200300400500600700Ion count rate (Hz)(b) Ion count rate at U = 0Figure 5.24: Ion transmission measurements of the cooler with helium gaspressure of 1.9× 10−2 mbar.163profile of the trapping potential and the position of the minimum potentialis dependent on the aperture voltage.When the aperture voltage Uap was switched to be lower than Uf , thetrapped ions would be ejected out of the RFQ cooler under the electricpotential gradient as shown in dashed lines in Figure 5.25. The ejectedions were detected by a CEM ion detector a few millimeters away from theaperture plate.The ion signal from the CEM detector was amplified by an Ortec VT120fast timing amplifier. The amplified ion signal was then sent to a LeCroyLRS 621AL discriminator to identify the ion signal with a threshold of-0.25 V. For each ion ejection, the amplified ion signal and the pulses outputfrom the discriminator were recorded by a Siglent SDS1104X-E oscilloscope.The switch voltage was also recorded by the oscilloscope. The data acqui-sition of the signals was done at 500 MS/s, corresponding to a 2 ns timeresolution.As an example, Figure 5.26 shows the ejection of 11 ions after the aper-ture voltage was switched to 0V. Before the ejection, the ions were trappedinside the cooler with the aperture set at 240 V. The repetition rate of theion ejection was 0.25 Hz to allow for 4 seconds of accumulation of cooledions (due to the low ion rate).The ejected ions have an average time of flight (ToF) tToF = 29 µs. Thespread of their ToF is σtToF = 11 µs. There was electronic noise in the ionsignal when the aperture voltage was switched. This noise was excludedand not counted as an ion signal in the data processing.Ion injection energyThe cooling and trapping of ions injected into the cooler was then testedand validated by varying the injection energy Einj of the incoming ions tostudy the buffer gas’s effect on slowing down the ions in the longitudinaldirection. For the helium gas pressure of 5.5× 10−3 mbar inside the cooler,the ion energy was varied by changing the ion source floating voltage Us inthe range from 30 V to 55 V. The results are shown in Figure 5.27.In the ideal case, in order to achieve the maximum ion trapping effi-ciency, the ion source floating voltage Us should be slightly above the float-ing voltage of the cooler Uf . If Us is too small, the ions won’t have enough1640 25 50 75 100 125 150 175 200Longitudinal position (mm)020406080100Axial DC Potential (V)Vap=200VVap=100VVap=60VVap=50VVap=0VVap= -100VVap= -300Vaperture position20 40 60 80 100 120 140 1603839404142434445Figure 5.25: Simulated DC potential along the RFQ cooler’s central axis forion trapping and ejection when the metal tube voltage was set at Utube =−150 V. The inset is a zoomed-in plot.165Figure 5.26: An example showing ion signal detected by the CEM iondetector from the ejected ions as a function of time of flight (ToF). Theions were trapped with the aperture voltage of 240 V. After the aperturevoltage was switched to 0 V, the ions started to fly out and hit the CEMdetector later. The ToF of the ions is obtained from the ion signals.energy to overcome the electric potential at the entrance of the cooler andwill be repelled; if Us is too large, the ions will have too much energy suchthat the buffer gas cannot sufficiently cool the ions and the ions will have around trip inside the cooler and exit through the entrance of the cooler.However, as shown in Figure 5.27 the setting of the optimum ion sourcefloating voltage to achieve the maximum number of trapped ion was exper-imentally found to be Us ≈41 V, which is a few volts lower than the floatingvoltage of the cooler Uf = 45.5 V. This is most likely due to the effect ofthe RF voltage which is oscillating. As a result, half of the cross-sectionalarea of the cooler’s entrance has an actual electric potential lower than Ufand allows the ions with lower energy to enter the cooler.The ion injection energy has a full-width of ∆Einj ≈ 6 eV at the half-maximum of the ion count per bunch.Later, the ion injection test was done at the higher helium gas pressureof 1.9× 10−2 mbar in the cooler. The results are shown in Figure 5.28.16630 35 40 45 50 55Ion energy (eV)01020304050Ion count per bunchFigure 5.27: Ion count of cooled and ejected ions as a function of injectionenergy at 5.2× 10−3 mbar helium buffer gas pressure.With the higher helium gas pressure inside the cooler, the ion coolingand trapping was more efficient. The optimum ion injection voltage forthe maximum ion trapping efficiency is Us ≈ 50 V. The full-width of theinjection energy at the half-maximum ion count per bunch is ∆Einj ≈ 20 eV.At these two tested gas pressures, the range of the ion injection energy∆Einj is proportional to the gas pressure. Therefore, at the designed op-erating pressure of 0.1 mbar, the cooler can achieve the required cooling ofions with injection energy in the range of ∆Einj ≈ 100.Effects of aperture voltage on ion ToFThe voltage of the aperture plate determines the DC potential near theexit of the cooler for the trapping and ejection of the ions, as shown inFigure 5.25. The aperture plate voltage therefore affects the kinetic energyof the ejected ions, which in turn determine the time of flight (ToF) of theions as below:• A higher trapping voltage on the aperture plate would push the posi-tion of the potential minimum further away from the aperture plate.16730 40 50 60 70 80 90Ion energy (eV)01020304050607080Ion count per bunchFigure 5.28: Cooling and trapping of ions with different injection energy at1.9× 10−2 mbar helium buffer gas pressure.As a result, the ions will have a longer path from the trapped positionto the CEM detector and correspondingly larger ToF.• A higher ejection voltage 1 on the aperture plate generates a less steeppotential gradient for the ions to fly out, hence the ions will experiencea smaller kinetic energy gain and have larger ToF to reach the CEMion detector.• On the contrary, a smaller trapping or ejection voltage will cause theion ejection to have a smaller ToF.Measurements were done with ion ejection at different trapping voltageUT and ejection voltage UE of the aperture plate. When the trappingvoltage is 110 V or lower, the average of the ion time of flight tToF is a fewmicroseconds, and the ToF spread σtToF is smaller than 1 microsecond asshown in the results in Figure 5.29.1The aperture voltage for ion ejection here is usually a negative value. So a higherejection voltage corresponds to a smaller absolute value of the voltage.16823456ToF mean t ToF (s)UT=110UT=100UT=90UT=80UT=70UT=60UT=500 100 200 300 400Ejection voltage |UE| (V)0.00.20.40.60.81.0ToF spread t ToF (s)Figure 5.29: Measured ion time of flight (ToF) mean tToF and ToF spreadσtToF as a function of the ejection voltage UE at trapping voltage UT from50 V to 110 V.169When the trapping voltage is 120 V or larger, both tToF (top plot)and σtToF (bottom plot) are significantly larger as shown by the results inFigure 5.30.20406080ToF mean t ToF (s)UT=180UT=170UT=160UT=150UT=130UT=1200 100 200 300 400Ejection voltage |UE| (V)10203040ToF spread t ToF (s)Figure 5.30: Measured ion time of flight (ToF) mean tToF (top plot) andToF spread σtToF (bottom plot) as a function of the ejection voltage UE atlarger trapping voltage UT from 120 V to 180 V.In order to understand the measured ToF of the ion ejection, SIMIONsimulations were performed for the cooler for comparison. The simulationsstarted with 1000 ions near the central axis of the cooler and the longitudinalposition of the minimum DC potential. The ions were cooled for 2000170microseconds to reach thermal equilibrium with the buffer gas before theion ejection was simulated. The ion ejection was simulated at these trappingvoltages UT and ejection voltages UE used in measurements. The results ofthe simulation are shown in Figure 5.31.102030ToF mean t ToF (s)0 100 200 300 400Ejection voltage |UE| (V)46810ToF spread t ToF (s)6080100120140160180Trapping voltage UT (V)Figure 5.31: Simulation of cooler’s ion ejection ToF as a function of theejection voltage UE at different trapping voltage UT . The trapping voltageUT is from 50 V to 180 V with a 10 V increment as represented by the sizeof the marker and the colorbar.For a fixed trapping voltage UT , the decrease of tToF and σtToF as afunction of the ejection voltage |UE | follow a similar trend in the simulationand experimental measurement. However, when UT ≥ 120 V, the measuredtToF and σtToF are larger than the simulation; when UT ≤ 110 V, the mea-sured tToF and σtToF are smaller than the simulation.After more careful troubleshooting and ion ejection measurements with171varied parameters, the discrepancy between the measured and simulatedion ToF was found to be caused by charge buildup on the surface of theelectrode holder as highlighted in Figure 5.32. The charge buildup wascaused by ion deposition on the surface of the insulating material.(a) Cooler with the tube set transparent (b) Electrode holder near the exitFigure 5.32: Charge buildup problem near the exit of the cooler. Thecharge buildup occurred after lots of ions deposit on the inner surface ofthe electrode holder (highlighted in blue).The deposited ions created charge buildup and distorted the potentialaround where the ions were trapped. The charge buildup has a potentialUCB ≈120 V (which is approximately Uf + V ). So that when the trappingpotential UT < UCB, the position of the trapped ions were pushed by thecharge buildup to be close to the exit of the cooler and cause the ion ejectionto have a smaller ToF. When UT > UCB, the position of the trapped ionswere pushed further away from the cooler exit and caused the ion ejectionto have a large ToF.When the trapping potential is about the same as the potential of thecharge buildup at 120 V, the ion bunch were observed to be split into twoas shown in Figure 5.33.The ion bunch 1 has an average ToF tToF = 6.7 µs, indicating these ions172Figure 5.33: Ion cloud split by the charge buildup, resulting in two ionbunches. The ejection voltage used in this measurement was UE = −20 V.were trapped close to the cooler exit, similar to the earlier measurementsshown in Figure 5.29. The ion bunch 2 has a much larger average and spreadof ToF, indicating these ions were trapped further away from the cooler exit,similar to the measurements shown in Figure 5.30.The trapping region that corresponds to the ion bunch 1 has very limitedvolume as the ion bunch 1 was found to have a maximum ion count ofaround 40. The ion bunch 2 can have up to 1000 ion count at a largertrapping voltage. The effect of the charge buildup was also less obvious atthe larger trapping voltages. So, the ion bunch 2 corresponds to a trappingpotential similar to the ones shown in Figure 5.25.The issue of the charge buildup can be resolved by removing as much aspossible of the electrode holder’s insulating material near the position of theion trapping as highlighted in Figure 5.32(b). The design of the electrodeholder was revised in such a way and shown in the mechanical drawings inAppendix B (drawing number LPT2CC04).If needed in the future, more detailed experiments with the ion ejectionwill be done with a retarding field analyzer and a emittance meter.173Ion storage lifetimeThe ion storage lifetime is important for the laser spectroscopy identificationof the barium ions in the laser spectroscopy ion trap. The ion identificationrequired the ions to be trapped for at least a few seconds. As a first stepof studying the storage lifetime of the trapped ions, experiments were donewith ions trapped in the cooler.The ion source was floated to 45 V for 10 seconds for the ions to enterthe cooler, then the ion source floating voltage was set to 0 V to preventfurther ions from entering the cooler. After a storage time ts, the ions wereejected from the cooler and detected by a CEM. For each ts from 1 secondto 285 seconds in log scale increment, 10 measurements were repeated andthe ion count in every bunch of ejected ions was recorded. The result isshown in Figure 5.34(a).The count of the ejected ions as a function of the storage time ts wasfitted to an exponential functionN = N0 exp(− tsτ). (5.7)The fitted initial ion count is N0 = 58.3 ± 0.8 and the storage lifetimeτ = 16.9± 0.6 s.Another set of measurements was done by fixing the ion storage timets = 1 s while increasing the ion accumulation time from 0.1 s to 140 s. Inthis case, the ion count in the ejected ion bunch will be affected by both therate of ions entering the cooler Rion and the storage lifetime of the ions inthe cooler τ . The measured results are shown in Figure 5.34(b) and fittedto a modified exponential functionN = Rionτ(1− exp(− taτ)). (5.8)The fitted ion count rate is Rion = 8.1± 0.1/s, and the ion storage lifetimeis τ = 13.5± 0.4 s.The shorter ion storage lifetime measured during the accumulation ofthe ions indicates saturation of ions in the cooler. The ion storage lifetimewas also found to be in the range of 7 < τ < 40 s depending on the voltageconfiguration of the cooler and the number of stored ions in the cooler.174101 102Ion storage time ts (s)10 510 410 310 210 1100101102Ion count per bunchFit: N0=58.3, =16.9measurement(a) Ion storage in the cooler10 1 100 101 102Ion accumulation time ta (s)10 1100101102Ion count per bunchFit: R=8.1, =13.5measurement(b) Ion accumulation in the coolerFigure 5.34: Measurement of ion numbers in the cooler as a function ofstorage time.175Simulations of ions trapped in the cooler with extended trapping timereveal the ions have stable ion temperature and confinement, no ion losswas caused by the RF heating. The loss of trapped ions in experimentsis likely caused by collisions with molecules or ions other than helium.Therefore, the storage lifetime of the ions depends on the vacuum in thesystem and the impurities in the helium gas. The helium gas used for theexperiment in this study has a purity of Ppurity=99.999%. The helium gaspurity can be improved by a gas purifier such as a SAES getter to makePimpurity = 1−Ppurity reach the ppb (parts per billion) level. The increasedhelium gas purity can improve the storage time of ions in the cooler and thelaser spectroscopy ion trap. For reference, barium ions have been observedto have a storage lifetime above 500 seconds in the previous generation oflinear Paul trap for barium tagging [GWD+07, Gre10] which had the heliumgas purified by a SAES purifier.The experiments with the ion cooler prototype demonstrated the successof its design with the novel tapered quadrupole electrodes in ion cooling,trapping and ejection.5.4 Ion temperature in the LPTThe ion time-of-flight (ToF) spread σtToF is related to the trapped ions’positional spread σx and velocity spread σvx as described in Section 3.6.The temperature of the trapped ions determines σx and σvx .Measurement of ion ejections without the effect of the charge buildupmay be obtained from the final setup of the LPT system with the revisedelectrode holder at the exit of the cooler. In an ideal case, the ion tempera-ture Tx in the longitudinal direction can be derived from the dependence ofthe ion ToF spread on the ion ejection electric field. Such dependence canbe seen in the simulations of the ion buncher in Section 3.6.In the following, the temperature of trapped 136Ba+ ions in the coolerwas obtained from simulations as discussed in Section 3.5.3 (the chargebuildup problem was not considered in the simulations). The simulationwas done at the designed cooler operating parameter of RF frequency fRF =1 MHz with 0.1 mbar of helium buffer gas. The result is shown in Figure 5.35.The ion temperatures are similar to the simulation results of the ion buncheras shown in Figure 3.31.The temperature Ty and Tz in the transverse directions of the ions cooled17660 80 100 120 140 160 180 200RF voltage V (V)300400500600700Ion temperature (K) TbuffergasTxTyTz0.2 0.3 0.4 0.5 0.6 0.7 0.8Stability parameter qFigure 5.35: Temperature of 136Ba+ ions trapped in the cooler with differentRF voltage obtained from simulation.and trapped inside the cooler was found to be below 400 K when the RFvoltage V is below 150 V. In the longitudinal direction, the ion temperatureTx is always close to the buffer gas temperature. These ion temperaturesmeet the requirement of the cooler in the LPT system.5.5 The final LPT systemThe machining of the LPT parts was completed as of November 2019. Theseparts include everything needed for the final LPT system except the coolerand the laser spectroscopy ion trap (LSIT). From the experience of the coolerprototype, the 3D printed parts were found to be good enough for the finalLPT in terms of vacuum compatibility and mechanical precision. Thereforewe decided to 3D print the parts for the cooler and the LSIT for the finalLPT system as well.The revised design of the electrode holder at the exit of the cooler wasused for the final LPT system to mitigate the charge buildup problem asdiscussed in Section 5.3.3.The final LPT system has been partially assembled at McGill University.The electronics, control and DAQ system described in this chapter will beused and further improved.177The LPT system will be tested and commissioned as a standalone setupfirst. Once completed, it will be installed in between the RF el and the MR-TOF mass spectrometer to systematically study the ions extracted from theRF funnel. The lasers and a fluorescent light detector will be installed at alater stage for the barium ion identification. If successful, the LPT can beused as a part of the setup for the barium tagging purpose of nEXO.178Chapter 6Conclusion and future workA linear Paul trap (LPT) has been developed for barium tagging in gaseousxenon. The LPT can be used for the nEXO if the barium ions can besuccessfully extracted from the liquid xenon through a cold probe or acapillary. Through barium tagging, nEXO can identify and consequentlyreject all non-ββ background, therefore greatly improve the sensitivity inthe search for 0νββ.For this study, the theory of ion trapping in an ideal radio frequencyquadrupole (RFQ) has been derived from first principles. An analyticalmethod was developed to describe the ion confinement and ion acceptancein the ideal RFQ. The mass resolving power and ion transmission efficiencyfor an ideal quadrupole mass filter (QMF) were solved analytically. Theanalytical results were also used to validate a simulation method.The validated simulation method was used to study the effect of thenon-ideal electric potential in a realistic RFQ. The impact of the electricpotential’s higher-order spatial harmonics was studied in detail. These sim-ulations led to the optimum design and operating parameters of an RFQ ionguide, a QMF and an RFQ ion cooler/buncher.The temperature of trapped ions in an LPT was studied via simulationand was found to be anisotropic. The ion temperature in the longitudinaldirection was found to be mostly close to the helium buffer gas temperature.In the transverse directions, the ion temperature is close to the helium gastemperature at a small stability value q ≤ 0.3; while, when the q valueapproaches 0.9, the ion temperature increases several fold. The trappingdepth, either in the longitudinal or transverse directions, was found tohave a small or non-existent effect on the ion temperature. However, thelongitudinal trapping potential was found to cause an expelling potential inthe transverse direction and can lead to ion loss if too strong. The effectof the expelling potential was observed in simulations and agreed with thetheory.179A LPT system was designed based on optimizations in simulations. Thedesigned LPT has phase independent acceptance large enough to capture ortransmit at least 99% of ions from the RF funnel.Practical considerations were taken into account for the mechanicaldesign of the LPT along with its vacuum system. The major componentsof the LPT consist of a QMF with a specially designed monolithic electrodeholder to reduce mechanical error during assembly, a cooler with a noveldesign of the quadrupole electrode to form the drag field and a laser spec-troscopy ion trap which can also work as an ion buncher.The mechanical drawings of the LPT system were sent for manufacturingat the Physics department machine shop of the Universite´ de Montre´al. Inthe meantime, prototypes of the QMF and the cooler were made to validatethe designs and for the development of the electronics, control and DAQ(data acquisition) systems.Three prototypes of the QMF were built and tested, each of whichshowed the expected ion transmission properties at the tested RF voltagesand frequencies. It was shown that the final prototype of the QMF, theQMF2.2, has the best achievable mass resolving power R ≈ 140 whichexceeds the requirement of R ≈ 80. When operating in ion guide mode,the absolute ion transmission efficiency was preliminarily measured to beT ≈ 89%. The electronics, control and DAQ systems were developed alongwith the QMF prototypes.A prototype of the cooler with the novel quadrupole electrodes wasmanufactured mostly via 3D printed parts. Measurements with the coolerrevealed its capability of working as a low-resolution QMF with a maximummass resolving power R ≈ 20 at helium gas pressure up to 1.9× 10−2 mbar.Successful ion cooling, trapping and ejection were demonstrated by thecooler prototype. The cooler can cool down injected ions with energy spread∆Einj ≈ 20 eV at 1.9×10−2 mbar helium gas pressure, and ∆Einj ≈ 100 eVat 0.1 mbar helium gas pressure.The ejected ions’ time-of-flight (ToF) of the cooler prototype was foundto agree with simulations qualitatively. The discrepancies were found to becaused by a charge buildup on the electrode holder at the exit of the cooler.A revised design of the electrode holder was 3D printed for the final setup ofthe LPT system. After modifications to the design, it is expected that thecharge buildup in the final LPT system is eliminated. The ion temperatureTx in the longitudinal direction can be obtained by analyzing the measuredand simulated ions’ ToF dependence of the electric field.180The ion storage lifetime in the cooler prototype was measured to beτ = 16.9 s which is long enough for the barium ion identification via laserspectroscopy. It is expected that the ion storage lifetime can be significantlyincreased by purifying the helium gas in the final LPT system.The final LPT system with improvements based on the prototypes isbeing set up at McGill University. The LPT will be first tested and com-missioned as a standalone system. Then the LPT will be combined withthe RF funnel, MR-TOF and the laser spectroscopy setup for a systematicstudy of barium ion extraction and identification. 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Other than validatingthe design of the QMF itself, these prototypes also facilitate the develop-ment of the RF, electronics, control and DAQ system for the experimentalsetup. The manufacturing, installation and tests with the first two proto-types QMF1 and QMF2.1 are described in this appendix. The final proto-type QMF2.2 is described in Section 5.2.1.A.1 Prototype QMF1The first prototype of the QMF (QMF1) was made mainly to test themachining procedure and the mechanical rigidity of the specially designedmonolithic QMF holder. The machining procedure and the rigidity hadbeen a concern about the design, while the machining of such a holder usinga CNC milling machine can be time consuming and expensive.A.1.1 QMF1 machiningAn acrylic rod was used to make this prototype to reduce the material cost,also to make the internal structure visible for demonstration purposes. Dueto the limitation in tools, 3/8 inch diameter rods were used for making thequadrupole electrodes and the QMF was scaled up by a factor of 6/5 in theradial direction compared to the design in Section 4.3. The machining wasdone by the author in the UBC Physics and Astronomy student machineshop using a manual vertical milling machine as shown in Figure A.1. Themain structure of the holder was formed by cutting the long and deep(29 mm) slots every 90◦.The openings at both ends of the holder were machined on a lathe, andthe remaining block of material along the central axis was removed from theopenings. The finished QMF holder is shown in Figure A.2(a). The elec-trodes were machined from commercially available tight-tolerance 3/8 inch194Figure A.1: Machining of the QMF1’s monolithic holder.stainless rods (McMaster-Carr 1255T16) and are shown in Figure A.2(a).The QMF1 after assembly is shown in Figure A.2(b).The mechanical rigidity of the QMF holder was inspected manuallyby applying a force to different positions of the holder and observing thedeformation. For the magnitude of force corresponding to the weight ofthe electrodes and force needed during the assembly, no deformation wasobserved. So the holder for QMF1 is rigid and strong enough for its purpose.In the later manufacturing of the QMF, a more vacuum friendly plasticmaterial PEEK (polyether ether ketone) will be used. PEEK has a largerrigidity than the acrylic, so the design of the QMF meets the mechanicalrigidity requirement. The mechanical tolerance of the QMF1 is discussedbelow.195(a) Parts for the QMF1(b) Assembled QMF1Figure A.2: The first prototype QMF1 before and after assembly.A.1.2 QMF1 mechanical precision measurementDue to the limited performance of tools used for holding the acrylic rodduring the machining process as shown in Figure A.1, the mechanical pre-cision of the holder is not expected to be great. A method for measuringthe positioning of the quadrupole electrodes in the QMF1 was devised andshown in Figure A.3(a).The measurement method makes use of the high precision (5 µm) read-out of a lathe to measure the position of the electrodes with a homemadeprobe. The probe is made of a long stainless steel rod with a smooth metalball of similar diameter soldered to one end of the rod. The probe is installedon the tool holder of the lathe.All the quadrupole electrodes were electrically connected by wires toone terminal of a multimeter, while the probe was connected to the otherterminal. The resistance is measured between the probe and the electrodes.The probe is manually controlled to move along with the lathe’s tool holder.Once the protruded metal ball on the probe touches the inner surface of196(a) Setup for measurement of quadrupole electrodes position0 50 100 150 200 250 300Longitudinal distance (mm)3.63.84.04.24.44.6Rod position (mm)Expectedrod0rod90rod180rod270(b)0 50 100 150 200 250 300Longitudinal distance (mm)7.88.08.28.48.6Rod spacing (mm)Expectedrod0 to rod 180rod90 to rod 270(c)Figure A.3: A setup for measurement of QMF electrode positioning andmechanical tolerance.197an electrode, the resistance measured by the multimeter would change frominfinity to zero and the multimeter would start beeping.The horizontal coordinates of the electrode x and y are displayed on thedigital readout of the lathe and recorded into an electronic spreadsheet. Forevery three electrodes on the same rotational angle, 16 measurements weredone along the longitudinal direction x. The QMF were rotated every 90◦to measure the position of all the electrodes. Before starting measurementfor each rotation angle, the QMF were rotated slightly and checked withthe probe to make sure of the exact alignment.The measured position of the inner surface of the electrodes is shown inFigure A.3(b). The value of the vertical axis is the distance of the electrode’sinner surface to the QMF’s central axis. For every segment of the electrode,three or four measurements were made at different longitudinal positions.In the ideal situation, all the measured positions of the electrodes’ innersurface should be rQMF10 = 4.21 mm as indicated by the black horizontalline in the same figure.The measured positions for 7 out of the 12 electrodes appear to be reliablebecause they each appeared to be along a straight line, also indicating these5 electrodes are aligned well with the QMF in the longitudinal direction x.If an electrode is not aligned well, the measured position will also reflect thecylindrical surface curvature of the electrodes.The measured positions of the other 5 electrodes don’t all appear tobe caused by the misalignment of the electrodes, so human error mighthave been introduced during the measurement. The probe is easy to deflectonce it touches the electrode, so the movement of the probe needs to becarefully and slowly controlled in order to get the exact coordinates of theelectrode’s inner surface. The measurement shown in Figure A.3(b) took2 hours to complete. A slower measurement is expected to avoid thesehuman introduced errors.The distance between each diagonal pair of electrodes is shown in Fig-ure A.3(c). Based on these results, the positioning of the QMF1 electrodeshas a mechanical tolerance of about 0.3 mm and cannot meet the require-ment as discussed in Section 4.1.3. The mechanical error mostly comes fromthe QMF holder, which had the mechanical precision limited both by thetools available in the student machine shop and the machining skills of theauthor. A QMF holder made by a professional machinist using a CNC millmay be able to meet the tolerance requirement.198A.1.3 Installation of QMF1 in test standThe QMF1 was installed into the 6-way cross vacuum chamber of the teststand as shown in Figure A.4. The ion source and the ion detectors wereinstalled in the same way as shown in Figure 5.5.Part of the QMF was positioned in the tube of the 8” to 4.62” CF flangereducer as shown in Figure A.4(b) and (c). The positioning of the QMFrelative to the vacuum chamber wasadjusted by the bolts attached to theQMF holder.The configuration is the same as during the ion source and detectors testdescribed in Section 5.1.3, except now the QMF1 is placed in between theion source and the two detectors. The position of the QMF’s exit was a fewmillimeters in front of the Faraday cup, and the CEM was still positioned afew millimeters behind the aperture on the bottom of the Faraday cup.After sealing the vacuum chamber and pumping for two days, the vac-uum level inside the chamber reached 4.8 × 10−6 mbar. The pressure is20 times higher than when the vacuum chamber is empty. The increasedpressure comes from the larger out-gassing rate of the acrylic material.However, the vacuum level is still suitable for the operation of a CEM iondetector for some ion transmission tests.A.1.4 Ion transmission testThe ion transmission test was done with the QMF1. Due to its rough me-chanical precision, the QMF was operated as an RFQ ion guide by applyingonly RF voltages.The tests were done at the early stage of setting up the electronic systemfor the test stand. A low-cost two-channel signal generator SainSmartMHS5200A was used to directly supply RF voltage of up to 10 V (20 Vpeak-to-peak) to the QMF. The dual channels were set to have synchronizedamplitude and the phases were set to be 180◦ different.To make the best use of the signal generator, the RF amplitude wasfixed to V = 10 V and the frequency fRF were used as a scan parameter.199(a) Side view (b) View from ion entrance(c) Position of major components in the vacuum chamberFigure A.4: Photos of the QMF1 installed in the vacuum chamber of thetest stand. See text for details.200The RF frequency fRF is related to the stability parameter q and the ionmass as defined in Eq. (2.8). The q value is calculated for the expected alkaliions from the ion source at a few RF frequencies as shown in Table A.1.The values between 0 < q < 0.908 are emphasized in bold font to indicatethat the ion can be transmitted. The maximum ion transmission is aroundq = 0.6 as shown in Figure 3.5.Table A.1: Stability parameter q for different ions at a few RF frequencies.Values between 0 and 0.908 are emphasized in bold font.XXXXXXXXXXXIonfRF (MHz) 0.2 0.3 0.4 0.5 1.0 1.1 1.47Li 19.6 8.73 4.91 3.14 0.79 0.64 0.4023Na 6.00 2.66 1.50 0.96 0.24 0.20 0.1239K 3.54 1.57 0.88 0.57 0.14 0.12 0.0785Rb 1.62 0.72 0.40 0.26 0.06 0.05 0.03133Cs 1.04 0.46 0.26 0.17 0.04 0.03 0.02The ion source was heated to 1.43 A and floated at 160 V. Ion transmis-sion measurements were done with the RF frequency scanned from 0.1 MHzto 1.4 MHz by manual adjustment through the signal generator (this wasbefore an automated control system had been built). The ion current in theFaraday cup was recorded and the results are shown in Figure A.5.The Faraday cup was used with the adapter of a 1.6 mm diameteraperture to allow 0.7% of ions passing through to the CEM, the same asdescribed in Section 5.1.3. The output signal from the CEM was amplifiedand sent to a digital oscilloscope. The ion count rate was displayed on theoscilloscope as the trigger rate of the ion pulse signals. The determinedion count rate is considered only rough estimations with large uncertaintiesbecause the oscilloscope trigger rate kept fluctuating. The result is alsoplotted in Figure A.5.The results indicate that ion transmission occurred between the RFfrequencies from 0.4 MHz to 1.1 MHz. Judging by the q value of ions in thisfrequency range, the ions transmitted are mainly 39K. A smaller percentageof the other ions may also have been transmitted. There is no significantamount of 7Li or 133Cs ions because their presence would result in ion signalsabove 1.4 MHz or below 0.4 MHz.201200 400 600 800 1000 1200 1400RF frequency (kHz)01234Ion current (pA)020040060080010001200Ion count rate (Hz)Figure A.5: Ion transmission test of QMSV1 with RF frequency scan.A.1.5 Summary for QMF1The mechanical rigidity and stability of the assembled QMF1 proved thatthe mechanical design for the QMF is viable. The design and drawings ofthe QMF were sent to the machine shop in the Physics department of theUniversitye´ De Montre´al for manufacturing.The ion transmission tests proved that the QMF1 can be operated as anRFQ ion guide even with its electrode’s positional mechanical precision of0.3 mm. Even though the QMF1 was operating at a low RF voltage (10 V),ion transmissions were observed at the expected frequencies and q values.Ion source #1 was identified as mainly emitting 39K ions.These ion transmission tests were done in a preliminary manner duringan early stage of setting up the electronics, signal processing and dataacquisition (DAQ) for the development of the LPT system. The control andDAQ system for the test stand was developed along with other prototypeslater, as described in the next sections.202A.2 QMF V2.1The next QMF prototype, QMF2.1, was built with the aim of a bettermechanical precision to reduce the electric potential distortion inside thequadrupole electrodes. It also had exactly the same quadrupole electrodegeometry as the formal design, as described in Section 4.3.A.2.1 QMF2.1 design and machiningFrom the author’s experience of machining the QMF holder for QMF1,higher mechanical precision is difficult to achieve for the design using amanual milling machine with the limited tools. So, a different design wasmade for the next QMF prototypes. The holding structure of these proto-types consist of separate parts and each of them is easier to machine. Thisdesign also leads to a reduced amount of material for the holders. So thevacuum friendly plastic PEEK was used as the material for these holders.Machining of the parts for QMF2.1 was done by the author in theTRIUMF ISAC-II machine shop. The machining procedure for one of theholders using a vertical milling machine is shown in Figure A.6. The fourround holes used for holding the electrodes were formed with a standard5/16 inch diameter end mill cutter. The position of these four holes wasprecisely determined by moving the holder along with the milling machine’sbed. The horizontal coordinates x and y were displayed on a digital readoutwith a resolution of 0.0002 inch (5 µm). The position readings are repeatableand hence, reliable according to multiple calibrations done to the edge ofthe holder. The position of the holes is estimated to have precision within10 µm.The holders were then machined on a lathe to have the material alongthe central axis removed. As much as possible of the material was removedto expose the surface of the electrode while making sure the holder can stillhold the electrodes accurately. The purpose is to make the holders havea large distance to the central axis where ions pass through, because suchinsulating materials are known to get charge buildup from ion depositionwhich distorts the electric potential in the center of the quadrupole elec-trodes.The finished holders were shown in Figure A.7(a) along with the elec-203(a) (b)Figure A.6: Maching of the holders for QMFV2.1.trodes and two aperture plates. The two aperture plates with the hole di-ameter of 4 mm and 2 mm each will be installed on the entrance of the QMFwith a 5 mm gap in between, similar to the design in Section 4.3.The assembled QMF2.1 with the electrical connections is shown in Fig-ure A.7(b).A.2.2 QMF2.1 mechanical precision measurementThe mechanical precision of the assembled QMF2.1 was measured by thespacing between the quadrupole electrodes with a digital caliper. The spac-ings between the inner surface of two diagonal rods were measured at thetwo ends of the QMF2.1 and compared todinner = 2rQMF2.10 = 7.02 mm; (A.1)the distance between the outer surface of the electrodes was also measuredat positions in between the holders and compared todout = 2rQMF2.10 + 4rQMF2.1e = 22.90 mm. (A.2)204(a) Parts machined for QMF2.1(b) Assembled QMF2.1Figure A.7: The QMF2.1 prototype before and after assembly.205The mechanical precision of QMF2.1’s quadrupole electrode positions wasdetermined to be 40 µm according to the maximum measured mechanicalerror. A more comprehensive measurement of the QMF2.1’s mechanicalprecision can be done as described in Section A.1.2 but was not done becausethe lathe in the TRIUMF ISAC-II machine shop doesn’t have a digitalreadout of the positional coordinates.The mechanical errors mainly come from the imperfect positioning ofthe electrodes in the circular shaped holes. The four holes were found to beslightly larger than the rods, hence human errors can be introduced duringthe assembling process.A.2.3 Installation of QMF2.1 in test standThe QMF2.1 was installed in the vacuum chamber of the test stand as shownin Figure A.8. The setup was similar to the QMF1 except the Faraday cupwas no longer used. Part of the QMF2.1 was positioned inside the 8” to4.5” CF flange reducer tube as shown in Figure A.8(b), so that the ions exitthe QMF2.1 right in front of the CEM ion detector.After sealing the vacuum flanges and starting pumping, the pressurereached 1 × 10−6 mbar in 5 hours. After two days, the ultimate pressurestabilized at 3.3 × 10−7 mbar, which is only 37% higher than when thevacuum chamber is empty. Compared to the acrylic used in QMF1, thePEEK used for QMF2.1 showed much lower out-gassing rate.A.2.4 Ion transmission testThe ion source was heated with 1.3 A of current and floated to 60 V. Iontransmission measurements were done for QMF2.1 at low RF amplitudesupplied by another low-cost two-channel signal generator FeelTech FY6600because the SainSmart MHS5200A had been repurposed as a counter tomeasure the ion count rate as described in Section 5.1.3.The two outputs from the signal generator were set to both have the am-plitude of V = 10 V while the phases are 180◦ different. The RF frequencywas scanned from 4.5 M Hz to 0.25 MHz . An ion transmission measurementwas first done with the RF-only voltage and shown in Figure A.9(a).206(a) Side view of quadrupole electrodes in vacuum chamber (b) View from ion entrance(c) Position of major components in vacuum chamberFigure A.8: QMF2.1 installed in the vacuum chamber of the test stand.2071 2 3 4RF Frequency (MHz)02004006008001000Ion count rate (Hz)(a)0.0 0.5 1.0 1.5 2.0 2.5 3.0q value of 39K02004006008001000Ion count rate (Hz)(b)Figure A.9: (a) Ion transmission test of QMF2.1 with frequency scan. (b)Same measurement re-plotted with the horizontal axis as the q value of 39K.208The qK value for each of the RF frequencies were calculated for39K ionsusing Eq. (2.8), and the ion count rate is also shown as a function of qK inFigure A.9(b). The ion transmission occurring at qK > 0.908 indicates thepresence of heavier ions such as Rb.A.2.5 QMF2.1 for mass measurement as a QMSThe QMF2.1 was then tested with DC voltages also applied to the quadrupoleelectrodes to make it work as a quadrupole mass spectrometer (QMS). Atfirst, the DC offset function of the signal generator FY6600 was used. Somepreliminary measurement results revealed that the DC voltages were noisyand unstable for this purpose. So, a dedicated two-channel DC power supplyKorad KA3303P was used later. The RF&DC mixing was done using aCmix = 10 nF capacitor and a Rmix = 1 MΩ resistor for each channelas shown in Figure A.10. The RF&DC mixing was done at the vacuumchamber’s feedthrough as shown in Figure A.8(c). The mixed voltage wasmonitored by an oscilloscope with a ×10 probe.Figure A.10: Photo of the RF&DC mixing boxes with the electronicsannotated.Measurements were done by scanning the RF frequency from 4.5 MHz to0.25 MHz (corresponding to ions of smaller mass to larger mass) at differentDC voltage U . The results are shown in Figure A.11(a) as a 2D plot.209Four distinctive stability diagram patterns can be recognized from the2D scan result shown in Figure A.11(a) corresponding to 23Na, 39K, 85Rband 133Cs. Note that the shape of the stability diagrams’ boundary isdifferent as compared to Figure 2.13 because the U/V was used in thevertical axis instead of a.The DC voltage U was obtained from the set value UPS in the powersupply KA3303A:U = ηDC ∗ UPS , (A.3)whereηDC =RscopeRDC +Rscope=1011(A.4)is a correction factor as the DC voltage was divided between the RF&DCmixing resistor Rmix = 1 MΩ and the load impedance Rscope = 10 MΩ ofthe oscilloscope Tektronix TDC2024C (with ×10 probes).The DC voltage corresponding to the upper tip of the stability diagramfor the mass scan is obtained as U/V = st, wherest =at2qt≈ 0.168. (A.5)The exact values of qt ≈ 0.706 and at ≈ 0.237 are given previously inEq. (2.43). The value of st is plotted as a horizontal dashed line in Fig-ure A.11(a) and its location matches the upper tips of the four ions’ stabilitydiagrams.The mass values of the measurements were calculated fromm = ηC1Ω2· 4eVr20qt, (A.6)where ηC is a calibration parameter. In the ideal situation of an RFQ withperfect mechanical precision and perfectly measured voltages,ηidealC = A22, (A.7)where A2 = 1.002 is the coefficient of the quadrupole term in the electricpotential in the center of the RFQ with round electrodes of re/r0 = 1.13.In the real world, ηC is determined experimentally to compensate for theactual values of the QMF geometry and the voltages. In mass spectrometry,21023Na 39K 85Rb 133CsMass peak00.1550.168U/V100101102103Count rate (Hz)(a) Mass spectrum at different U values0 25 50 75 100 125 150 175Mass (u)100101102Ion count rate (Hz)23Na 39K 85Rb 133Cs(b) Mass spectrum at U/V = 0.155Figure A.11: Mass measurement using QMF2.1 as a quadrupole massspectrometer.211this process is usually done by calibration using the known mass of one ormore ions.The above measurements for QMF2.1 were calibrated with a one-pointcalibration to match the second tip in Figure A.11(a) to the actual mass of39K. The calibration parameter ηC = 0.925 was obtained and used to cal-culate the mass values shown in the horizontal axes of both Figure A.11(a)and (b). The relatively large deviation of ηC from 1 are likely to be causedby the limited accuracy of the RF voltage measured by the oscilloscope,which is specified to be 3%.The measurement with U/V = 1.55 near the upper tip of the stabilitydiagram is shown in Figure A.11(b) as a mass spectrum. Besides 39K whichwas used for the calibration, the other peaks in the measured mass spectrummatch well with the actual mass of these corresponding ions annotated inthe figure.A.2.6 Mass measurement with square wave RF signalIn all the ion transmission tests described above, the RF signal used was si-nusoidal. Square wave RF signal is an alternative and has been successfullyused in ion traps [SGSP99, DSB+04] and RFQ cooler/buncher [BSB+12].Such square wave RF signals can be generated by switching the outputbetween preset voltages; it has the advantage of wide frequency range witha relatively high amplitude. In comparison, analog sinusoidal RF signalsare more limited by the maximum power and the frequency range of the RFamplifier.The ion motion in a square wave driven RFQ was found to still havestable and unstable regions in the (q, a) parameter space [RHH73], thefirst stable region is similar to the sinusoidal case except the q values aresmaller. For example, when a = 0, the ion motion is stable in a smallerrange 0 < q < 0.7125 [DK06].Measurements were done with the QMF2.1 using square wave voltagesfrom the signal generator FY6600. The purpose is to test the functionalityand performance of the QMF for a possible square wave RF generator to bepermanently used in the future.The maximum RF amplitude of V = 10.2 V was configured for the212square wave from the function generator FY6600. The same frequencyrange of 0.25 MHz to 4.5 MHz was used for the mass scan. The results areshown in Figure A.12. In this case, the tip of the stability diagram for massscan corresponds to U/V = 0.214.The mass values shown in the horizontal axes of Figure A.12 are calcu-lated fromm = ηC1Ω2· 4eVr20qst, (A.8)where qst ≈ 0.554 is the q value at the upper tip of the ion’s stability di-agram for a square wave. The same value of ηC = 0.925 obtained in theabove measurement with sinusoidal wave was used here. The mass spectrummeasured with U/V = 0.200 is shown in Figure A.12(b) and all the peaksagree well with the actual mass of these ions.A.2.7 Mass measurement at higher RF amplitudeThe mass measurements described above have a maximum mass resolvingpower at FWHM (full-width at half-maximum) RFWHM = m/∆mFWHM ≈20, which is mainly limited by the relatively low RF amplitude V ≈ 10 Vof the signal generator. The low RF amplitude corresponds to a smallRF frequency, hence reduced RF cycles when the ions travel through theQMS. A low RF frequency also leads to a smaller ion acceptance. For thesereasons, a higher RF amplitude is better suited for the QMS operation whenthe RF frequency sweeping is used, as described in this study.An RF amplifier Aigtek ATG2022H was acquired for this work. Theamplifier has dual channels; each channel can deliver up to 200 Vp-p witha maximum power of 50 W. The -3dB frequency bandwidth is DC to 1 MHz.From this point forward the RF signals from the function generatorFY6600 are amplified by the ATG2022H before being mixed with the DCvoltages and sent to the quadrupole electrodes. The voltage gain of the am-plification is set to 30 times for both channels. The frequency-dependenceof the gain was tested in the following way: the function generator outputwas set to 1 V and the amplified signal was measured by an oscilloscope.The measured RF voltage and the gain are shown in Figure A.14. The gainis close to the set value of 30 only at low frequency. At higher RF frequency,the gain started to drop and reached 21.2 (30/√2) around 1 MHz. This is21323Na39K 85Rb 133CsMass peak00.2000.214U/V100101102103104Count rate (Hz)(a) Mass spectrum at different U/V values0 50 100 150 200Mass (u)100101102103Ion count rate (Hz)23Na39K 85Rb 133Cs(b) Mass spectrum at U/V = 0.207Figure A.12: Mass measurement using QMF2.1 as a QMS with square waveRF signal.214(a) Front and back view (b) Internal viewFigure A.13: Photos of the RF amplifier Aigtek ATG2022H.expected for a -3dB frequency band of the amplifier.0 1 2 3 4 5RF frequency (MHz)1020RF amplitude(V)1030230GainFigure A.14: RF amplitude (and gain) of the RF amplifier at 1 V input from0.01 to 5 MHz.The gain of the RF amplifier was found to be not a simple mathematicalfunction of the frequency. Therefore it was decided to measure the RFamplitude while operating the QMS.A set of measurements was done with QMF2.1 at higher RF amplitudeusing the RF amplifier. The RF amplitudes of the function generator wereset to be VFG = 4 V for both channels. The RF voltage was amplified by215a set gain of 30 times. The RF frequency was scanned from 1.8 MHz to0.8 MHz.The ion source was still heated to 1.3 A but floated at a lower voltage of24 V so that the ions would have smaller velocity while flying through theQMF and experience more RF cycles. The anode voltage of the ion sourcewas set to -170 V to obtain a maximum ion count rate. The measuredresults are shown in Figure A.15.The mass values of the transmitted ions, as identified on the horizontalaxes of the plots of Figure A.15 are calculated using Eq. (A.6) and themeasured RF voltage V at each RF frequency. The calibration parameterηC = 1.052 was found using the mass of39K using one-point calibration.The difference in the value of ηC compared to the above measurementswith low RF voltage most likely originates from the measurement of theRF amplitude. These low voltage RF were measured at the output ofthe function generator, which is slightly higher than the amplitude afterRF&DC mixing at the electrical feedthrough.A mass spectrum measured at U/V = 0.164 is shown in Figure A.15(b).The mass peaks for potassium isotopes 39K and 41K were identified; theheight of these two peaks matches their natural abundance of 93% and 7%.The rubidium isotopes 85Rb and 87Rb can also be identified; the ratio oftheir peaks also matches their natural abundance of 72% and 28%.The maximum mass resolving power is found to beRFWHM = m/∆mFWHM ≈ 51according to the 39K peak. The mass resolving power was likely limited bythe QMF2.1’s mechanical precision of 40 µm. The next prototype QMF2.2was built with the aim of better mechanical precision.216(a) Mass spectrum at different U values20 40 60 80 100 120 140Mass (u)100101102103Ion count rate (Hz)23Na 39K 85Rb 133Cs(b) Mass spectrum at U/V=0.164Figure A.15: Mass measurement with QMF2.1 at higher RF amplitude.217Appendix BMechanical drawings of theLPTThe mechanical drawings of the linear Paul trap designed in Chapter 4 areshown here. Note that all the drawings have been scaled from their originalsize to 11 inch ×8.5 inch. The drawings were made by X. Shang at McGillUniversity based on the 3D Solidworks models of this study. A few changesand many improvements have also been contributed by Shang in the 3Dmodels and the drawings.218219,,,219220,,,220221,,,221222,,,222223,,,223224,,,224225,,,225226,,,226227,,,227228,,,228229,,,229230,,,230231,,,231232,,,232233,,,233234,,,234235,,,235236,,,236237,,,237238,,,238239,,,239240,,,240241,,,241242,,,242243,,,243244,,,244245,,,245246,,,246247,,,247248,,,248249,,,249250,,,250251,,,251252,,,252253,,,253254,,,254255,,,255256,,,256257,,,257

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