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Diffusion and surface trapping of 8Li in rutile TiO2 and the comparison on 8Li and 9Li spin relaxation… Chatzichristos, Christos Aris 2020

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Diffusion and surface trapping of 8Li inrutile TiO2 and the comparison on 8Li and9Li spin relaxation using β -NMRbyChristos Aris ChatzichristosDiploma, National Technical University of Athens, 2013A 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)February 2020c© Christos Aris Chatzichristos 2020The following individuals certify that they have read, and recommend to theFaculty of Graduate and Postdoctoral Studies for acceptance, the dissertation enti-tled:Diffusion and surface trapping of 8Li in rutile TiO2 and the comparison on 8Liand 9Li spin relaxation using β -NMRsubmitted by Christos Aris Chatzichristos in partial fulfillment of the require-ments for the degree of Doctor Of Philosophy in PhysicsExamining Committee:Robert F. Kiefl, Physics and AstronomyCo-supervisorW. Andrew MacFarlane, ChemistryCo-supervisorBrian Wetton, MathematicsUniversity ExaminerJohn D. Madden, Electrical and Computer EngineeringUniversity ExaminerAdditional Supervisory Committee Members:Douglas Bonn, Physics and AstronomySupervisory Committee MemberJoerg Rottler, Physics and AstronomySupervisory Committee MemberMark Halpern, Physics and AstronomySupervisory Committee MemberiiAbstractIt is well established that the properties of many materials change as their thicknessis shrunk to the nanoscale, often yielding novel features at the near-surface regionthat are absent in the bulk. Even though there are several techniques that can studyeither the bulk or the surface of these materials, there are very few that can scanthe near-surface region of crystals and thin films versus depth. Beta-detected NMR(β -NMR) is capable of this and therefore has been established as a powerful toolfor material science. This thesis aims to further develop the capabilities of β -NMR.The first part of this thesis demonstrates that by comparing the spin-lattice relax-ation rates (SLR) of two radioactive Li isotopes (8,9Li) it is possible to distinguishwhether the source of SLR in a given situation is driven by magnetic or electricinteractions. This is an important development for β -NMR, since there are instanceswhere it is problematic to distinguish whether the measured relaxation is due tomagnetic or electric fluctuations. Using this method, it was found that the SLR inPt is (almost) purely magnetic in origin, whereas the spin relaxation in SrTiO3 isdriven (almost) entirely by electric quadrupolar interactions.The second part of this thesis traces the development of α-radiotracer, that usesthe progeny α-particles from the decay of 8Li, in order to directly measure thenanoscale diffusivity of Li+ in Li-ion battery materials. To develop this technique,Monte Carlo simulations of the experimental configuration were carried out, a newapparatus and a new α-detector were designed and used for experiments on rutileTiO2. In rutile, the measurements revealed that Li+ gets trapped at the (001) surface,a result that helps explain the suppressed intercalation of Li+ in bulk rutile. Moreover,the diffusion rate of Li+ in rutile was found to follow a bi-Arrhenius relationship,with a high-T activation energy in agreement with other reported measurementsand a low-T component of similar magnitude with the theoretically calculateddiffusion barrier as well as the activation energy of the Li-polaron complex foundwith β -NMR below 100 K.iiiLay SummaryBeta-detected NMR (β -NMR) is capable of studying both the magnetic and electricproperties of materials at the nanoscale (billionth of a meter) and therefore hasbeen established as a powerful tool for material science. The first part of thisthesis demonstrates that by comparing the β -NMR signal of two radioactive lithiumisotopes it is possible to distinguish whether the dominant interaction in a givensituation is magnetic or electric in origin. This development can aid the identificationand fabrication of materials with useful properties for a wide range of applications.In addition, in this thesis the α-radiotracer technique for studying nanoscalelithium diffusion was developed. Studying diffusion is crucial for identifying bettermaterials for next-generation lithium-ion batteries. This technique uses nuclearphysics to study how fast lithium ions move inside materials and also what happenswhen these ions reach a surface.ivPrefaceI was the lead investigator for the projects located in Chapters 3.3, 5 and 6, where Iwas responsible for all major areas of concept formation, data analysis, as well as themajority of manuscript composition. For the project located in Sections 3.1- 3.2.3, Iwas responsible for the data analysis and manuscript composition and I took part inthe data collection, while Rob Kiefl and Andrew MacFarlane were responsible forthe concept creation.Most of the results presented in Chapters 3 and 6, as well as some work presentedin Chapter 5 have been published in peered-reviewed articles, either in journalsor conference proceedings. In these articles I am listed as the first author and theco-authors took part in the data collection and manuscript revision processes. Moredetails on the division of work between myself and the co-authors are given belowfor each publication. The relevant papers are the following:• A. Chatzichristos, R. M. L. McFadden, V. L. Karner, D. L. Cortie, C. D. P.Levy, W.A. MacFarlane, G. D. Morris, M. R. Pearson, Z. Salman, and R. F.Kiefl,“Determination of the nature of fluctuations using 8Li and 9Li β -NMRand spin-lattice relaxation”, Phys. Rev. B 96, 014307 (2017) [1]; whichcontains a version of Sections 3.1 to 3.2.3. I carried out the data analysisin collaboration with R. M. L. McFadden and prepared the manuscript. Ipresented the findings of this paper in an oral presentation (invited talk) at the“Applications of Nuclear Physics” Mini-Symposium of the American Physi-cal Society, Division of Nuclear Physics meeting, Vancouver, BC, Canada(October 2016).• A. Chatzichristos, R. M. L. McFadden, V. L. Karner, D. L. Cortie, C. D. P.Levy, W. A. MacFarlane, G. D. Morris, M. R. Pearson, Z. Salman, and R. F.Kiefl, “Comparison of 8Li and 9Li spin relaxation in SrTiO3 and Pt: A meansto distinguish magnetic and electric quadrupolar sources of relaxation” inProceedings of the 14th International Conference on Muon Spin Rotation,Relaxation and Resonance (µSR2017) [2]; which contains a version of Sec-tions 3.3 and 3.4. I carried out the data analysis and prepared the manuscript.I presented the findings of this paper on a poster at the 14th InternationalvConference on Muon Spin Rotation, Relaxation and Resonance (µ-SR 2017),Hokkaido University, Sapporo, Hokkaido, Japan (July 2017).• A. Chatzichristos, R.M.L. McFadden, M.H. Dehn, S.R. Dunsiger, D. Fuji-moto, V.L. Karner, C.D.P. Levy, I. McKenzie, G.D. Morris, M.R. Pearson,M. Stachura, J. Sugiyama, J.O. Ticknor, W.A. MacFarlane, and R.F. Kiefl,“Bi-Arrhenius Diffusion and Surface Trapping of 8Li+ in Rutile TiO2”, Phys.Rev. Lett. 123, 095901 (2019) [3]; which contains a version of Sections 6.1to 6.4. I carried out the simulation study, developed the data analysis tools,conducted the data analysis and prepared the manuscript. I presented thefindings of this letter in an oral presentation at the 28th annual symposium ofthe Hellenic Nuclear Physics Society, Thessaloniki, Greece (June 2019).The publishers of the aforementioned papers have granted permission for thismaterial to be incorporated into this thesis.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Importance of nanomaterials . . . . . . . . . . . . . . . . . . . . 11.1.1 Experimental techniques to study nanoscale phenomena . 21.2 Li-ion Diffusion - Solid State Li-ion Batteries . . . . . . . . . . . 61.2.1 Importance of Li-ion Batteries . . . . . . . . . . . . . . . 61.2.2 Experimental Methods for Studying Solid State Diffusion 91.2.3 Studying nanoscale Li diffusion . . . . . . . . . . . . . . 111.3 Organization of this Thesis . . . . . . . . . . . . . . . . . . . . . 122 Beta-detected Nuclear Magnetic Resonance . . . . . . . . . . . . . 142.1 Physics of β -NMR . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Experimental Details of β -NMR at TRIUMF . . . . . . . . . . . 162.2.1 Polarized Beam Production and Delivery . . . . . . . . . 172.2.2 β -NMR and β -NQR Spectrometers . . . . . . . . . . . . 222.2.3 Beam Implantation Profiles . . . . . . . . . . . . . . . . 262.3 Spin Interactions of β -NMR probes in a lattice . . . . . . . . . . 282.3.1 Magnetic Interactions . . . . . . . . . . . . . . . . . . . 29vii2.3.2 Electric Quadrupolar Interactions . . . . . . . . . . . . . 302.4 Types of Measurements . . . . . . . . . . . . . . . . . . . . . . 322.4.1 Resonance Spectra . . . . . . . . . . . . . . . . . . . . . 322.4.2 Spin Lattice Relaxation Spectra . . . . . . . . . . . . . . 362.5 Enhancing the capabilities of β -NMR using α detection . . . . . 383 Using α-tagged 9Li β -NMR to Distinguish the Source of Spin LatticeRelaxation in 8Li β -NMR . . . . . . . . . . . . . . . . . . . . . . . . 413.1 Isotopic Comparison Method . . . . . . . . . . . . . . . . . . . 423.2 Experimental Demonstration of the Isotopic Comparison Method 433.2.1 Platinum . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2.2 Strontium Titanate . . . . . . . . . . . . . . . . . . . . . 463.2.3 Ratio of Relaxation Rates . . . . . . . . . . . . . . . . . 503.3 Enhancing the Effective Asymmetry of 9Li Using α-tagging . . . 523.3.1 αLithEIA Method . . . . . . . . . . . . . . . . . . . . . 533.3.2 Experimental Testing of the αLithEIA Method on ZnS(Ag) 563.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 Principles of Interstitial Diffusion in Solids . . . . . . . . . . . . . . 624.1 Macroscopic Theory of Solid State Diffusion . . . . . . . . . . . 624.1.1 Fick’s Laws . . . . . . . . . . . . . . . . . . . . . . . . 634.1.2 Temperature Dependence of Diffusion: Arrhenius Law . . 644.2 Microscopic Theory of Diffusion . . . . . . . . . . . . . . . . . 654.2.1 One Dimensional Random Walk . . . . . . . . . . . . . 664.2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . 684.2.3 Einstein-Smoluchowski Law . . . . . . . . . . . . . . . 704.2.4 Isotopic mass effect on diffusion . . . . . . . . . . . . . 745 Principles of Studying Nanoscale Lithium Diffusion Using the α-Decayof 8Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.1 Basic Principles of the 8Li α-radiotracer Technique . . . . . . . . 765.2 Simulations of α-Detected Lithium Diffusion . . . . . . . . . . . 815.2.1 Temporal Evolution of the Diffusion Profiles . . . . . . . 825.2.2 Geant4 Simulations . . . . . . . . . . . . . . . . . . . . 855.3 Calculating the α-radiotracer signal . . . . . . . . . . . . . . . . 965.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996 Measurements of 8Li Diffusion in Rutile using α Detection . . . . . 1016.1 Rutile - General Characteristics . . . . . . . . . . . . . . . . . . 1016.2 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . 103viii6.3 Experimental Results on Rutile TiO2 Using the α-radiotracer Tech-nique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.4 Arrhenius Fits and Discussion . . . . . . . . . . . . . . . . . . . 1086.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1137 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . 1147.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147.1.1 Isotopic comparison for distinguishing magnetic and elec-tric relaxation in β -NMR . . . . . . . . . . . . . . . . . 1157.1.2 Establishing the α-radiotracer technique for studying nanoscaleLi diffusion . . . . . . . . . . . . . . . . . . . . . . . . . 1167.1.3 Study of the Li+ motion in rutile TiO2, using the α-radiotracertechnique . . . . . . . . . . . . . . . . . . . . . . . . . . 1167.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119AppendicesA The cryo-oven . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140B Geant4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145B.1 Getting started . . . . . . . . . . . . . . . . . . . . . . . . . . . 145B.2 The project code tree . . . . . . . . . . . . . . . . . . . . . . . . 145B.3 Running the simulation . . . . . . . . . . . . . . . . . . . . . . . 147B.4 Structure of the Geant4 project . . . . . . . . . . . . . . . . . . . 148B.4.1 LiDiffusion.cc . . . . . . . . . . . . . . . . . . . . . . . 148B.4.2 LiDetectorConstruction/LiDetectorMessenger . . . . . . 148B.4.3 PhysicsList . . . . . . . . . . . . . . . . . . . . . . . . . 149B.4.4 LiActionInitialization . . . . . . . . . . . . . . . . . . . 149B.4.5 LiPrimaryGeneratorAction/LiPrimaryGeneratorMessenger150B.4.6 LiRunAction/LiEventAction/LiSteppingAction . . . . . . 151B.4.7 Output files . . . . . . . . . . . . . . . . . . . . . . . . . 151B.4.8 Macro (input) file . . . . . . . . . . . . . . . . . . . . . 152ixList of Tables1.1 Comparison of classical NMR with β -NMR . . . . . . . . . . . . 42.1 Intrinsic nuclear properties of radioisotopes used in β -NMR . . . 163.1 The asymmetry of each decay mode of 9Li . . . . . . . . . . . . . 543.2 Fits of the untagged, α-tagged and no-α-tagged spectra of 9Liβ -NMR at 308 K in ZnS(Ag) . . . . . . . . . . . . . . . . . . . . 60xList of Figures1.1 Li-ion battery market expansion (2000-2015) . . . . . . . . . . . 71.2 Schematic of a (C/LiCoO2) Li-ion battery . . . . . . . . . . . . . 81.3 Direct and indirect methods for studying solid state diffusion . . . 91.4 The steps for measuring diffusion with the (radio)tracer method . 112.1 Anisotropy of the direction of the emitted β -particles . . . . . . . 152.2 Energy distribution of the β - and α-particles coming from the decayof 8Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Internal view of the 520 MeV main cyclotron vault at TRIUMF . . 182.4 Schematic of the polarizer and the section of the beamline leadingto the β -NMR and β -NQR spectrometers . . . . . . . . . . . . . 192.5 Schematic of the spin polarization method for 8Li . . . . . . . . . 202.6 Schematic of the high-field β -NMR spectrometer . . . . . . . . . 232.7 Image of the β -NMR sample holder . . . . . . . . . . . . . . . . 242.8 Over-view of the β -NQR spectrometer . . . . . . . . . . . . . . . 252.9 Image of the β -NQR sample holder . . . . . . . . . . . . . . . . 262.10 Tracks of 8Li implanted in TiO2, as simulated by SRIM-2013 . . . 272.11 8Li beam implantation profile in TiO2 as simulated by SRIM-2013 282.12 The energy splitting of the magnetic sub-levels of 8Li, due to theaxial part of the Hamiltonian . . . . . . . . . . . . . . . . . . . . 322.13 Resonance spectrum for 8Li in a Bismuth single crystal at 294 K . 332.14 Resonance spectrum for 8Li implanted in a epitaxial Ag film grownon MgO at 155 K . . . . . . . . . . . . . . . . . . . . . . . . . . 352.15 SLR spectrum of 8Li implanted in a single crystal of bismuth at at295 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.1 Resonance spectra in Pt foil at 300 K and 1.90 T with 8Li and 9Li 453.2 SLR spectra for 8Li+ and 9Li+ in Pt foil at 300 K and 6.55 T . . . . 463.3 Measured SLR rates for 8Li implanted in Pt . . . . . . . . . . . . 473.4 Crystal structure of SrTiO3 . . . . . . . . . . . . . . . . . . . . . 483.5 SLR spectra of 8Li and 9Li in single crystal SrTiO3 at 300 K . . . 49xi3.6 Field dependence of 1/T1 for 8Li and 9Li in SrTiO3 (sample S1) at300 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.7 SLR spectra of 8Li and 9Li in SrTiO3 at 10 mT and 300 K . . . . . 513.8 Ratios of 9Li to 8Li 1/T1 relaxation rates in Pt and in the two SrTiO3samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.9 Properties of the principle β -decay modes of 8Li and 9Li . . . . . 533.10 Theoretical initial β -asymmetry detected in coincidence/absence ofan α , for varying α-detection efficiency . . . . . . . . . . . . . . 553.11 Resonance spectrum for 8Li in ZnS(Ag) at 10 K . . . . . . . . . . 573.12 Crystal structure of ZnS(Ag) . . . . . . . . . . . . . . . . . . . . 583.13 Temperature dependence of the fraction of the relaxing componentand the relaxation rate in ZnS(Ag) . . . . . . . . . . . . . . . . . 593.14 Untagged, α-tagged and no-α-tagged spectra of 9Li β -NMR at310 K in ZnS(Ag) . . . . . . . . . . . . . . . . . . . . . . . . . . 593.15 The normalized decay curves of the light emitted from a ZnS(Ag)/6LiFscintillator illuminated with 5.5 MeV α-particles . . . . . . . . . 604.1 Density profiles with an accumulative wall . . . . . . . . . . . . . 694.2 Density profiles with a reflective wall . . . . . . . . . . . . . . . 714.3 Image of n individual random steps on a lattice . . . . . . . . . . 724.4 Interstitial atom’s jumping process . . . . . . . . . . . . . . . . . 735.1 Schematic of the ultra-high vacuum sample region with the cross-section of the ring detector . . . . . . . . . . . . . . . . . . . . . 775.2 Schematic of the α-detection geometry . . . . . . . . . . . . . . 785.3 Simulated counts at the α-detector versus time . . . . . . . . . . . 805.4 Calculated depth profiles of Li ions versus time . . . . . . . . . . 845.5 Simulated fraction of Li ions trapped at the front surface of a TiO2sample versus time . . . . . . . . . . . . . . . . . . . . . . . . . 855.6 Geometry simulated by Geant4 . . . . . . . . . . . . . . . . . . . 885.7 Simulation of the distribution of the energy deposited at the α-detector at different times . . . . . . . . . . . . . . . . . . . . . . 905.8 The normalized α-spectrum, as simulated with Geant4 in TiO2, fordifferent α-detector energy thresholds . . . . . . . . . . . . . . . 915.9 The normalized α-signal, as simulated with Geant4 in TiO2, fordifferent diffusion rates . . . . . . . . . . . . . . . . . . . . . . . 925.10 The normalized α-signal, as simulated with Geant4 in TiO2, for areflective surface . . . . . . . . . . . . . . . . . . . . . . . . . . 935.11 The normalized α-signal, for various trapping probabilities at thesurface of the sample . . . . . . . . . . . . . . . . . . . . . . . . 94xii5.12 Simulated α-signals for beam energies of 10 keV and 25 keV . . . 955.13 The energy deposition profile of α- and β -particles inside the α-detector for various detector thicknesses . . . . . . . . . . . . . . 965.14 Probability of α-detection versus decay depth . . . . . . . . . . . 985.15 Comparison of calculated and simulated normalized Y nα (t) signals 996.1 A 3-D view of the rutile TiO2 unit cell . . . . . . . . . . . . . . . 1026.2 Comparison of the measured normalized α-yield for the (110)- and(001)-oriented rutile . . . . . . . . . . . . . . . . . . . . . . . . 1056.3 Comparison of the measured normalized α-yield between unrotatedand rotated (001)-oriented rutile . . . . . . . . . . . . . . . . . . 1066.4 Fits of the measured normalized α-yield for the (001)-orientedrutile at various temperatures . . . . . . . . . . . . . . . . . . . . 1076.5 Arrhenius plot, comparing reported Li diffusivity in rutile TiO2 . . 1096.6 Arrhenius fit to the diffusion rates of 8Li found in this study . . . . 110A.1 Figure showing the sample holder used in the cryo-oven assembly 140A.2 Images of two SolidWorks cuts of the lower region of the cryo-ovenassembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143A.3 Images of two SolidWorks overviews of the cryo-oven assembly . 144xiiiAcknowledgmentsTrue to the spirit of sir Isaac Newton’s famous quote about standing on the shouldersof giants, this work would be completely impossible without the help and guidanceof my supervisors, Prof. Rob Kiefl and Prof. Andrew MacFarlane. Their expertiseand knowledge in both experimental and theoretical issues, but also their enthusiasmand vision were crucial for the completion of most aspects of this work. I want alsoto thank the other members of my PhD committee: Douglas Bonn, Mark Halpernand Joerg Rottler for taking the time to read the thesis and for their insightfulcomments.A major part of this thesis revolves around the incarnation of the α-radiotracertechnique, which, among other things, required the design, fabrication and assemblyof a novel, completely custom-made cryostat: the chimeric cryo-oven. This projectwas benefited greatly by the expertise and knowledge of Gerald Morris, who aidedus in all phases of the cryo-oven’s development. I want to thank Mike McLay forhis great work in translating the doodles of the various parts of the cryo-oven Roband I made to meaningful 3D blueprints. Last but not least, I want to acknowledgeRahim Abasalti and Deepak Vyas for their really herculean labor and their immensecontribution to the assembly, mounting and preparation of the cryo-oven.I want also to say a great thank you to all my colleagues in the β -NMR group:Victoria Karner, Ryan McFadden, Martin Dehn, Derek Fujimoto, John Ticknor, IainMcKenzie, Monika Stachura, Sarah Dunsiger and others. This whole endeavor wasa highly collaborative process with everybody taking shifts for data acquisition andhelping with most aspects of analysis, interpretation, etc. Special thanks to Victoriaand Ryan, who gave me the introductory crash course on the ins and outs of β -NMR.I would also like to express my gratitude to TRIUMF staff members Matt Pearson,Phil Levy, Bassam Hitti, Donald Arseneau and Suzannah Davie for their aid and alltheir meaningful contributions during the β -NMR and α-radiotracer experiments.xivDedicationTo all the people that guided and helped me, my spouse and my parents most of all.Κτῆμα τε ἐς αἰεὶ μᾶλλον ἢ ἀγώνισμα ἐς τὸ piαραχρῆμαxvChapter 1Introduction1.1 Importance of nanomaterialsNew experimental tools and techniques capable of studying nanoscale phenomena,are constantly revealing new and potentially useful properties. The differencebetween nanomaterials (e.g., particles or thin-films) and their bulk forms stems fromfinite size effects and interfaces which can give rise to new and unusual properties.When one (or several) of a material’s dimensions approaches the nanoscale, certainunderlying quantum effects cannot be neglected any more, as the classical limitgets progressively less valid. This gives rise to novel material properties (optical,electromagnetical, chemical, etc), that are much less important on a macroscopiclength scale. For instance, the 2D electron gas (2DEG) [4], realized at the interfaceof certain heterostructures, exhibits extremely high electron (hole) mobility andcan also give rise to the quantum Hall effect, where the conductance can onlytake quantized values [5]. In addition, the material properties are in many casessize-dependent in this scale. This permits the fine-tuning of certain properties(e.g., electrical conductivity and magnetic permeability) by changing the size of theparticle or the thin-film. An important example of this tunability relates to quantumdots, which are small semiconductor particles having a nanoscale diameter. Whencertain quantum dots are illuminated they will emit light, the frequency of whichdepends on the particle’s size [6].Apart from the rise of quantum mechanics at this scale, a second reason behindthe importance of the nanoscale is the large surface-to-mass ratios of nanomaterials.Indeed, a spherical nanoparticle with a radius of 100 nm would have a surface-to-mass ratio 10,000 times larger than a 1 mm radius single-crystal of the samesubstance. A greater surface area increases the chemical reactivity of the substance,which can greatly improve, for instance, the properties of catalysts used in chemistryand chemical engineering.Moreover, the developments in nanoscale fabrication and experimental tech-niques revealed a plethora of interesting surface phenomena. A few examples in-clude the novel material class of topological insulators [7], which have topologically-protected conduction states at their surface while being bulk insulators, as well asthe fabrication of magnetic surfaces, interfaces and thin films [8].1The above-mentioned phenomena are not just of pure scientific interest, butthey rather have very important applications. For instance, quantum tunneling isresponsible for the development of flash memory for computers [9] and topologicalinsulators might be used in the future for better magnetic (memory) devices [10].1.1.1 Experimental techniques to study nanoscale phenomenaMany interesting and important nanoscale phenomena have been discovered by theparallel development of a large number of experimental methods capable of studyingthem. For instance, nuclear magnetic resonance (NMR) [11] is sensitive to localelectromagnetic fields and resonant X-ray diffraction (XRD) [12] can effectivelyprovide site-specific chemical information, but both lack the sensitivity to exploresurface or near-surface phenomena, however useful they may be in studying thebulk of materials.Another very important class of experimental techniques includes many thatcan probe the surface properties of thin films and single crystal samples. Grazingincidence X-ray diffraction and neutron reflectivity, are two complementary tech-niques [13] that illuminate the sample with different probes and use their diffractionto extract properties of the surface and near-surface region. Atomic force microscopy(AFM) [14] uses piezoelectric elements to image the surface of solids with sub-nanometer resolution, as well as to perform force spectroscopy. Scanning tunnelingmicroscopy (STM) [15] is capable of imaging surfaces with single-atom resolution.It makes use of the quantum tunneling effect, by bringing a conducting tip close tothe sample surface and then observing the tunneling current versus applied voltageand tip position. Angle-resolved photoemission spectroscopy (ARPES) [16] canbe used to explore the electronic structure of solids (i.e., their k-space/momentum-resolved band structure), using the electron photoemission upon illumination withsoft X-rays.Despite the large number of available techniques for studying nanoscale materi-als, very few are able to study the near-surface region (1 nm to 100 nm) of a sample.This is a very interesting region, where one can study how the material propertieschange versus the distance from the surface [17–19] (or an interface between twoheterostructures [20]). Among the techniques mentioned above, only grazing inci-dence X-ray diffraction and neutron reflectivity are able to probe this near-surfaceregion and both of them are reciprocal-space probes as opposed to real-space probes.Indeed, there are only two general purpose real-space techniques capable of studyingthe near-surface region of thin-films and single crystals, β -detected NMR (β -NMR)and low energy muon spin rotation/relaxation (LE-µSR), accompanied by somespecial-case techniques, such as conversion-electron Mo¨ssbauer spectroscopy [21].In recent years, β -NMR has been established as a powerful tool of material2science due to its inherent sensitivity to magnetic and electronic properties of theprobe’s local environment [22]. It has been used extensively to study, amongst othertopics, metals [23, 24], insulators – both classical [25] and topological [26] –, su-perconductors [27], semiconductors [28], lithium diffusion [29, 30], soft condensedmatter [31], antiferromagnetism [17], as well as recently ionic liquids [32].Despite its wide range of applications and its demonstrated success in studyingthe nanoscale, β -NMR is still not very well known or widely used. The primaryreason is that it requires an extensive infrastructure, including a particle acceleratorcapable of delivering an intense beam of spin-polarized radionuclides. There arevery few centers in the world with such capabilities and all of them focus primarilyon nuclear or particle physics research. These include TRIUMF in Canada [33],ISOLDE in Switzerland [34, 35], Osaka [36] and Moscow [37]. From these, thecenter with the most developed β -NMR program is the ISAC facility at TRIUMF,Canada’s national laboratory for nuclear and particle physics.A major limitation on the proliferation of the β -NMR technique is that ISAC is asingle-user facility, meaning that only one experiment may be using the radioactivebeam at any given time. Typically, the beamtime dedicated to β -NMR experimentsat TRIUMF is limited to 5 weeks/year, but at the time of writing it is significantlylower, due to extended shutdown periods each year for helping with the ARIELupgrade [38]. As a consequence, the β -NMR beamtime at TRIUMF has to be usedvery effectively, which sometimes makes it difficult to obtain beamtime for controlmeasurements. This underlines the need for more β -NMR user facilities.The principle success of TRIUMF’s low-energy incarnation of β -NMR [33,39] is the ability to study thin films, surfaces, and interfaces — situations whereconventional NMR is ill-suited. This stems from β -NMR’s high sensitivity (pernuclei) relative to conventional NMR; typically only∼ 108 nuclei (instead of∼ 1017)are required for a discernible spectrum (see Tab.1.1).3Table 1.1: Comparison of classical NMR with β -NMR. Nuclear Polarization refersto the percentage of the spin-probes in the sample that are spin-polarized, whereasSensitivity refers to the number of spin-probes needed for a spectrum. The muchgreater sensitivity of β -NMR, permits the study of thin-films.NMR β -NMRNuclear Polarization  1% ∼ 70-80%Nuclear Probesstable nuclei with spin,e.g.,1H, 13C, 57Fe, 17O, etcunstable nuclei with spin,e.g.,8Li+, 9Li+, 31Mg+, etcSensitivity >1016 probes ∼ 108 probesDetection MethodInduced EMF in a coil fromprecession of nuclear magnetizationAnisotropic β -decayIn classical NMR, the nuclear spin polarization is achieved by applying a strongmagnetic field on the sample. This field splits the degeneracy of the nuclear spinstates, making the state with the nuclear spin parallel to the field energeticallyfavorable by an energy difference –for spin 1/2 nuclei – of ∆E = 2µB, where µ isthe nuclear magnetic moment. At a temperature T, the ratio of the occupation of thetwo spin states of a spin-1/2 nucleus is given by a Boltzman distribution:N+/N− = exp(−∆E/kBT ) (1.1)where N+ and N− are the populations of the ± nuclear spin states at the axisdefined by the applied magnetic field and kB is the thermodynamic Boltzmannconstant. Almost always ∆E kBT , which leads to the approximation:N+/N− ∼ 1− ∆EkBT (1.2)It is important to note that unlike conventional NMR, where the thermodynamicBoltzmann factor (∆E/kBT ) determines the polarization, the initial non-equilibriumnuclear polarization in β -NMR is close to unity and independent of the sampletemperature and/or magnetic field. Consequently, measurements can be madeunder conditions where conventional NMR is difficult or impossible e.g., at hightemperatures, low magnetic fields or in thin films. The intensity of the implantedbeam (typically ∼107 s−1), is such that the concentration of the nuclear probesis so small that there is no interaction between probes and thus no homo-nuclearspin-coupling.The only other real-space technique with equivalent sensitivity over a compara-ble material length scale (viz. 10 nm to 200 nm) [40] is low-energy µSR [41–43].4µSR is a sister technique to β -NMR, in the sense that both rely on the same un-derlying physical principle (i.e. parity violation in weak nuclear interactions, seeSect. 2.1). However, they operate in complementary time-windows due to thedifferent probe lifetimes (1.21 s for 8Li+ vs. 2.2 µs for µ+). As a result, the twotechniques are sensitive to spin relaxation rates of completely different frequencies.Broadly speaking, the time range of each technique is defined as [τβ/100−100τβ ],where τβ is the lifetime of the radioactive probe, which means that µSR is mostsuitable to study electromagnetic spin relaxation rates at the MHz scale, whereas8Li β -NMR can be used to probe spin relaxation rates at the sub-Hz to kHz range.Thus, both techniques have leveraged the nuclear physics of β -decay to investigatetopical problems in condensed matter physics, including magnetic surfaces, thinfilm heterostructures and near surface phenomena.This thesis aims to further enhance the capabilities of 8Li β -NMR, by developingnew tools and methods that use the α-emissions of 9,8Li in conjuncture with theβ -NMR signal coming from the β -decay.The first part of the thesis (Ch. 2-3) describes the isotopic comparison method,which compares the spin-lattice relaxation (SLR) rates of two radioactive isotopes(in this case 9Li and 8Li) under identical experimental conditions to distinguishthe source of SLR in a given situation, enhancing thusly β -NMR’s capabilities ofstudying the nanoscale. Indeed, using this method we were able to show that theSLR rate in Pt is caused by primarily magnetic interactions, whereas in SrTiO3 themain effect driving the SLR is electric quadrupolar in origin. One major limitationwhen using the 9,8Li isotopic comparison method is that the effective β -decayasymmetry of 9Li is very low. This is due to the fact that 9Li has three differentdecay paths, two of which have opposite asymmetries that nearly cancel. Thisresults in a very small β -NMR signal which makes the 9Li β -NMR measurementsvery time-consuming. To partly overcome this issue, we developed a method fordistinguishing the decay paths of 9Li by using the α-particles coming from thedecay of 9Li. This method is termed α-detection for Lithium-9 Enhanced InitialAsymmetry (αLithEIA) and can in principle increase the effective asymmetry of9Li by a factor of ∼3.The second part of this work (Ch. 4-6) describes the development (from firstprinciples to full material studies) of the 8Li α-radiotracer method, which is capableof studying nanoscale Li diffusion using the α-decay of 8Li. Indeed, we employedthis method to study 8Li diffusion in rutile TiO2 (see Ch. 6), which is a materialwhere Li diffusion is very fast and quasi one dimensional. Rutile has potential usesas an anode in Li-ion batteries. Based on our measurements, we were able to showthat Li+ gets trapped upon reaching the (001)-surface of rutile, which is very hardto establish with other techniques and might be causing the reported difficulty ofLi+ intercalation in bulk rutile. Furthermore, we found that the diffusivity of Li+ in5rutile follows a bi-Arrhenius relationship. At high-T , the corresponding Arrheniuscomponent is in agreement with what other techniques (including β -NMR) havefound. Below ∼200 K, a second Arrhenius component becomes dominant, whichhas an activation energy similar to what was found with β -NMR below 100 K as partof a Li-polaron complex [29], as well as close to the activation energy calculatedtheoretically [44].This is the only direct method for studying nanoscale Li diffusion and thereforemight be proven very useful in understanding the motion of Li+ interstitials throughvarious materials and their interfaces, as well as in the pursuit of next-generationLi-ion battery materials, the importance of which is described in the next part ofthis introduction.1.2 Li-ion Diffusion - Solid State Li-ion Batteries1.2.1 Importance of Li-ion BatteriesThe current global fossil-fuel-based economy poses a series of economical, politicaland environmental risks to the planet, which include the depletion of key energysources, the risk of politically-induced fuel price crisis (similar to the 1973 oilcrisis), as well as the potentially devastating results of the ongoing (man-made)climate change. Indeed, the global CO2 emissions have increased dramatically overthe past few decades, despite the efforts by the international community to containthem (Kyoto 1997 protocol, Paris 2015 accords). To avoid (or at least mitigate) theabove crisis, a strong turn towards clean energy solutions is required, both in theenergy production and in the transportation sectors. A key issue in both cases is tofind feasible ways to store electrical energy.Given that the most mature renewable energy technologies are wind and solar,which are generally not readily available when the demand is high, highly efficientenergy storage systems integrated with the electric grid are required for a real shifttowards renewable energy. Electrochemical storage systems (i.e., batteries andcapacitors) are able to store the surplus energy from renewable energy power plantsto be used when the electricity demand is higher than the supply [45]. Batteries inparticular are highly efficient in storing energy and have long lifetimes, with Li-ionbatteries being able to provide a higher energy return factor than other types [46].Turning to the transportation sector, sustainable transport requires the replace-ment of the internal combustion engine (ICE) by zero (or low) emission vehicles,such as electric vehicles (EVs) and hybrid electric vehicles (HEVs). These requirepower sources with high output and energy density, as well as fast charging times.Li-ion batteries play already a key role as power sources in HEVs and EVs, as theyare considered the best available option [47]. Indeed, over the last few years Li-ion6powered vehicles moved very successfully from proof-of-principle prototypes towidely used commercial products (see Fig. 1.1) [48, 49].Figure 1.1: Li-ion battery market expansion (2000-2015). Adapted from Ref. [50].Conventional Li-ion batteries have a graphite anode, a lithium metal oxidecathode (e.g., LiCoO2, LiMO2) and a liquid or polymer gel electrolyte, which isa solution of lithium salt in an organic solvent (such as LiPF6 in EC-DMC). Theworking principle of such a battery involves the (reversible) transport of Li+ betweenan anode through the electrolyte towards a cathode. A schematic of a conventionalC/LiCoO2 battery can be seen at Fig. 1.2.When the battery is charged, the lithium ions are stored in the anode. Duringdischarging, lithium ions diffuse through the electrolyte and separator diaphragmand into the cathode, where Co ions change valence to accommodate the oxidizedLi+. The resulting charge imbalance drives the flow of electrons in the externalcircuit through the load. Note that in order to avoid short circuiting the currentthrough the load circuit, the electrolyte has to be able to conduct the Li ions butat the same time be an electrical insulator. During charging, a voltage is appliedto drive electrons back to the anode which causes the lithium to diffuse from thecathode back into the anode.These conventional Li-ion batteries work with a typical voltage of ∼4 V and aspecific energy 100 Whkg−1 to 150 Whkg−1 [51].Conventional Li-ion batteries present, however, a number of challenges. Theseinclude low energy density, high cost, as well as relatively fast aging [53]. Thesecharacteristics ultimately relate to the motion of Li+ interstitials in such materials, interms of how fast Li+ diffuses, but also to the extent of Li uptake and intercalation,the effect of Li concentration on its motion, as well as its fate as it diffuses to7Figure 1.2: Schematic of a (C/LiCoO2) Li-ion battery. Adapted from Ref. [52].the interfaces of such materials. In particular, diffusion of lithium ions insidematerials used in Li-ion batteries is important, since it ultimately determines thecharging/discharging rate of the battery.The need to overcome these drawbacks, as well as to enhance Li-ion batteries’relative advantages, motivates the search and study of novel electrode materialswith superior characteristics. To that end, present-day research efforts focus onfinding materials to replace current electrode and electrolyte materials with othershaving better characteristics in terms of cost, energy/power, safety and lifetime.Replacement materials for the graphite and LiCoO2 electrodes should have a higherenergy density and/or lower cost, whereas a replacement of the liquid electrolytesolution with a solid electrolyte would result in safer Li-ion battery systems, since theliquid electrolytes tend to be more susceptible to fires and hydrogen gas liberation.To be able to identify potential Li-ion battery materials with superior char-acteristics, experimental techniques capable of studying Li diffusion in differentsystems and under different conditions are required. Sect. 1.2.2 describes the mostcommonly used methods for studying Li diffusion in scales of µm and above andSect. 1.2.3 the techniques capable of investigating nanoscale Li diffusion.In addition, understanding the characteristics of the Li motion across interfaces8of materials is also of great importance, as in an actual battery Li+ will have to beable to seamlessly move across material boundaries to diffuse from one electrode tothe other across the electrolyte.1.2.2 Experimental Methods for Studying Solid State DiffusionMany experimental techniques have been used to study diffusion in solids. Theseare generally split into direct and indirect methods (see Fig. 1.3).Figure 1.3: Direct and indirect methods for studying solid state diffusion withtheir ranges in terms of diffusivity (D) or mean residence time (τ¯). Adapted fromRef. [54] with the addition of β -NMR and α-radiotracer (in blue).The direct methods rely on Fick’s laws (see Ch. 4) and relate directly tothe macroscopically defined diffusion coefficient D. In this sense, they can bethought of studying macroscopic (or long-range) diffusion. Important examplesof such techniques include the radiotracer method [55, 56], secondary ion massspectroscopy (SIMS) [57], electron microprobe analysis (EMPA) [58] and fieldgradient NMR [59]. Spreading resistance profiling (SRP) is used to study dopantdiffusion in semiconductors. It can be considered to be a direct method as it cangenerate a depth profile for the spreading resistance, even though it requires a9transformation to convert that into a concentration profile. The technique developedas part of this thesis, namely the nanoscale 8Li α-radiotracer method, also belongsin this category.Even though most NMR techniques study diffusion indirectly (see below),field-gradient NMR (FG-NMR) is considered as a direct method. It employs a fieldgradient (either static or pulsed) over the sample and permits thusly the measurementof macroscopic diffusion without the need of invoking an atomistic model for thediffusion. FG-NMR, just like all NMR techniques – with the exception of β -NMR – require big samples and therefore is not applicable for nanoscale diffusionmeasurements, as it is limited by the gradient (dB0/dz) that can be produced.The radiotracer method is the most commonly used technique for studyingself diffusion in solids, provided that radioactive isotopes of the diffusing elementwith suitable half-lives exist. This technique studies diffusion by introducingtracer quantities of the radioisotope at one side of the sample (see Fig. 1.4). Aftersome time, the sample is cut in slices and the diffusion coefficient is extracted bymeasuring the radioactivity of each slice. Even though this technique is suitable overa large range of diffusion coefficients, it is not applicable for nanoscale diffusion.Moreover, it cannot study Li diffusion, because of the lack of suitable radioisotopes– the Li radioisotope with the longest half-life is 8Li, but its 0.8 s half-life is notsufficient for this method –.There are variations of the above technique, which use non-radioactive tracerswith mass spectroscopy detection instead of radioactivity and are therefore able tostudy Li diffusion. They follow the steps outlined at Fig. 1.4, but they measure thedensity of the imported tracer in each slice by some other means, e.g., by opticalabsorption [60].The rest of the direct techniques are of a narrower scope and applicability. SIMSuses sputter profiling and is therefore capable of studying low diffusivities oversmall diffusion scales. Auger electron spectroscopy (AES) is only able to studythe diffusion of foreign atoms, as it cannot distinguish between different isotopes.Rutherford back scattering (RBS) is well suited for the study of heavy solutes inlight solvent, in contrast with nuclear reaction analysis (NRA), which can study lightsolutes, provided that there is a suitable nuclear reaction with a narrow resonance.On the other hand, indirect methods are techniques for studying diffusion thatare not directly based on Fick’s laws. They rather infer the diffusion rate of atomsin a solid by measuring phenomena related to the atomic motion. They directlymeasure atomistic quantities such as correlation and relaxation times(rates), line-widths, etc. From these, the hop rate τ−10 is extracted, which in turn is translated intoa macroscopic diffusion rate using Einstein-Smoluchowski’s law (see Sect. 4.2.3).Most of the indirect methods are based on nuclear or radioactive signals andprobes. One important exception is impedance spectroscopy (IS) [61], which10Figure 1.4: The steps for measuring diffusion with the (radio)tracer method. Firstthe tracer is deposited at one surface, then the sample is annealed for some time.Finally, it gets sliced in thin layers and the tracer quantity of each slice is measured.From that, the diffusion coefficient can be extracted. Adapted from Ref. [54]. Incontrast, both β -NMR and the 8Li α-radiotracer method are non-destructive to thesample.measures the complex electric impedance of the material under study versus ACfrequency. Mo¨ssbauer spectroscopy (MBS) is not widely applicable because ofthe small number of Mo¨ssbauer isotopes. The most important of them is 57Fe,which makes possible to study Fe diffusion with MBS [62]. Quasi-elastic neutronscattering (QENS) [63] can be also used to study fast diffusion rates, provided thatthe material of interest has isotopes with large quasi-elastic scattering cross-sections.Nuclear magnetic resonance (NMR) along with spin-lattice relaxation spec-troscopy (SLR) [54] are widely used to infer diffusion rates. They are applicable ina broad range of diffusivities and for a large selection of diffusing species. Thesetechniques employ the nuclei of certain lattice atoms as spin-probes and they usethe line-width of the resonance and/or the spin-lattice relaxation rate to extract thediffusion rate of interstitial ions.1.2.3 Studying nanoscale Li diffusionNanostructured Li-ion battery materials (nanoparticles, nanofibers, thin-films, etc)have been found to exhibit substantially increased mass transport capabilities [64–67] compared to their bulkier forms, leading to enhanced electrode charge-dischargerates. This stems from the shorter diffusion lengths for Li+ transport betweenthe different parts of the battery, but also because of the much larger electrode-electrolyte contact areas, that allows for a highly increased lithium intercalation.Even though there are many techniques capable of studying Li diffusion, veryfew of them are applicable to study Li diffusion at the nanoscale – or in nano-sizedsamples – and all of them are indirect. They include impedance spectroscopy11(IS) [68], low-energy muon spin rotation (LEµSR) [69] and β -detected nuclearmagnetic resonance (β -NMR) [22, 29].In the literature, there are many cases of materials that the various methods yielddiffusion rates differing by many orders of magnitude under the same conditions,e.g., in rutile TiO2 [29, 60, 68, 70–73], or in LiCoO2 [74–76]). A direct methodapplicable to the nanoscale could potentially shed light on these discrepancies.To that end, we developed a spin-off of the classical radiotracer method, namelythe 8Li α-radiotracer method, which uses the attenuation of the (grand)daughterα-particles coming from the radioactive decay of 8Li, in order to study nanoscaleLi diffusion.This direct method for studying Li diffusion differs from conventional radio-tracer diffusion experiments in several key areas: it is non-destructive to the sample;it is sensitive to atomic motions over the nanometer length scale [56, 76–78]; and itis amenable to the use of short-lived radioisotopes with relatively short half-livesτ1/2 (Li has no radioactive isotopes with τ1/2 1s). It makes use of a focused 3 mmdiameter low-energy 8Li+ ion beam to inject a target material with the radiotraceratoms. Both during and following the beam pulse, the temporal evolution of theα-decay signal is monitored, whose yield and shape are correlated to the rate ofarrival at the crystal surface and the details of the diffusion boundary conditions(e.g., a reflective or absorptive surface).Such a technique has been developed in Japan for micrometer Li diffusion [79–83] and recently also for nanometer-scale studies [56, 76–78]. As part of thisthesis, a new incarnation of the above technique has been developed (Ch. 5 andCh. 6), sharing many characteristics with its Japanese version, but also having keydifferences (see Ch. 5). This method is demonstrated not only to be able to extractthe nanoscale diffusion rate of Li ions, but also to indicate what happens to Li ionswhen they reach the sample surface.1.3 Organization of this ThesisThis thesis presents the recent developments on using 8Li as a probe in condensedmatter physics. For the case of β -NMR, we show how comparing 8Li and 9Lione can determine the source of spin relaxation. In addition, we show that theα-particles emitted as part of the decay process can be used to track the motion of8Li or to enhance the beta-decay asymmetry of 9Li.Chapter 2 is intended as an introduction to β -NMR. It includes an outline ofthe physical basis of the method, a synopsis of the experimental infrastructureemployed, as well as a presentation of the different types of measurements that areused in this work.12Chapter 3 then presents the development of the αLithEIA isotopic comparisonmethod, which makes use of 8Li and 9Li beams to identify the source of spin-latticerelaxation in β -NMR. This new tool for β -NMR can be very useful in cases that itis not clear whether the primary source of relaxation in a sample is of electric ormagnetic origin. However, 9Li β -NMR is much more time-consuming and tedious(compared to 8Li), due to the suppressed asymmetry signal associated with thatisotope. To amend this issue, the αLithEIA method was developed, which tags the9Li β -decays in coincidence/anti-coincidence with an α-particle and is shown toenhance the 9Li β -NMR signal by a factor of ∼ 2 (and theoretically it can enhanceit by a factor of ∼ 3).Chapters 4–6 outline the other application of using the α-decay signal in β -NMR, namely studying Li diffusion at the nanoscale (i.e., the 8Li α-radiotracermethod).Chapter 4 provides the theoretical background of interstitial diffusion theory,both from a macroscopic and a microscopic point of view.The basis of the 8Li α-radiotracer method is covered in Chapter 5, whichpresents the calculations and Monte Carlo simulations which were carried out aspart of this work to establish the feasibility of the technique, as well as to providethe tools needed for the extraction of the diffusion coefficient from the experimentaldata.Chapter 6 follows the development of the 8Li α-radiotracer technique by pre-senting the study of Li diffusion in rutile TiO2. With these measurements, we wereable to extract both the diffusion coefficient of Li in various temperatures, as well asthe energy barrier for Li diffusion. In addition, our measurements show that Li+ hasa high probability (≥ 50%) to get trapped upon reaching the (001)-surface of rutile.Finally, Chapter 7 summarizes the above results and identifies possible futurework to further develop this study.This thesis contains material taken from three of the author’s papers [1–3].Chapter 2 and 3 are partially taken from References [1, 2], while Chapter 5 and6 contain material from Reference [3]. This introductory Chapter also partiallyincorporates material from all these papers.13Chapter 2Beta-detected Nuclear MagneticResonanceThis Chapter is intended to be a short introduction to the β -NMR technique, bydiscussing the physics behind it (Section 2.1), the details of the radioactive, spin-polarized beam production and delivery to the spectrometers (Section 2.2), theinteractions of the spin-probes with their environment once inside the materialunder study (Section 2.3), as well as the experimental modes used to infer theseinteractions (Section 2.4). Finally, Section 2.5 presents the merits of using theinformation from the α-decay in parallel to the β -decay in order to enhance thecapabilities of β -NMR, which is the primary focus of this thesis.2.1 Physics of β -NMRThe basis of β -NMR is the parity-violating (nuclear) weak interaction, whereby thedirection of the emitted electron (positron) from the decaying nucleus is correlatedwith the nuclear spin polarization at the time of decay:W (θ) = 1+βapcos(θ) (2.1)where β = ν/c is the velocity of the high energy electron (positron) normalizedto the speed of light, p is the magnitude of the nuclear polarization vector, θ is theangle between the nuclear polarization and the electron (positron) velocity and ais the asymmetry parameter depending on the properties of nuclear β -decay (seeFig.2.1).The resulting anisotropic decay pattern for the high energy electron allows oneto monitor the nuclear polarization from highly polarized 8Li+ beams (or otherβ -NMR nuclear probes) implanted in the sample. In particular, the asymmetry inthe count rate at time t between two opposing β -detectors is proportional to thecomponent of nuclear polarization along the direction defined by the two detectors:A(t) =NB(t)−NF(t)NB(t)+NF(t)= A0 pz(t) (2.2)14Figure 2.1: Anisotropy of the direction of the emitted β -particle based on thepolarization of the parent nucleus. The asymmetry of β -counts at two opposingdetectors (B and F) can be used to monitor the degree of nuclear polarization versustime. Adapted from Ref. [22].where NB(t) and NF(t) are the counts measured in the backward and forwarddetectors, pz(t) is the component of nuclear polarization along the z-axis definedby the detectors, and t is the time of decay after implantation. The detectors aregenerally positioned so that z is along the direction of initial polarization.Note that the asymmetry in the count rate has a maximum value of A0 at t = 0which is reduced relative to the theoretical asymmetry a, as calculated from thenuclear properties, owing to instrumental effects such as the finite solid anglesubtended by the detectors and scattering of the β -particles before reaching thedetectors. Note also that pz(t) and thus A(t), are time dependent, reflecting the factthat the nuclear polarization is subject to spin relaxation processes in the sample(see Sec. 2.3), which in fact is the quantity of interest in a β -NMR experiment.By changing the energy of the beam (between 0.1 keV to 30 keV) the meanimplantation depth can be tuned in a range of approximately 10 nm to 200 nm. Thus,it is possible to perform depth-resolved experiments, which makes this techniqueideal for the study of near-surface phenomena [19] (see Sec. 2.2.3).The most commonly used β -NMR isotope at TRIUMF is 8Li, which has alifetime of τ =1.21 s, spin I = 2, gyromagnetic ratio γ =6.3015 MHzT−1, and arelatively small –but non-zero– electric quadrupole moment Q = +32.6 mb [84].The fact that 8Li is not a purely magnetic probe makes it possible to study electricquadrupolar phenomena. Moreover, the theory of nuclear β -decay predicts that thetheoretical asymmetry parameter a (Eq. 2.1) of 8Li is about 1/3.Typical radionuclides that have been used as β -NMR probes can be seen at15Tab. 2.1. This thesis focuses exclusively on 8Li and 9Li.Table 2.1: Intrinsic nuclear properties of radioisotopes used in β -NMR. Ipi isthe nuclear spin (and parity), µ is the magnetic moment, and Q is the electricquadrupole moment. When Q is non-zero, the probe is sensitive to the local electricfield gradient else it is sensitive to the local magnetic field alone.Ipi τβ (s) µ (µN) Q (mb)8Li 2+ 1.2096(5) [85] +1.653560(18) [86] +32.6(5) [84]9Li 3/2− 0.2572(6) [87] +3.43678(6) [86] -31.5(5) [84]11Be 1/2+ 13.74(8) [88] -1.6813(5) [89] -31Mg 1/2+ 0.236(20) [89] -0.88355(15) [90] -8Li β -decays to the first excited state of 8Be, which in turn decays with a shortmean lifetime of 3.01×10−22 s into two α-particles, emitted at an angle of 180o inthe center-of-mass frame of the 8Be daughter nucleus (Eq.(2.3)).83Li→84 Be∗+ e−+ ν¯e84Be∗→ 2α (2.3)The average energy of the emitted β is∼6 MeV, but due to the three-body decaykinematics, it varies continuously between 0 MeV to 12.5 MeV, with the upper limitdefined by the Q-value of the decay and the mass of 8Be (Fig. 2.2a). The meanenergy of the emitted α-particles is 1.6 MeV with a full width at half maximumof 0.6 MeV due to lifetime broadening. Because of quantum mechanical mixingof the first and the higher excited states of 8Be [91], the energy spectrum of eachα-particle has an asymmetric high energy tail (Fig. 2.2b).2.2 Experimental Details of β -NMR at TRIUMFThis section describes the process of how TRIUMF produces and delivers to theexperimental spectrometer(s) a clean, highly-polarized, intense beam of the re-quired isotope (Sect. 2.2.1). The next part then gives a detailed description of thecharacteristics of the high-field and low-field β -NMR spectrometers (Sect. 2.2.2).16(a)0 2 4 6 800.511.5E (MeV)RelativeYield(b)Figure 2.2: Energy distribution of the β -particles (left) coming from the decay of8Li (Adapted from Ref. [92]) and that of the α-particles (right) coming from thesubsequent decay of 8Be [93].2.2.1 Polarized Beam Production and DeliveryAt TRIUMF, the first step towards the production of the spin-polarized radioactivebeam of the specific β -decaying isotope required for the β -NMR experiments (e.g.,8Li, 31Mg, etc) is to extract and accelerate a beam of negatively charged hydrogenions. An ion source ionizes hydrogen gas to produce 11H− ions, which get initiallyaccelerated by a sixty-meter-long electrostatic linear accelerator. This beam is theninjected into the main 520 MeV cyclotron (see Fig. 2.3).The main cyclotron accelerates the negatively charged hydrogen ions using analternating electric field with a frequency of 23 MHz and a 4000-Tons six-sectornormally-conducting magnet made of iron. The magnet produces a magnetic fieldof 0.56 T using 18500 A of current. The Lorentz force produced by the combinationof the electric and magnetic fields accelerates the ions up to 75% of the speed oflight after 1500 revolutions in an outward spiral trajectory inside the cyclotron tank.At the end of the acceleration process, the ion beam passes through a thin carbonfoil, which strips the two electrons from the negatively charged ions to produce bareprotons. This process reverses the charge of the accelerating beam, which, in turn,reverses the direction of the cyclotron motion and guides the ion beam out of thecyclotron and into the post-acceleration beamline. This extraction process increasesthe efficiency of the main cyclotron significantly, relative to extracting a beam ofbare protons, allowing for a more intensive beam to be produced (up to ∼100 µA).The beam of 520 MeV protons is then directed towards the Meson Hall, or the17Figure 2.3: Internal view of the 520 MeV main cyclotron vault at TRIUMF. Noticethe size of the vault compared to the man at the center. Adapted from Ref. [94].ISAC facility. In the Meson Hall, it is used to treat ocular melanoma of cancerpatients, by selectively depositing the proton’s energy in the tumor [95], or itbombards a suitable solid target, which creates a secondary beam of pions whichfurther decays into a muon. The muons are used to study the properties of materialswith the µSR technique [96].When a beam of a specific isotope is requested, the proton beam is directedtowards the Isotope Separation and Acceleration (ISAC) facility [97], where itcollides with a carefully engineered target and creates a variety of isotopes [98]by nuclear spallation. The target is chosen to enhance the production and releaseof the desired isotope(s) and is made, amongst others, by Tantalum (Ta), SiC orUranium(U)/Uranium Carbite(UC). Ta is the optimal target for production of alkalimetals such as 8Li. The target is kept at a temperature of ∼ 2000 to 2500 ◦C anda positive voltage of 28 to 30 kV. The high temperature allows the newly formedisotopes to diffuse fast to the surface of the target and exit the target vessel througha tube of 1-2 mm diameter. For the production of short-lived isotopes – with halflife ∼10 µs – keeping this diffusion time short can be critical. This is yet anothermotivation for trying to understand diffusion in solids (see Ch. 4-6).The positive potential of the target (relative to the grounded extraction electrode)accelerates the ions leaving the target via the small diameter tube and forms anintense beam with a cross section of a few mm, moving towards a high-precisionmagnetic mass separator. The magnetic field bends the trajectories of the differentisotopes by a different angle (depending on their momentum/mass) thereby ensuringthat only ions of the desired isotope reach the experiment. 8Li is very easy to mass-separate, since there are no stable isotopes with a similar mass. Using a typicalproton beam current of 40 µA, the ISAC target can seamlessly provide a 8Li+ beam18with an intensity of 108 ions/s.Before reaching the β -NMR experimental spectrometer(s), the radioactive beamis (nuclear-)spin-polarized in flight by a single-frequency dye laser [39]. Thistechnique is able to polarize the 8Li ions to a ∼ 70% [99] level.Figure 2.4: Schematic of the polarizer and the section of the beamline leading tothe β -NMR and β -NQR spectrometers. Note that the neutralizer cell is currentlyfilled with Rb vapor, not Na. Adapted from Ref. [39].The first step towards producing a highly spin-polarized beam is to neutralizethe beam by passing it though a cell filled with Rb vapor, held at a temperatureof ∼250 ◦C. The Rb vapor is confined in the cell using a recycling jet target.Alkali metals can easily lose one electron to the positively charged ions, thus theyeffectively neutralize the beam with a typical efficiency of 50%. The neutralizedbeam then drifts by 1.9 m inside the optical pumping region, whereas the part ofthe beam that failed to get neutralized is dumped into a Faraday cap by a pair ofelectrostatic reflection plates. A small longitudinal magnetic field (∼2 mT) is presentinside the optical pumping region, for the purpose of defining the polarization axis.The neutralized beam overlaps with a laser beam created by a dye laser source,counter-propagating in the polarizer beamline. The laser light is circularly polarizedwith respect to the polarization axis – which is also the axis the beam is propagatingon – and is tuned to the Doppler-shifted D1 transition close to 671 nm for 8Li (seeFig. 2.5). This wavelength corresponds to the energy needed for the transition fromthe ground state 1S1/2 to the first excited state 2P1/2 of 8Li. Because the groundstate of 8Li is split into two hyperfine levels, as is the case with many alkali isotopes,19to achieve a high level of polarization, the light has to be tuned at both sub-levels’wavelengths. For the polarization of other isotopes/elements the laser frequenciesare tuned using similar considerations.Figure 2.5: Schematic of the spin polarization method for 8Li. The upper imageshows the ground state and first excited state with their hyperfine splittings. Thelower image shows the pumping of the positive helicity (σ+) after a few cycles ofoptical pumping. Adapted from Ref. [100].The Zeeman splitting due to the small longitudinal magnetic field further splitsthe hyperfine levels to 2F+1 sub-levels mF , where F= I+J. In the lower part ofFig. 2.5, the effect of illumination with σ+ (i.e., positive helicity) circularly polar-ized laser light is shown. The only allowed absorptions are the ones that increase themagnetic quantum number mF by +1. This effect populates gradually the highestangular momentum state of +5/2, with both nuclear and electronic polarizationhaving their highest permitted value. Spontaneous fluorescence – which is allowedfor transitions satisfying ∆mF =0,±1 – has a lifetime of 27 ns, thus, the drifting timethrough the optical pumping region (∼2 µs) allows for the 8Li atoms to go throughdozens of pumping cycles of laser light absorption and fluorescent emission.The net effect of this process is that the 8Li nuclei are nuclear-spin-polarized20and the majority ends up at the nuclear spin state mI=+2 (-2 for the σ− helicity).This corresponds to a high level of nuclear polarization (∼70 %), which is definedas:P(t) =1I ∑mImI pmI (t) (2.4)where mI is the nuclear spin quantum number, P(t) is the nuclear polarization(along the axis defined by the external magnetic field) versus time and pmI (t)corresponds to the population of the mI-th state at that point in time.After the polarizing process, the neutral beam enters into a cell filled with coldHe gas that re-ionizes the ions with an efficiency of about 1/2. The part of thebeam that is left neutral is dumped at the neutral beam monitor (NBM) – made by athin palladium foil – which measures its intensity and polarization. The re-ionizedpart of the beam is steered by an electrostatic bender, which delivers it to eitherthe high-field or the low-field spectrometer, referred usually as the β -NMR andβ -NQR spectrometers, respectively. These electrostatic benders do not affect thepolarization direction, which is determined by the helicity of the laser beam directedalong the axis of the polarizer.The section of the beamline after the polarizer leading to the β -NMR spec-trometer includes three Einzel lenses, which electrostatically focus the beam to thecenter of the sample. Two sets of manual slits can be used to collimate the beam andreduce the beam spot on the sample. The size and the position of the beamspot is avery sensitive function of a large number of parameters. The beam is accelerated toa certain energy when extracted from the ISAC target and its trajectory and size areinfluenced by the upstream elements (Einzel lenses, magnetic quadrupoles), as wellas the magnetic field B0 inside the spectrometer. If the beam reaches the spectrome-ter perfectly centered on the magnet axis of the superconducting solenoid, then theEinzel lenses and the bias voltage of the sample should not affect its trajectory. Inreality it is very hard to tune the beam to arrive perfectly, so the combined effect ofthe magnetic field and the electrostatic bias results into a spiraling motion leadingto a beam spot which depends slightly on the magnetic field and the decelerationpotential.To ensure a good beamspot in all required magnetic fields and electric biases,before the start of the experiments, the ISAC operator tunes the EM elements ofthe beamline with 7Li+ until good beam transport is achieved and then fine-tunesthe last downstream elements using a 8Li+ beam until a centered, adequately smallbeamspot is achieved in all required conditions. For each condition (magneticfield and electrostatic bias) the beamspot is imaged by a charge-coupled device(CCD) camera lying outside vacuum behind an optical port, using a scintillating21sapphire at the sample position. The camera images the front surface of the sampleusing a mirror installed inside the main chamber which bends the optical path byapproximately 90◦.2.2.2 β -NMR and β -NQR SpectrometersFor all the experiments related to this thesis, TRIUMF’s dedicated β -NMR andβ -NQR spectrometers were used. In both spectrometers, the sample material is keptunder ultra high vacuum (UHV) at the end of ISAC’s beamline. A superconductingmagnet can generate a magnetic field of up to 9 T around the sample of the high-field β -NMR spectrometer, pointing along the probe ions’ spin polarization. Aregular conducting Helmholtz magnet is capable of creating a field of 0-20 mT(i.e., 0-200 Gauss) at the low-field β -NQR spectrometer. The platforms on whichthe spectrometers are positioned are electrically isolated from ground so they canbe raised to several kV, effectively increasing the electric potential of the samples.This way, the mean implantation depth of the radioactive beam inside each samplecrystal can be controlled by the applied platform voltage [101]. In this section, thecharacteristics of these two instruments will be presented. As part of this thesis, anupgraded version of the β -NQR cryostat – the so-called “cryo-oven” – was designedand fabricated. Its specific characteristics can be found in Appendix A. The β -NMR spectrometerThe high field spectrometer, usually referred to simply as the “β -NMR spectrome-ter”, has a longitudinal geometry, in the sense that the nuclear polarization and thedirection of the external magnetic field all coincide with the direction of the beam’smotion and they are perpendicular to the surface of the sample under study.The β -NMR apparatus is a cold finger UHV-compatible cryostat from OxfordInstruments (see Fig. 2.6), capable of maintaining a vacuum of up to 10−10 Torr.This level of vacuum is achieved via a system of load locks that allow for changingthe sample without venting the main chamber. To change the sample, the cryostat isdriven back by a motor that stretches a bellows, out of the bore of the superconduct-ing magnet, until the sample holder port of the cryostat lies directly below the loadlock and gate valve shown in the figure below.The sample is mounted outside vacuum on an aluminum sample holder, at theend of a stainless steel rod (see Fig. 2.7). The typical size of β -NMR-compatiblesamples is 8 mm x 10 mm x 0.5 mm. Thin films, or samples of smaller sizesare mounted on a substrate (usually made of sapphire due to its high thermalconductivity and natural scintillation properties) by using UHV-compatible silverpaint. The sample is then attached on the sample holder by two small Al clamps.22Figure 2.6: High-field β -NMR spectrometer. The ion beam enters from the leftside, passing through the hole of the upstream scintillation detector and reaches thesample sitting at the center of the deceleration space with the polarization parallelto its momentum. A cryopump is used to maintain the UHV vacuum in the samplespace and a cold finger cryostat to control the sample temperature. Adapted fromRef. [33].The sample holder is mounted into the cryostat using a load lock so the mainchamber can be maintained at UHV during sample changes. After the sample ismounted, the cryostat is driven back in the bore of the magnet. Typically a samplechange requires between 45 min and one and a half hour.The sample is positioned at the center of a superconducting solenoid magnetcapable of generating a uniform static magnetic field B0 of up to 9 T. Since thenuclear polarization is longitudinal, the β -detectors are positioned upstream anddownstream of the sample and initial polarization direction (see Fig. 2.1). Both areplastic scintillation detectors. The downstream detector – the so-called “Forwarddetector” – lies outside vacuum behind a thin stainless steel window. The β particleseasily pass through the window without much attenuation compared to their averageenergy (see Fig. 2.2a), but all α-particles are blocked. The upstream detector –the “Back detector” – has a hole in the middle to allow for the incoming beam toreach the sample. Due to the focusing effect of the strong magnetic field, the Backdetector has to be positioned outside the bore of the magnet in order to allow theβ particles to reach it, instead of moving further upstream by passing through itscentral hole. The two detectors cover very different solid angles in the absence of a23Figure 2.7: β -NMR sample holder. At the rightmost side the end of the sample rodis visible, on which the copper part of the sample holder is screwed. The sample ismounted on the sample holder using two small clamps.magnetic field, but they have similar effective solid angles, when a high magneticfield is applied.A non-resonant helical transmission line can generate a transverse radio-frequency(RF) field B1(t) in a range of frequencies ω up to 45 MHz. The direction of B1(t) isperpendicular to both the static magnetic field B0 and the polarization of the beam.It is used for frequency-resolved experiments (see Sec. 2.4.1).The temperature of the sample is controlled using a combination of liquid heliumflow and a resistive heater. A Lakeshore temperature controller can automaticallymaintain the sample at a constant temperature by balancing the cooling power ofliquid helium with the appropriate amount of current passing through the heater.This allows one to control the sample temperature in the range 3.5 K to 318 K. Thewhole apparatus sits on a platform, on which a biasing voltage of up to ±30 kV canbe applied. The β -NQR spectrometerThe low-field β -NQR spectrometer differs from the high-field β -NMR spectrometerin several ways. In contrast with β -NMR, the geometry of the β -NQR spectrometeris transverse. The beam is implanted into the sample with its momentum normal tothe sample’s surface, but its nuclear polarization is perpendicular to its momentum,i.e., parallel to the surface of the sample-material. This is evident from Fig. 2.4 sincethe polarization is longitudinal in the polarizer. The two 45o benders in Fig. 2.4 areelectrostatic and thus do not influence the direction of nuclear polarization.The static applied field B0 can be completely absent, permitting for zero-fieldexperiments. For low-field measurements, B0 is oriented (anti)parallel to the po-larization of the beam. It is generated by passing current through a Helmholtz24coil, which creates a uniform magnetic field around the sample (of 0-20 mT) at thesample position.A small vertical Helmholtz coil can generate a transverse RF field B1(t) in arange of frequencies from 0-2 MHz. The direction of this field is perpendicular toboth the polarization of the beam and its momentum. In other words, if the beammomentum vector defines the z-axis of a lab-frame, then the polarization vectorwould be on the y-axis and the RF magnetic field on the x-axis (see Fig. 2.8).As discussed in Sec. 2.1, the β -detectors should be positioned along the linedefined by the initial nuclear polarization. Thus, in the case of the β -NQR spec-trometer, they are to the left and right of the sample, lying outside vacuum behindtwo thin stainless steel windows. They are naturally referred to as the “Left” and“Right” detectors, respectively. They are fast plastic scintillators which convert theenergy of the incoming β -particles into light, which is turned into electrical signalby a photomultiplier tube (PMT).Figure 2.8: Over-view of the β -NQR spectrometer. The ion beam enters from thetop of the figure (parallel to the z-axis), with its polarization along the y-axis, onwhich the two β -detectors are situated. Note that the coordinate system in thisover-view is slightly unusual, as the z-axis is generally defined to be along thedirection of the magnetic field and the nuclear polarization. Adapted from Ref. [33].The samples are loaded into the cryostat from above, through a vacuum load-lock, using a long (∼1.6 m) stainless steel rod and a process similar to that forβ -NMR. The β -NQR sample holder has four sample positions (see Fig. 2.9). Eachposition can be filled with a sample of dimensions 12 x 12 x 0.5 mm3 a priori,25reducing thus the number of sample changes needed during an experimental runperiod. Sample changes need to be done at ∼300 K. Each sample change typicallytakes about 1-2 hours. The rod can be turned by hand from outside vacuum aroundthe x-axis of Fig. 2.8, thus introducing an angle between the sample surface andthe beam axis. This allows for angle-resolved experiments, which are crucial forstudying electric quadrupolar phenomena [29].Figure 2.9: The β -NQR sample holder (made of copper) with four samples mounted.In contrast to the β -NMR one, it has four sample stations, which reduces signifi-cantly the number of sample changes needed during an experimental run period.Similarly to the β -NMR spectrometer, the temperature of the sample is con-trolled by a combination of liquid helium flow and a heater, using a Lakeshorecontroller to stabilize or change the temperature effectively. The lower thermal massof this cryostat permits a much faster temperature change (tens of K/min) comparedto the β -NMR cryostat (1-2 K/min). The β -NQR cryostat has a temperature rangebetween 4.5 K and 300 K.The β -NQR platform can be raised to a high voltage of 30 kV allowing the en-ergy of implantation to be adjusted between 0.1 keV and 30 keV which correspondsto a mean implantation depth between about 10 and 200 nm.2.2.3 Beam Implantation ProfilesChanging the energy of the ion beam through varying the β -NMR/NQR platformsbias voltage allows for depth resolved measurements. The higher the energy ofthe ion beam, the deeper the incoming ions stop on average. The energy rangeaccessible with ISAC’s ion beams permits for a mean implantation depth of 10 nmto 200 nm. This can be used to study, among other things, how the temperature ofphase transitions depends on proximity to the sample’s surface [19], the dynamicsclose to a buried interface of two heterostructures [20], studying the vortex latticeat a superconducting substrate [102], or long-range magnetic effects in thin films26caused by the proximity to a magnetic [103] substrate. In all these cases, it isparamount to know the depth profile of the β -NMR ion probes upon implantation.The ion implantation (statistical) profile can be simulated using the Stoppingand Range of Ions in Matter (SRIM) software. SRIM is a well established Monte-Carlo-based algorithm used extensively in condensed matter physics. It generatesthe statistical stopping distribution of the incoming ion beam by simulating thetrajectory of each implanted ion individually (see Fig. 2.10). Each ion follows adifferent path due to stochastic processes, such as Rutherford scattering to differentlattice ions resulting into abrupt changes in the direction of motion, as well asdifferent momentum transfers between the incoming ions and the lattice.Figure 2.10: Example of 8Li tracks after implantation in rutile TiO2, as simulatedby SRIM-2013.When the energy of the ion beam is increased, both the mean implantation depthand its standard deviation (also referred to as “ion straggle” in this context) of thestopping distribution increases. As an example, Fig. 2.11 depicts the depth profilesfor 8Li+ ions implanted in TiO2 for different beam energies.Comparisons between SRIM and experiments [104] show that for Li ions,81% of the simulated profiles that were compared to experiments differ from theexperimental points by less than 10% of the value predicted by SRIM.A known issue with SRIM predictions is that it treats all target materials asbeing amorphous. In some instances, though, the crystalline structure of the targetcould provide some channeling paths for the incoming ions along specific directions.If such a direction differs from the implantation angle below a critical angle, thenthe channeling effect cannot be neglected and the actual implantation profile couldhave a large tail towards the bulk of the crystal, which would be absent in the SRIMcalculation.270 20 40 60 80 100 120 140 160 180 200 220 240Depth (nm)Stoppingprobability(nm−1)SRIM|10 keVSRIM|25 keVFigure 2.11: 8Li beam implantation profile in TiO2 as simulated by SRIM-2013 fora beam energy of 10 and 25 keV. By increasing the energy of the beam, both themean implantation depth and the ion struggle increase.The critical angle of channeling is given by [105]:ψ ≈√(Z1e)(Z2e)E(d/2)(rad) (2.5)where Z1 is the atomic number of the ion beam, Z2 that of the target material,e2 =1.44×10−5 MeVA˚, E is the energy of the beam (in MeV) and d is the effectivedistance of the target’s atoms along the channeling pathway (in Angstroms).The effect of channeling of lithium ions implanted along the c-axis of rutileTiO2 will be discussed at Sect. 6.3.Apart from the implantation profile itself, SRIM can also provide estimates onother processes, such as the mean number of ion/hole pairs created on the track ofan ion, as well as the damage inflicted on the sample’s lattice due to the implantationprocess.2.3 Spin Interactions of β -NMR probes in a latticeAfter implantation, the β -NMR probe stops at well defined interstitial position(s)where there is a minimum in the potential energy surface. The nuclear spin polar-ization will evolve in time according to a spin Hamiltonian which describes theinteraction between the nuclear spin and the surrounding crystalline lattice.There are two basic kinds of interactions described by the corresponding spinHamiltonian. These can be generally categorized into magnetic (Sect. 2.3.1) and28electric quadrupolar interactions (Sect. 2.3.2), with the latter being applicable onlyfor nuclear probes with a non-zero electric quadrupole moment.2.3.1 Magnetic InteractionsThe nucleus, which is a compound system consisting of neutrons and protons, has acharacteristic magnetic moment µ proportional to its total angular momentum J:µ = γJ = γ h¯I (2.6)where γ is a scalar quantity called the gyromagnetic ratio, h¯ is the reduced Plankconstant and I is a dimensionless angular momentum (operator).As discussed in Sec. 2.2.1, the application of a magnetic field B lifts the degen-eracy of the magnetic quantum number m to form the nuclear Zeeman states. TheHamiltonian defining this process is given by:Ĥm =−µ ·B (2.7)Without loss of generality, one can set B = B0zˆ, so Eq. 2.7 can be rewritten as:Ĥm =−γ h¯B0Îz (2.8)Thus, the eigenvalues of Eq. 2.8 are the eigenvalues of Iˆz, multiplied by aconstant, i.e.:Em =−γ h¯B0m, m =−I,(−I+1), ...,(I−1), I (2.9)The nucleus can change its spin state by absorbing energy equal to the energydifference ∆E of the initial and final states:h¯ω = ∆E (2.10)This change can only happen if the interaction causing it contains a non-zero ma-trix element connecting the initial and final states. Experimentally, this is achievedby applying a (weak) oscillating magnetic field B1(t) = B1cos(ωt)xˆ perpendicularto the static field B0zˆ. The Hamiltonian governing this perturbation is:Ĥpert =−γ h¯B1cos(ωt)Îx (2.11)The matrix element of this Hamiltonian connecting states m and m′ is:〈m′|Ĥpert |m〉 ∝ 〈m′|Îx|m〉 (2.12)which is non-zero only if m′ = m±1.29This defines the amount of energy ∆E that should be given to the nucleusto change its state, as well as the corresponding resonance frequency ω0 of theoscillating B1 field given by Eq. 2.11:h¯ω0 = ∆E = γ h¯B0⇔⇔ ω0 = γB0(2.13)After the spin-probes stop in the sample, they interact with the net magneticfield Bnet at their stopping position, given by:Bnet = B0+Bint (2.14)where B0 is the (static) external field and Bint is the internal magnetic field dueto the surrounding material at the implantation site of the nuclear probe.Thus, their resonance frequency of absorption will differ from Eq. 2.13 and willbe instead:ωL = γBnet (2.15)where ωL is the Larmor resonance frequency of the nuclear spins in their crystalenvironment.In β -NMR, the net magnetic field is extracted by sweeping the frequency ωof B1. When it coincides with the Larmor frequency of the nuclear spin-probesunder the influence of Bnet , the nuclear spins will start to precess around Bnet (inthe rotating reference frame) and thus the polarization present will be diminished.In Sec. 2.4.1 the details of extracting Bint using this effect will be presented.2.3.2 Electric Quadrupolar InteractionsThe nucleus of each isotope consists of Z protons and N=A-Z neutrons, whereZ is the atomic and A the mass number uniquely defining the isotope. Becauseof the fact that the nucleus is not a point-like object, but rather a complicatedsystem of its constituent nucleons, the nuclear structure can be understood – froman electromagnetic point of view – as a series of electric and magnetic multipoles,that depend on the spin of the nucleus.For the simple case of I=1/2 the nucleus can be thought as an electric monopoleand a magnetic dipole (i.e., it has charge +Z and magnetic moment µ), but for I>1/2,there is also a non-zero electric quadrupole moment eQ, due to the fact that thedistribution of charge for high-spin nuclei is non-spherical. As a result, nuclei withspin I≥1 will interact with an electric field gradient (EFG) present at the nuclearposition.30The EFG is defined by a matrix, the EFG tensor, with each of its elementscalculated by:Vi j =∂ 2V∂xi∂x j(2.16)where V is the electromagnetic potential at the site of the probe nucleus, mea-sured in Volts. Vi j is measured in V/m2.This general tensor can be diagonalized by rotating the frame to the so-calledprincipal axis system (PAS). In this frame of reference, the EFG tensor has only threenon-zero elements across its diagonal, namely Vxx, Vyy and Vzz with det(V ) = 0.These three axes are selected so that |Vyy| ≤ |Vxx| ≤ |Vzz|. Vzz is then called theprincipal component of the EFG tensor.In the PAS frame, the Hamiltonian describing the interaction of the electricquadrupole moment of the nucleus with the applied EFG is given by:Ĥq =3eQVzz4I(2I−1)h¯ [I2z −13I(I+1)+η3(I2x − I2y )] (2.17)where I is the spin of the nucleus and η is the asymmetry parameter of the EFG.η takes values between 0 and 1 and is zero in the case of axial symmetry. It isdefined by the diagonal terms of the EFG tensor by Eq. (2.18).η =Vxx−VyyVzz(2.18)In the case of the spin-2 8Li, Hamiltonian (2.17) reduces to:ĤI=2q = hνq[I2z −2]+ηνq[I2x − I2y ] (2.19)where νq is the quadrupole (transition) frequency given by:νq =e2Qq8h=eVzzQ8h(2.20)where eq=Vzz is the principal component of the EFG tensor. Even at zeromagnetic field, the nuclear spin sub-levels |m > are split due to Hq. For spin-2,there are two resonant frequencies, one at ν = νq stemming from the |±1 >→ |0 >transition and one at ν = 3νq corresponding to the |±2 >→ |±1 > transition (seeFig. 2.12).The position of the resonant frequencies in a given crystal environment can bestudied with the β -detected nuclear quadrupole resonance (β -NQR) technique withzero applied field (B0 = 0). A β -NQR resonance spectrum (see Sec. 2.4.1) would31Figure 2.12: The energy splitting of the magnetic sub-levels of 8Li, due to the axialpart of Hamiltonian (2.19). Adapted from Ref. [106].reveal 2I satellite resonances due to the single quantum transitions (∆m =±1) and2I-1 resonances due to the double quantum transitions (∆m =±2).A change in the strength of the EFG would shift the resonant frequenciesof the β -NQR spectrum, whereas a departure from axial symmetry (i.e., η > 0)would introduce a mixing of the |±2 > states with a relevant frequency splitting∆±2 ∼ 3η2νq. Thus, β -NQR can in principle be used to study any structural phasetransitions in a material.2.4 Types of MeasurementsRegardless of the chosen spin-probe (e.g., 8Li, 9Li, 31Mg), or the spectrometer usedfor a specific β -NMR or β -NQR study, there are two basic types of measurementsthat can be employed for the study of the local properties of crystals with the β -NMR technique: Either the frequency of the transverse RF field is scanned to revealthe resonance(s) of the nuclear probes in their crystal environment using (usually) acontinuous ion beam, or the time evolution of the β -decay asymmetry is registeredin the absence of a RF field while using a pulsed ion beam. The former type isgenerally refereed to as a “frequency scan”, whereas the latter is known as a “SLRrun”.If a typical ion beam rate of 106 8Li+/s is assumed, both types of measurementswould require 15 min to 40 min for acquiring a run with reasonable statisticaluncertainties. For other isotopes this time can be much longer (see Chap. 3).2.4.1 Resonance SpectraBefore discussing fluctuations and spin relaxation it is important to note that thequasi-static (or time-averaged) parameters of the spin Hamiltonian (e.g., local32Figure 2.13: (a) Combined resonance spectrum for 8Li in a Bismuth single crystal at294 K with four single quantum transitions (SQT) at ν =±νq and ±3νq, interlacedby three sharper double quantum transitions (DQT). Due to the high degree ofnuclear polarization, most 8Li probes start at the |m = ±2 > state and thus the“outer” resonances corresponding to the |±2 >→ |±1 > transition (e.g., the onesat ν =±3νq for the SQT) have a larger amplitude than the inner ones, reflecting thelarger probe population of the state. (b) The same spectrum, but helicity-resolved.The fact that the position of the resonances is reversed for the two helicities indicatesthat the resonances are quadrupolar, as the sign of the frequency ν of each resonance(compared to the Larmor frequency) depends only on the nuclear spin state whichis reversed for the opposite helicity. Thus the spectra of the two helicities arereflections of each other from the Larmor frequency. Adapted from Ref. [22].magnetic field or electric field gradient) are directly obtained from the β -NMRresonance spectra. Such spectra may be observed by implanting a continuous beamof highly polarized probe nuclei in the sample in the presence of a static externalmagnetic field B0 applied along zˆ. One then monitors the lifetime averaged detectorasymmetry:〈A(ω)〉=∫ ∞0exp(−t/τ)Apz(ω, t)dt (2.21)as a function of the frequency (ω) of the smaller RF magnetic field B1cos(ωt)oriented perpendicular to B0 and the initial polarization direction zˆ. In contrast toclassical NMR, the external field B0 is not used to create the nuclear polarization,33but rather simply to hold the polarization and control the resonance frequency.The RF frequency ν =ω/2pi is varied in steps dν in a range around the expectedresonance frequency. Every frequency step lasts typically for at least one nuclearlifetime (e.g., at least for a second for 8Li), in order to avoid the spin-probe’smemory effect, namely having at a given time in the sample probes from theprevious frequency steps still present. That would create an unwanted mixingbetween the measurements of different frequencies. The actual lingering timeat each frequency bin is selected by balancing the trade off between the need tominimize the aforementioned memory effect and the time investment required forthe acquisition of a resonance spectrum.A reduction in the time integrated asymmetry 〈A(ω)〉 occurs when the frequencyof B1 matches the Larmor resonance frequency (Eq. (2.15)) just like in continuouswave (CW) NMR. This reduction is caused by the resonant precession of the nuclearspins around B1 (in the rotating reference frame), which results into depolarizationof the spins.Once a frequency scan is complete, the helicity of the laser is reversed andthe frequency scan is repeated with the opposite spin polarization (see Fig. 2.14).The asymmetries of the two helicities will have opposite signs and the combinedasymmetry spectrum can be obtained by averaging the absolute values of them:A¯ =A+−A−2(2.22)where A¯ is the combined asymmetry and A± is the asymmetry for each laserhelicity.This process removes many sources of systematic error associated with the β -NMR measurement. For instance, in the geometry of the β -NQR spectrometer, anysource of β -scattering between the sample and the L/R detectors that is asymmetricbetween the two detectors will shift the asymmetries of both helicities by a constant,frequency-independent value, but the combined asymmetry will be completelyunaffected by such an effect, as the constant would cancel out.A resonance measurement consists typically of several (∼2-20) scans for eachhelicity, in order to minimize the statistical uncertainties. If any given frequencyscan is identified as problematic (e.g., due to a proton trip or problems with thelaser), it can be individually disregarded before averaging all scans into a singleresonance spectrum.The resonance shift compared to that of a reference material allows for thestudy of the local internal magnetic field Bint . In β -NMR, the most commonly usedreference material is MgO, which is a non-magnetic insulator with very few nuclearmoments and also νq = 0, thus the 8Li β -NMR resonance is very narrow and moreor less unshifted except for a small chemical shift. Therefore, for MgO it holds34Figure 2.14: (a) Combined resonance spectrum for 8Li implanted in a epitaxialAg film (19 nm) grown on MgO at 155 K. There is one resonance from MgO andtwo from the thin Ag film. The resonance labeled O is attributed to 8Li in theoctahedral interstitial site in Ag, whereas the resonance labeled S is due to 8Li in thesubstitutional site in Ag. All these sites have cubic symmetry and therefore show noquadrupolar splitting. (b) The same spectrum, but helicity-resolved. The position ofthe spectra is the same for both helicities, which indicates that they are not due toquadrupolar effects. Adapted from Ref. [22].approximately that νMgO ∼ γB0. As a result, the resonance shift ∆ν of a materialrelative to MgO yields:∆ν = ν−νMgO = γBnet − γB0 = γBint (2.23)where Eq. (2.14) is used to get the last equality.The position of the resonance(s) is determined by the quasi-static or timeaveraged parameters of the spin Hamiltonian. Nevertheless, as in conventionalNMR, the shape of the β -NMR resonance may still be sensitive to the fluctuationsin the local environment. For example in the fast fluctuation limit the line width isnarrowed and varies as ∆2τc where ∆ is the static line width and τc is correlationtime for fluctuations in spin Hamiltonian terms responsible for the line narrowing.Coupling between the nuclear quadrupole moment and any local electric fieldgradient will lead to additional structure (e.g., quadrupolar splittings, see Sec. 2.3.2),as in conventional NMR. By gradually varying the frequency of the RF field, a35spectrum of all the resonances of the system can be generated.The power of the RF field is what destroys the nuclear polarization, so itsvalue can affect (in a well-defined way) the resonance spectrum. If the RF poweris small, it won’t be enough to depolarize all spin-probes at resonance, so theasymmetry peak will have a small amplitude and will be hard to resolve. This effectsuppresses the β -NMR signal, so it would make a resonance measurement moretime consuming. On the other hand, a strong RF would depolarize the probes onresonance completely, but it would also partially destroy the polarization of probesbeing slightly off-resonance. This effect artificially broadens the resonance and isgenerally referred to as “power broadening”. Due to these effects, sometimes it isnecessary to measure the same resonance using different RF powers, in order to beable to distinguish the underlying resonance width from the power broadening.2.4.2 Spin Lattice Relaxation SpectraInformation on the fluctuations of the electromagnetic fields in a material of interestis obtained through measurements of the (longitudinal) spin-lattice relaxation (SLR)rate in the absence of a RF magnetic field. Note that this type of measurement doesnot provide any spectral resolution of the fluctuations driving the SLR.The SLR may be studied by implanting a series of beam pulses into the sampleand then monitoringA (t), which is the convolution of the asymmetry A(t− t ′) withthe beam pulse N(t ′) where t ′ is the time of arrival for a given probe and t− t ′ is thetime spent in the sample before its β -decay:A (t) =∫ t−∞N(t ′)A(t− t ′)dt ′ (2.24)In general the SLR rate, usually denoted as 1/T1 (with T1 being the longitudinalspin-lattice relaxation time), originates from fluctuations in the local magnetic field(and the EFG for nuclei with I≥1) occurring in a direction perpendicular to thenuclear polarization axis. They can be arising from fundamental processes such asphonon scattering, magnon scattering, conduction electron scattering, diffusion, etc.The measured SLR relaxation rate depends mostly on those fluctuations thathave an appreciable spectral density close to the Larmor frequency. Since νL isrelated to the total magnetic field (see Eq (2.15)), the SLR relaxation rate dependson the applied magnetic field B0, so the latter can be varied to allow for the study offluctuations of different frequencies.In the case that the SLR is caused by fluctuating magnetic dipolar interactions,the 1/T1 is often approximated using the Bloembergen-Purcell-Pound (BPP) theory.The ansatz of BPP is that the fluctuations driving the relaxation are described by a36autocorrelation function which is proportional to exp[-t/τc], where τc is the corre-lation time. The Fourier transform of the autocorrelation function is a Lorentziandescribing the the spectral density of the fluctuation. The observed relaxation rate isproportional to the magnitude of the spectral density at the Larmor frequency. Inthe case of the magnetic dipole interaction driving the SLR, 1/T1 is given by:1T1= Kτc1+ω20τ2c(2.25)where K is a constant depending on the nuclear probe and the dipole-dipoleseparation distance.In the simplest case of all nuclear probes being at equivalent environments witha well defined 1/T1 rate, the SLR rate can be extracted by fitting the SLR spectrumusing a single exponential:A(t− t ′) = exp[−(t− t ′)/T1] , (2.26)Substituting this into Eq. (2.24) and assuming a square beam pulse during thetime interval [0,∆], one obtains a form for the asymmetry during and after the pulse,by averaging over all arrival times:A (t) ={A0 τ′τβ1−exp(−t/τ ′)1−exp(−t/τβ ) t ≤ ∆A(∆)exp[−(t−∆)/T1] t > ∆, (2.27)where τβ is the radioactive lifetime, 1/τ ′ = 1/τβ + 1/T1 and A0 is the initialasymmetry at the time of implantation. Note that the SLR spectrum has two distinctregions (see Fig. 2.15): during the beam pulse (0 < t < ∆) the asymmetry relaxestowards a dynamic equilibrium value [107]:¯A =A01+ τβ/T1, (2.28)Note that this equilibrium value is the “baseline” (i.e., off-resonance) asymmetryof the resonance spectrum (see Sec. 2.4.1).After the beam pulse (t > ∆)A (t) decaystowards the Boltzmann equilibrium value, which is essentially zero on our scale.There is a pronounced kink inA (t) at t = ∆ when the beam pulse ends. This is alsothe time with the highest event rate and smallest statistical uncertainty in A (t). Thestatistical uncertainties of β -NMR measurements are governed by Poisson statistics,so the relative uncertainty of a measurement decreases by the square root of thenumber of counts. Because of the radioactive decay law (see Eq. (2.29)), the numberof decay events are maximum when the largest number of radionuclides are presentin the sample, which is exactly at time t=∆.37dN(t)dt= N(t)exp(−t/τβ ), (2.29)where N(t) is the number of nuclides present in the sample at time t.The beam-on time ∆ is set to several nuclear lifetimes (typically ∆∼ 4τβ ) and thebeam-off region should be adequate for all nuclear probes to decay before anotherbeam pulse arrives (∼ 10τβ ). After the end of the measuring period describedabove, the laser helicity is reverted and another beam pulse is implanted, having theopposite nuclear polarization. As in the case of a frequency scan (see Sec. 2.4.1),the SLR spectra of the two helicities can be combined using Eq. (2.22) to generatethe combined SLR histogram.Figure 2.15: SLR spectrum of 8Li implanted in a single crystal of bismuth at 295 K.The combined asymmetry is shown in (a). In (b), the helicity-resolved SLR spectraare shown. Adapted from Ref. [22].2.5 Enhancing the capabilities of β -NMR using αdetectionAccording to Eq. (2.3), each 8Li decay leads to the production of a β - and two α-particles (plus an electron antineutrino). So far, 8Li β -NMR has ignored completelythe subsequent α-decay of 8Be and any information that it could be carrying. One38aim of this thesis is to couple the information from the α- and β -particles in orderto expand the capabilities of β -NMR.As mentioned in Sect. 2.1, 8Li has a non-zero electric quadrupole moment.This makes it possible to study with the same probe both magnetic and electricquadrupolar phenomena (see Sect. 2.3.1 and Sect. 2.3.2). On the other hand, incertain instances it might be hard to identify the relative contributions of theseunderlying interactions to the spin lattice relaxation rate.As part of this thesis, we showed that it is possible to resolve the primary sourceof relaxation in a material, by performing an isotopic comparison using two differentβ -decaying isotopes of the same element (in this case 8Li and 9Li). The differentnuclear characteristics of the two isotopes (spin, electric quadrupole moment) resultin different SLR rates for the two isotopes in the limits of either purely magnetic orelectric quadrupolar interactions (see Ch. 3).A major limitation when using the isotopic comparison of 9Li/8Li is that thesignal from 9Li is heavily suppressed, i.e., 9Li has a smaller effective β -decayasymmetry. Indeed, we would typically spend many hours acquiring a single 9LiSLR spectrum, compared to 15-20 min for 8Li, and still our uncertainties weredominated by the 9Li measurements. Even though we don’t need to compare the1/T1 rates for 9Li and 8Li for every condition, but only for a few temperatures andfields, it would still be advantageous to enhance the 9Li signal.The reason for the lower 9Li asymmetry is that 9Li can β -decay into threedifferent energy levels of 9Be, two of which have opposite asymmetries that nearlycancel when weighted by their branching probabilities. Nevertheless, two of thethree main decay channels of 9Li further decay into two α-particles, which makes itpossible to differentiate between the different channels by tagging the β -particlesin coincidence or in anti-coincidence with an α . In this thesis we designed asystem that allows for the aforementioned α-tagging (see Sect. 3.3.2). The effectiveasymmetry of an α-tagged measurement was found to be 3 times larger than thatwithout α-detection.Thus, we showed that by coupling the signal of the β and α-decays of 9Li, wecan amend considerably the limitations of using the isotopic comparison method in8Li β -NMR, establishing it as a new tool for β -NMR.In addition, being able to detect the α-particles coming from the decay of 8Liallows us to measure the rate with which lithium ions diffuse inside materials. Tomeasure lithium diffusion, a short beam pulse of 8Li ions gets implanted in thesample material. The energy of the beam defines the initial depth profile of the ions.Upon implantation, the lithium ions start diffusing through the sample and decayat random times (following the decay rate of 8Li) to 8Be and then immediatelyinto two α-particles. The α-particles attenuate inside the material very fast, so thehighest energy α-particles can come only from decays close to the surface.39In order to establish the feasibility of this technique and identify the optimumdetection geometry a detailed Monte Carlo simulation of the geometry and physicsof the experiment using the Geant4 simulation package was carried out. Geant4was developed at CERN, and it is a very powerful and flexible toolset for particle-material interactions (see Ch. 5).Based on the aforementioned simulations, we designed a new cryostat that issuitable for this study (see Appendix A). The new cryostat has a nominal temperaturerange of 5 K to 400 K (compared to 3.5 K to 300 K of the old β -NQR cryostat),thus is really a cryo-oven (which is of course an oxymoron). Being able to reachhigher temperatures can be critical for studying slow diffusion in materials wherethe diffusion is too slow at room temperature, since the diffusion rate increasesexponentially with temperature in materials that follow Arrhenius law.Following the commission of this new spectrometer, we conducted proof ofprinciple experiments on rutile TiO2 (see Ch. 6), where we were able to directlymeasure the diffusion rate of lithium at a nanometer scale over a wide temperaturerange. With this measurement we showed that the temperature dependence of thediffusion is bi-Arrhenius, with a second, previously unknown Arrhenius componentbelow∼200 K and also proved that Li+ gets trapped upon reaching the (001)-surfaceof rutile, which explains the reported suppressed Li intercalation in this material.In the remaining chapters of this thesis, these two applications of the α-detectionsystem coupled to 8Li β -NMR will be presented.40Chapter 3Using α-tagged 9Li β -NMR toDistinguish the Source of SpinLattice Relaxation in 8Li β -NMRAs 8Li has a non-zero electric quadrupolar moment, a key issue in any 8Li β -NMRexperiment is to identify the source of spin-lattice relaxation (SLR) and in particularwhether the fluctuations driving the SLR are magnetic or electric in origin. Unlikethe positive muon, µ+ (I = 1/2), 8Li (I = 2) is not a pure magnetic probe and itsrelaxation is sensitive to both fluctuating magnetic fields and electric field gradients(EFG’s).In some cases, the primary source of relaxation may be inferred. For example,in simple metals the observed relaxation is linear in temperature [107] as expectedfrom the Korringa relaxation [108], which originates from the scattering of the spinof the conduction electrons at the Fermi surface from the nuclear spin, mediatedthrough the (magnetic) contact hyperfine interaction.However, in more complicated instances, such as heterostructures comprised ofmagnetic and non-magnetic layers, it becomes difficult to determine the contributionof each type of relaxation. LaAlO3/SrTiO3 multilayers are particularly illustrativeof this point; the bulk layers are non-magnetic insulators, while there is evidence ofmagnetism at their interfaces [20].In conventional NMR it is possible to differentiate between relaxation mecha-nisms by isotopic variation of the nuclear probe (if suitable isotopes exist), sincethe absolute relaxation rates for each isotope scale according to their nuclear mo-ments. For two isotopes with significantly different nuclear moments (e.g., 6Liand 7Li [109]) the ratio of the relaxation rates should be distinctly different in thelimits of either pure magnetic or pure electric quadrupolar relaxation. Here we testthe feasibility of isotope comparison applied to β -NMR — using 8Li and 9Li, twoβ -radioactive isotopes.The stopping sites of 8Li and 9Li are often interstitial rather than substitutionalas in the case of conventional NMR. However, we expect that both implanted 8Liand 9Li will probe the same sites. Measurements on 9Li are more time consuming41than for 8Li. This is related to the fact that 9Li lies one neutron further away fromthe valley of stability, consequently the beam intensity in this experiment was about10 times lower and 9Li has a more complicated β -decay scheme, which results in aβ -decay asymmetry about 3 times smaller than for 8Li, as will be discussed below.Measurements reported here were made in Pt metal, where the spin relaxationrate of 8Li (9Li) is dominated by Korringa scattering [110], which is magnetic, andin strontium titanate (SrTiO3), which is a non-magnetic insulator with a large staticelectric quadrupolar interaction for implanted 8Li. SrTiO3 is a common substratematerial but also has interesting properties on its own which have been studiedextensively with a wide variety of methods including β -NMR. Although we expectthe quadrupolar fluctuations in EFG caused by lattice vibrations to dominate thespin relaxation, there are also potential magnetic sources of relaxation that couldcontribute as explained below.In the following sections we first summarize the theoretical considerationsbehind the isotopic variation method. This is then followed by the experimentalresults along with a discussion.3.1 Isotopic Comparison MethodThe magnitudes of each contribution to the spin-lattice relaxation rate (SLR) for agiven probe nucleus scale according to their nuclear properties; namely, their spin,I, magnetic moment, µ , and electric quadrupole moment, Q. Measurements ofSLR rates for two different isotopes under identical experimental conditions (i.e.,magnetic field, temperature, etc.) can be compared through their ratio, R:R(I, I′)≡ 1/T1(I)1/T1(I′)=1/T M1 (I)+1/TQ1 (I)1/T M1 (I′)+1/TQ1 (I′), (3.1)where I and I′ denote the spin quantum number of each isotope and 1/T M1 (I),1/T Q1 (I) are the SLR rates due to magnetic and electric quadrupolar interactions,respectively.Two limits are of interest here: when the relaxation is solely due to eithermagnetic or quadrupolar interactions within the host-sample.In the former case, Eq. (3.1) reduces to the ratio of pure magnetic relaxation,RM, which in the limit of fast fluctuations (i.e., τ−1c  ω0, where τc is the NMRcorrelation time and ω0 is the Larmor resonance frequency, see Sect. 2.3.1) is:RM(I, I′)=(µ/Iµ ′/I′)2=(γγ ′)2, (3.2)42where µ and γ are the magnetic moment and gyromagnetic ratio of each isotope.Note that the fast fluctuation limit ensures that 1/T1 is independent of ω0, whichsimplifies the ratio considerably.In the other case, Eq. (3.1) yields the ratio of relaxation rates in the purequadrupolar limit, RQ:RQ(I, I′)=f (I)f (I′)(QQ′)2, (3.3)where Q are the nuclear quadrupole moments, and [111]f (I) =2I+3I2(2I−1) (3.4)Thus, given the nuclear moments of each isotope, one can calculate the ratioof relaxation rates when either mechanism is dominant. Using Eqs. (3.2) and (3.3),along with the nuclear spins and moments for 8Li and 9Li (see Table 2.1), we findthe limiting cases for T−11 (9Li)/T−11 (8Li): 7.67964(16) and 2.1362(4) for RM andRQ, respectively.The difference between these limits is not as pronounced as for 6Li and 7Li [109],where RM and RQ differ by a factor of ∼90 [112]. Nevertheless, 8Li and 9Li aresufficiently different that the nature of fluctuations and resulting spin relaxation(magnetic versus electric quadrupolar) may be differentiated by such a comparison.3.2 Experimental Demonstration of the IsotopicComparison MethodTo demonstrate the comparison of 8Li and 9Li in β -NMR, two very differentmaterials were selected.The first is Pt which is a d-band metal with a face cubic centered (fcc) crystalstructure in which the 8Li resides at a site with little or no quadrupolar interaction(i.e., the octahedral interstitial site, at which the EFG is zero by symmetry). In thistest case we expect the relaxation to be predominantly magnetic, as 1/T1 is knownto depend linearly on temperature [107], as expected from Korringa scattering.SrTiO3 on the other hand is a non-magnetic paraelectric – on the verge ofbeing ferroelectric [113] – insulator with few nuclear moments and no conductionelectrons (so no Korringa relaxation). Previous work in SrTiO3 shows that 8Liexperiences a large quasi-static quadrupolar interaction [25]. Thus in this case, weexpect quadrupole fluctuations to play a more important role. Nevertheless, it isstill unclear to what extent magnetic relaxation can be neglected in SrTiO3, as therecould be defect-related magnetic effects present.43For example, O vacancies in SrTiO3 result in two Ti3+ ions which are typicallyparamagnetic. In principle, the resulting paramagnetic defects would have lowfrequency magnetic fluctuations which will contribute to the SLR of the implantedLi nucleus in SrTiO3, as dilute paramagnetic effects are well known to affect the1/T1 in both solids and liquids.3.2.1 PlatinumThe sample was a high-purity (99.999%) Pt foil with dimensions 12x12 mm2and thickness of 0.1 mm. It was cut from the same initial foil that was studiedwith β -NMR by Ofer et al. [110]. The implantation energy was 18 keV, whichcorresponds to a range of 42 nm and a straggle of 23 nm for both isotopes of lithium.The implantation profile was estimated using the Monte-Carlo-based SRIM-2013software [104].The resonance spectra (see Sect. 2.4.1) at 1.90 T and 300 K for both 8Li and 9Lican be seen in Fig. 3.1. To account for power broadening, the spectra were fit to aVoigt line shape [101] simultaneously, sharing a common (normalized) resonancefrequency and Gaussian width.For 8Li, the intrinsic Gaussian FWHM was found to be 0.85(10) kHz. Thisvalue is somewhat smaller than those reported previously in Pt [114], where powerbroadening was not taken into account. By comparing the 8Li resonance in Ptwith that in the standard β -NMR reference material MgO [115], the Knight shiftis estimated as −309.8±1.9 ppm, in agreement with previous measurements. For9Li in Pt at 1.90 T and 300 K, a single resonance at 33.18685±0.00028 MHz withFWHM of 6.8±0.6 kHz is observed.The fact that 8Li and 9Li resonance measurements in Pt exhibit a single narrowline below 300 K, indicates that both isotopes occupy a single site with a vanishing(static) EFG [101, 114]. The spectrum is simpler than in other metals, wheremultiple Li+ sites are found below 300 K [23, 24, 116–120].Given the simplicity of the spectrum, we expect SLR in Pt to follow a singleexponential form (see Eq. 2.27).The SLR rates for 8Li+ and 9Li+ implanted at 300 K were measured in magneticfields of 1.90 T and 6.55 T — the latter shown in Fig. 3.2. For both isotopes thelength of the beam pulse (∼ 3.3τβ ) and the total observation time (∼ 9.9τβ ) werechosen to minimize the statistical uncertainties.Temperature dependent SLR of 8Li+ in Pt has been studied previously by Ofer etal. [110] between 3 K to 295 K at 4.10 T, where the SLR rate was found to increaselinearly with temperature, implying Korringa relaxation [108]. This relation holdsfor high magnetic fields and different implantation energies.440.0050.06ν9Li0ν8Li0= 2.771246(24)γ9Liγ8Li= 2.77122(6)〈A〉11960 11975 120000.0020.006ν˜ ≡ ν ×γ8LiγLi[kHz]〈A〉Figure 3.1: Resonance spectra in Pt foil at 300 K and 1.90 T with 8Li (top) and 9Li(bottom). The frequency has been normalized to the gyromagnetic ratio of eachisotope.The temperature-dependent 8Li+ SLR rates at various magnetic fields are shownin Fig. 3.3, including our measurements, as well as results from Ofer et al. [110].The 8Li SLR rate at 6.55 T is in good agreement with the Korringa fit by Oferet al. [110], extrapolated to 300 K, whereas the measured SLR rate at 1.9 T islower by about 10%. It is unlikely that this is a real effect since any additionalsource of relaxation would increase the relaxation at the lower magnetic field whichis opposite to what is observed. The slight reduction in 1/T1 measured at 1.9 Tsuggests there may be a small systematic error related to the fact that the beamspot is a bit larger at that magnetic field (thus there could be a small non-relaxingbackground signal from 8Li+ stopping outside the sample) and the ratio betweenthe β -rates in the two detectors is different compared to the higher field. However,it should be noted that the measured 8Li SLR rates in Pt foil appear to increaselinearly with temperature, independent of implantation energy and applied magneticfield.450 2 4 6 8 1000.0180.0370.0550.073Beam ON OFF8LiTime (s)Asymmetry0 0.851 200.0030.0060.0090.012Beam ON OFF9LiTime (s)(a) (b)Figure 3.2: SLR spectra for 8Li+ (left) and 9Li+ (right) implanted in Pt foil withan energy of 18 keV at 300 K 6.55 T. The solid orange lines are fits to Eq. (2.27).Note the different time scales, which reflect the lifetime of each radionuclide. Theabsolute SLR rate for 9Li+ is 1.60(10) and 0.2368(26) for 8Li+.The ratios of T−11 (9Li)/T−11 (8Li) at 6.55 T and 1.90 T are in good agreementwith each other and we find a relaxation rate ratio, RPt, of 6.8(4) and 5.9(9) at 6.55 Tand 1.90 T, respectively.3.2.2 Strontium TitanateSrTiO3 was chosen for this study since it is a non-magnetic insulator, and a materialwhere the 8Li relaxation is expected to be dominated by electric quadrupolar interac-tions. It has been studied extensively with low-energy 8Li β -NMR [18, 106, 121].SrTiO3 is a cubic perovskite at 300 K (Fig. 3.4). Implanted 8Li occupies threeequivalent interstitial non-cubic sites [19], namely the face-centered sites in the unitcell centered at Sr2+. At 300 K, the EFG is axially symmetric, with the main axisalong Sr-8Li-Sr.Two SrTiO3 samples were studied in this experiment. Both were 10x8x0.5 mm3single crystals with the (100) orientation, i.e., with the a cubic axis perpendicularto the face of the sample. Both samples were epitaxially polished (0.2 nm RMSroughness). Sample 1 (S1) was left bare, while sample 2 (S2) was capped with30 nm of LaTiO3 1. At the implantation energy of 18 keV, a negligible fraction of8Li+ ions stop in the LaTiO3 film, or the near surface region. This was checked forboth 8Li and 9Li by using SRIM-2013 [104].1Note that the thin layer of LaTiO3 was for a different experiment, but had no effect on the currentexperiment, since it was so thin that most of the 8Li passed through it.460 50 100 150 200 250 3000.× 10−4 K−1 s−14.10 T6.55 T1.90 TTemperature (K)T−1 1(s−1)Figure 3.3: Measured SLR rates for 8Li implanted in Pt. The relaxation rateincreases linearly with temperature, appearing insensitive to both implantationenergy and magnetic field strength, consistent with a Korringa mechanism [108].Measurements from this work are highlighted in colored disks, while black diamondmarkers indicate data from earlier measurements on Pt foil [110]. The solid orangeline is Korringa fit to all the SLR rates in Pt and differs somewhat from the result ofOfer et al. due to the additional data points from this work.Figure 3.5 shows the SLR spectra for 8Li and 9Li at 300 K at various magneticfields between 0 mT to 15 mT applied along the (100) cubic crystallographic axis ofsample S1. It is evident from the data that the relaxation is more complex than in Ptsince a single exponential fails to describe the decay of spin polarization.The spectra were best fitted with a two-component exponential function, butgiven that one of the relaxation rates is found to be nearly zero, a phenomenologicalrelaxation function of the following form [121] was used:A(t− t ′) = f exp[−λ (t− t ′)]+(1− f ) , (3.5)where f is the fraction of the relaxing asymmetry (0≤ f ≤ 1) and λ ≡ 1/T1.Since f is approximately field-independent in our range of fields, the SLRspectra for 8Li and 9Li were fit globally, sharing a common f , which turned out tobe 0.347(3).One might expect this since a magnetic field applied along the (100) directionbreaks the local symmetry between the 3 otherwise equivalent sites. More specifi-47Figure 3.4: The crystal structure of the cubic perovskite SrTiO3. A Sr2+ (yellowsphere) lies at the center of the unit cell, with O2− ions (green spheres) at thecorners of the cube and Ti4+ ions (red spheres) at the octahedral holes created bythe oxygens. Implanted 8Li+ occupies one of the face-centered sites of the unit cell.Adapted from Ref. [122].cally the EFG tensor is axially symmetric about one of the three orthogonal cubicaxes, with only a small asymmetry factor η ∼ 0.01 [123] (see Sect. 2.3.2). Thus theapplied magnetic field is either along the EFG axis or perpendicular to it. Howeverf was about the same in zero-field (ZF) and given that the two 90 degree sites don’tcontribute to the ZF signal and that f is about the same at ZF, the more complexrelaxation function observed in SrTiO3 must be unrelated to the angle between themagnetic field and the symmetry axis of the EFG.Consequently there must be an additional source of fluctuations affecting theSLR for all 3 sites in the same way but in an inhomogeneous manner either in timeor space. Previous studies have found that the relaxing fraction f is also temperatureindependent [121]. This suggests that the origin of the relaxing component shouldbe structural, associated with defects close to about one third of the implanted Li,with the rest being in a non-relaxing environment away from such centers.Regarding the relaxation function, note that this is an unfamiliar regime, wherethe Zeeman interaction is smaller than νQ = 153.2kHz (see Sect. 2.3.2) over the fullrange of fields, since even for our highest field measurement at 15 mT, (γ/2pi)B =94kHz. At high fields (several Tesla), previous work suggests that f = 0 [25]. Inthe high-field limit, the relaxation of any β -NMR experiment approaches zero,because the Zeeman splitting becomes larger than the fluctuation rate terms givingrise to relaxation, which in turn converts the relaxing component into a non-relaxingone, leading to f = 0. There is likely some change that will happen around 50 mT,where the Zeeman interaction really starts to take over. In addition, the fraction fis expected to be independent of the sample orientation, since SrTiO3 is a cubic480 2 4 6 8 1000.0250.050.0750.10Beam ON OFF8Li0 mT5 mT10 mT15 mTTime (s)Asymmetry0 0.851 200.0050.010.0150.02Beam ON OFF9Li3.6 mT10 mTTime (s)(a) (b)Figure 3.5: SLR spectra of 8Li (left) and 9Li (right) in single crystal SrTiO3(sample S1) at 300 K. The solid orange lines are a global fit to Eqs. (2.24) and (3.5)where a common parameter f is shared between all spectra, as it was found to beapproximately field-independent in this field range.material.In sample S1, the SLR rate for 8Li is found to vary weakly with applied magneticfield below 15 mT, reaching a plateau below 5 mT (see Fig. 3.6). It is likely, butunclear due to the limited statistics, that a similar behavior occurs for 9Li. At 300 K,the ratio of the 9Li/8Li SLR rates for SrTiO3, RSTO, was found to be 3.7(7) at 10 mTand 2.4(5) at 3.6 mT.For comparison, the SLR rate of 8Li and 9Li was also measured in a secondSrTiO3 sample (S2) (see Fig. 3.7).The fit for sample S2 using Eq. 3.5 does not include a small but very fast relaxingcomponent that can be identified at the early part of Figure 3.7. Given the largestatistical uncertainties of the 9Li spectrum, it would be over-fitting to add an extrafast relaxing component, in addition to the two terms of Eq. 3.5. Moreover, this veryfast relaxing component is not present in other studies on SrTiO3 (including ourcurrent measurements for sample S1), which means that this extra fast relaxationis most probably related to lithium stopping outside the sample (e.g. due to beaminstabilities or backscattering off the target material). In low magnetic field, havinga percentage of the incoming lithium stopping outside the sample often manifests asa small but very fast relaxing component, irrelevant to the material under study.The ratio of relaxation rates in sample S2 at 10 mT was found to be 2.4(5).490 5 10 150.0128Li9LiMagnetic Field (mT)T−1 1(s−1)Figure 3.6: Field dependence of 1/T1 for 8Li and 9Li in SrTiO3 at 300 K. The(orange) triangle represents a linear interpolation at 3.6 mT from the 2.5 mT and5 mT 8Li measurements.3.2.3 Ratio of Relaxation RatesThe ratio of relaxation rates in platinum RPt = 6.82(29), which is the weightedaverage of the measurements at 6.55 T and 1.90 T. Note that this value is somewhatless than expected from the pure magnetic limit RM (Fig. 3.8).The reason for this discrepancy could be the non-zero temperature. All mea-surements were taken at 300 K where the lithium ions could have some quadrupolarcontribution due to local vibrations and scattering of phonons which leads to afluctuating EFG. However 1/T1 is very linear in temperature, whereas any suchcontributions would have a stronger temperature dependence. It would be interestingto repeat the measurements at a lower temperature to check if RPt is closer to themagnetic limit or not. In any case, an electric quadrupolar contribution to 1/T1cannot be very large in Pt at 300 K.We also reported a value of RSTO in two samples of SrTiO3. In the first sample,the weighted average RSTO of the measurements at 3.6 mT and 10 mT yielded 2.9(4).This value is close, but not within experimental error of the quadrupolar limit ofRQ≈ 2.14. After taking into account the measurement on the second SrTiO3 sample,which was 2.4(5) at 10 mT, the weighted ratio of relaxation rates in SrTiO3 is foundto be 2.7(3), closer to the quadrupolar limit. Still there is a small disagreementwhich suggests some small magnetic contribution to 1/T1.This small magnetic relaxation may be related to the observed non-exponential500 4 1000. ON OFF8LiTime (s)Asymmetry0 0.851 200.0040.0750.0110.015Beam ON OFF9LiTime (s)(a) (b)Figure 3.7: SLR spectra of 8Li (left) and 9Li (right) in SrTiO3 (sample S2) at 10 mTand 300 K. The solid orange lines are a global fit to Eqs. (2.24) and (3.5) where acommon parameter f is shared between both spectra.decay of polarization. The relaxing fraction f has been found in previous studies aswell and it is approximately temperature independent [121] and independent of theangle between the magnetic field and the crystallographic axis [123]. This suggeststhat it could be due to the dynamics associated with defects close to some of theimplanted Li. These fluctuations would be primarily paraelectric [123], but a smallportion could be magnetic in origin. For example any O vacancies a few latticesites away would give rise to paramagnetic Ti3+ ions, in addition to paraelectricfluctuations. A typical level of oxygen vacancies in SrTiO3 of about 1−2% wouldresult in lithium having such a defect for a nearest or next-nearest neighbor 20−30%of the time, which could explain the fraction f ∼ 0.3.As to whether these defects are primarily intrinsic to the crystal or caused bythe beam implantation, note the following: There is no doping taking place with 8Li,since each ion decays into a β and two α-particles (plus an electron antineutrino).9Li, on the other hand, decays into the stable ground state of 9Be half of the time,so some Be doping should be expected. Based on the total number of implanted9Li+, the typical width of the beamspot and the implantation depth simulated bySRIM-2013, this doping is calculated to be in the order of parts per billion (50ppb for Pt, 130 ppb for SrTiO3). Such a small concentration should not create anyconsiderable effect.In addition, a typical lithium ion can create multiple Frenkel pairs (∼ 80 defect-s/ion in Pt, ∼ 90 defects/ion in SrTiO3), so their concentrations are expected to behigher. Using again SRIM, their concentration was calculated in the order of 100512468 MagneticLimitQuadrupolarLimitPt2.7(3)SrTiO3(S1)SrTiO3(S2)RFigure 3.8: Ratios of 9Li to 8Li 1/T1 relaxation rates in Pt (weighted average of allmeasurements) and in the two SrTiO3 samples. The red line represents the weightedaverage of the measurements in both SrTiO3 samples.ppm in SrTiO3 and ∼ 50 ppm in Pt. These numbers correspond to the upper limit ofFrenkel pairs present, as in reality some would anneal away quickly. Note that thesedefects would be formed gradually with time, i.e., they would affect primarily themeasurements taken last. Such time-dependence was not observed, though. Also,here we assume that there is no recombination of the created Frenkel pairs, which,given the temperature of the experiments, should not be the case. Therefore, theseconcentrations represent the upper boundary of the extrinsic defects.In comparison, the intrinsic defects in SrTiO3 (primarily O vacancies) are in the∼ 1% range, orders of magnitude higher than even the upper limit of extrinsicallycaused defects. From that, we conclude that the small magnetic part of the relaxationin SrTiO3 was not caused by the beam implantation.3.3 Enhancing the Effective Asymmetry of 9Li Usingα-taggingFig. 3.2 and 3.5 reveal that the uncertainty associated with the 9Li measurementswas much larger than 8Li, even though we typically spent ∼ 10 times more time for52the acquisition of the 9Li spectra. The figure of merit for a β -NMR measurementis A2N, where A is the observable asymmetry and N is the total number of decayevents — both factors for 9Li are significantly reduced relative to 8Li. Since 9Lilies further away from the valley of nuclear stability, it has a shorter half-life andfewer ions were extracted from the ion source and delivered to the spectrometer(∼106 s−1 vs ∼107 s−1 for 8Li+). This in turn reduced the factor N for 9Li. Also,the asymmetry for 9Li is much smaller than for 8Li (see Sec. 3.3.1). As a result,about 90% of the data acquisition was spent on 9Li, since these results dominatedour uncertainty in the ratio of the relaxation rates.To further develop the capabilities of the isotopic comparison method in β -NMR,we designed a system that was able to enhance the effective initial asymmetry of 9Liby a factor of ∼ 2. That system, coupled with the development of certain aspects ofTRIUMF’s ISAC facility that increased the intensity of 9Li+ beams by a factor of∼ 10, suppressed greatly the uncertainty of 9Li measurements, making the figure ofmerit for 9Li comparable to 8Li.In this section the basis of the new system is presented (Sec. 3.3.1), followed bythe proof-of-principle experiments (Sec. 3.3.2).3.3.1 αLithEIA MethodThe reduction in asymmetry for 9Li compared to 8Li is attributed to 9Li’s morecomplicated β -decay scheme (see Fig. 3.9).Figure 3.9: Properties of the principle β -decay modes of 8Li and 9Li [124]. Whereas8Li decays always to the first excited state of 8Be, 9Li can decay to the ground stateor one of the excited states of 9Be.In particular, 9Li has three main decay channels, two of which have oppo-site asymmetries that nearly cancel after weighting by the branching probabilities(Eq. 3.6).53a= ags pgs+a1 p1+a2 p2 = 0.505(−0.4)+0.34(0.6)+0.1(−1) =−0.096, (3.6)where ags,pgs are the asymmetry and branching probability of the 9Li β -decayto the ground state and ai,pi are the values associated with the decay to the i-thexcited state of 9Be (see Tab. 3.1).Thus, most of the observed asymmetry derives effectively from the weakestdecay mode which has a branching probability of only 0.1 but a large theoret-ical asymmetry parameter a = −1.0. The relevant branching probabilities andasymmetries of each decay mode are reported in Fig. 3.9.Table 3.1: The asymmetry (a) of each decay mode of 9Li [124]. The total asymmetryfor 9Li is the sum of the asymmetry weighted by the relevant probability of eachdecay mode.9Be state Probability Ipi a Decay modeground state 50.5% 3/2− -2/5 stable2429.4 MeV 34% 5/2− 3/5 n+2α2780 MeV 10% 1/2− -1 n+2αFrom Tab. 3.1, note that by tagging the 9Li β -events according to whether an αis emitted or not, it would be possible to distinguish between the decays going tothe ground state of 9Be versus the excited states and isolate their contributions. Anefficient α-detection system can enhance the initial asymmetry of the 9Li spectrasignificantly, since it should be possible to register two spectra in parallel, one withthe β detected in coincidence with an α , and one without an α . Using Eq. 3.6, theformer spectrum would have an initial asymmetry of a = 0.236, twice as big as thatobtained without α-detection. The latter spectrum would have an increased initialasymmetry of a = 0.4 if the α-detection efficiency is close to 100%, making theeffective asymmetry of the measurement 3 times larger than that without α-detection(see Fig. 3.10).If the α-detection efficiency is lower than 100%, then some of the β -particlescoming from a decay to an excited state of 9Be would mistakenly be added to theanti-coincidence spectrum (i.e., β -with-no-α of Fig. 3.10). This would effectivelylower the asymmetry of the anti-coincidence spectrum. The asymmetry of theanti-coincidence spectrum versus the α-detection efficiency is given by Eq. 3.7:540 0.2 0.4 0.6 0.8 1alpha detection efficiency-0.4-0.3-0.2- coincidencebeta with no alphaFigure 3.10: Theoretical initial β asymmetry detected in coincidence with an α(blue), or in the absence of an α (red), for varying α-detection efficiency. Theasymmetry of the anti-coincidence spectrum depends on the α-detection efficiencybecause coincidence events with undetected α-particles would falsely mix the twospectra.abα(e) =ags pgs+a1 p1 ∗ (1− e)+a2 p2(1− e)pgs+ p1(1− e)+ p2(1− e) =−0.202+0.158(1− e)0.505+0.44(1− e) , (3.7)where e is the efficiency of the α-detector (0≤ e≤ 1).Also note, that when the 9Li decays into an excited state of 9Be, it subsequentlyemits two α-particles, at an angle of 180o. This means that an efficient α-detectorcan register all decays that emit an α , by just covering half the solid angle, namely2pi , instead of the total 4pi .Thus, our proposed detection system consists of a hat-like detector over thesample with a hole for the beam to enter, as well as a system of lenses to guidethe produced photons from the scintillator to a photomultiplier tube (PMT) outsidethe ultra high vacuum (UHV). Most of the α-particles will only have to cross afew nanometers of material (depending on the implantation depth) to reach thescintillator, so they will escape the sample without much attenuation. This has beenconfirmed with a Monte-Carlo Geant4 simulation [125].55This detection system will be referred henceforth as αLithEIA (α-detection forLithium-9 Enhanced Initial Asymmetry), spelled “alı´thea” from the Greek word for“truth” (αλη´θεια2). αLithEIA was incompatible with the current β -NMR andβ -NQR cryostats (see Sec. 2.2.2), so we designed a new apparatus (the so-called“cryo-oven” (see Appendix A)) that would permit us to detect at the same time theα- and the β -particles, but also was designed to be an upgrade over the old β -NQRcryostat.3.3.2 Experimental Testing of the αLithEIA Method on ZnS(Ag)We decided to perform a proof of principle experiment of αLithEIA, using ZnS(Ag)as the sample. The main advantage of this is that it ensured a very high efficiencyfor detecting the α-particles, given that all the 8Li+ stopped directly in the ZnS(Ag)scintillator. In other words, the ZnS(Ag) served both as the sample and the detector.ZnS is a cubic material in the sphalerite form. When doped with Ag, theresulting compound is known to be an extremely bright scintillator [126] (∼ 95,000photons/MeV). Commonly it is used in a polycrystalline form, sprayed as a powderon a surface. The incoming α-particles produce a high number of photons, withan intensity peak at a wavelength of 460 nm [127], caused by the recombinationbetween shallow donors and simple silver substitutional acceptors [AgZn] [128].The small thickness of the powder suppresses significantly the scintillation due toincoming β -particles, which attenuate in a much longer range (see Sect.,making this material ideal for the α-detection system of this study.In the first part of this experiment, we studied with 8Li β -NMR a polycrystallinepowder of ZnS doped with 6% Ag, deposited on a (fairly transparent) Al2O3substrate. The light emitted from the α-particles propagating in the ZnS paste,was passing through the transparent substrate and was guided to a photomultipliertube (PMT) lying outside vacuum, by a system of two convex lenses. As the sameα-detector assembly was used for the study of 8Li+ diffusion, a more detaileddiscussion on the geometry of the detector can be found in Ch. 5. As a high α-detection efficiency is highly advantageous (see Fig. 3.10), the energy threshold ofthe PMT was set just above the noise level.We acquired a 8Li resonance spectrum at 10 K (Fig. 3.11) and in additionverified that the scintillating properties of this material are unchanged in a widerange of temperatures (5-310 K), by imaging the scintillation produced by theimplanted beam with a CCD camera.The resonance spectrum was fitted best with two Lorentzian curves, both cen-tered at the same frequency (129.197(4) kHz). The two curves had very different2αλη´θεια has an interesting etymology: αλη´θεια > α+λη´θη = something that should notbe forgotten, i.e., the truth.56125 130 135 1400.0830.114Frequency (kHz)Figure 3.11: Resonance spectrum for 8Li in ZnS(Ag) at 10 K.widths: The narrow one had a FWHM of 0.64(5) kHz, whereas the wide line had aFWHM of 2.61(25) kHz. The amplitude of the narrow line was twice as big as thatof the wide line.Based on the unit cell structure of ZnS(Ag), we expect 8Li ions to inhabit thetetrahedral interstitial sites, with four sulfur nearest neighbors (Fig. 3.12), as bothAg+ and Zn2+ have a (−1)e relative charge and this Li+ is attracted to these centers.The multicomponent nature of the resonance is most likely related to the disor-dered nature of the sample since it is ∼ 6% Ag doped with other defects present aswell. We attribute the narrow resonance to lithium ions being in a slowly relaxingenvironment, whereas the wide resonance should be due to 8Li being at positionsthat lead to fast but inhomogeneous relaxation. Such centers of relaxation could becomplexes of Ag+ with other paramagnetic centers and other defects, which canproduce a small EFG at the 8Li site. Near the lithium implantation site, there are sixpossible sites for doped Ag ions to be. Thus, at a 6% doping level, there is a 31%probability of lithium to have at least one Ag ion as a (next-nearest) neighbor. Thismodel is supported by the fact that the amplitude of the narrow resonance is twiceas large as the wide resonance, i.e., 1/3 of the resonance amplitude is due to the fastrelaxing component and 2/3 due to the slow component.In addition to the resonance spectrum, the temperature dependence of SLR(Fig. 3.13) further supports the aforementioned model. The SLR spectra were fittedbest with a two-component fit, with one of the two 1/T1 rates being systematicallycompatible with zero at all temperatures, therefore it was fixed to zero, leading to afitting just like that of Eq. 3.5.Evidently, both the fraction f of the relaxing component and the relaxation rate57Figure 3.12: The crystal structure of ZnS(Ag). The yellow spheres represent S,the gray Zn and the green Ag ions. Implanted 8Li+ is expected to occupy one ofthe tetrahedral interstitial sites, with four sulfur nearest neighbors. Adapted fromRef. [129].1/T1 are temperature-independent in the region 5 K to 225 K. In this temperaturerange, the fraction of the relaxing component is ∼ 0.3, further supporting the viewthat the relaxation is due to electric quadrupolar interactions of 8Li with Ag atomspresent at the vicinity. The relaxation rate 1/T1 is ∼ 1 s−1 in this temperaturerange, but it increases up to 3 s−1 at 310 K. In the region 225 K to 310 K, thefraction f or the relaxing component increases as well, up to ∼ 0.6. This behavioris consistent with lithium becoming gradually mobile above 225 K, thus increasingthe probability of being near at least one Ag ion for some time during its lifetime.Turning to the 9Li measurement, we measured an SLR spectrum at 310 K. Incontrast to Sect. 3.2, this time we registered three spectra in parallel. Except forthe regular (untagged) SLR spectrum without any consideration to the emittedα-particles, we also registered each count to either the coincidence (α-tagged) orthe anti-coincidence (no-α-tagged) spectrum, depending on whether an α was alsodetected accompanying the β . These three spectra (with fits to Eq. 3.5) can be seenin Fig. 3.14.From Fig. 3.14, note that both the α-tagged and the no-α-tagged spectra havea significantly increased initial asymmetry compared to the untagged spectrum,without any loss in statistics, since each untagged count is also stored in some ofthe two tagged spectra as well. The no-α-tagged spectrum had an initial asymmetrytwice as big and the α-tagged spectrum four times as big as the untagged spectrum.580 50 100 150 200 250 30000.250.50.751Temperature (K)f0 50 100 150 200 250 30001234Temperature (K)T−11(s−1)(a) (b)Figure 3.13: Temperature dependence of the fraction of the relaxing component f(left) and the relaxation rate 1/T1 (right) in ZnS(Ag).0 0.2 0.4 0.6 0.8 1 1.2 1.4−0.15−0.1−0.0500.05ON OFFTime(s)AsymmetrySpectrumUntaggedα-taggedno-α-taggedfitsFigure 3.14: Untagged (black circles), α-tagged (red squares) and no-α-tagged(blue triangles) spectra of 9Li β -NMR at 310 K in ZnS(Ag).Tab. 3.1 indicates that the probability of 9Li to decay with an α is 49.5%.Therefore, if our system could detect all α-decays, the α-tagged and no-α-taggedspectra would have roughly equal counts. Based on the number of counts stored atthe two histograms, our α-detection efficiency was ∼50%.59Table 3.2: Fits of the untagged, α-tagged and no-α-tagged spectra of 9Li β -NMRat 308 K in ZnS(Ag).Spectrum f slow 1/T1 (s−1) fast 1/T1 (s−1) χ2Untagged 0.50(4) 0.41(15) 19(6) 1.15no-α-tagged 0.480(20) 0.34(8) 18(3) 1.10α-tagged 0.192(9) 0.000(5) 100(68) 1.06From Tab. 3.2 it is evident that the fit of the anti-coincidence (i.e., no-α-tagged)spectrum agrees well with the fit of the untagged spectrum, but with uncertainties inits fit-parameters smaller by a factor of∼2. The fit for the coincidence spectrum doesnot agree with either of the other two, due to the effect of rate-dependent distortion.Indeed, there is a pile-up effect present, due to which the energy threshold of theα-detector effectively decreases.ZnS is known to emit light with a pulse having a long tail in time (Fig. 3.15).Figure 3.15: The normalized decay curves of the light emitted from a ZnS(Ag)/6LiFscintillator illuminated with 5.5 MeV α-particles. Adapted from [130].The decay time of ZnS(Ag) phosphors is longer for α-particles than γ’s andcan be as long as 100 µs [131]. This can cause significant pile-up even for smallα-rates, that artificially allows low-energy α-particles to pass over the detector’sthreshold. In this case, as the energy threshold was set just above noise, the pile-upeffect resulted in noise being miscounted as α-events, thus mixing decays without60an α into the coincidence spectrum in a time-depended manner.Even though this time-dependent distortion effectively rendered the α-taggedspectrum unusable, we were able to increase the efficiency of the 9Li measurementby a factor of ∼ 2, not very far from the maximum expected theoretical gain of ∼ 3(see Sect. 3.3.1). In Sect. 7.2, some ideas as to how to overcome this issue in thefuture will be presented.3.4 ConclusionsWe have measured the ratio between 1/T1 of 9Li and 8Li in Pt and SrTiO3 in orderto help identify the nature of the fluctuations responsible for the spin relaxation(i.e., if they are magnetic or electric quadrupolar). In Pt, the relaxation is singleexponential and the ratio RPt was found to be very close to, but slightly less than,the pure magnetic limit. This is consistent with Korringa relaxation being dominantas suggested by the linear temperature dependence in 1/T1 reported previously.Nevertheless the small reduction in RPt relative to the pure magnetic limit meansthat excitations causing a fluctuating EFG may provide a small contribution to theobserved spin relaxation.In SrTiO3 at 300 K the results confirm that the dominant source of relaxationis electric quadrupolar. However, the relaxation function is more complicatedinvolving a relaxing part and a non-relaxing part. This suggests there is someinhomogeneous source of fluctuations/spin relaxation, possibly due to nearby de-fects. The ratio RSTO is close to, but slightly larger than, the pure quadrupolar limit,indicating that there may be some small magnetic contribution. However, the mainsource of spin relaxation is quadrupolar. This is consistent with expectations giventhe large quasi-static nuclear quadrupole interaction.Most importantly, we have demonstrated that the method of isotope comparisoncan be used in β -NMR to distinguish the nature of the fluctuations responsible for1/T1. This represents an important new tool for β -NMR, since in many systemsthere is uncertainty in the source of relaxation that cannot be removed simply byvarying experimental parameters.To further develop the method of isotopic comparison in β -NMR using 8Liand 9Li, we designed and tested a system of tagging the β -particles coming fromthe decay of 9Li in coincidence with/without an α-particle. This system wasdemonstrated to increase the effective asymmetry of 9Li by a factor of ∼2. Thisdevelopment, coupled with the tenfold increase in 9Li+ rate that was achievedrecently at TRIUMF, increases the figure of merit of the 9Li β -NMR measurementsby a factor of ∼ 40, making it comparable with that of 8Li.61Chapter 4Principles of Interstitial Diffusionin SolidsDiffusion of ions in solids is a mass transport mechanism of great significance formany branches of material engineering, fabrication and science. Its applicationsinclude – amongst others – diffusion hardening (e.g., steel), sintering and corrosionof metals, semiconductor doping, as well as solid state batteries, a subset of which(namely, the search for better Li-ion battery materials, see Sect. 1.2) is the mainfocus of the second part of this thesis.In this Chapter, a short outline of the theory of interstitial (self)diffusion willbe presented, both from a macroscopic (Sect. 4.1) and a microscopic (Sect. 4.2)point of view. The following two Chapters introduce the 8Li α-radiotracer methodboth from a theoretical and computational (Ch. 5), as well as from an experimentalperspective (Ch. 6).4.1 Macroscopic Theory of Solid State DiffusionIn contrast to the cases of a species moving (diffusing) through a gas or a liquid,the diffusion in a (crystalline) solid is restricted and is based on one (or several)hopping mechanism(s).We consider first solute atoms which occupy interstitial positions in the crystallattice. In such a case, diffusion is mediated by the interstitial ions hopping fromone energetically favorable site to the next, by passing over an energy barrier. Thelattice atoms are not permanently displaced due to this process. This is called aninterstitial mechanism for diffusion. This is conceptually the simplest process, butalso the most relevant to this work, since Li diffusion in Li-ion battery materials isusually based on such a mechanism.If the solute atoms are similar in size to that of the host crystal atoms, then theycan substitute a lattice ion and form a substitutional solution. The diffusion of suchatoms relies on the existence of vacancies close to the solute atom, since in this caseit can jump to the neighboring vacancy. This process is called a vacancy mechanismfor substitutional diffusion and it is generally slower than the simple interstitial case62(under the same thermodynamical conditions).Several other cases exist in between the two mechanisms discussed above, suchas divacancy substitutional diffusion (requiring two vacancies close to the soluteatom), or interstitialcy (i.e., indirect interstitial) diffusion, which is a collectivemotion of an interstitial solute atom and an interstitial host atom.The scope of this chapter is to provide the theoretical framework for the studyof Li interstitial diffusion in Li-ion battery materials, so only this case will bepresented.4.1.1 Fick’s Laws4.1.1.1 First lawFrom a macroscopic point of view, the diffusion of a species (e.g., atoms of aspecific element or isotope) in a solid is described by Fick’s laws. The first law isdefined by the formula:j =−D5 c, (4.1)where j is the flux, i.e., the number of atoms passing through a cross-sectionalarea per unit time, D is the diffusion coefficient tensor and c is the concentration ofthe diffusive species, namely the number of atoms per unit volume. The dimensionsof the diffusion tensor are length squared over time, so it is measured in units ofm2/s (or cm2/s).In isotropic media, such as cubic crystals, icosahedral quasi-crystals, or amor-phous metals, the diffusion tensor reduces to a scalar D (i.e., is direction-independent),but generally is a symmetric tensor of rank 2. Every such tensor can be written indiagonal form, so Eq. 4.1 reduces to a system of three equations after rotating thecoordinate system to that of the orthogonal principal directions (x1, x2, x3):j1 =−D1 dcdx1 , j2 =−D2dcdx2, j3 =−D3 dcdx3 (4.2)where D1, D2, D3 are called the principal diffusivities. Generally these threecan be unequal and the diffusion along a random direction (a1,a2,a3) is given by:D(a1,a2,a3) = D1cos2(θ1)+D2cos2(θ2)+D3cos2(θ3) (4.3)where θi denotes the angle between ai and the principal axis-i.In the case of systems with uniaxial symmetry, such as tetragonal, hexagonalor trigonal latices, two of the principal diffusivities are equal and the diffusioncoefficient depends only on the angle Θ between the direction in question and theaxis of symmetry:63D(Θ) = D1sin2(Θ)+D3cos2(Θ) (4.4) Second lawThe second Fick’s law can be derived from the first law (Eq. 4.1), if coupled to aconservation (or continuity) equation:∂c∂ t=−5 j (4.5)Equation 4.5 holds if the species undergoing diffusion is conserved (i.e., doesn’ttake part in chemical reactions, radioactive decay etc). In such cases, couplingEq.4.1 and Eq. 4.5 leads to Fick’s second law:∂c∂ t=5(D5 c) (4.6)If the diffusivity does not depend on the concentration – which is always thecase for tracer studies like the present –, then the diffusion equation along directionx is given by:∂c∂ t= D∂ 2c∂x2(4.7)Eq. 4.7 is usually refereed to as the diffusion equation. This can be solvedanalytically under given initial and boundary conditions. Infinite medium solutionIn the case of diffusion in an infinite 1D medium, starting with a concentrationc(0,0) at x = 0 at time t = 0 and c(x,0) = 0 elsewhere, Eq. 4.7 yields the followingsolution:c(x, t) =c(0,0)2√piDtexp(−x2/4Dt) (4.8)The characteristic length scale of this equation is the so-called diffusion length√2Dt.4.1.2 Temperature Dependence of Diffusion: Arrhenius LawGenerally speaking, the diffusivity (also known as diffusion rate) is temperature andpressure dependent. Temperature especially has a pronounced effect on the diffusionrate, which is suppressed at low temperatures and rises rapidly with increased T . It64is found empirically that in many cases (but not always) the diffusion rate over acertain temperature interval follows Arrhenius law:D(T ) = D′0exp(−∆H/kBT ), (4.9)where ∆H is the activation enthalpy of diffusion (in units of eV/atom), kB is thethermodynamic Boltzmann constant and D′0 is the so-called pre-exponential factor.The latter can be written as:D′0 = D0exp(∆S/kB), (4.10)where ∆S is the diffusion entropy and D0 can be calculated using the Einstein-Smoluchowski relation (see Sect. 4.2.3). Using the thermodynamic equation atconstant pressure:∆G = ∆H−T∆S, (4.11)where ∆G is the Gibbs free energy of activation, Eq. 4.9 can be written in termsof D0 directly as:D(T ) = D0exp(−∆G/kBT ), (4.12)Note that the thermodynamic Gibbs free energy of activation corresponds tothe activation energy EA defined by the characteristics of the particle’s microscopicmotion (see Sect. 4.2.3).An Arrhenius diagram depicts the logarithm of the diffusion rate (in units ofm2/s or similar) versus inversed temperature (in K−1), i.e., the graphical represen-tation of the function log10[D(1/T )]. If the system under study follows Arrheniuslaw, then its Arrhenius diagram will be linear, with the slope being −∆H/kB (orequivalently −EA/kB).4.2 Microscopic Theory of DiffusionThe macroscopic theory of diffusion follows the example of classical thermody-namics, both in the concepts it employs (Gibbs free energy, diffusion entropy, etc),as well as in the way of viewing the corresponding phenomena. This parallelismextends also to the fact that the laws of both macroscopic theories can be derivedfrom microscopic statistical arguments. The microscopic (or atomistic) path todiffusion is the subject of this section, which aims to recover Fick’s second law andto relate the diffusion pre-exponential factor D0 and Gibbs free energy for diffusion∆G with microscopic parameters of the lattice and its vibrations.65To that end, the problem of the random walk of a particle in a (rigid) latticewill be considered, both in the case of diffusion in an “infinite” lattice, as well as inthe case of diffusion close to a surface. The connection of this construct to Fick’sformalism is achieved by using Einstein-Smoluchowski’s law (Sect. 4.2.3).4.2.1 One Dimensional Random WalkBefore discussing the problem of the random walk of a particle in a lattice, it isimportant to address a common misconception about the term “random” in thiscontext. In this classical treatment, the individual particles (ions, molecules, atoms,etc) do not move randomly at any given point in time, but rather they followa trajectory governed by the potential landscape of the lattice around them, aswell as their momentum. The randomness of their long-range motion stems fromthe stochastic collisions between the diffusive species and their lattice neighbors.These collisions change the magnitude and direction of the particles’ momentum,making their apparent motion to seem random, when viewed in isolation from theirenvironment.The general formulation of the random walk problem contains a number npof particles, each undergoing a series of N random displacements r1,r2,...,rN witheach step being independent of the previous ones both in direction and magnitude,but with the probability of ri lying between r and r+dr governed by a pre-defineddistribution function τi(r). The diffusion process is then given by calculating theprobability W(R)dR that the particle will be at the interval [R,R+dR] after Ndisplacements [132].Studying the one-dimensional random walk of a particle is the simplest case ofthe aforementioned scenario and is very relevant to this work, since the diffusion oflithium in rutile TiO2 – which was selected to be the test case of the technique forstudying nanometer scale lithium diffusion presented in this thesis – is known to behighly one dimensional (see Sect. 6.1).In the case of an one dimensional random walk, we assume that each particleis displaced by steps of equal length l, but with each step having equal probability(1/2) to move the particle to the left or to the right. After N steps, the possiblepositions of a particle starting at 0 is:−N,−N+1, ...0,N−1,N, (4.13)where the x-coordinate of the i-th point is given by i · l.To calculate the probability W (m,N) of the particle being at point m after Ndisplacements, note that this would be given simply by multiplying the numberof different paths that would lead the particle to point m after N steps with the66probability of each path, normalized to the total number of possible paths with Nsteps.Since the probabilities of stepping to the left and to the right are both 1/2, allpaths should be assigned equal probability of (12)N . Also, to arrive at point m theparticle should undergo (N+m)/2 steps to the right and (N-m)/2 to the left. This isclearly a Bernoulli situation with:W (m,N) =N![(N−m)/2]![(N+m)/2]!(12)N, (4.14)In the limit of N→ ∞ and m N Eq. 4.14 can be simplified using Stirling’sformula:log(n!) = (n+12)log(n)−n+ 12log(2pi)+O(n−1)(n→ ∞) (4.15)to give:log(W (m,N))'−12log(N)+ log(2)− 12log(2pi)−m2/2N (4.16)which, for large N, leads to the asymptotic formula:W (m,N) =√2/piNexp(−m2/2N) (4.17)The natural choice for the coordinate for 1D-diffusion is the displacementx = m · l. The probability W (x,N)∆x is connected to Eq. 4.17 by:W (x,N)∆x =W (m,N)(∆x2l)=1√2piNl2exp(−x2/2Nl2)(4.18)The final step is to insert time into the probability equation, as diffusion isof course a time-dependent process. Assuming that the particle undergoes n dis-placements per unit time, we can define the diffusion rate as D = 12 nl2 using theEinstein-Smoluchowski relation (see Sect. 4.2.3). After time t, there is (by con-struction) a number of N = nt displacements, so Dt = 12 Nl2. Equation 4.18 can berewritten thusly:W (x, t)∆x =12√piDtexp(−x2/4Dt)∆x (4.19)67A comparison with Eq. 4.8 shows that this statistical analysis of the microscopicinterstitial motion recovers the 1D solution of Fick’s law (in the limit of a largenumber of particles and steps).4.2.2 Boundary ConditionsEquation 4.19 determines the probability of a particle starting at x = 0 at time t = 0to be in the interval [x,x+∆x] at time t. It supposes an infinite medium in bothdirections. A very important question, with very relevant consequences for thiswork, is how Eq. 4.19 would be affected under certain boundary conditions.In reality, no medium is infinite. Particles diffusing in a crystal (e.g., lithiumions in rutile TiO2, see Ch. 5-6) would reach after some time one of the surfaces ofthe crystal. There, they would either get trapped – thus, their diffusion would cometo a stop –, or they would get reflected. Here these two boundary conditions arepresented. Absorbing surfaceTurning first the case of an absorbing surface at a distance of m = m1 steps awayfrom the starting point m = 0. This trapping surface is supposed to absorb allparticles that reach it, which means that any diffusing particle that arrives there isnot allowed to move any more. Due to that, there are two important questions toanswer in this case, namely what is the probability of a particle arriving at somepoint m after N displacements and also what is the rate of absorption at the surfacem = m1.To calculate the probability W (m,N;m1), the only difference with the simplecase of an infinite medium presented at Sect. 4.2.1 is that all paths that arrive topoint m after passing through point m1 should be excluded. Every such forbiddenpath can be thought as a path to the image point (2m1-m), thus the probability ofarriving at point m under the absorbing boundary condition can be written in termsof Eq. 4.17 as:W (m,N;m1) =W (m,N)−W (2m1−m,N)=√2/piN[exp(−m2/2N)− exp(−(2m1−m)2/2N)](N→ ∞)(4.20)By introducing the coordinate x and the diffusion rate D, one can write Eq. 4.20in a form similar to Eq. 4.19:68W (x, t;x1) =12√piDt[exp(−x2/4Dt)− exp(−(2x1− x)2/4Dt)] (4.21)This formula is valid for all x < x1. For x > x1 the probability is zero byconstruction (all particles will get absorbed before reaching that point).0 2 4 6 8 10x (m) (a.u.)t = 1 sect = 2 sect = 3 sect = 10 sect = 500 sect = 1000 secFigure 4.1: Examples of density profiles W (x, t) starting with all particles at thex = 0 surface at t = 0 (i.e., W (0,0) = 1) and diffusing with D =1 m2 s−1. Thereis an accumulative wall at x = 10 m which traps all particles reaching it. As aresult, the density tends to even out in the 0≤ x < 10 space and also gradually getsuppressed (e.g., see the profiles for t = 500 and t = 1000 sec), as more and moreparticles end up trapped at the wall.Turning to the second question raised upon defining the current boundarycondition, namely what is the rate of absorption at the trapping surface, note thatthis is given by the probability a(m1,N) of arriving at the point m=m1 after exactlyN steps, excluding all paths that pass through m1 at an earlier step.Using similar arguments as above, this probability is given by:a(m1,N) =m1NW (m1,N)=m1N√2piNexp(−m21/2N)(N→ ∞)(4.22)Finally, by the usual change of variables, one finds the rate of absorption:69dadt(x1, t) =x1t12√piDtexp(−x21/4Dt) (4.23) Reflecting surfaceIn the case of having a reflecting surface at point m = m1, then the probability ofarriving at a point m after N displacements is given by simply adding the probabilitygiven by Eq. 4.17 with that for arriving at the “image” of m relative to the boundarysurface, 2m1−m, again as it would have been calculated in the absence of a boundarycondition. Therefore:W (m,N;m1) =W (m,N)+W (2m1−m,N)=√2/piN[exp(−m2/2N)+ exp(−(2m1−m)2/2N)](N→ ∞)(4.24)Which leads after changing the variables to the diffusion equation:W (x, t;x1) =12√piDt[exp(−x2/4Dt)+ exp(−(2x1− x)2/4Dt)] (4.25)Notice that the only difference of Eq. 4.25 with Eq. 4.21 is the sign of the termcorresponding to the image point 2x1− x.4.2.3 Einstein-Smoluchowski LawEinstein-Smoluchowski law connects the macroscopically defined diffusion rate D(see Sect. 4.1.1) with the microscopic random motion of the diffusing particles, viathe relation:D =12dt〈R2〉, (4.26)where d is the dimensionality of the diffusion – i.e., (one half of) the possiblediffusion paths on the lattice – and 〈R2〉 is the mean square displacement of theparticle after stepping at random n-times in a time interval t.R is by construction:R =n∑i=1ri (4.27)Thus:700 2 4 6 8 10x (m) (a.u.)t = 1 sect = 2 sect = 3 sect = 10 sect = 500 sect = 1000 secFigure 4.2: Examples of density profiles W (x, t) starting with all particles at thex= 0 surface at t = 0 (i.e., W (0,0) = 1) and diffusing with D=1 m2 s−1. In contrastto Fig. 4.1 there is now a reflective wall at x= 10 m. This change has minimal effectat early times, when the density tends to even out in the 0≤ x < 10 space, but at thelong time limit W (x, t) does not get suppressed.R2 =n∑i=1r2i +2n∑i=1n∑j=i+1rir j (4.28)Which leads to the following expression for 〈R2〉:〈R2〉=n∑i=1〈r2i 〉+2n∑i=1n∑j=i+1〈rir j〉 (4.29)If the steps are uncorrelated (i.e., the walk is truly random), then the secondterm averages to zero. Using this fact, one can define the degree of randomness ofthe walk using the correlation factor f , defined as:f = limn→∞〈R2〉∑ni=1〈r2i 〉= 1+2 limn→∞∑ni=1∑nj=i+1〈rir j〉∑ni=1〈r2i 〉(4.30)For a true random walk, f = 1. For dilute interstitial diffusion (which is thefocus of this thesis), usually this is the case. This is so, because most neighboringinterstitial sites are empty, hence a jump towards any one of them is equally probableand independent of the previous hop.71Figure 4.3: Image of n individual random steps ri on a lattice, leading to a totaldisplacement R. Adopted from [54].If the average step length is l, then:n∑i=1〈r2i 〉= nl2 (4.31)Combining Eq. 4.30 and Eq. 4.31 with Einstein-Smoluchowski law (Eq. 4.26),yields:D =f nl22dt(4.32)To write Eq. 4.32 in a form independent of the number of steps n, one can definethe mean residence time τ of a particle at a certain lattice position. The inverseof the mean residence time, τ−1 is called the hop rate. By the construction of theproblem, the particle undergoes n displacements in time t, so τ = t/n. Substitutingthis to Eq. 4.32:D =f l22dτ(4.33)This is the most commonly used form of the Einstein-Smoluchowski expres-sion, since it connects the macroscopically defined diffusion rate with microscopicproperties of the lattice and the particle motion that can be found experimentally orthrough simulations.72Physically, a particle in a lattice hops from an energetically favorable site tothe next, by overcoming the potential barrier separating the two. The height ofthe barrier is equal to the energy difference EA between the particle’s equilibriumposition and that of the barrier’s saddle point (see Fig. 4.4). EA is called theactivation energy.Figure 4.4: Interstitial atom’s jumping process. It hops from site A to site B bymoving through the saddle point. The energy difference between the energeticallyfavored initial (and final) sites with the saddle point defines the energy barrier EA.This is also the difference of Gibbs free energy GM between these points, assumingthat the process is reversible (and thus ∆S∼ 0). Adapted from [54].The hopping over the barrier is possible because at a non-zero temperaturethe lattice atoms vibrate around their equilibrium positions and this fluctuation ofthermal energy provides the particles with enough kinetic energy to overcome thebarrier. The frequency of this vibration τ−10 is of the order of the Debye frequency,with typical values of 1012 Hz to 1013 Hz. The hop rate would be then equal toτ−10 ≡ v0 – also known as the attempt frequency – multiplied by the probabilityof having enough energy to overcome the barrier. This probability follows theBoltzmann distribution, so [133]:τ−1 = τ−10 exp(−EAkBT) (4.34)Note that Eq. 4.34 leads to Arrhenius law (Eq. 4.9) by substituting τ−1 inEq. 4.33. Then:73D =f l22dτ=f l22dτ−10 exp(−EAkBT)= D0exp(− EAkBT )(4.35)4.2.4 Isotopic mass effect on diffusionAs a final note on the theory of interstitial diffusion, the isotopic effect will bebriefly discussed. Since in this work Li diffusion in solids is studied through theradioactive decay of 8Li, it is important to address how different the diffusion rateof 8Li(compared to the stable 6,7Li) is expected to be under identical circumstances.In the general case, let there be two isotopes of the same element, with relevantmasses mα and mβ . Due to the different masses, the two isotopes are expected todiffuse through the lattice with different diffusion coefficients Dα and Dβ .To quantify the above discussion, Eq. 4.34 will be used, with the notationalsubstitution τ−1 ≡ ω and τ−10 ≡ v0:ωα,β = v0α,β exp(−EAα,βkBT)≡ v0α,β exp(−∆HMα,βkBT) (4.36)where ∆HM is the activation enthalpy of diffusion defined at Eq. 4.9. The rele-vant quantities for the two isotopes are to be distinguished by the use of subscripts,e.g., v0α ,v0β , ∆HMα , ∆HMβ .Under identical thermodynamical conditions, the diffusion barrier is expected tobe independent of the mass of each isotope, since all isotopes of the same elementhave by definition equal charges. This leads to EAα = EAβ = EA, or equivalently:∆HMα = ∆HMβ = ∆HM (4.37)The only exception to the above statement is hydrogen (and to a lesser extenthelium). For these very light nuclei, quantum tunneling and zero-point motion can-not be neglected and since these effects are mass dependent, the effective diffusionbarrier for, e.g., 1H and 2H are expected to be different. For heavier ions, theseeffects are negligible.Using Eq. 4.36 to form the ratio of the jump rates, taking into account Eq. 4.37:ωαωβ=v0αv0β(4.38)74According to the classical rate theory [134], the attempt frequency v0 is thevibration frequency of the atom at the direction of the jump attempt. The vibrationfrequency can then be related to the isotopic mass by imposing Einstein’s modelfor the vibrational frequencies of atoms in a crystal, which treats each atom as anindependent (quantum) harmonic oscillator. v0 turns out to be proportional to theinverse square root of the mass, thus:DαDβ=v0αv0β≈√mβmα(4.39)For the case of 8Li vs 6,7Li, using Eq. 4.39 the diffusion coefficients for thethree isotopes yield: D6 = 1.15D8 and D7 = 1.07D8. In other words, accordingto Einstein’s simple model, the diffusion coefficient extracted from 8Li diffusionshould be 10-15% lower than what would be for 6,7Li under identical circumstances.Indeed, for the system studied in the following Chapters, namely Li diffusion inrutile TiO2, Eq. 4.39 has been found experimentally to be valid, within error [70]and no isotopic effects on the diffusion barrier were reported.75Chapter 5Principles of Studying NanoscaleLithium Diffusion Using theα-Decay of 8Li5.1 Basic Principles of the 8Li α-radiotracer TechniqueAs part of this thesis we have developed a novel method for directly measuring therate with which Li ions diffuse inside materials and potentially across their interfaces,namely the 8Li α-radiotracer method. The primary practical interest is in regardto Li-ion batteries, where Li diffusion determines the charging/discharging rate ofthe battery and thus is a very important characteristic of all the key componentsof a Li battery, i.e., the anode, cathode and electrolyte (see Ch. 1.2). In addition,this method can determine whether Li+ gets reflected or trapped upon reaching thesurface of the sample, a fact that is very hard to establish with other techniques. Thesurface boundary condition for the Li+ motion can critically affect the ease of Liintercalation in a given material.The method is a variation of the classical radiotracer method and uses theattenuation of the progeny α-particles from the radioactive decay of 8Li, to studynanoscale Li diffusion. As explained in Ch. 2, 8Li decays to one β -particle, two α’sand an electron antineutrino. The energy of an α-particle is attenuated significantlyover a depth of 100 nm, which can be comparable to the diffusion length of 8Liwithin its radioactive lifetime. The energetic β -particles, on the other hand, are muchmore weakly attenuated and thus can serve as a convenient way to normalizationfor the overall 8Li decay rate.To measure Li diffusion, a short beam pulse of low energy (0.1− 30 keV)8Li+ ions is implanted close to the surface of the sample (at an average depth of∼100 nm) housed in an ultra-high vacuum cold finger cryostat [33, 123]. Theenergy of the beam defines the initial Li+ implantation profile. Upon arrival, the 8Li+start to diffuse through the sample and undergo β -decay to 8Be which then decay(immediately) into two energetic α-particles, each with mean energy of 1.6 MeV.Due to their rapid attenuation inside the sample, the highest energy α-particles76escaping the sample originate from 8Li+ that have diffused back to the surface. Tofurther amplify the sensitivity to 8Li+ near the surface, the α-detector is placed at agrazing angle, θ ≤ 4.4o, relative to the surface, as shown in Fig. 5.1.Figure 5.1: Schematic of the ultra-high vacuum (10−10 Torr) sample region showingthe cross-section of the ring detector. The α-particles originating at depth d thatreach the α-detector traverse distance d/sinθ [77] through the sample. Not toscale.The α-detector in our setup is an Al ring, whose inside surface is cut at ∼45◦and coated with a thin layer of Ag-doped ZnS, a well known scintillator sensitive77to α-particles [131]. Note that the results of the simulation study presented hereare not heavily dependent on the type of the detector, as long as its thickness issufficient to stop α-particles and at the same time thin enough to keep the β energydeposition inside it to a minimum (see Sect. light from the ZnS(Ag) scintillator is collected in the forward directionusing two 5 cm  convex lenses (see Fig. 5.2) which focus the light onto thephoto-cathode of a fast photomultiplier tube (PMT). The first lens is attached atthe radiation shield of the cryostat – inside the ultrahigh vacuum region –, whereasthe second lens and the PMT are positioned outside the vacuum chamber, behinda transparent viewport. A stainless steel tube that is housing the PMT is attachedaround the optical viewport in order to block all ambient light from reaching thephotomultiplier. The PMT pulses have a large signal to noise ratio (> 10) and passthrough a timing filter amplifier to be discriminated, so that only the top 1/3 ofpulses above the noise level are counted.Figure 5.2: Schematic of the α-detection geometry, not to scale. The 8Li+ beamcomes from the left of the figure and reaches the 7x7x0.5 mm3 sample after passingthrough a pinhole Al collimator, which ensures a centered beam. The α-particlesthat reach the ZnS scintillator produce light (drawn here as black arrows), whichpropagates through a series of two focusing lenses towards the face of a photomul-tiplier tube, lying outside vacuum inside a stainless steel housing that blocks allambient light. The lens closest to the sample is placed about one focal length awayfrom it (∼5 cm).To ensure a stable, well defined 8Li+ beamspot at the center of the sample, an78aluminum mask can be placed in front of the sample-detector region, having a smallhole (2 mm to 3 mm diameter) at its center to allow for the beam to enter. Thissystem does not allow for beam near the edge of the sample, for that could changeartificially the diffusion signal.This system was incompatible with the geometry of the β -NQR cryostat andfor this reason a new custom cryostat was designed, the so-called “cryo-oven”(see Appendix A). The new cryostat has a nominal temperature range of 5-400 K(compared to 3.5-300K of the old β -NQR cryostat). Being able to reach highertemperatures can be critical for studying slow diffusion, since the diffusion rateincreases exponentially with temperature in materials that follow Arrhenius’ law.Thus, if the diffusion rate is prohibitively slow in a material at room temperature,one can try to measure it at an elevated temperature.Using the system described above, the diffusion rate of Li inside the sample isdirectly related to the time it takes to reach the surface, which in turn relates to theα-rate as a function of time. This method has intrinsic time- and length-scales ofτ1/2∼1 s and d∼100 nm, respectively, which leads to a theoretical sensitivity to thediffusion rate D from 10−12 to 10−8 cm2 s−1. This technique thus covers an optimalrange of D for battery materials. However, our effective sensitivity limit is closerto 10−11 cm2 s−1, determined by experimental factors such as the finite acquiredstatistics and the existence of small distortions from nonlinear detector response.790 1 2 3 4 5BeamOn OffTime (s)NαFigure 5.3: Simulated raw counts at the α-detector versus time. There is a suddenchange when the beam goes off at 1 s. Prior to that, the number of 8Li ions presentat any given moment in the sample was approaching transient equilibrium, whereasafter the end of implantation, the remaining 8Li is left to decay away.In situations where Li+ is not mobile, the probability of detecting an α is time-independent and the measured α-counts follow the decay rate of 8Li (Fig. 5.3).This can be monitored conveniently using the high energy β -particles from the 8Lidecay, which are weakly attenuated over these distances. Thus, the ratio of countsYα = Nα/Nβ is constant in time. On the other hand, when Li+ is mobile, the ratio istime-dependent when the mean diffusion length in the 8Li lifetime is comparable tothe mean depth of implantation, reflecting the fact that the 8Li+ depth distribution(and hence the probability of detecting an α) is evolving in time.The information on Li diffusion comes from the time evolution of the α-signal.The absolute α-to-β ratio, i.e., the baseline ratio of Yα , in the absence of diffusion,depends on experimental factors such as detector efficiencies, therefore in orderto account for these systematics, each α-spectrum is self-normalized to start fromunity at time zero, i.e., Y nα (t) = Yα(t)/Yα(0).In Sect. 5.2.2, simulated normalized α-signals Y nα (t) for several diffusion rateswill be presented. By fitting the experimentally acquired normalized α-signalY nα (t;T ) at a specific temperature to a library of simulated (or calculated) normalizedsignals, the diffusion rate of Li at that temperature can be extracted (see Chapt. 6).A technique similar to the one discussed here has been developed by Jeonget al. [81] for Li+ diffusion on micrometer and, recently, by Ishiyama et al. [56]80on nanometer length scales; however, the technique reported here differs in afew key ways. In particular, the 8Li implantation rates accessible at TRIUMF(typically 106-107 8Li+/s) are 1-2 orders of magnitude larger [56], which allowsthe α-detector to be placed at a grazing angle θ (≤4.4◦ versus 10(1)o [78]). Thisdetector configuration significantly decreases the α-counts, but greatly enhancesthe sensitivity to the near-surface region. Using this development, we were ableto perform many α-radiotracer measurements at various temperatures in a limitedamount of time, which permitted for the first time the extraction of the Li diffusionactivation energy (see Ch. 6).5.2 Simulations of α-Detected Lithium DiffusionIn this section the aforementioned technique for studying lithium diffusion usingthe (subsequent) α-decay of 8Li will be simulated under various conditions. Thissimulation study is a very important first step towards the realization of the 8Liα-radiotracer method, because it can provide valuable information on choosing thebest possible experimental configuration (e.g., the geometric characteristics of theα-detector, its width, positioning etc), on how to avoid possible pitfalls (such asa high degree of β -contamination in the α-detector) and also how the normalizedα-signal Y nα (t) changes under various diffusion conditions (faster or slower diffusionrate, different boundary conditions at the sample surface). In all the simulationspresented in this Chapter, the sample material is TiO2, because this material wasselected as the first case to be studying experimentally (see Ch. 6).In order to extract the Li diffusion rate with the α-radiotracer method, weperformed numerical solutions to Fick’s laws in 1D to generate the time-evolveddepth distribution of 8Li+, accounting for the boundary conditions of the crystalsurface and the initial 8Li+ stopping profile as simulated by the SRIM Monte Carlopackage [104]. The normalized α-signal Y nα (t;D) is then obtained using MonteCarlo techniques aided by the Geant4 codebase [125, 135, 136].The procedure issummarized as follows:1. The implantation profile of the 8Li+ beam for an energy E between 0.1 keVto 30 keV for a rutile TiO2 target is generated using the SRIM Monte Carlopackage [104].2. A custom code was used to diffusively evolve the initial implantation profilewith time for various diffusion constants that span five orders of magnitude(10−7 cm2 s−1 to 10−12 cm2 s−1) using Fick’s law (see Sect. 4.1.1). Twodifferent boundary conditions were considered, namely Li+ trapped (with acertain probability) or reflected at the surface of the sample.813. Finally, these temporally evolved profiles of the 8Li+ depth distribution wereimported into the Geant4 simulation package. Using Geant4, all the relevantgeometrical structures (sample material, detectors, etc), physical interactions(EM, radioactive decays, etc) as well as the energy/momentum distributionsof the α- and β -particles were defined and the α-counts versus time at theα-detector were generated.5.2.1 Temporal Evolution of the Diffusion ProfilesThe first part of this simulation study is to generate the beam implantation profilesof the 8Li beam for given beam energies inside the sample (see Fig. 2.11). This isdone with SRIM, as discussed at Sect. 2.2.3.Generally speaking, by increasing the energy of the beam, the mean depth of theions increases, as does the width of their spatial distribution (i.e., the ion struggle).Different beam energies might be ideal for studying different scenarios, thereforea careful analysis of the optimum beam energy prior to the actual experiment isvery important. If the diffusion is very slow, close to the detection limit, it isadvantageous to decrease the energy of the lithium beam in order to bring most of8Li closer to the surface. On the other hand, if the diffusion is very fast, a narrowdistribution of lithium close to the surface could lead to most of the change in theα-detected signal happening at the very early times after the beam implantation,possibly lying outside our detection capabilities. In such a case, a higher beamenergy would be necessary.The next step after simulating the beam implantation profile is to study itstime evolution due to lithium diffusion. Only diffusion on the axis of implantation(defined here as the z-axis) is considered, because our detection scheme is sensitiveonly to changes in the depth distribution of lithium.To observe any effect on the α-detection probability due to diffusion in thexy-plane, the lithium ions would have to be able to reach the edge of the sample inthe time scale of a few half-lives of 8Li (∼6 s). The required diffusion rate for thisto happen would be six or seven orders of magnitude larger than a typical diffusionrate of a Li-ion battery material at room temperature. Such an effect is thereforeneglected and the xy-distribution of lithium in the sample is considered constant,defined by the properties of the beam. Typically a beam spot of 1 mm to 3 mmdiameter is considered, having a Gaussian density distribution. This requirement isrealized in the experimental configuration by adding a blocking Al mask in front ofthe sample, as discussed in Sect. 5.1.To generate the temporally-evolved depth distribution of the Li ions after asmall time interval ∆t, the initial implantation spectra are split in infinitesimal bins∆z (1 nm wide) and then each bin is let to diffuse independently using the one82dimensional bulk diffusion equation (Fick’s second law):c(z, t) =c(z′,0)2√piDtexp[−(z− z′)24Dt], (5.1)where D is the diffusion rate and c(z, t) is the concentration of Li at depth z andtime t, due to the diffusion of the initial concentration of the bin at z′. The diffusionlength L traveled by a 8Li+ in time ∆t is:L =√2D∆t. (5.2)The new depth distribution after time ∆t is given by superimposing all thepoint-sources by summing over the diffused profiles of each bin:c(z, t+∆t) =∫ ∞0c(z′, t)√piD∆texp[−(z− z′)24D∆t]dz′. (5.3)Where c(z, t + ∆t) is the new concentration of lithium at depth z, D is thediffusion rate and c(z′, t) is the previous concentration of lithium at depth z′.Apart from the non-zero width of the initial depth distribution, a second compli-cation arises from the fact that not all ions are implanted simultaneously, but ratherover some time period [0,∆], where ∆ is typically set to 1 s. During that time, theinitial depth distribution of Li is continuously replenished from the incoming beam.For t > ∆, the beam implantation comes to an end the remaining 8Li are left todiffuse and decay. The algorithm employed here accounts for both the initial depthdistribution and the non-zero beam implantation period to calculate numerically thetemporal evolution of the depth profiles.In addition to the aforementioned bulk-diffusion considerations, one has toimpose some boundary conditions at the two surfaces of the target crystal. Since atthe surfaces of different materials Li ions might interact with different potentials(e.g., potential wells or barriers), two cases of boundary conditions were studied,corresponding to the cases presented at Sect. 4.2.2.In this section, only the effect of the front sample surface will be presented. Thisis by far the most important surface, since virtually all of the α-signal comes fromdecays happening close to it. Also, for typical samples with a thickness of∼0.1 mmto 1 mm (see Sect. 6.3), this is the only relevant surface, because the lithium ionswon’t have time to reach the back of the crystal in a few lifetimes of 8Li.The first boundary condition to be considered is for Li ions to be reflected at the(front) surface of the target. This is probably the most common case in materialsthat contain Li in their crystal structure; the reason being that all the trap sites for Liions are occupied. The time-evolved distributions of Li ions for different diffusionconstants were calculated, using as a starting point the implantation profile for a83beam implantation energy of 25 keV (Fig. 5.4). The acquired temporally-evolvedspectra for a diffusion constant of D = 10−10cm2/s are presented at Fig. 5.4.0 200 400 600 80000.0020.0040.0060.0080.01depth (nm)Relative8Liyieldversusdepth(nm−1)time (s)as implanted1 s2 s3 s4 s5 sFigure 5.4: Calculated depth profiles of Li ions versus time (using Eq. 5.3) for abeam pulse of 1 s in TiO2, diffusion constant D =10−10 cm2 s−1, a beam energy of25 keV and with Li ions getting reflected at the surface of the material.It is evident that the depth profile of the 8Li concentration is relatively narrowupon implantation, but it broadens significantly after a few seconds. For fasterdiffusion rates of 10−9 cm2 s−1 to 10−8 cm2 s−1, the 8Li ions will get distributedover a wide depth range of several µm after a few seconds.The second boundary condition we consider is that Li ions get trapped (with100% probability) when reaching the front surface of the sample. This leads to anincreasing percentage of Li at the surface over time. The percentage of Li trappedat the front surface of the sample versus time for various diffusion rates is depictedat Fig. 5.5:840 1 2 3 4 5020406080100BeamOn Off10−12 cm2/s10−11 cm2/s10−10 cm2/s10−9 cm2/sTime (s)Surface%ofLiFigure 5.5: Simulated fraction of Li ions trapped at the front surface of a TiO2sample versus time, for a beam pulse of 1 s and an initial beam energy of 25 keV.From Fig. 5.5, it is evident that for a fast diffusion constant of 10−9 cm2 s−1,more than 80 % of the remaining Li is trapped at the front surface of the sampleafter 5 s. Of course Li diffuses in both directions through a random walk, butbecause it gets trapped at the surface most of it will end up there, after a fewlifetimes. For slower diffusion constants these effects get gradually smaller. Fordiffusion constants in the order of 10−12 cm2 s−1, virtually no Li ions will havereached the surface after 5 s. The accumulation of Li at the surface for diffusionrates >10−10 cm2 s−1 approaches saturation after a few Li lifetimes, while slowerdiffusion rates result in a more linear accumulation. To study slower diffusion rates10−12 cm2/s to 10−11 cm2/s, one could decrease the energy of the beam, in orderto move the distribution closer to the front surface.Having generated the depth profiles of Li ions as functions of time, diffusionconstants and boundary conditions, the next step is to import them into the Geant4simulation, in order to study the detector response at each scenario (i.e., time, D,Ebeam).5.2.2 Geant4 SimulationsGeant4 is a well established software toolkit used for the simulation of particle-material interactions [125], such as the passage of particles through matter, detectorresponse to particle beams, etc. Geant4 is based structurally on the computerlanguage C++ and utilizes a Monte Carlo algorithm and a random number engine. Itis designed for use in a very wide range of applications, spanning from high energyand accelerator physics, to nuclear physics, space applications, as well as medical85physics and radiation safety. Indeed, it is applicable in an energy range of meV(such as ion-DNA interactions [137]) to TeV (LHC-related applications).Geant4 offers an extended toolkit covering all aspects of a particle-detector sim-ulation. By using many predefined libraries and functionalities, the user can definethe geometry of the simulation (materials, shapes, dimensions, relative positions,static EM fields), can choose from a number of physical models that determinethe processes relevant to the project (including electromagnetic and hadronic in-teractions, decays etc.), the type, energy and the momentum (distributions) of theprimary beam particles, the shape and position of the beam. The user can also definethe desired output type at the end of the simulation process, such as the energydeposited in the detector, the number of particles reaching the detector, etc.Because of its object-oriented and modular architecture, the user is free tochoose, load or customize only the tools needed for their specific project. This allowsfor a comprehensive structure that promotes the understanding of the code, lying faraway from black-box-type simulations. In addition, its multi-threading capabilitiesallow for a nearly linear scale up of the number of simulated events [136].The code of this simulation study is intended to be very adaptive to differentscenarios, therefore the user is allowed to define most aspects of it (using a macrofile or the command prompt) at the beginning of each simulation. This way, the usercan choose the energy of the 8Li beam (that corresponds to a SRIM implantationprofile to be loaded, see Sect. 5.2.1, the position and size of the (Gaussian) beamspot,the sample’s (and substrate’s, if applicable) material and dimensions, the lithiumdiffusion rate that will be simulated, as well as the size, width and depth of theα-detector around the sample. The physical processes and the energy distributionsof the decay products of 8Li are defined internally, so the user would have to re-compile the code, if any changes are needed on these aspects. More details on thestructure of this Geant4 project can be found in Appendix B.The α-detector response is simulated for every required point in time the fol-lowing way: First, the depth profile of 8Li is calculated for that time-bin, using thealgorithm outlined in Sect. 5.2.1. The user defines how many Monte Carlo events(i.e., simulated 8Li decays) are required per time-bin. These decays are generated ata random depth that follows the aforementioned depth distribution and their decayproduct energies (one β , two α and one antineutrino) following the distributionsof Fig. 2.2. The initial direction of each particle is random, except for the twoα-particles that are emitted always back-to-back due to momentum conservation.Note that the parity violation of the β -decay is not implemented and is neglected asirrelevant to this study.Each decay product is tracked individually through the geometry of the simula-tion. At each step of their trajectory all relevant physical processes are taken intoconsideration and they ultimately define stochastically the new position, direction86and momentum of that particle’s trajectory. As a result, both α- and β -particlesscatter and lose energy while inside the sample, but over very different length scales,as explained in Sect. 5.1. Part of their energy loss can be transformed into secondaryparticles (such as low energy photons or electrons) that are also tracked in turn.Any particle that reaches the detector(s) will interact with it and deposit energy.When the total energy deposited in the detector is above a user-defined threshold,then Geant4 registers a count in that detector. Note that all energy deposited ismeasured, which means that if there is a lot of contamination from other particlespecies, the detector signal could be potentially swamped. The issue of a possibleβ -contamination at the α-detector is studied in Sect. Physical processesIn this simulation study, the main relevant physical processes are the electromagneticinteractions and the radioactive decays. Since the energies of all particles thatparticipate in this experiment are not higher than a few MeV, many high energynuclear processes are forbidden (e.g., pair productions). The only nuclear process(other than β -decay and α-decay) that can in principle be present is low energyelastic nuclear scattering. Note that Geant4 automatically selects only the relevantprocesses in each scenario based on the energy and type of particles. For instance,if the hadronic interactions are “turned on” in a simulation concerning only leptons,they will have no effect at all.Because of the vast energy interval of possible Geant4 applications (meV-TeV),there are multiple libraries (called PhysicsLists) that the user can select to simulatethe electromagnetic and nuclear interactions with different degrees of accuracy basedon the energy scale [138]. Some of them focus on enhanced accuracy and others onbetter CPU performance. In this study, the PhysicsLists used were selected basedonly on accuracy considerations. This is because the simulation time-investmentscales very rapidly with energy, so in this “low energy” range it is still rather small.For electromagnetic interactions, the library G4EmLivermorePhysics was se-lected, because it is more accurate than the usual G4EmStandardPhysics in theMeV energy range. The hadronic interactions were included by importing thelibraries G4HadronElasticPhysics and G4HadronPhysicsQGSP BERT, which usethe Bertini model as a basis of simulating hadronic interactions. The various as-pects of ion nuclear decay physics were covered by importing the relevant libraries(G4DecayPhysics, G4RadioactivePhysics, G4StoppingPhysics, G4IonPhysics).Note that Geant4 failed to generate the energy distribution of the α-particles ofFig. 2.2b, because the quantum mechanical mixing of the two first excited states of8Be was not implemented. To solve this issue that was leading to an unphysicallynarrow energy spectrum for the α-particles, the energy distribution of Fig. 2.2b was87imported by hand. All other aspects of the simulation, such as the energy spectrumof the β and neutrino particles, the momentum distributions etc., were verified to besimulated correctly by Geant4. GeometryFigure 5.6: Geometry simulated by Geant4. It contains a sample (optionally ona substrate) at the center of the coordinate axes, with the ring α-detector aroundits beam-facing surface, plus a β -detector behind a collimator at the back side ofthe sample, used for calibration and testing purposes. The collimator is placedto simulate the stainless steel foil in front of the β -NMR β -detectors that blocksall alphas while allowing the beta particles to reach the detectors without muchattenuation.The geometry of the simulation study can be seen at Fig. 5.6. It is comprised of fourdifferent parts:1. The sample under study (the light blue square at the center of the coordinatesystem). The user can define its size, thickness and material.2. At the back side of the sample there can optionally be a substrate. The user isfree to define its properties as well, but by default it is made of vacuum (i.e.,it does not exist).3. Around the front surface of the sample, there is a thin scintillating ring (madeby default of ZnS(Ag)) for α-detection. This ring is placed in front of the88sample, in the sense that the ending point of the ring parallel to the z-axisis at the same plane with the surface of the sample. The ring has a defaultwidth of 10.4 mm diameter, a width of 0.4 mm and a depth of 0.1 mm. Itis angled internally to match the specifications of the actual experimentaldetector. The fact that its depth is very small, makes it sensitive selectivelyto α-particles, which deposit all their energy very rapidly, while making itcompletely insensitive to β -particles (see Sect. As with most otheraspects of this geometry, the user is free to modify the characteristics of theα-detector.4. A second detector is placed at a distance behind the sample, lying behind athin stainless steel collimator, which blocks all α-particles from reaching thisβ -detector. It is used to normalize the α-detector counts, but its purpose ispurely for calibration and testing. α-detector signalThe distribution of the energy deposited at the α-detector at different points in timeis depicted at Fig. 5.7, as simulated by Geant4 for a diffusion rate D =10−9 cm2 s−1and the two aforementioned boundary conditions at the sample’s surface. Notethat this is the total energy deposited in the α-detector, by all incoming particles(i.e., α-particles, β -particles and secondary photons and electrons produced by theprimary decay products of 8Li).891 2 3 4 5Trapping SurfaceEnergy Deposition (MeV)Relativeα-yieldtime (s)0.01 s1 s5 s1 2 3 4 5Reflective SurfaceEnergy Deposition (MeV)time (s)0.01 s1 s5 s(a) (b)Figure 5.7: Simulation of the distribution of energy deposited at the α-detectorper decaying 8Li ion at different times for a 8Li beam energy of 25 keV, with8Li diffusing with a diffusion rate D =10−9 cm2 s−1 and Li ions getting trapped(Left) or reflected (Right) at the sample’s surface. In the trapping case, there is asignificant increase with time in α-yield at high energies, as an ever increasingpercentage of Li+ gets trapped at the surface and can therefore reach the detectorwithout attenuation, accompanied by a decrease in yield at the low-energy part ofthe spectrum, as the percentage of Li+ deep into the sample decreases accordingly.The situation is the opposite for the reflective case.When 8Li gets trapped at the sample surface (Fig. 5.7(a)), the yield of highenergy (>1.5 MeV) α-counts increases with time, because most of the 8Li willreach the surface after a few lifetimes and will get trapped. The situation is morecomplicated when Li gets reflected at the surface (Fig. 5.7(b)), because the 8Li ionswill initially diffuse (and) towards the surface, but then the distribution will startdiffusing primarily towards the bulk of the sample.In order to suppress the background, namely the part of the signal that has noconnection or dependence on lithium diffusion, an energy threshold has been setat 2 MeV. As a result, the α-detector registers a count if at some time (interval)the total energy deposited in the detector is at least 2 MeV. This threshold hasbeen selected in order to maximize the signal-to-background ratio of the detector,while maintaining significant count rates, since it allows to measure ∼30 % of theunattenuated α-particles. A larger threshold value would reduce the counts of thedetector, without any significant effect on the quality of the signal. On the otherhand, a smaller threshold gradually decreases the quality of the signal (see Fig 5.8).900 1 2 3 4 50.60.811. OffTime (s)Yn α(t;Eth)Threshold0.25MeV0.5MeV1MeV2MeV3MeVFigure 5.8: The normalized α-spectrum, as simulated with Geant4 in TiO2, for dif-ferent α-detector thresholds, a beam pulse of 1 s, diffusion constant D=10−9 cm2/sand an initial beam energy of 25 keV, using the accumulative boundary condition.Each simulated point is generated by 107 8Li decays. Evidently, the energy thresh-old does not affect the temporal evolution of the Y nα signal, but rather differentdetector thresholds can enhance or suppress the amplitude of Y nα . The error bars aresuggestive of what they would be in an actual experiment, based on the statistical√N factor. They are the smallest at the beam off time (t = 1 sec), when the numberof 8Li+ decays is at maximum (see Fig. 5.3) and they increase substantially afterthat.Even though the qualitative characteristics of the normalized α-signal (e.g., therate of reaching saturation) are unaffected by varying the energy threshold of theα-detector, a threshold between 1 MeV to 2 MeV results into the greatest percentagechange of the normalized α-counts with time and thus to the clearest α-signal.With those considerations in mind, the normalized α-signal with an α-energythreshold of 2 MeV is plotted in Fig. 5.9 as a function of time for lithium trappingat the surface of the sample material:911 2 3 4 511.21.41.6BeamOn Off10−9 cm2/s10−10 cm2/s10−11 cm2/s10−12 cm2/sTime (s)Yn α(t;D)|trappingFigure 5.9: The normalized α-signal, as simulated with Geant4 in TiO2, for a beampulse of 1 s, different diffusion rates and an initial beam energy of 25 keV, usingthe accumulative boundary condition. Each simulated point is generated by 107 8Lidecays.Comparing Fig. 5.5 and Fig. 5.9, it is clear that the simulated spectrum looksqualitatively very similar to the time evolution of the percentage of 8Li decayingnear the surface. There is a ‘kink’ at t = 1s, when the beam pulse is switchedoff. Before that time, the initial beam profile gets continuously replenished bythe beam, antagonizing the fast lithium diffusion. After the beam is switched off,lithium diffuses in both directions, but because it gets trapped at the surface mostof it will end up there, after a few lifetimes (see Fig. 5.5). For fast diffusion (e.g.,D >10−10 cm2/s) this leads to a saturation of the signal for t > 3s. For a slowerdiffusion rate (D <10−10 cm2/s) the saturation regime is out of the range of afew lithium lifetimes, so the signal increases much more linearly and never reallysaturates. The situation is very different when lithium gets reflected at the surfaceof the sample (Fig. 5.10).920 1 2 3 4 BeamOn Off10−9 cm2/s10−10 cm2/s10−11 cm2/s10−12 cm2/sTime (s)Yn α(t;D)|reflectiveFigure 5.10: The normalized α-signal, as simulated with Geant4 in TiO2, for abeam pulse of 1 s, different diffusion rates and an initial beam energy of 25 keV,using the reflecting boundary condition. Each simulated point is generated by 1078Li decays.This boundary condition leads to a more complicated dependence of the α-signalon the diffusion rate. If the diffusion rate is lower than 10−11 cm2/s, then the Lihaven’t had time to reach the surface of the sample after 5 s, so the α-signal increasesvery similarly to Fig. 5.9. For D =10−10 cm2/s, the signal initially increases, as thepeak of the 8Li distribution moves towards the surface, but decreases rapidly at latertimes, after most of 8Li has bounced on the surface and moves towards the bulk. Ifthe diffusion rate is faster than 10−9 cm2/s, then the initial upturn will end almostinstantaneously and the α-signal will be monotonically decreasing.A comparison between Fig. 5.9 and Fig. 5.10 indicates that the simulated α-signal is very different depending on what happens to Li when it reaches the samplesurface. Therefore, if the boundary condition for Li reaching the surface of amaterial is not known a priori, it can be deduced by comparing the experimentallyacquired α-signal with Fig. 5.9 and Fig. 5.10.In the general case, where there is a trapping probability Ptr between 0−1 for8Li+ to get trapped at the surface, then for the same diffusion constant the α-signalY nα (t) evolves from its shape with a reflective boundary condition towards its shapewith a fully trapping surface, when Ptr gradually increases to 1. This is depicted inFig. 5.11 for a diffusion rate D =10−9 cm2/s.930 1 2 3 4 500.511.5BeamOn OffTime (s)Yn α(t;Ptr)Trapping %1%10%30%50%90%Figure 5.11: The normalized α-signal, for a beam pulse of 1 s, a diffusion con-stant D =10−9 cm2/s and an initial beam energy of 25 keV, using various trappingprobabilities at the surface of the sample.Regardless of the nature of the boundary condition at the surface, the α-signalgets increasingly further away from the simple exponential decay of 8Li withincreasing diffusion rate. With an accumulating boundary condition (Ptr ∼ 1), fasterdiffusion results in increasing α-to-β ratio, while a reflecting boundary condition(Ptr ∼ 0) leads to α-counts that decrease with time faster than a simple exponential,as Li bounces off the surface and starts diffusing towards the bulk of the sample.In both cases, a diffusion constant larger than 10−12 cm2/s results in an observ-able variation from the simple exponential decay line – which is the flat line atthe Y nα (t) spectrum. Studying slower diffusion is possible by decreasing the beamenergy and thus moving the lithium closer to the surface (see Fig. 5.12).940 1 2 3 4 50.90.9511.05BeamOn Off25keV10keVTime (s)Yn α(t;Ebeam)|reflectiveFigure 5.12: Simulated α-signals for a diffusion rate D =10−11 cm2/s and beamenergies of 10 keV and 25 keV, using the fully reflecting boundary condition. Eachsimulated point is generated by 107 8Li decays. The (black and red) lines are justguides for the eye.From the above figure, it is evident that changing the energy of the beam canhave a significant effect on the quality of the Y nα (t) α-signal. For a beam energyof 25 keV, the signal for a diffusion rate D =10−11 cm2/s changes by less than1 % over a period of 5 s, making the measurements of such an effect very timeconsuming and marginal. On the other hand, reducing the beam energy to 10 keVresults into a signal that changes by∼10 % over the same time period, thus allowingfor an easier measurement of slow diffusion. β -contaminationSince the β -particles are insensitive to nanoscale Li diffusion, having a lot ofenergy deposited by β -particles inside the α-detector could result to a diffusion-independent background in the α-detector signal, which potentially could swampthe actual α-signal.For this reason, the thickness of the α-detector has to be as thin as possible inorder to minimize the contamination from β -particles, while being thick enough tostop the α-particles completely. Fig. 5.13 shows how much energy gets deposited byα- and β -particles inside the α-detector – made of ZnS(Ag) – for various detectorthicknesses of the order of 0.01 mm to 1 mm.950 1 2 3 4 500.51·104alpha energy depositionEnergy Deposition (MeV)RelativeYield ZnS:Ag depth0.01 mm0.1 mm1 mm0 0.2 0.4 0.6 0.8 100.511.52·105beta energy depositionEnergy Deposition (MeV)ZnS:Ag depth0.01 mm0.1 mm1 mm(a) (b)Figure 5.13: The energy deposition profile of α- (Left) and β -particles (Right)inside the α-detector for various detector thicknesses.From Fig. 5.13(a) one can see that an α-detector thickness in the range 0.1 mmto 1 mm is adequate to stop completely the α-particles, while a detector thicknessof 0.01 mm is not enough to completely stop the α-particles with energy >4 MeV.Turning to Fig. 5.13(b), the energy deposited in the α-detector by the β -particlesis virtually zero for a detector thickness of 0.01 mm, while for a thickness of 0.1 mm,it is still limited around 100 keV. For larger detector thickness, in the order of 1 mm,the β -particles can deposit as much as 1 MeV in the α-detector, which could leadto background.The above considerations identify an α-detector thickness of 0.1 mm as theoptimum value if the detector is made of ZnS. If another material is used, then thisstudy has to be repeated, since different material densities and Z-values result indifferent stopping profiles for the incoming α- and β -particles.5.3 Calculating the α-radiotracer signalA major limitation of the simulation study presented in the previous section is thetime investment required for the simulation of a given scenario. This is true evenafter complete parallelization of the Geant4 code, which provides almost linearscale up with the number of processors [136].Computationally speaking, each such simulation can be divided into two parts:The calculation of the 8Li depth profile versus time and the Monte Carlo simulationfor every given point in time.96The time needed for the first part increases exponentially with the diffusionrate D. For slow diffusion rates (D between 10−12 cm2/s to 10−11 cm2/s), all 8Liions are distributed in the first few hundred nanometers from the surface, but forfaster diffusion (D between 10−9 cm2/s to 10−8 cm2/s)), the depth profile of 8Lispans over tens of micrometers after a few seconds. Consequently, calculating thedepth profiles of 8Li for slow diffusion is almost instantaneous, but can take up toseveral hours for fast diffusion (using a machine deploying an Intel(R) Core(TM)i7-4710HQ processor, CPU @ 2.50GHz, 4 Core(s), 8 Logical Processor(s)).In addition to the time consuming calculation of the depth distributions, runningthe Geant4 simulation typically takes a few hours to complete a crude time-scanof 15-20 data points with 107 Monte Carlo statistics per point. This raises the totaltime investment significantly. For example, if a fine grid of data points is used overa period of, e.g., 5 s after the beam implantation, for a multitude of diffusion ratesspanning at least 3-4 orders of magnitude, the required computational time for theGeant4 simulations is prohibitively long.To amend this problem, a method was developed to bypass the Geant4 simulationcompletely. The idea is that the result of the Monte Carlo simulation can becalculated (instead of simulated) by multiplying each bin of the depth profile of 8Li(such as Fig. 5.4) with the probability of detecting an α for that depth (see Fig 5.14).97Figure 5.14: Probability of α-detection versus decay depth, as simulated by Geant4in TiO2. Pndet is the probability of detection normalized to 1 at the surface, namelyPndet(d) = Pdet(d)/Pdet(0). At the main plot, the logarithm of Pndet is plotted over arange of 120 µm. There is a very sharp decrease over the first 100 nm and a veryslow probability reduction after that. For clarity, the first 1 µm from the surface isplotted at the insert, on a linear scale.For the case of TiO2, this distribution can be fitted effectively with a phenomeno-logical sum of two exponentials:Pdet(z;Eth) = c1exp(a1z)+ c2exp(a2z) (5.4)with parameters: c1 = 0.01774, a1 =−0.003809 nm−1, c2 = 0.0133 and a2 =−1.288 ·10−6 nm−1. Evidently, the fast decaying component dies out after the first500 nm, while the slowly decaying component has a range on the milimeter scale.The fast component is related to the high energy tail of Fig. 2.2b and the fact that theα-detector ring has a non-zero width, whereas the slow component is most probablydue to α-particles emitted at a low angle relative to the surface of the sample thatreach the detector after they scatter close to the surface. This scattering results in asmall d/sinθ parameter, with only a weak increase of the attenuation with decaydepth.Note that the probability Pdet of α-detection is a function of both the decaydepth and the energy threshold of the detector, as well as the properties of the98material (density, Z-value, etc), therefore this simulation has to be run once permaterial under study.Then, the α-counts versus time are given by:Nα(t) = N(t)∫ ∞0C(z, t)Pdet(z;Eth)dz (5.5)where N(t) is the number of α-decays versus time, C(z, t) is the concentrationof Li at depth z and time t and Pdet(z;Eth) is the probability of detecting an α abovea given energy threshold Eth at that depth. A comparison between the simulated andcalculated α-signals under the same condition can be seen in Fig. 5.15.0 1 2 3 4 50.911. OffTime (s)Yn α(t)calculated signalsimulated signalFigure 5.15: Comparison of the calculated (continuous red line) and simulated(black points) Y nα (t) α-signals for a diffusion rate of D = 10−10 cm2 s−1.Evidently, this method succeeds in reproducing the Geant4 simulation. Themerit of this approach is that it allows for the calculation of normalized α-signals inonly a fraction of the time required for the full Geant4 simulation. This developmentwas exploited for extracting the diffusion rate of Li from experimental data (Ch. 6).5.4 ConclusionsIn summary, the subsequent α-decay of 8Li can be used to extract the nanoscalediffusion rate of Li in solids.The very different attenuation profiles of the emitted α- and β -particles insidematerials allow for the monitoring of the nanoscale 8Li+ motion towards/away from99the sample surface by measuring the ratio of the α-to-β counts versus time.A detailed simulation study of the above concept was carried out. The implan-tation profile of 8Li+ for various beam energies was simulated by SRIM. Theseprofiles were evolved in time for different diffusion coefficients D in accordancewith Fick’s second law, in order to calculate the 8Li depth distribution versus timeunder the boundary condition of 8Li getting either trapped (with a certain proba-bility) or reflected upon reaching the sample surface. Then, for each of the abovedepth distributions, the output of the α- and β -detectors was simulated by Geant4.By analyzing the various simulations presented in this Chapter, a favorableexperimental geometry was identified, which was used as a basis for the design ofthe new cryo-oven (see Appendix A).The combined graphs of the normalized α-signals Y nα versus time for variousdiffusion coefficients showed that it is possible to determine the boundary conditionat the sample surface based on the time evolution of Y nα .It was also identified that it is possible to accelerate the compilation of a li-brary of normalized Y nα (t;D) by performing a series of calculations that are able toreproduce the Geant4 simulations in a fraction of the required time.In the next Chapter, this theoretical/numerical study is put to the test, by per-forming proof-of-principle 8Li α-radiotracer measurements in rutile TiO2.100Chapter 6Measurements of 8Li Diffusion inRutile using α DetectionIn this study we employ the α-radiotracer method to extract the diffusion rateof isolated ion-implanted 8Li+ within ∼120 nm of the surface of oriented single-crystal rutile TiO2, at the dilute limit of Li concentrations. The α-particles fromthe 8Li decay provide a sensitive monitor of the distance from the surface andhow the depth profile of 8Li evolves with time. The main findings are that theimplanted Li+ diffuses and traps (with a probability ≥50 %) at the (001) surface.The T -dependence of the diffusivity is described by a bi-Arrhenius expression withactivation energies of 0.3341(21) eV above 200 K, whereas at lower temperatures,it has a much smaller barrier of 0.0313(15) eV. The low-T behavior is discussed inthe context of the recently reported Li-Ti3+ polaron complex [29].6.1 Rutile - General CharacteristicsMany transition-metal oxides are good materials for lithium-ion batteries [139]; theircapability of assimilating the extra charge from lithium ions, their open-structures,and relatively low production cost all make them attractive intercalation electrodes.In particular, the TiO2 polymorphs have garnered significant attention to this end inrecent years.Rutile is the most stable TiO2 polymorph (Fig 6.1), which crystallizes in abody-centered tetragonal structure made up of two Ti and four O atoms per unitcell [140] (lattice parameters a = b = 4.59A˚ and c = 2.95A˚). The crystal’s macro-structure is comprised of stacked TiO6 octahedra, which share edges with theirneighbors in the c-direction and corners in the basal ab-planes. This fortuitousatomic arrangement gives rise to open channels parallel to its c-axis, providing a 1Dpathway for interstitial ion transport. In this sense, rutile is structurally similar tomany so-called superionic conductors which also posses crystallographic tunnelsthat confine the motion of mobile ions.Regarding the surface properties of rutile, there have been extensive studies oftheir properties [141–143], but the focus was never on the (001) surface, as it is not101stable [144–146]. Rather, a reduction of the surface Ti atoms’ coordination fromsixfold to fourfold reduces the energy configuration of the surface. As a result, the(001) surface tends to reconstruct at higher temperatures, which could give rise totrapping centers for Li+.Figure 6.1: A 3-D view of the rutile TiO2 unit cell [147] (left) and the rutilelattice [148] (right), with the possible sites for lithium intercalation marked withpink (octahedral sites) and blue (tetrahedral sites).It has been known for some time [60, 70] that Li+ diffusion down these intersti-tial channels is extremely fast, greatly surpassing all other interstitial cations [149],with a room temperature diffusion coefficient exceeding many modern solid-stateLi electrolytes [150]. A major limitation, however, for the use of rutile as a Li-ionbattery material is its limited Li uptake at room temperature [151, 152]; however,the discovery that using nanosized crystallites mitigates this issue [65] has led torenewed interest in its applicability [139].There are a number of poorly understood aspects of rutile lithiation, includingthe cause of the limited Li+ uptake, as well as why reported Li diffusion ratesdiffer by orders of magnitude, even under the same experimental conditions [29,60, 68, 70–73, 153–155]. As pointed out recently [29], there is large scatter in thereported diffusion coefficients [29, 60, 71–73]. Such systematic discrepancies arenot unprecedented (see e.g., LiCoO2 [74–76]), but rather unexpected in a relativelysimple ion conductor.The 1D ionic transport of Li+ is characterized by an activation energy of∼0.33 eV, as evidenced by both techniques sensitive to microscopic [29, 155]and macroscopic [60, 68] Li+ motions. Surprisingly, these results greatly con-trast the large body of theoretical work that predicts significantly lower barriers of0.03-0.05 eV [44, 148, 156–163], suggesting a key ingredient is missing from thecalculations.At low temperatures (below 100 K) a second Arrhenius-like component with an102energy barrier of ∼0.027 eV was recently reported using β -NMR [29], but it wasattributed to the local dynamics of Li+-polaron complexes, rather than long-rangediffusion.This complex is formed when Li+ comes into proximity with an electron polaronTi´Ti, namely a localized electron on a Ti site (i.e., a Ti3+ site in a lattice of ideallyTi4+). The electron polaron is mobile along the c-axis stacks of TiO2 octahedrawith an activation energy of 0.033 eV, according to DFT calculations [157].Interestingly, DFT calculations of the Li-polaron complex hopping, yield anactivation energy of 0.29 eV, very close to the value of 0.33 eV found at high-T inall relevant experiments.Using the results of the current study, we were able to suggest a novel inter-pretation of the Li+ dynamics in rutile TiO2 that can answer some of the abovequestions.6.2 Experimental DetailsThis experiment was performed using the facility infrastructure at TRIUMF [33],in Vancouver, Canada. The α-radiotracer method used a low-energy radioactiveion beam of 8Li+ to introduce Li into a target material. The ions are implanted withtypical energies≤30keV, whose precise value can be tuned with a decelerating biasapplied to an electrically isolated spectrometer platform [33, 123, 164]. The incidentenergy is used to influence the 8Li+ stopping distribution, providing depth resolution,with a typical mean range of ∼100 nm at these energies. The incoming 8Li+ beamwas implanted in rutile targets housed in an ultra high vacuum cold finger cryostatwithin a dedicated low field spectrometer [33, 123]. The samples were commer-cial chemo-mechanically polished (roughness <0.5 nm) single crystal rutile TiO2substrates (CRYSTAL GmbH) with typical dimensions of 7 mm×7 mm×0.5 mm.The surfaces were free of macroscopic defects under 50x magnification.6.3 Experimental Results on Rutile TiO2 Using theα-radiotracer TechniqueUsing the α-radiotracer technique, we performed measurements on rutile TiO2at various temperatures with two beam energies (10 and 25 keV) and two sampleorientations.It is the 1D character of lithium diffusion in rutile TiO2 (the tensor D is veryanisotropic, see Sect. 4.1.1) that makes it an ideal test case for this technique.As Li+ is known to diffuse primarily along the c-axis of rutile, if the c-axis is103oriented parallel to the surface (perpendicular to the beam), then the 8Li+ motionshould not change the initial implantation profile. Since the ab-plane diffusivityDab10−12 cm2 s−1, Y nα (t) is expected to be time-independent. On the other hand,if the c-axis is perpendicular to the surface, then the depth distribution of lithiumshould evolve rapidly with time, since Dc 10−12 cm2 s−1. Depending on theboundary condition at the surface, we expect either a spectrum like Fig. 5.9, Fig. 5.10,or something in between.In Fig. 6.2 we compare the measured normalized α-yield, Y nα , for the c-axisparallel and perpendicular to the surface. As expected, the time spectrum for the(110) orientation of TiO2 rutile (c-axis parallel to the surface) is completely flatat 294 K, indicating that the ab-plane diffusion rate is lower than the detectionlimit ∼10−12 cm2 s−1 (Fig. 5.9), consistent with other studies reporting an ab-planediffusion rate of 10−15 cm2 s−1 or lower [60].In contrast, the spectrum from the (001)-oriented crystal increases rapidly withtime. Most of the percentage change in the signal happens during the first 2 secfollowed by a much more gradual increase. This is the signature of fast diffusion(D >10−9 cm2/sec), approaching the saturation regime of Fig. 5.9.Next, we rotated the (001)-oriented rutile by θ ∼ 10o – introducing an anglebetween the incoming beam and the crystallographic c-axis – and repeated themeasurement under otherwise identical experimental conditions (294 K, 25 keV).The two Y nα (t;θ) signals can be seen Fig. 6.3. The comparison of the two cases canbe used to decide whether channeling (see Sect. 2.2.3) can be neglected or not, asthe critical angle for channeling in rutile is calculated (using Eq. 2.5) to be 6.9o.Evidently, the two spectra look very similar – inside their experimental uncertainty–, but the signal of the rotated sample evolves more linearly close to the end ofthe beam pulse, where the number of alphas collected is maximum. This slightdifference might be due to the reduction of a (small) pile-up effect at the detector,as the rotation of the sample reduced the effective solid angle of the detector andthus the probability of two light pulses of lower energy than the threshold to arriveat the same time at the PMT tube and be counted as a single alpha of energy abovethe threshold (similarly to the case discussed in Sect. 3.3.2).From the above test, we conclude that channeling cannot be very significantunder these experimental conditions, therefore the SRIM implantation profile wasused to generate a library of calculated normalized α-spectra versus diffusion ratein a range [10−12 - 10−8] cm2/sec, as described in Sect. 5.2. Exploiting the factthat the calculated α-signals can be generated significantly faster than the simulatedones, this library contained 100 calculated spectra for each order of magnitude ofdiffusion coefficient in the above-mentioned range.After these tests at room temperature, we also performed a temperature scan inthe range of 60 K to 370 K with beam energies of 10 keV and 25 keV. In Fig. 6.4 are1040 1 2 3 4 50.811. OffTime (s)Yn α(t)Orientation(110)(001)fitsFigure 6.2: Comparison of the measured normalized α-yield Y nα (t;D,Ptr) for the(110)- and (001)-oriented rutile TiO2 and fits (orange lines) for a beam energy of25 keV and a surface trapping probability Ptr = 1. The increasing signal with time inthe (001) crystal is consistent with the anisotropy of the Li diffusion coefficient [60]and indicates that Li diffuses fast along the c-axis and gets trapped upon reachingthe sample surface. Each spectrum took roughly 40 min, with an α-rate of∼ 20,000ions/pulse. Each beam pulse was 1 s long and was repeated every 15 s.a few examples of experimental data for the (001) orientation with a beam energyof 25 keV, with the corresponding fits.In this temperature range, the value of the normalized α-signal Y nα (t;T ) at 5 secincreases with temperature up to ∼300 K, but then evidently starts getting reducedagain, even though it is clear that the high temperature spectra reach saturationmuch earlier. This apparent signal reduction in the high-T region is due to theself-normalization process, which normalizes the spectrum to its (extrapolated)value at time zero. For a beam energy of 25 keV and diffusion rates of up to∼10−9 cm2/sec, Li+ does not move fast enough to reach the surface at the very firsttime-bins. For faster diffusion, though, this is no longer true. For D>10−9 cm2/sec,the percentage of Li near the surface (and therefore the probability of α-detection)is much higher even at very early times, so the whole spectrum gets normalized to ahigher value, hence the apparent signal suppression.This effect is also present in the simulations. The turning point in the diffusioncoefficient, above which the signal is getting suppressed is 3.1×10−9 cm2/sec in1050 1 2 3 4 50.811. OffTime (s)Yn α(t)Sample’s rotationUnrotated10oFigure 6.3: Comparison of the measured normalized α-yield Y nα (t;θ) betweenunrotated and rotated (by 10o) (001)-oriented rutile TiO2 at 294 K.the current experimental configuration. This is the slowest diffusion rate whichenables the peak of the implantation profile of the 25 keV beam of Fig. 2.11 to reachthe surface during the first 10 ms time-bin.To fit the data, we used a custom C++ code applying the MINUIT [165] min-imization functionalities of ROOT [166] to compare the Y nα signals to the libraryof calculated spectra. The free parameters of the fit were the diffusion rate D andthe trapping probability at the (001) surface, Ptr. All Y nα (t;D,Ptr) spectra at bothimplantation energies (10 keV and 25 keV) were fitted simultaneously with a sharedPtr value. For the (001) orientation Y nα increases rapidly with time, approachingsaturation, indicating that lithium diffuses fast along the c-axis and gets trapped at(or within few nm of) the surface (see Fig. 5.9). For Ptr ≥50 %, the global χ2 valueis completely insensitive to Ptr, but for Ptr <50 %, the quality of the fits deterioratesrapidly. This can be understood by revisiting Fig. 5.11, where it is evident that theY nα signal is virtually unaffected by the trapping probability Ptr, for Ptr ≥50 %.This is the first unambiguous evidence for Li trapping (with at least 50 %probability) at the (001) surface. There is no evidence of Li de-trapping up to 370 K,since at that temperature Y nα (t;T ) reaches saturation after ∼2 s and any Li surfacede-trapping would lead to an observable decrease of Y nα (t;T ) at later times. Thenon-zero trapping probability is most likely related to the reported difficulty ofintercalating Li into rutile, as the Li ions would tend to stick at or near the surface1060 1 2 3 4 50.511.522.5 BeamOn OffTime (s)Yn α(t)Temperature260 K274 K310 K340 KfitsFigure 6.4: Examples of measured normalized α-yield Y nα (t;D,Ptr) for the (001)-oriented rutile TiO2 and fits (orange lines) for a beam energy of 25 keV and asurface trapping probability Ptr = 1. For increasing temperature, Y nα (t;D) saturatesmore and more rapidly, indicating that above room temperature, most of Li getstrapped at the (001)-surface during its lifetime. Above room temperature, the c-axis normalized spectra Y nα (t;T ) get progressively suppressed, as the normalizationfactor Yα(t = 0;T ) increases substantially due to fast diffusion. Each spectrum tookroughly 40 min, with an α-rate of ∼ 20,000 ions/pulse. Each beam pulse was 1 slong and was repeated every 15 s.rather than diffuse towards the bulk.This view is in agreement with the suggestion of Hu et al. [65] that there mightbe a surface storage mechanism for Li+ in nanosized rutile, with a calculated surfaceLi content of roughly 50 % of its maximum surface capacity in storing Li+. Inaddition, Zhukovskii et al. [167] indicated using ab initio simulations that it mightbe energetically favorable to store Li+ at the surface of metal/Li2O nanocomposites.It is worth mentioning, however, that both these studies were focusing on nanometer-sized systems, whereas the current study used macroscopic single-crystals, with Liimplanted near a (001)-oriented surface.It is not clear whether the Li+ surface trapping is caused by an electrostaticpotential well (similar to H in Pd [168]), a partially reconstructed surface [141] (seeSect. 6.1), or by a chemical sink due to a solid state reaction at the surface (e.g.,forming another phase such as cubic LiTiO2). With X-ray diffraction, it was found107that about 40 % of the (001) surface of a polished and annealed rutile sample wasflat and the remaining 60 % consisted of (011)-oriented facets [169], which couldprovide trapping centers for Li+.6.4 Arrhenius Fits and DiscussionTurning to the values of D(T ) extracted using the above analysis (see Fig. 6.5), theyreveal a bi-Arrhenius relationship of the form:D(T ) = DH exp [−EH/(kBT )]+DL exp [−EL/(kBT )] , (6.1)where Ei is the activation energy and Di is the pre-factor of each component (i=H/L). These were found to be EH = 0.3341(21)eV and DH = 2.31(18)×10−4 cm2 s−1for the high-T component and EL = 0.0313(15)eV and DL = 7.7(7)×10−10 cm2 s−1for the low-T component, respectively.This extracted EH is in excellent agreement with values deduced by othertechniques [29, 60, 68] and the diffusion rates at high temperatures are very similarto the ones found in rutile nanorods using impedance spectroscopy [68], but threeorders of magnitude lower than what is found with β -NMR [29], even though thecurrent measurements agree with β -NMR at low temperatures (T ∼ 150K), as canbe seen at Fig. 6.5.From the values of the energy barriers and the pre-exponential factors of thebi-Arrhenius fit, one can easily see that the energy barriers are reported with a muchsmaller relative uncertainty, compared to the pre-factors. This stems from the factthat a change in the trapping probability Ptr in the range discussed above (Ptr ≥50 %),results in the bi-Arrhenius plot of D(T) (shown in Fig. 6.5) to be shifted verticallyby a constant amount. The lower the value of Ptr, the faster the diffusivity, while theenergy barriers of the two components are virtually unaffected. Also, note that theuncertainties of the Arrhenius parameters reported here are purely statistical, but thesystematic sources of uncertainty, such as possible discrepancies between the variousparameters of the Geant4 simulation with the actual experimental configuration,are taken into account in the level of fitting each individual normalized α-yield, byintroducing a stretching factor which is able to stretch (or suppress) the measuredY nα (t) signals by ∼10 %. This factor increases the available phase-space of each fit,therefore increases the uncertainty of the diffusivities of each individual fit.An additional reason for the higher uncertainty of the pre-exponential factors,is that both data sets acquired with beam energies of 10 and 25 keV yield virtuallythe same bi-Arrhenius activation energies and they are in agreement at high-T ,but the low-T component of the 10 keV data is shifted lower by about an order ofmagnitude (see Fig. 6.6).108Detection limit (25 keV)EA ≈0.35eV EA ≈0.32eV4 6 8 10 12 14−10.5−10−9.5−9−8.5−8−7.5−71000/T (K−1)log 10(D/[cm2s−1])α-radiotracer|25 keVβ-NMRImp. SpectroscopyFigure 6.5: Arrhenius plot, comparing reported Li diffusivity in rutileTiO2 [29, 68]. The solid black line is the bi-Arrhenius fit of Eq. 6.1 withPtr =100 %. The red line is the high-T Arrhenius component found withβ -NMR [29]. The blue dashed line is the fit of the present data using the twoArrhenius components found with β -NMR [29], assuming that only a fraction f ofthe β -NMR fluctuations corresponds to a hop. which f was the only free parameterand yielded f =28.1(17)×10−6For trapping probability Ptr <100 %, the apparent gap narrows and for Ptr =50 %is about half an order of magnitude wide. The persistency of the gap suggests that itmight be related to either a discrepancy between SRIM and the actual implantationprofiles (e.g., due to channeling [170]), or due to some small random disorder closeto the surface parameterized by some energy scale (∆). At higher temperatures whenkT >> ∆ its effect would diminish. Both these effects would affect the (closer tothe surface) 10 keV data more than the 25 keV and would become irrelevant at fastdiffusivities, explaining the agreement of the two sets at high temperatures and whythe diffusion seems slower at low-T for the 10 keV data. The insensitivity of theα-yield Y nα (t;θ) to the angle of sample’s rotation θ (see Fig. 6.3), suggests that apossible channeling effect cannot be very large, which makes the latter explanationof a near surface random disorder more probable.109Detection limit (25 keV)Detection limit (10 keV)4 6 8 10 12 14−11.5−11−10.5−10−9.5−9−8.5−81000/T (K−1)log 10(D/[cm2s−1])α-radiotracer|25 keVα-radiotracer|10 keVFigure 6.6: Arrhenius fit to the diffusion rates of 8Li found in this study in (001)-oriented rutile TiO2 assuming a fully trapping (Ptr = 100%) sample surface and theimplantation profile provided by SRIM for a beam energy of 10 and 25 keV.A bi-Arrhenius relationship for diffusivity is not uncommon; in vacancy ionconductors [171], it may occur from a crossover between a region at high-T , wherevacancies are thermally generated, to a region at lower T with a shallower slope.As α-radiotracer is always only measuring the diffusion of Li+, rather than the netionic conductivity, the origin of the two Arrhenius components can’t be the same asabove. While we cannot be conclusive about it, we consider some possibilities.Using molecular dynamics (MD) in rutile TiO2, Gligor et al. [172] found abi-Arrhenius behavior above room temperature for high Li concentrations, butreported an Arrhenius relation with EA ∼0.22 eV for low concentrations (LixTiO2with x ≤ 0.06). Gilgor et al. attributed the additional Arrhenius component athigh concentrations to Li+ blocking the c-axis channels, resulting in the collectivemotion of several Li+ moving together at high temperatures, whereas individual Li+ions are thought to be moving at lower temperatures. In spite of the bi-Arrheniusfinding, this MD study contrasts ours, as our measurements were taken in ultradilute concentrations of Li+ and the second Arrhenius component was found farbelow room temperature.110To provide a possible explanation of the two Arrhenius components, we considera recent β -NMR experiment on rutile [29], which also used an implanted 8Li+ beamon similar crystals. The β -NMR measurements revealed two peaks in the relaxationrate 1/T1, one below 100 K and one above 200 K.Below 100 K, a 0.027 eV barrier was attributed to dynamics of electron-polaronsin the vicinity of the implanted ion [157, 173]. Electron-polarons in rutile TiO2are manifested as Ti3+ centers, produced by an electron trapped at a normally Ti4+ion. These extra electrons can be provided either by oxygen vacancies [174], bydoping with Li [173], F or H, by laser illumination [175], or due to electron/holeexcitations produced by the implantation process of an ion beam. When Li+ ionsare introduced in rutile at low temperatures (∼20 K), the interstitial ion has beenshown [173] to occupy a position neighboring a Ti3+ center, forming therefore aneutral paramagnetic bound Li-polaron complex.In principle, the dynamics of these complexes might not be diffusive, e.g. the8Li+ is static and the polaron is thermally trapped by the Li and cycles throughtrapping and detrapping. Nonetheless, our current measurement really shows thatthere is some long range diffusion of 8Li+ at low-T , with a barrier significantlydifferent than high-T . While our EL is of a similar magnitude to that found withβ -NMR, it is also compatible with the diffusion barrier predicted from theory forisolated Li in rutile [148, 156, 157, 159–161, 163, 172]. The α-radiotracer cannotdistinguish whether Li moves either as a simple interstitial, or as part of a Li-polaroncomplex, it would only identify their weighted average contribution to the motionof 8Li+. The similarity of the observed activation energy at low temperatures tothe theoretical value suggests that a small fraction of the Li+ interstitials does notcombine with a polaron, but rather diffuses as a simple ion. If this fraction is small,that would explain why the low-T pre-factor is so much smaller than the high-T .It seems possible that the larger activation energy observed above 200 K mayinvolve diffusion of a more complex object, possibly a Li-polaron complex, or itcould be related to a dissociation energy of Li+ with the polarons, which are knownto form Coulomb bond defect complexes. Indeed, theory predicts a diffusion barrierof 0.29 eV for the Li-polaron complex and a disassociation energy of 0.45 eV [157],both comparable to the high-T barrier found here. The Li-polaron complex is overallelectrically neutral, so its movement should contribute to the diffusivity of Li butnot to the ionic conductivity (charge transport). An electric field would not cause itto move - unless it was strong enough to destabilize the complex (strong potentialgradient). Thus, if it is a neutral Li-polaron complex moving at high-T , one wouldexpect the impedance measurement to yield a very different Arrhenius slope. Thisspeaks in favor of the Li-polaron dissociation case, but without further investigationwe cannot exclude the possibility of Li-polaron diffusion, or a combination of thetwo aforementioned mechanisms.111The much larger pre-factor above 200 K, compared to low-T , is also quiteintriguing and is further evidence that these are two very different mechanisms fordiffusion of Li in rutile. Indeed, DH , when written in terms of frequency, yieldsτ−1H ∼2×1012 s−1, which is in the 1012-1013 s−1 range one would normally expectfrom phonons driving a thermally activated motion. Note that this frequency is∼5000 times smaller than what was found with β -NMR [29], as well as with opticalabsorption [60].The above explanation on the nature of the two Arrhenius components relies onthe assumption that the Li-polaron complex is stable or metastable at high temper-atures. In the absence of Li doping, the intrinsic Ti3+ polarons [174] are unstableat high temperatures, with the corresponding electron paramagnetic resonance(EPR) signals vanishing above 20 K [175]. The Li-polaron complex is shown to bemore stable, with the EPR signal visible up to 50 K [173] (above which the signalbroadens) and the corresponding β -NMR signal being present up to 100 K [29].There are no studies reporting the existence of Li-polaron complexes above100 K, but the fact that the dissociation energy for the complex is expected tobe large (0.45 eV [157], an order of magnitude larger than the 0.025 eV thermalenergy at room temperature), suggests that these complexes might be present at hightemperatures and the EPR signal might be broadened by some other process, ratherthan the dissociation of the complex. For instance, its broadening could reflect themobilization of the complex, or internal excitations (i.e., polaron hopping locallywhile still overall bound to Li+.One could argue that the Li-polaron complex is unbound in equilibrium at hightemperatures (above 100 K), but the ion beam implantation process creates a numberof electron/hole pairs in the vicinity of the 8Li+, which may form the complex withone of the free electrons at higher temperatures for a short time. However, thisexplanation would suggest that the high-T Arrhenius component is beam-induced,rather than inherent to the crystal and would therefore fail to explain the agreementof EH with techniques that don’t use an ion beam.An alternative explanation regarding the nature of the two Arrhenius compo-nents, would be the opposite of the one discussed above, namely having Li-polaroncomplex diffusion at low-T and simple interstitial motion at high-T. This scenariohas the merit of not requiring the Li-polaron complex to be stable at high tempera-tures and also explains why the low-T Arrhenius component found with β -NMRhas the same activation energy as the one found here. However, it does not provideany insight as to why the energy barrier predicted by DFT for simple interstitialmotion is ten times smaller than EH and (coincidentally?) agrees with EL, whereasthe predicted values for Li-polaron complex diffusion and dissociation are similarto EH and an order of magnitude larger than EL.1126.5 ConclusionsIn summary, we used the radioactive α-decay of 8Li to study Li diffusion in a singlecrystal rutile TiO2 in the range of 60 K to 370 K.This work established the feasibility of the α-radiotracer technique, as wellas its applicability to answer science-relevant questions about the Li+ motion ina Li-battery material. It is the first time that α-radiotracer is used to extract theactivation energy of the Li+ motion in a sample. Moreover, it is demonstrated to bethe only technique to date that can clearly identify the surface boundary conditionof the Li+ motion.In rutile TiO2, the nanoscale Li diffusion rate was found to exhibit bi-Arrheniusbehavior. We report a high-T activation energy of EH = 0.3341(21) eV, in agree-ment with measurements carried out with different techniques [29, 60, 68]. Atlow temperatures, a second Arrhenius component was revealed, with an activationenergy of EL = 0.0313(15)eV. We suggest that this might be related to a smallfraction of the Li+ that does not bind to a Li-polaron complex but rather hops as asimple interstitial with an activation energy near theoretical calculations. In addition,we found evidence that Li traps at the (001)-surface, which could contribute to thereduced Li uptake at room temperature. We believe that this technique can shednew light on the Li motion in Li-ion battery materials and across their interfaces.113Chapter 7Conclusions and Future Work7.1 SummaryIn this work, we have enhanced the ability to study nanoscale phenomena using theradioactive β - and α-decays of 8Li and 9Li.We showed that by comparing the β -NMR spin lattice relaxation rate (SLR) of8Li and 9Li under the same experimental conditions it is possible to distinguish thesource of relaxation in a given sample (Ch. 3). As a proof of principle, we studieda Pt foil and a SrTiO3 single-crystal. The SLR in Pt was shown to be primarilymagnetic in origin, whereas in SrTiO3 the main source of relaxation was identifiedto be electric quadrupolar. To further develop the isotopic comparison method ofβ -NMR, we showed that the 9Li β -NMR asymmetry can be increased by a factorof at least ∼2, by tagging the decay branches of 9Li in coincidence/anti-coincidencewith an emitted α-particle. The isotopic comparison method can greatly enhance thecapabilities of β -detected NMR, by providing a tool capable of lifting any ambiguityas to what is the dominant source of spin-lattice relaxation in a given situation.In the second part of this work, we developed from first principles the α-radiotracer technique and we used it to study the Li motion in rutile TiO2.Detailed simulation studies (Ch. 5) were carried out in order to guide the designof a dedicated spectrometer (Appendix A) and also as a basis of the data analysistools required for extracting the diffusion rate of Li+ from the raw α-rate signal.Using the α-radiotracer technique, we studied the Li+ motion in rutile TiO2, inwhich we showed that Li+ gets trapped upon reaching the (001) surface of rutile andthat the diffusion rate of Li+ versus temperature follows a bi-Arrhenius relationship(Ch. 6). The high temperature component was in agreement with other techniquesand the previously unknown low temperature component yielded important novelinformation on the nature of the Li motion. We believe that the α-radiotracertechnique is very effective in studying nanoscale Li+ diffusion, as it is the onlyexperimental technique that directly measures the Li+ diffusion rate in this scaleand can clearly extract the surface trapping probability of Li+, which has importantimplications on the applicability of a given material as part of a Li-ion batterysystem.1147.1.1 Isotopic comparison for distinguishing magnetic and electricrelaxation in β -NMRAs there are instances where the source of the SLR in a β -NMR measurement isnot easily identifiable as either magnetic or electric quadrupolar, we developed theisotopic comparison method using 8Li and 9Li probes. The basis of the methodis that the SLR of the same element should scale between isotopes based on theirdifferent nuclear characteristics, such as spin and quadrupole moment. For the caseof 9Li and 8Li, we showed that the ratio of the relaxation rates of the two isotopesin the limit of fast fluctuations is ∼ 7.7 for pure magnetic interactions and ∼ 2.1 forpurely electric quadrupolar.Following this theoretical result, we performed β -NMR measurements with bothisotopes in two very different materials, namely Pt and SrTiO3. Pt is a metal in whichion-implanted Li feels a very small, if any, quadrupolar interaction, so the primarysource of relaxation was expected to be through the (magnetic) Korringa hyperfineinteraction to the conduction electrons. Indeed, the ratio of the relaxation rates for9Li and 8Li yielded 6.82(29), close to the limit of purely magnetic interactions.In contrast, the same ratio in SrTiO3 was 2.7(3) , very close to the limit of thespin fluctuations being caused by purely electric quadrupolar interactions. That wasexpected from a non-magnetic insulator such as SrTiO3 with very few availablenuclear moments. Indeed, previous measurements with 8Li β -NMR had found allthe usual signs of a dominant electric quadrupolar interaction at the stopping site of8Li. Hence, we showed that using the isotopic comparison technique with β -NMRusing 8Li and 9Li as probes, it is possible to infer the primary source of spin latticerelaxation in a given situation.The 9Li β -NMR measurements dominated both the time investment and theuncertainty of the isotopic comparison. A large part of the reason behind thisis that 9Li decays into three possible states of 9Be, two of which are associatedwith opposite β -decay asymmetry and thus suppress the average β -NMR signalsignificantly. To increase the usability of the isotopic comparison, we invented amethod (termed αLithEIA) of partially resolving the decay branches of 9Li, bytagging each β -decay in coincidence/anti-coincidence with an α-particle. The α’sare emitted from decays to one of the excited states of 9Be, whereas a decay tothe ground state emits only a β (plus an electron antineutrino for lepton numberconservation). Using the αLithEIA method, we were able to increase the effectiveasymmetry of 9Li β -NMR by a factor of ∼2.1157.1.2 Establishing the α-radiotracer technique for studyingnanoscale Li diffusionSo far, the α-decay of 8Li had been completely neglected as a means of studyingnanoscale phenomena in solids. In this work, we use the rapid attenuation of α-particles in solids to extract the long-range diffusion rate of Li+ in a direct manner,with the so-called α-radiotracer technique. This allows one to study at the sametime both the long-range motion of Li+ (using the α-decay) and its local hop-rate(using β -NMR) with a single ion-implanted radioactive probe (8Li).To achieve that, we had to study all aspects of the new technique, in order to findan experimental geometry that is compatible with the constrains of both β -NMRand the α-radiotracer. This was done using a combination of the Monte CarloGeant4 simulation package and custom codes. The findings of that work were anindispensable guide for the design, development and testing of a novel spectrometerwith the specified characteristics, the so-called cryo-oven.In addition, the aforementioned simulation study doubled as the basis of thedata analysis tools required for the extraction of the diffusion rate and the surfacetrapping probability of Li+ from the raw experimentally-acquired α-rate.The conclusion of the simulation study regarding the optimum α-detectiongeometry was the placement of a thin Al ring just upstream of the sample surface,coated with a layer of ZnS(Ag) paste (ideally ∼0.1 mm thick). This geometrymaximized the quality of the normalized α-signal without diminishing the effectivesolid angle of the detector. The light produced from α-particles reaching theZnS(Ag) ring was then guided through a system of lenses to a photomultiplier tube,whose energy threshold was selected again in order to maximize the quality of thesignal, based on the simulation.Moreover, the simulation study showed that it is possible to determine theprobability of Li+ trapping at the surface of the sample, in addition to extracting thediffusion rate and energy barrier of the Li+ motion. These findings then were put tothe test by studying the diffusion of Li+ in single crystal rutile TiO2.7.1.3 Study of the Li+ motion in rutile TiO2, using the α-radiotracertechniqueRutile TiO2 is an interesting 1D ion conductor with several open questions regardingthe Li interstitial motion through its c-axis and the fate of Li+ upon reaching itssurface. The much slower Li diffusion in the ab-plane makes it an ideal test case forα-radiotracer. Indeed we found no evidence of Li motion perpendicular to the c-axisabove the theoretical detection limit of this technique (i.e., Dab10−12 cm2 s−1),as expected from other studies showing a much lower diffusion rate. In contrast,116the c-axis diffusion rate was much larger than 10−12 cm2 s−1 and we were able toget full temperature scans between 60 K and 370 K with two beam energies (10 keVand 25 keV).These data where then fitted with the model developed through the simulationstudy (Sect. 7.1.2) and yielded information on both the Li (001)-surface trappingprobability and the temperature dependence of the diffusion rate. Based on theanalysis, we found that Li+ has a probability to get trapped upon reaching the(001)-surface of rutile larger or equal to 50 %, with no evidence of de-trapping upto 400 K. In addition, we showed that the temperature dependence of the diffusionrate follows a bi-Arrhenius relationship, rather than a simple Arrhenius.The Arrhenius component of the diffusion rate that is dominant at high tem-peratures was found to have an activation energy of 0.3341(21) eV, in excellentagreement with other techniques, including our recent β -NMR measurement.At lower temperatures – below 200 K – a second, previously unknown, Arrhe-nius component was revealed, having a diffusion barrier of 0.0313(15) eV. Ourrecent 8Li β -NMR measurement in rutile [29] had found a second fluctuation mech-anism below 100 K that followed Arrhenius law with a similar activation energy0.027 eV, but was attributed to local dynamics with the electron-polaron. As α-radiotracer is insensitive to local effects, we concluded that part of the β -NMRsignal should be related to long-range Li motion. In addition, the similarity ofthe low-T activation energy with the theoretical diffusion barrier of the simple Li+interstitial, suggests that at low-T a portion of Li+ does not combine with a polaronand moves as a simple interstitial.Overall, the work on rutile has showed the capability of the α-radiotracertechnique to study the nanoscale Li motion, but also its power to extract newinformation about diffusion and surface trapping of Li+ in technologically relevantmaterials.7.2 Future WorkMoving forward, there are advances to be made in both the isotopic comparisonand the α-radiotracer methods, as well as interesting new research pathways andmaterial properties to be studied using the aforementioned techniques.As for the isotopic comparison method, the most impactful immediate advanceis to remove the distortions of the α-tagged spectrum (see Sect. 3.3). This isexpected to further increase the enhancement of the 9Li asymmetry by an additionalfactor of ∼ 2. To achieve this, one option would be to move away from the ZnS(Ag)scintillator and design a different α-detector with suppressed noise and better timeresponse (i.e., rapid light output). Such an example would be a Si-strip detector.117Nonetheless, given how successful the current very simple setup is, there is a bigincentive to look for other options for cleaning up the α-tagged spectrum. One suchoption would be to add an additional non-polarized pulse in the sequence of positiveand negative helicities (i.e., having a pulse sequence of P+→ P0→ P−). As theunpolarized beam should result in zero β -asymmetry, any variation of the α-taggedspectrum from the A = 0 line would be the product of distortion. One could thensubtract the unpolarized spectrum from the helicity-resolved ones and recover theactual time-evolution of the undistorted spectrum. The obvious drawback of thisoption is that it effectively reduces the time used for the actual SLR measurementby 1/3, so it should only be used if its resulting increase in asymmetry sufficientlycompensates for that.Turning to the Li diffusion measurements, we propose to further advance andestablish the capabilities of the α-radiotracer technique by studying the motion of8Li+ across an interface of two Li-ion battery materials, as well as by increasing themaximum temperature of the cryo-oven. By changing the thermometer lead wiresto higher-T materials, it should be possible to increase the maximum temperature ofthe cryo-oven to ∼600 K.Regarding the study of the Li+ motion through an interesting heterostructure,we propose to use a thin-film of graphene on rutile TiO2 as our proof-of-principleexperiment. On one hand, it has been shown that capping rutile with graphenesubstantially enhances (up to 100%) Li-ion insertion capacity [176, 177], which wasattributed to achieving better conductivity and to a structure that supports insertionreactions [178]. Based on our result that Li+ traps at the surface of rutile TiO2, wethink that this increase may, to some extent, be due to a more favorable boundarycondition (i.e., a more reflective boundary) for Li intercalation at the surface of rutile.α-radiotracer is the only technique capable of directly extracting this information,so we propose to study how the graphene cap changes the Li+ surface trappingprobability. 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Thegeometry of this new apparatus is designed around the α-detection system discussedin Ch. 5.Starting from the sample itself, it sits on an Aluminum sample holder, whichcontains three sample positions and two thermometers, one Pt resistor and oneSi diode (see Fig. A.1). The Pt resistor can measure temperatures up to 873 K(600 ◦C), but does not work below 30 K, whereas the Si diode is functional at lowtemperatures but unable to measure above 523 K (i.e., 250 ◦C).Figure A.1: Figure showing the sample holder used in the cryo-oven assembly. Thetop picture (F) is the side facing the beam and the lower one (B) the back of thesample holder. At the right hand side, the two thermometers are visible, both the Ptresistor (top picture) and the Si diode (lower picture). To their left, the three samplepositions can be seen. On the middle position, there is an Al2O3 crystal coatedwith ZnS(Ag) and capped with a 3 mm  Al collimator. It is used for 8Li+ beamdiagnostics. On the top and bottom positions, the (001) and (110)-oriented rutileTiO2 samples are sitting on the 7x7 mm2 slot, which has a circular gap around it forthe light emitted from the ZnS(Ag) to go through. On the front of the two samples(top picture), there is an Al ring coated with ZnS powder. At the rightmost sample,this ring is visible, as the Al collimator was not installed yet at the time the picturewas taken. The Al collimator is then installed on top of the Al ring, as shown in theleftmost sample position. It allows only the centered part of the 8Li+ beam to reachthe sample.140The sample-mounting concept and process is identical to the conventional β -NQR cryostat, as discussed in Sect. 2.2.2. The only difference is that the samplesintended to be studied with the α-radiotracer technique should be mounted on aspecial 7x7 mm2 base, which sits on the regular 12x12 mm2 slot. On the upstreamside of the sample, a ring painted with ZnS(Ag) and a Al mask with a pinhole arealso mounted (see Fig. A.1).The Al collimator, mounted ∼1 mm upstream of the sample, stops the 8Li beamfrom getting implanted off-center, as discussed in Ch. 5.The light produced from the ZnS(Ag) ring passes through the peripheral holearound the 7x7 mm2 base and can reach the photomultiplier tube (PMT) lyingoutside vacuum after getting focused on its front surface by a system of two convexlenses. The first lens is mounted on the radiation shield of the cryo-oven, whereasthe second one is mounted outside vacuum, at the front of the PMT housing. Thislight-collection system is depicted on Fig. 5.2.The lower region of the cryo-oven, with the α-detection, the cooling and heatingsystems visible, can be seen on Fig. A.2a. In the two cuts shown in that figure, onecan identify the following parts:1. The front (Fig. A.2a) and back (Fig. A.2b) sides of the three sample positionson the Aluminum sample holder at the end of a hollow stainless steel rod.2. The position of the Si diode (Fig. A.2a) and Pt resistor (Fig. A.2b) thermome-ters occupying the topmost (blind) position on the sample holder.3. Vertical-cut views of the radiation shield installed around the sample region.4. View of the beam collimator positioned on the radiation shield upstream ofthe sample (Fig. A.2a).5. Images of the optical viewport with a convex lens positioned on the radiationshield at the back (i.e., downstream) side of the sample.6. The Al wedge piece on which the sample holder sits.7. The electric heater.8. The liquid He inlet tube.9. The liquid He exhaust tube.In addition to the two above-mentioned thermometers on the sample holder,there is a thermometer mounted on the Al wedge piece, one on the copper heaterand one on the radiation shield, all Pt resistors.141Fig. A.3a shows the whole cryo-oven assembly both externally (Fig. A.3a) andas an isometric cut (Fig. A.3b). The system’s key components are enumerated asfollows:1. The radiation shield installed around the sample holder.2. View of the beam collimator positioned on the radiation shield upstream ofthe sample.3. Images of the optical viewport with a convex lens positioned on the radiationshield at the back (i.e., downstream) side of the sample.4. View of one of the two windows allowing the β -particles to reach the twobeta detectors positioned at the left and right sides of the sample (Fig. A.3a).5. The liquid He inlet tube.6. The liquid He inlet port.7. The liquid He exhaust tube.8. The liquid He exhaust port.9. The upper stainless steel flange with the rotatable bellows allowing the cryo-oven alignment with the beamline.10. The ports of the electrical feedthroughs used for the thermometers and theheater positioned on the cryostat and the radiation shield.11. The port of the electrical feedthroughs passed through the hollow sample rodfor the thermometers on the sample holder.12. The load lock valve that allows the differential pumping during the samplechange process.142(a) (b)Figure A.2: Images of two SolidWorks cuts of the lower region of the cryo-ovenassembly with the various parts enumerated. See text for details.143(a)(b)Figure A.3: Images of two SolidWorks overviews of the cryo-oven assembly withthe various parts enumerated. See text for details.144Appendix BGeant4This Appendix discusses how the Geant4 project used in this thesis is structured.The assumption is that it runs on a Unix-type operating system (e.g., Linux), asGeant4 is almost incompatible with Windows.As discussed in the Ch. 5, the Geant4 code doesn’t have to be used per se inorder to compile the library of Y nα (t) versus D, but rather Geant4 is required in orderto provide the probability of alpha detection versus depth, each time some aspectof the experiment (such as sample properties, detector geometry, energy threshold)changes. Nonetheless, the project presented here can generate both the α-detectionprobability vs depth and the actual Y nα (t;D) signal, simply by changing the inputmacro file.B.1 Getting startedAn overview of Geant4 can be found here: https://geant4.web.cern.ch/Upon downloading the source code, detailed instructions on how to buildand install Geant4 can be found here: http://geant4-userdoc.web.cern.ch/geant4-userdoc/UsersGuides/InstallationGuide/html/installguide.htmlAfter the Geant4 software is installed, the novice user can familiarize themselvesby going through the example projects provided by the Geant4 development teamand are distributed as part of the Geant4 source code.B.2 The project code treeTurning now to the project at hand: The classes defining the Geant4 project devel-oped as part of this thesis are very long. If printed here, they would cover more than80 pages. Instead, they can be found and downloaded at the link: https://github.com/arishadj/LiDiffusion-v2 together with all secondary files needed for theexecution at https://github.com/arishadj/LiDiffusion-v2-build.The following supposes that the source code of the project is copied in afolder named LiDiffusion-v2, where the main class file LiDiffusion.cc and the145CMakeLists.txt lay, as well as two secondary folders, src and include. src containsthe .cc files of the Geant4 code and include the header files with the extension .hh:LiDiffusion-v2/LiDiffusion.ccCMakeLists.txtsrc/LiActionInitialization.ccLiDetectorConstruction.ccLiDetectorMessenger.ccLiEventAction.ccLiPrimaryGeneratorAction.ccLiPrimaryGeneratorMessenger.ccLiRun.ccLiRunAction.ccLiSteppingAction.ccPhysicsList.ccinclude/Analysis.hhLiActionInitialization.hhLiDetectorConstruction.hhLiDetectorMessenger.hhLiEventAction.hhLiPrimaryGeneratorAction.hhLiPrimaryGeneratorMessenger.hhLiRun.hhLiRunAction.hhLiSteppingAction.hhPhysicsList.hhThe CMakeLists.txt file allows the compilation of the project. For more detailson how to build and compile a Geant4 project, see:http://geant4-userdoc.web.cern.ch/geant4-userdoc/UsersGuides/InstallationGuide/html/quickstart.htmlThe compiled code lies on a second folder LiDiffusion-v2-build, which alsoincludes the shell environment setup script for Geant4 geant4.sh and all othersecondary files needed for this project:LiDiffusion-v2-build/LiDiffusiongeant4.shstandard t.macinit vis.mac146vis.macProbVsDepth.Ccmake install.cmakeMakefileDiffusionProfiles/EnergyHistograms/beta.txtAlphaEnergySpectrum.txtWilkinson raw data.txtAlphaSpectrum.Calpha spectrum paramsSRIM/TiO2 25keV.txt...etc...TiO2 25keV/...etc...ROOTfiles/NormalizedSignal.CThe function of each file, both from the source code and the secondary auxiliaryfiles, will be discussed below.B.3 Running the simulationTo run the compiled code, simply open a terminal in the LiDiffusion-v2-build folderand execute:1 >> s o u r c e g e a n t . sh2 >> . / L i D i f f u s i o nThis initializes the Geant4 code. After the end of this process, the user can applycommands that either define parameters of the simulation, or start the simulationitself. Because the simulation at hand is rather complex and is designed to beflexible, a large number of parameters have to be defined by the user prior to thesimulation start. This is taken care by a macro file, such as standard t.mac. To run amacro file, the user should use the command:1 I d l e> / c o n t r o l / e x e c u t e s t a n d a r d t . mac / / o r n a m e o f a n o t h e r m a c r oThe standard t.mac has the visualization capabilities of Geant4 turned off, inorder to speed up the performance. If the user wants to visualize the geometry ofthe simulation, the macro file init vis.mac should be used instead.147B.4 Structure of the Geant4 projectB.4.1 LiDiffusion.ccLiDiffusion.cc is the main class of the project. It defines the random number gen-erator, chooses between single-threaded and multi-threaded programming stylesand instantiates the runManager, by creating instances of the main classes requiredfor the simulation. These are the LiDetectorConstruction, which defines the ge-ometry of the simulation (see Sect. B.4.2), the PhysicsList that defines the modelsof the physical interactions to be used in the simulation (Sect. B.4.3) and the Li-ActionInitialization class (Sect. B.4.4), which instantiates all other parts of thesimulation.B.4.2 LiDetectorConstruction/LiDetectorMessengerThe LiDetectorConstruction class defines all the information regarding the materi-als and the geometry of the simulation. The user can change most aspects of thegeometrical structures (materials, sizes etc) using the macro file, which passes theseinputs to LiDetectorConstruction through the LiDetectorMessenger class. LiDetec-torMessenger contains the definition of the available commands for changing thegeometry of the simulation.The header file LiDetectorConstruction.hh contains the definitions of all publicand private functions, as well as all protected objects that are used during thesimulations (e.g., the G4LogicalVolume structures).The .cc file contains the definition of all relevant methods. MakeMaterials()instantiates the material manager and defines the default material of each component(e.g., sample, detector etc). Construct() is the main function that defines and createsthe geometry of the simulation, as well as which LogicalVolumes correspond towhich detector. This source code also contains a number of auxiliary functions thatare used to define the material of each component, e.g., SetSampleMaterial(...) setsthe material of the sample, as defined by the user through the macro file.Finally, the class LiDetectorConstruction containts the methods required inorder to import the SRIM implantation profile and calculate the temporally-evolveddepth profile. These methods are called at the LiPrimaryGeneratorAction class thatdefines the properties of the particle beam, but are placed in LiDetectorConstruction,in order to allow for parallel execution of the simulation, as LiDetectorConstructionis shared between all CPU cores, in contrast with the LiPrimaryGeneratorActionwhich is defined locally in each core.The method CheckDepthDistribution() checks if the depth distribution for theuser-defined diffusion rate, boundary condition and point in time already exists in148the folder .../LiDiffusion-build/DiffusionProfiles/NAMEofSRIMfile/For instance for a diffusion rate 10−9 cm2 s−1, a trapping boundary conditionat the sample’s surface, SRIM file TiO2 25keV and time 0.1 s, it will seek the file.../LiDiffusion-build/DiffusionProfiles/TiO2 25keV/D1e-09t0 1accumulative.txt. Ifit exists, then it will just get imported. If not, then the method: MakeDepthDistribu-tion(...) will be invoked.MakeDepthDistribution(...) creates the time-evolved profiles starting from theSRIM file. It calls the method ReadSRIMProfile(...) to import the SRIM profile andthen calculates the depth distribution for the requested point in time by steppingfrom time t = 0 and calculating all intermediate distributions with a finite differencesnumerical code. Until the BeamOff time, the initial implantation distribution getsreplenished at each time increment and the depth profile is evolved in time usingFick’s Second law (Eq. 5.1).After the final depth profile is calculated, it is used by the method SetCoeffi-cients(...) to define the coefficients needed to sample the profile by the randomnumber generator of the LiPrimaryGeneratorAction class.B.4.3 PhysicsListPhysicsList class defines which models will be imported in order to simulate all rele-vant physical interactions. The classes G4EmLivermorePhysics and G4EmExtraPhysicsare used for the electromagnetic interactions, as they are considered the most accu-rate models in the MeV energy region [179]. G4HadronElasticPhysics defines thenuclear elestic scattering and G4HadronPhysicsQGSP BERT imports the Bertinimodel for the hadronic interactions, as it is deemed the most suitable for energiesbelow ∼10 GeV [180].B.4.4 LiActionInitializationLiDetectorConstruction and PhysicsList are the two classes that are shared byall CPU cores, i.e., they are running in the master thread. LiActionInitializationinstantiates both the master thread - with the method BuildForMaster() - and eachparallel thread with the method Build().Each thread creates instances of the classes LiRunAction, which defines theoutput of the simulation (see Sect. B.4.6), the LiPrimaryGeneratorAction class,which defines the properties of the “particle gun” (see Sect. B.4.5), LiEventAc-tion (Sect. B.4.6), which collects all relevant information after the simulation ofeach “event” (i.e., each 8Li decay) and the LiSteppingAction, which collects theenergy deposited in the detectors during each “step” of each particle’s trajectory(Sect. B.4.6).149In a nutshell, the structure of the simulation is the following: Each thread createsthe primary particles (i.e., the products of an 8Li decay) with an initial position,momentum, energy and direction defined by the user or through statistical distri-butions and random number generation. Each of these particles is propagated inthe geometry of the simulation in small steps. At each step, all relevant physicalinteractions are applied in a stochastic manner. They can generally result in energyloss, momentum/direction changes and secondary particle creation. At the end ofeach step, the energy deposited at the detectors is collected by the class LiSteppin-gAction, which passes it to the class LiEventAction. The above tracking processcontinues until each primary particle loses all its energy, or propagates out of thedefined simulation geometry. Then, the same process is repeated for each producedsecondary particle.LiEventAction accumulates the energy deposited in the detectors during thewhole tracking of all particles created by a single 8Li decay and passes the totalenergy deposited at that decay event to the LiRunAction class which registers it inan output ROOT file.B.4.5 LiPrimaryGeneratorAction/LiPrimaryGeneratorMessengerLiPrimaryGeneratorAction defines all properties of each primary particle to betracked in the simulation geometry in the main method GeneratePrimaries(...). The(x,y) coordinates of the decay are generated by a Gaussian distribution, with centerand sigma values defined by the user in the macro file through the LiPrimaryGenera-torMessenger class. The decay depth follows the distribution defined by the methodMakeDepthDistribution(...) of the LiDetectorConstruction class (see Sect. B.4.2).If the user wants to generate the probability vs depth instead of simulating theY nα (t) signal, then they should pass a negative value for time through the macro file.Then LiPrimaryGeneratorAction will generate a flat distribution of depths in thefirst 20 µm.The beta particle coming from the 8Li beta decay is assigned a random energyfollowing the experimental beta energy distribution by calling the method Make-BetaDistribution() which reads the auxiliary file beta.txt (Sect. B.2) and creates arandom number following that distribution. Its momentum direction is completelyrandom. To take into account the spin polarization of the beam, a custom codeshould be inserted here. As this is inconsequential for the diffusion study, it isomitted.For each decay event, a pair of alpha particles emitted back-to-back at thesame coordinates as the beta are also generated. Their energy follows the energydistribution defined in the auxiliary file AlphaEnergySpectrum.txt. This file is createdusing the experimental values of the spectrum from [93] by running the ROOT file150AlphaSpectrum.C. As the file AlphaEnergySpectrum.txt is already defined, the userdoes not have to recompile it each time.B.4.6 LiRunAction/LiEventAction/LiSteppingActionLiSteppingAction gets the information of the current coordinates of the trajectory ofeach propagating particle, checks if it is inside some detector defined at LiDetector-Construction (Sect. B.4.2) and if it is, passes the energy deposited in that detectorby the tracked particle during the current step of its trajectory to the LiEventActionclass.LiEventAction class sums at the end of each decay event the energy deposited inthe detectors by all/some particles in all the steps of their trajectories and fills thehistograms defined in the LiRunAction class with the relevant values.LiRunAction defines what information will be stored in the output ROOT file.It creates a number of histograms to be filled (e.g., the histogram of the 8Li depthdistribution in the sample), as well as a “Ntuple” structure, which contains “leafs”with the relevant information of each decay event (e.g., decay depth and energydeposited at the detector).B.4.7 Output filesThe output of the simulation is a .root file. If the simulation was aiming to get theprobability of detecting an alpha versus decay depth, then one has to run the ROOTfile ProbVsDepth.C with ROOT, in order to generate the relevant .txt file from theraw .root output file.If the chosen output is the Y nα (t) signal, then upon completion of all simulations(one for each chosen instance in time), the .root files should be moved to theROOTfiles folder and the code NormalizedSignal.C should be executed with ROOTin order to generate the normalized alpha signal from the raw energy deposition atthe detector.To change the detector threshold, one has to manually change the value definedin the ProbVsDepth.C and NormalizedSignal.C files, but can use the same .rootfiles in order to re-calculate the Y nα (t) signal. The detector threshold can be definedeither directly by an energy value (e.g., 2 MeV), or indirectly by choosing whichpercentage of the highest energy alphas should be detected (e.g., the 30 % of thehighest energy alphas corresponds to an energy threshold of roughly 2 MeV).151B.4.8 Macro (input) fileThe macro files with the extension .mac define all parameters of a simulation. Forinstance, the following commands define the verbose levels of the simulation:1 / c o n t r o l / v e r b o s e 02 / c o n t r o l / s a v e H i s t o r y3 / run / v e r b o s e 04 / c o n t r o l / c o u t / i g n o r e T h r e a d s E x c e p t 05 / e v e n t / v e r b o s e 06 / t r a c k i n g / v e r b o s e 0The following commands then define the various geometrical characteristics ofthe sample, (optional) substrate and alpha detector:1 / L i D i f f u s i o n C o d e / geomet ry / c r e a t e S a m p l e M a t e r i a l 3 . 1 5 Mg 1 F 223 / L i D i f f u s i o n C o d e / geomet ry / se tSampleWid th 7 mm4 / L i D i f f u s i o n C o d e / geomet ry / s e tSampleDep th 500 um5 / L i D i f f u s i o n C o d e / geomet ry / s e t S u b s t r a t e W i d t h 7 mm6 / L i D i f f u s i o n C o d e / geomet ry / s e t S u b s t r a t e D e p t h 500 um7 / L i D i f f u s i o n C o d e / geomet ry / s e t D e t e c t o r W i d t h 0 . 4 mm8 / L i D i f f u s i o n C o d e / geomet ry / s e t D e t e c t o r D e p t h 0 . 1 mm9 / L i D i f f u s i o n C o d e / geomet ry / s e t D e t e c t o r R a d i o u s 5 . 7 2 mm10 / L i D i f f u s i o n C o d e / geomet ry / s e t F i l e n a m e MgF2In this case, the sample would be a 7x7x0.5 mm3 MgF2. The substrate hasnominally the same size, but as its material is not defined in the macro file, it takesits default value of being made of vacuum, i.e., it does not exist. The alpha detectoris a ring (with its internal side cut at a 45o angle) and the above commands defineits radius, width and depth. As its material is not defined in this macro file, it takesits default value of being made of ZnS(Ag).The characteristics of the particle beam and the requested instance in time to besimulated are defined by the following commands:1 / L i D i f f u s i o n C o d e / geomet ry / L i I o n s P e r P u l s e 10000002 / L i D i f f u s i o n C o d e / geomet ry / setBeamOnTime 1 s3 / L i D i f f u s i o n C o d e / geomet ry / se tSRIMf i l ename TiO2 25keV4 / L i D i f f u s i o n C o d e / geomet ry / s e t D i f f u s i o n R a t e 1e−115 / L i D i f f u s i o n C o d e / geomet ry / s e t B o u n d a r y C o n d i t i o n r e f l e c t i v e6 / L i D i f f u s i o n C o d e / geomet ry / se tT ime 0 . 1 s152The above commands define that the ion beam will contain 106 Li ions, which istaken into account during the beam implantation time, defined to be 1 s. The SRIMfile to be imported is the TiO2 25keV (i.e., a 25 keV 8Li+ beam implanted into TiO2)and this initial depth distribution is to be developed in time with a diffusion rateof 10−11 cm2 s−1 and a reflective boundary condition at the surfaces of the sample,until the depth distribution at the requested point in time (t =0.1 s) is calculated.Finally, the following commands update the parameters with the user-definedvalues and instantiate the simulation, in this case with 107 8Li decays.1 / L i D i f f u s i o n C o d e / geomet ry / d i a g n o s t i c s 12 / L i D i f f u s i o n C o d e / geomet ry / u p d a t e34 / run / beamOn 10000000To generate a full Y nα (t;D) signal, the above commands should be re-definedin succession for each required point in time. To generate the probability vs depthdistribution, then the /LiDiffusionCode/geometry/setTime command should be setto some negative value (e.g. −1). In this case, it is recommended to run very longsimulations, at least 106 decay events per nanometer of the profile.153


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