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Quantum tunnelling of muonium in copper(I) chloride MacFarlane, W. Andrew 1992

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QUANTUM TUNNELLING OF MUONIUM IN COPPER(I) CHLORIDE By W. Andrew MacFarlane B. Sc. University of Victoria, 1990  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA  October 1992  ©  W. Andrew MacFarlane, 1992  In presenting this thesis in  partial fulfilment of the requirements for an advanced  degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of  -  The University of British Columbia Vancouver, Canada Date  DE-6 (2/88)  Abstract  The unusual behaviour of paramagnetic muonium centres in the binary semiconductor Copper(I) Chloride at low temperatures lacks a satisfactory explanation. In this thesis, after a brief review of the experimental results to date, a model for the spin dynamics of the muon as it undergoes local quantum tunnelling motion amongst four wells arranged in a tetrahedron is presented. Calculations based on this model suggest that a level iSR crossing resonance associated with the tunnel splitting could be observed in a 1 experiment.  11  Table of Contents  Abstract  11  List of Tables  V  List of Figures  vi  Acknowledgements  vi’  1  Introduction  1  2  A Short Introduction to pSR  4  2.1  The Physical Basis of The 1 SR Technique  2.2  Spin Hamiltonians  2.3  2.4 3  2.2.1  Free Muonium  2.2.2  Muonium Interacting with Nuclear Surrounding Spins  2.2.3  Muonium in a High Magnetic Field.  2.2.4  Interaction with the Lattice  tSR Observables 1 2.3.1  Muon Spin Polarization  2.3.2  Initial States  Muon Level Crossing Resonance  15  Diffusive Phenomena and Muon Spin Dynamics 3.1  Classical Diffusion in a Discrete Lattice  :iii  17 17  4  5  6  3.2  From Classical Hopping to Quantum Diffusion  18  3.3  The Effects of Muon Motion on Its Spin Dynamics  22  3.3.1  Zero Applied Field  23  3.3.2  Transversely Applied Field  24  3.3.3  Longitudinally Applied Field  27  3.3.4  Level Crossing Resonance  27  Muonium in Copper(I) Chloride  29  4.1  Properties of CuC1  29  4.2  Muonium in CuC1  30  4.2.1  High Transverse Field Measurements  30  4.2.2  tLCR Measurements  32  4.2.3  Longitudinal Field T 1 Measurements  34  Tunnelling in a Tetrahedron  41  5.1  General Considerations  41  5.2  The Model Hamiltonian  43  5.2.1  46  High Field Approximation  5.3  Calculation of The Muon Polarization Function  47  5.4  Finite Lifetime Damping  48  Results, Discussion and Conclusion  50  6.1  Results and Discussion  50  6.2  Conclusion  57  Bibliography  58  A Note on Signs and Units  61  iv  List of Tables  1.1  Approximate Crossover Temperatures  4.1  Low Temperature Formation Probabilities for Muonium in CuC1  4.2  Nuclear Hyperfine Structure of Muonium in CuC1  5.1  Muonium Frequencies Corresponding to the Intersections of Two Breit— Rabi Diagrams  2 .  .  .  .  30 33  44  A.1 Magnetons  61  A.2 Gyromagnetic Ratios  62  A.3 Isotope Data for CuC1  62  v  List of Figures  2.1  Surface Muon Beam Production  5  2.2  The Elements of the 1 SR Method  6  2.3  Breit—Rabi Diagram for Isolated Muonium  10  2.4  Level Crossing Resonance  16  4.1  CuCl Crystal Structure  29  4.2  T’ and r vs. Temperature  37  4.3  A Successful LF Fit using Celio’s Model  38  4.4  Low Temperature Failure in Fitting LF Data  39  4.5  An Unexplained Peak in Tj’  40  5.1  Intersections of Two Breit—Rabi Diagrams  45  6.1  Predicted Tunnelling Resonances for Four Wells  6.2  Resonances from Interaction with a Single Nucleus  6.3  Predicted Time Dependent Muon Spin Polarization  54  6.4  Predicted Tunnelling Resonances for Two Wells  55  6.5  Effects of Level Population  56  vi  .  .  .  51. 53  Acknowledgements  For their patient and invaluable help in the production of this work I thank my su pervisors Rob Kiefi and Philip Stamp.  For their advice and helpful suggestions, I  also thank Jürg Schneider, Chao Zhang, Kim Chow, Syd Kreitzman, Tanya Riseman, LSR group at Gerald Morris, Jess Brewer, and the rest of the 1  TRIUMF.  For financial  assistance, I acknowledge NSERC and UBC. For the encouragement and support that brought me to this point, I thank my parents Bill and Lynda. And finally, for their love and support during my studies (and pleasant diversions therefrom), I thank Karen and Luca.  vii  Chapter 1  Introduction  The motion of impurities in solids, especially quantum motion, has long interested condensed matter physicists because it both confirms and extends our knowledge of quantum transport generally, and because impurity effects on the properties of solids have such an enormous practical significance. This thesis deals with the motion of impurities under conditions where quantum effects are important, i.e. quantum diffu sion, in particular, the motion of muonium (Mu), a light hydrogen-like atom, as an interstitial impurity in the semiconductor CuC1. The conditions for quantum motion are highly sensitive to the impurity mass, m, as a result of two important quantum mechanical phenomena: zero-point motion and sub-barrier tunnelling. For the zero-point motion of a particle in a harmonic potential, 1 and the energy as the mean square deviation about the potential minimum goes as m m. The probability amplitude for tunnelling decreases roughly exponentially with m  as can be seen from the Schrödinger equation, given that the particle’s energy lies below the potential barrier. These examples show that lighter particles exhibit enhanced quantum effects, making the positive muon (,ic) and its neutral charge state, muonium  (,icFej, at a mass of  the mass of atomic hydrogen, ideally suited to experimental  investigation of quantum behaviour of both charged and neutral impurities. Furthermore, quantum effects in the diffusion rate are most apparent at low tem peratures because they result from coherent particle motion which is suppressed at higher temperatures by scattering processes such as collisions with phonons. Thus we 1  Chapter 1. Introduction  2  expect a crossover temperature, T*, below which quantum tunnelling will be dominant and above which phonon—assisted quantum diffusion will eventually give way to the high temperature classical limit. Experimentally, for muons or muonium, the crossover temperatures are a significant fraction of the Debye temperature and thus are easily accessible; approximate results in a wide range of solids are given in table 1.1. The T* (K) 50 5 100 20 50 70 90 30  in Cu in Al in Fe Mu in s N 2 Mu in KC1 Mu in NaC1 Mum GaAs 1 in CuC1 Mu —  a 0.4 0.6-0.7 —  5.7 3.3 3.0 2.7  Reference [1,2] [3,2] [4] [5] [6] [7] [8] [9,10]  Table 1.1: Approximate crossover temperatures, T*, and the temperature dependence (T) of the hop rate for quantum motion of muons or muonium atoms (Mu) in several solids. crossover temperature corresponds to a minimum of the diffusion coefficient, D, as a function of temperature. Another parameter describing the nature of the quantum diffusion is the shape of the rise in D(T) below T*. Commonly, this rise is fitted to a power law D(T)  T; the exponents alpha for some materials are given in table 1.1.  Clearly, the phenomenon of muon(ium) quantum diffusion is quite universal, but does vary significantly with host material. This thesis presents a model for the quantum motion of muonium in CuC1 below T*, and the resulting influence on the spin dynamics of the muon. The remainder is divided iSR technique intended for into several chapters. Chapter 2 is a brief overview of the 1 those unfamiliar with its details.  Chapter 3 is a review of diffusive phenomena in  solids with emphasis on light interstitial impurity diffusion at low temperature, and it  Chapter 1. Introduction  3  also contains a summary of how motional effects are measured with tSR. Chapter 4 comprises a review of the (notably anomalous) experimental results for muonium in CuC1. In Chapter 5 the model for local quantum tunnelling of muonium in CuC1 is presented, and its effects on the SR spectra are predicted. In Chapter 6 the results of this model are presented and discussed.  Chapter 2  A Short Introduction to SR  Muon Spin Rotation/Relaxation ( iSR) is a technique in materials research that flour 1 ishes at the interface of particle, condensed matter, and chemical physics. The basis of SR and related methods, the asymmetry of the muonic decay, allows one to measure the ensemble average magnetic interaction of the muon magnetic moment with a sample of interest. Though SR is frequently applied to both gases and liquids and sometimes using negative muons, the discussion herein will be restricted to implantation of positive muons in solid crystalline targets.  2.1  The Physical Basis of The SR Technique  In this section the salient features and physical basis of the technique are outlined. For more a complete description, the reader is referred to [11,12]. The essential feature of the muon decay that connects the muon spin and the decay positron, a precondition for the very existence of SR, is the asymmetry of the weak decay of the muon, i.e. the preferential emission of the decay positron in the direction of the muon spin. This is a consequence of the parity violation of the weak decay, predicted by Lee and Yang [13], first measured in /3-decay by Wu et al. [14], and in muon decay by Garwin et al. [15]. It allows one to monitor the muon spin polarization in the sample as a function of time after implantation. SR requires high flux beams of spin polarized muons that can be focussed to a beam 1 ), thereby minimizing background. The 2 spot roughly the size of the sample (a few cm  4  Chapter 2. A Short Introduction to pSR  Cyclotron  5  Production Target  (proton accelerator)  (typically Be  K ETRI OOMeV 5 +  p ;  Pion Production: +  +  +  +  p+p-41T+p+n p+n+n+n  zi,  Surface Pion Decay: iT —*/ +1)  T26flS  KE4. 1MeV p29.8MeV/c Muon Helicity —1  Figure 2.1: Surface Muon Beam Production. most intense continuous beams are produced at the meson factories,  TRIUMF,  and  PsI  in Switzerland. The production scheme for a beam of positive muons is summarized in fig. 2.1. Muons originating from pion decay at the production target surface are commonly called surface muons. They have a stopping range of only  2 0.15g/cm  and are frequently used in iSR because of they are 100% spin polarized and possess a narrow range distribution. In a typical iSR experiment, a beam of surface muons is trained, through a thin intervening scintillator, on the target sample. When the muons incide on the target they decelerate, at first slowly, but as their speed approaches that of the orbital electrons in the target, the deceleration increases. The entire process of thermalization is quite fast (i—’  iris [16]), and little of the spin polarization is lost. The muon’s magnetic moment  Chapter 2. A Short Introduction to ,uSR  6  Muon Decay: , T,2.2/1S 7 L->e+ve+i  Positron Counter  [0,52.3MeV] e  +  Target Sample  B, I //“e  Muon Counter  iSR method. Note: the three body decay of the muon Figure 2.2: The elements of the 1 results in a range of positron energies. Im.et: Standard polar plots of the asymmetry in the probability of positron emission. The polar angle 0 is the angle between the muon spin direction and the direction of positron emission as in the figure, and the scale is arbitrary. The left panel shows the asymmetry averaged over all positron energies, and the right shows the asymmetry for the maximum positron energy, 52.3MeV. subsequently interacts with the net local magnetic field, the sum of internal fields and, if present, an externally applied field. When the muon eventually decays, the emitted positron may be detected by one of an array of counters surrounding the sample. Muon implantation and decay are represented in fig. 2.2. There are basically two modes of SR data acquisition: time differential and time integral. In the former, one measures the time dependence of the spin polarization of the muon over a time range of up to  16is and with a typical time resolution of  two nanoseconds. In the latter, one measures the time integrated polarization which  Chapter 2. A Short Introduction to SR  7  amounts to a time average weighted by the muon lifetime, ri,. In time differential SR one histograms the measured time interval between the incidence of a muon and the detection of an acceptable positron for each counter. In the longitudinal field (LF) geometry, the external magnetic field is applied parallel to the initial muon spin. The time histogram, after subtracting a time independent background, is of the form Noe_t/ [1 +  aPz(t)]  ,  (2.1)  where N 0 is a normalization factor and a is the asymmetry parameter, giving the amplitude of the z-component of the muon polarization Pz(t). In the absence of any internal fields or hyperfine interaction, Fz(t) = 1; however, in general Pz(t) may exhibit both oscillation and relaxation (see  § 2.3). In the transverse field geometry the field is  applied perpendicular to the muon spin’. The histogram, in this case, is of the form Noe_t/T [i +  AR(t) cos (wit +  (2.2)  where the number of terms in the sum depends on the type and number of post— thermalization muon states. R(t) describes the relaxation associated with the oscilla tion of frequency w, and  is the initial phase.  Time integral tSR measures the behaviour of the muon effectively averaged over all time. Because the requirements for a good positron detection are relaxed much higher incident muon rates can be employed. The particular time integral method of interest in this thesis is the integral measurement of a muon level-crossing resonance, jtLCR (see §2.4). Actually the muon spin is more often rotated 90 degrees by application of crossed electric and 1 magnetic fields prior to implantation, and the field is applied in the direction as in the LF case.  Chapter 2. A Short Introduction to RSR  2.2  8  Spin Hamiltonians  In this section the general Hamiltonian governing the muon’s spin dynamics in various situations is presented. Each Hamiltonian is based on the general spin—spin interaction between two distinguishable particles with spins HhYP  where  i=  {[3(  -  (  and S: +  .  2)63(},  (2.3)  r joins the two spins, and (2.4)  = —I’o’y172;  The yj’s are the (signed) gyromagnetic ratios (see Appendix A), and  is the vacuum  magnetic permeability. For a spherically symmetric distribution of one of the particles about the other, only the last term in the braces, known as the Fermi contact term, survives averaging. In this case H’’ reduces to (2.5) where A, the hyperfine parameter, is A  =  cvp(O);  (2.6)  p(r) is the probability density of one of the particles about the other. For an asymmetric distribution, rather than including the detailed form of equation 2.5, it is common to include the dipolar (non-contact) interaction terms by introducing the hyperfine tensor, HhYP  2.2.1  =  S  .  A  52.  (2.7)  Free Muonium  Since the first excited spatial state of an isolated muonium atom lies  1.2 x 10 K 5  above the ground state, and we are interested in much lower temperatures, we make  Chapter 2. A Short Introduction to SR  9  the assumption that Mu is in its is spherically symmetric ground state. Therefore, in the presence of an applied magnetic field B, the spin Hamiltonian for muonium is H/IU  =  + Hhyp,  (2.8)  where .  (2.9)  ,  —  and (2.10) The eigenvalues of this Hamiltonian are found to be 1 E  =  3 B  =  2 \/A + (B) A, B 2__T+ 2 + (B) 2 A —  2 (2.11) where  =  —  7e. The two eigenstates corresponding to S  =  +1 have the muon  and electron spins parallel and either aligned or anti-aligned with the field. The two eigenstates corresponding to S  =  0 are field-dependent linear combinations of the  two states where the electron and muon spins are anti-aligned. A graph of these energy levels vs. field strength, i.e. a Breit—Rabi diagram is shown in fig. 2.3.  2.2.2  Muonium Interacting with Nuclear Surrounding Spins  In semiconductors and insulators, a positive muon often forms neutral paramagnetic iSR and muonium centres. These centres can be stable long enough to be observed by 1  Chapter 2. A Short Introduction to ,uSR  10  9-  0 Cl) C  >  a) C  bJ  0.0  0.5  1.0 1.5 2.0 2.5 Magnetic Field (Units of A/a)  3.0  Figure 2.3: Breit—Rabi diagram for isolated muonium. The signs at the right are the Paschen—Back limit approximate eigenstates, me, mn). are found to resemble vacuum muonium in some cases but are often quite different. Because the magnetic moment of the electron is roughly 200 times that of its muonic counterpart (by virtue of its small mass), the direct muon—nuclear coupling is almost always negligible in comparison to indirect coupling through the muonium electron. Furthermore, the electron Zeeman interaction may be anisotropic (the muonic and nuclear interactions do not exhibit this), so the electron Zeeman term contains the tensor gyromagnetic ratio, g.  The Hamiltonian for muonium interacting with the  nuclear spins is thus (2.12)  Chapter 2. A Short Introduction to 1 uSR  11  where •g B  B.  —  (2.13)  A is the muonium hyperfine tensor (describing the spin—spin interaction between the muon and the muonium electron), A is the nuclear hyperfine (nhf) tensor (describing the interaction between the muonium electron and the nuclei), and  Q  is the nuclear  electric quadrupole (neq) tensor. The neq term represents the interaction of the nuclei  2  with the electric field gradient, which is non-zero in any non—cubic material, but also contains a contribution due to the presence of the muon(ium). For the particular case where both the nhf and neq tensors, for the ith nucleus, are axially symmetric about a common  axis, the nuclear interaction part of the  Hamiltonian is + A(SS + SS) + Q[(S) 2  -  + 1)],  (2.14)  where (xi, yi, zi) is the direction between the muon and the nucleus i. 2.2.3  Muonium in a High Magnetic Field  The muonium electron Zeeman interaction dominates the Hamiltonian of the previous such that S is a good quantum number, and the  section for high fields (B >> two subspaces corresponding to S  =  +1/2 can be treated independently with an  interaction entering in first order in the inverse electron Zeeman frequency w [17]. To find the effective Hamiltonian on the two electron spin manifolds, one can use an , one finds 1 operator perturbation expansion [18]. To first order in w 1 H  =  (T  H  1)  +  H  IH )  (2.15)  A nucleus has non—zero electric quadrupole moment only for S,, > 1/2, so, in particular, this term 2  is zero for a spin 1/2 nucleus as can be seen in equation 2.14.  Chapter 2. A Short Introduction to SR  1 H  =  (  H  )  -  12  H  t)(t IH ),  (2.16)  where H t and H 1 are the two effective Hamiltonians and H is the full muonium Hamil tonian including interaction with the nuclei. 2.2.4  Interaction with the Lattice  Relaxation phenomena depending on, for example, temperature or dopant concentra tion are not contained in the above Hamiltonians . To include such effects, one must 3 include interactions with other relevant degrees of freedom of the crystal; for instance, inclusion of interactions with both the conduction electrons and phonons has led to success in explaining the low temperature diffusion of  in Copper [19]. For muo  nium important effects on the electron spin (other than nhf effects) may come from, for example, exchange interactions with paramagnetic dopant atoms, or interactions with charge carriers. The full Hamiltonian describing the interaction with the lattice would have the form H=Ho+HL+V.  (2.17)  0 is the spin Hamiltonian as in the preceding sections, HL is the Hamiltonian for the H lattice, and V is the interaction between the two. Examples of HL and V can be found in [19,20,21]. 1f interactions with many nuclei are included, the observables may appear to be relaxing. 3  Chapter 2. A Short Introduction to SR  2.3  13  SR Observables  2.3.1  Muon Spin Polarization  The observables in tSR experiments are all derived from the muon spin polarization,  P(t)  (a),  =  (2.18)  where ö is the vector of Pauli spin operators, appropriate to the spin-1/2 muon. In the Schrödinger picture, the expectation value is obtained as  P(t)  =  Tr(p(t)),  (2.19)  where the p(t) is the density operator (state) of the system. The time dependence of the density operator is governed by the Landau—von Neumann equation,  d =  subject to the initial condition  p(O)  =  —[H,p],  po  (see  §  (2.20)  2.3.2). For a time independent Hamil  tonian, like that of the above section, the time dependence may be solved analytically,  p(t)  (2.21)  =  In this thesis, the observables of interest are the longitudinal spin polarization, with the direction of both B and S , 1  P(t)  =  (oj,  (2.22)  and its time averaged value weighted according to the muon lifetime (Laplace transform at 100  -  =  —  j  P(t)e_t/n1dt.  (2.23)  Chapter 2. A Short Introduction to uSR  2.3.2  14  Initial States  It is commonly assumed that the non-muon degrees of freedom are at thermal equilib rium, at some temperature T, when implantation occurs. Denoting the Hamiltonian in the absence of the muon as P0  ft, the initial state is [I +  =  ],  (2.24)  For temperatures much larger than the transition energies of H, this reduces to the following. At the temperatures under consideration, it is often reasonable to assume that, initially, the N nuclei surrounding the muon or muonium are unpolarized, and that the muonium electron is also unpolarized. If the initial muon spin polarization is  , then the initial density operator for the spin system is P0  =  21e® flN(2S  +1)In®2  [i+P],  (2.25)  where I... are the identity operators on the appropriate subspaces. When interaction with the lattice is important, it is assumed that the lattice (R) and the system consisting of the muon(ium) and N nuclei (S) are initially uncorrelated, so POPO®POS.  (2.26)  Since the macroscopic lattice is never of experimental interest, the reduced density matrix for the spin system is defined as Ps  =  TrR(p),  (2.27)  where the trace is taken only over R’s degrees of freedom. From this definition, an equation of motion for  PS  can be found using equation 2.20.  Chapter 2. A Short Introduction to 1 uSR  2.4  15  Muon Level Crossing Resonance  Muon Level Crossing Resonance (tLCR) is a form of cross-relaxation or double res onance involving muon spins, first suggested by Abragam [22]. In general, it can be explained as follows. In zero or first order (in the interaction with the  spin), there  exist longitudinal fields for which the non—interacting muon(ium) + nuclear spin sys tem energy spectrum contains degeneracies, see fig. 2.4. From perturbation theory, these degeneracies are removed in the presence of interaction with an accompanying mixing of the previously degenerate states [23, pp 191-193]. Normally a large magnetic field applied parallel to the muon spin polarization leads to a time—independent P; however, the above mixing of states with different S, causes a coherent oscillation of the muon spin polarization that can be observed provided zSB) < —O.1r , 1 h  (2.28)  where z(B) is the avoidance between the levels. In time differential tSR, the resonance usually appears as a peak in the depolarization rate, but the oscillation, in some cases, may be fast enough to observe [24]. In an integral measurement, the resonances appear as dips in P(B) (see fig. 2.4). Usually, the field is modulated to reduce systematic errors, so the measured quantity is approximately P. The method, thus, provides a way to probe the details of the energy level structure of the spin system. If the couplings are anisotropic, varying the field direction relative to the crystal axes provides information about the symmetry of the muon(ium) site. tLCR has been applied to accurately measure the neq couplings in Copper [25] and 1 the nhf and neq couplings for muonium in semiconductors, e.g. [26] (LCR results for muonium in CuCl are discussed in  §  4.2.2). In both cases, these measurements gave  detailed information about the muon(ium) site, the electronic structure of muonium,  Chapter 2. A Short Introduction to ,uSR  16  Magnetic Field Figure 2.4: Level Crossing Resonance. Top: The lines correspond to unperturbed Zeeman energy levels, and the curves to the energy eigenvalues assuming a small mixing of the levels. Bottom: If these levels correspond to different states of the muon spin, and one is 100% populated away from the crossing, then the dip in FZ(B) takes this form. and the muon’s dynamics.  Dynamical effects on 1 iLCR are SR signals including 1  discussed briefly in the next chapter.  Chapter 3  Diffusive Phenomena and Muon Spin Dynamics  3.1  Classical Diffusion in a Discrete Lattice  Classical continuous diffusion, familiar from fluid and heat dynamics, is embodied in Fick’s Law, p  =  p, 2 DV  (3.1)  where p is the diffusing quantity (e.g. fluid velocity, solute concentration, or thermal energy), and D is the diffusion constant (n.h. [D]  =  Time 2 Lerigth ) 1 . In the case of  interest here, the solute is muonium and the solvent a crystalline solid, so the set of possible solute locations, instead of a continuum, forms a discrete array of sites within the crystal lattice. To connect the continuous limit with this discrete picture, consider the 1 dimensional version of equation 3.1: Dp(z,t) _DDp,t)  (.32  8z2  —  Replacing the derivatives with approximate expressions, we find (for Lit, Lz p(z,t + z.t) w h ere A  =  Ap(z + Az,t) + (1— 2A)p(z,t) + Ap(z  —  z,t),  —*  0) (3.3)  Dt — (z) 2  In the discrete case the concept of concentration becomes tenuous at the microscopic level, so, instead, we interpret p(z, t) as the probability of finding a solute particle at  17  Chapter 3. Diffusive Phenomena and Muon Spin Dynamics  18  z at time t . Equation 3.2 therefore expresses a random walk process wherein, for 1 each small time step / t, a particle at z has probability A of hopping to either of its neighbours (and, therefore, a probability of (1  —  2A) of not hopping). Also from  equation 3.2, we get D  =  hni  t,z—+O  Lt  where the limit is assumed finite. Physically, however, L z is the small but finite lattice constant, a; furthermore, it is conventional [11] to choose A as a purely geometric factor representing the number of neighbouring solute sites in a particular lattice, so all the interesting thermal dependence of D is tied up in / t, the mean time between hops, which is more usually labelled  Thop.  With these definitions, we have Aa 2  (3.4)  Thop  The dependence on temperature of  Thop  is quite complex and is the subject of the next  section.  3.2  From Classical Hopping to Quantum Diffusion  At sufficiently high temperatures, the diffusing particle is pictured to be hopping be tween potential wells, as the result of coupling with a heat bath. Provided Ea > kT, one expects [27] that the hop rate, r,, increases exponentially with temperature in accordance with an Arrhenius law, Tho’P = vae_ET,  (3.5)  1 T his interpretation with equation 3.1 requires that the motions of different solute particles are uncorrelated, i.e. there are no solute-solute interactions. This doesn’t compromise the experimental validity of the discussion, since all pSR experiments are performed in the limit of extreme dilution there is usually only a few muons in the sample at any time. —  Chapter 3. Diffusive Phenomena and Muon Spin Dynamics  where  Va  19  is an attempt frequency (of the order of a phonon frequency), and Ea  S  the activation energy. For light interstitials, Ea is often determined by the energy of the lattice distortion caused by the presence of the interstitial [28] rather than the magnitude of the classical barrier. On the other hand, at the lowest temperatures, one expects quantum tunnelling motion, governed by the band structure for an impurity in the Bravais lattice of inter stitial sites, E(k), to dominate. Neglecting coupling to the phonon bath, such motion will be described by a wavefunction composed of a linear combination of Bloch waves, and the motion, rather than random hopping between localized states as in the previ ous section, will be that of a wave in a dispersive medium. For this case a more general definition of the diffusion constant is required [29], D where A is, as in  (•.)  §  =  A  j((t)  v(O))dt,  (3.6)  3.1, a geometric factor, ‘(t) is the particle velocity kE(k), and  is the thermal equilibrium expectation value.  This form of D coincides with  that of equation 3.4 in the classical case. For a non-interacting impurity, the velocity autocorrelator is finite and time independent, so D is infinite. Scattering off other crystal defects or elementary lattice excitations will introduce time-dependence into (7(t) (O)), and thereby limit D. In the case of scattering from diffuse impurities D .  can be expressed for the long—time limit as [30,31] D where  Tfree  (3.7)  = A(V >Tfree, 2  is the mean free time between scattering events. The diffusion constant  is still determined by a random process, but the characteristic time is no longer the residence time in a single well,  ,,. 0 rh  For temperatures much larger than the impurity  bandwidth, this case yields a temperature independent T  —*  0 plateau in D which  Chapter 3. Diffusive Phenomena and Muon Spin Dynamics  20  depends on the scattering centre concentration, x. For weak scattering interaction, D  ‘-‘  , and for strong interaction, D 1 x  -‘  [31]. These effects may be visible  in the low temperature SR data as sample or dopant concentration effects. Without impurities, the behaviour of D(T) is thus increasing at both high and low temperatures, so a minimum of D(T) must exist at some temperature T*, as mentioned in chapter 1. Between the two extreme limits above, theorists have described many diffusive pro cesses involving other components of the crystal in attempt to connect the low and high temperature regimes and predict T*, e.g. [28]. In metals, where conduction elec trons at the Fermi surface play an important part in determining impurity diffusion coefficients, successful treatments have been made by Kondo [19] and Yamada [32]. However, we will focus here on muonium in semiconductors and insulators, in which lattice vibrations play the dominant role in limiting D. Phonon influenced diffusive processes are generally divided into two classes: coher ent, in which the motion of an impurity from one site to a neighbouring site involves no change in the phonon occupation numbers (i.e. the mean free path exceeds the lattice spacing), and incoherent, including one, two and many phonon dissipative processes that act to reduce the mean free path and destroy coherent motion. It should be noted that for atomic impurities, the energy bandwidth for coherent motion, L, is small com pared with the Debye energy, so interaction with the phonons is relatively strong and cannot be treated perturbatively. Furthermore, single phonon processes are kinetically restricted, and below a threshold bandwidth, play no part in dissipative scattering (they are still important, though, in determining the static lattice distortion around the impurity analogous to the polaron effect for electrons). Thus all of the analysis of phonon scattering until recently has focussed on two phonon processes. These descriptions began with Andreev and Lifshitz [30]. The subsequent treat ments, beginning with Flynn and Stoneham [33], were based on the theory for motion  Chapter 3. Diffusive Phenomena and Muon Spin Dynamics  21  of small polarons, and led to the following results. For temperatures small compared to the Debye temperature, GD, scattering is primarily with long wavelength acoustic phonons, and in this limit two phonon scattering causes oscillations in the energy dif ference between corresponding levels in adjacent sites which limits coherent diffusion, a phenomenon called “dynamical destruction of the band”. The temperature dependence of the incoherent and coherent diffusion constants (Dh and D respectively) is found to be [20,34] (3.8)  (-p)  Dh  TT, 2 e_  (3.9)  where (T) is the polaron exponent which is weakly temperature dependent. Both G(T) and  arise from coherent “fluctuational preparation of the barrier”, i.e. time  variation of the intersite potential barrier due to motion of the nearby nuclei. The pro cesses of “dynamical band destruction” and “fluctuational barrier preparation” compete in their effect on the diffusion coefficient. In the case where z is large enough to allow one phonon dissipative processes, for probabilistic reasons, they will be dominant. Using a Debye density of states, applicable for low energy phonons, single phonon processes lead naturally to [21] D  F(T), 3 ()  (3.10)  where F(T) is a slowly varying Debye function. This result is quite different from the previous ones and is closer to the observed behaviour of muonium in insulators and semiconductors (recall table 1.1). At sufficiently low temperature, any interstitial impurity will eventually exhibit quantum diffusion. For all but the lightest atomic impurities, however, the diffusion rates are very slow. The mass of the muon is such that it exhibits fast diffusion in  Chapter 3. Diffusive Phenomena and Muon Spin Dynamics  22  the quantum regime; moreover, the SR technique allows detailed measurement of the diffusion (see  §  3.3).  3 behaviour of the muonium diffu In an effort to explain the approximately Tsion constant in insulators and semiconductors, the above single phonon results were used [21]. A numerical error in [21], however, has thrown doubt on the applicability of this result to Mu. Currently, the most succesful theory [35] for Mu quantum dif fusion in insulators balances the effects of “dynamical band destruction” and “barrier preparation”, using phonon dispersion curves from neutron scattering experiments, to match the T behaviour over the appropriate T regime. Moreover, a recent report of 57 behaviour in solid Nitrogen [5], tends to favour this approach. T  3.3  The Effects of Muon Motion on Its Spin Dynamics  In this section the influence of motion of the muon on SR spectra is discussed. The most important dynamically dependent feature of tSR spectra is the spin relaxation, so a theoretical connection between the relaxation and the motion is required for quanti tative estimation of parameters characterizing the motion. It should be noted, however, that ,uSR provides no way to distinguish motion of the muon from time variation in the local field experienced by it. Other experimental information (such as NMR data) is required to eliminate the latter possibility and, thereby, establish the link between the muon’s spin relaxation and its motion. Muonic spin depolarization is most often due to stochastic interaction between the magnetic moment of the muon and electronic and nuclear moments in its surround ings. Some examples of such processes are collision with charged excitations, magnetic dipolar interaction with randomly oriented nuclear spins, and nuclear hyperfine inter action between the Mu electron and nearby nuclear spins. Naturally these interactions  23  Chapter 3. Diffusive Phenomena and Muon Spin Dynamics  depend on motion of the muon. For random hopping motion, as presented in relevant parameter is  Thop.  §  3.1, the  At low temperature where quantum motion is important,  the parameters characterizing the motion are the muon(ium) bandwidth A and  Tfree  the mean free time between collisions with phonons, impurities, charge carriers, etc.. 3.3.1  Zero Applied Field  In zero field, assuming an isotropic Gaussian distribution of local fields (due to static nuclear moments within the crystal) characterized by width u, we have for the static muon case the semi-classical Kubo-Toyabe relaxation function, Pz(t)  + (1  =  —  2 The distinctive feature of this function is that it attains a minimum at t returns to an asymptotic value of  ,  (3.11)  /2. t 2 u2t2)e_J  =  then  corresponding to the average fraction of the spin  polarization which is energetically locked along the direction of the local field compo nents parallel to the quantization axis. flayano et al. [36] analyzed the zero-field case in the presence of hopping motion of the muon; they found that the  tail is suppressed even for slow hopping. Their  analysis was based on a “strong collision” model in which it is assumed that: • The magnetic field experienced by the muon changes abruptly at the time of a hop. • The time between hops,  Thop,  is much longer than the time taken for a hopping  transition. • The field after a hop is independent of the field before the hop, so the sequence of fields is uncorrelated.  24  Chapter 3. Diffusive Phenomena and Muon Spin Dynamics  Such a model, particularly the last assumption, may not be justifiable in certain cases. Firstly, if the return probability for a muon at a particular site is non-zero, then the sequence of fields will be correlated. Also if the initial and final sites of a hop are likely to share at least one neighbouring nuclear dipole, then the fields will also be correlated. This case has been discussed briefly by Celio [37]. The validity of the second assumption depends on the nature of the dynamics: defining the time for a transition to occur as Tmin  (problematic for the case of quantum tunnelling), the model is valid provided  Thop  is in a time range: (3.12)  Tmin <Th p <T,t, 0  where the 1 iSR signal is affected by the motion, but not so that the details of the transition dominate the depolarization. The “strong collision” model, when it applies, provides a means of calculating the dynamic relaxation function, Fz(t), from the static one, Ptat(t), even for the fully quantum mechanical spin interaction.  In general this requires, however, numerical  solution of the following implicit definition for Pz(t) [1] Pz(t)  =  It  e_t/ThopPtat(t) + Thop  P:tat(s)e_8/ThOPFZ(t  —  s)ds.  (3.13)  0  The result of using the quantum mechanical Psat(t)  in  the strong collision model  differs from the Kubo-Toyabe relaxation function importantly in the large t behaviour; the time-asymptote is replaced by oscillation due to flip-flop terms in the muon—nuclear dipolar interaction. Such differences are found to be most conspicuous for low-spin systems involving one or two nuclei [38]. 3.3.2  Transversely Applied Field  SR Consider a muon diffusing in a lattice containing unpolarized nuclear spins; the 1 signal undergoes motional narrowing as in NMR: a fast moving muon experiences the  25  Chapter 3. Diffusive Phenomena and Muon Spin Dynamics  randomly oriented fields at many sites, so the time-averaged field is small, reducing the depolarization. Before presenting the dynamic relaxation function, the phenomenon of motional narrowing may be neatly summarized by the result of a Redfield Theory calculation (section 5.12 of Slichter [39]), (3.14)  =  =+ where  -yb,  is the muon gyromagnetic ratio,  is the Larmor frequency of the muon in  the external field, H? is the second moment of the ith component of the magnetic field at the muon, and r 0 is a correlation time for the motion which here corresponds to the hopping time  Thop.  2 whose This expression gives the transverse relaxation time T  inverse is proportional to the width (in Fourier Space) of the muon frequency peak at Thus equation 3.14 trivially expresses motional narrowing: faster motion (smaller ) results in a larger 0 r  ‘2.  The two terms in this equation can be interpreted as follows:  the first term (1/Ti) is a consequence of dephasing of the muons’ spins due to variation ) represents the 1 of the z-component of the magnetic field, and the second term (1/T non-secular broadening, i.e. it is the effect of the finite lifetime of the spin eigenstate caused by transitions in the approach to equilibrium with the other degrees of freedom in the surrounding crystal. Returning to the relaxation function, Hayano et al. [36] find for this case a nuiner ically evaluated function (based on the “strong collision” model) that is well approxi mated by: R(t)  (3.15)  = e_j2Th2_t/T0_1+t/T0).  This function can be derived under the assumption of smooth time variation of the field. It interpolates analytically between the static and fast-motion limits:  TO  >> t  =  G(t)  26  Chapter 3. Diffusive Phenomena and Muon Spin Dynamics  is Gaussian, and 0 r << t  =  G(t) is exponential, repectively. Finally, it can be shown  [361 that the transverse field relaxation function is much less sensitive to motion than that of zero-field, making zero-field measurements the better choice for investigating diffusion. Alternatives to the “strong collision” model using Redfield and master equation theories of spin relaxation have been developed by Celio, Meier and Yen [40,411. They both replace interaction with environmental moments by a stochastically time-varying effective Zeeman interaction, H  =  (3.16)  6exe (t),  where i(t) is a vector whose components fluctuate randomly between two values +T, and 6 is a coupling constant. This treatment assumes the coupling with the lattice is weak relative to the energies of the uncoupled system, i.e. that the Hamiltonian 3.16 is a small perturbation of the free Hamiltonian. Furthermore r, the correlation time for the reservoir corresponding to  Thop  is assumed much smaller than the spin relaxation  time T . The stochastic character of the motion assumed in this treatment is evident in 1 the use of exponential correlation functions for the components of T(t). For muonium diffusing in a transverse applied field, this approach yields the polarization function Px(t)  where C.’jj are  the  =  [cos2 (coswi te_12t 2  +  3 2 sin t 4 e_14t (cosoi  +  muonium transition frequencies,  initial state, and the relaxation rates,  te_A34t) 34 + cos (3.17)  te23t)], 23 cosw  /3  is  the  mixing  angle specifying the  depend, in a complicated way, on  and /3; the exact expressions can be found in [41, pp 45-46].  ex, Tc, 6  27  Chapter 3. Diffusive Phenomena and Muon Spin Dynamics  3.3.3  Longitudinally Applied Field  In longitudinal field, the situation is similar to zero field, but because the total local field distribution is no longer isotropic (it is stretched along the z-direction), making the ‘conserved’ z-component of the initial polarization larger than of P’(t) is field dependent, i.e. it ranges from  ,  the time-asymptote  to 1 as o varies from 0 to oo. The  relaxation function in this case is [1] pz(t) = 1  —  (1  —  /2 t 2 e_  cosot) +  ‘0  wo  f  22 e /  rdT. 0 sinw  (3.18)  0  In weak longitudinal fields (WLF), it should be noted, the quantum mechanical effects mentioned for the zero field case are less important, so the WLF configuration is preferable since parameter estimation of  Thop  will be less sensitive to the model of  static relaxation [1]. The result of the Redfield theory calculation, as discussed in the previous section, for diffusing muonium, is pz(t) =  (t) 1 p  —  (t) 4 (t) + cos (2)[p 3 p  —  (t)} + 2 p  , t t)e24 24 sin (2)2 cos (w  where /3 is the mixing angle, and the oscillation frequency  24  (3.19)  is usually too high to  be measured. The functions p(t) have decaying exponential time dependence that is determined by a 4 x 4 relaxation matrix, R b(S, r, ,8, w(B)) [41, pp 46-47]. 3 3.3.4  Level Crossing Resonance  The effect of stochastic motion on level crossing resonances is generally a broadening and shallowing of the dips in P(B) due to disruption of the oscillation in spin polar ization that the muon undergoes at the LCR field. This effect can be seen, for instance, in figure 5.1 of [1]. Quantitative theories including such motion have been given, for neq LCR, by Kreitzman [42], for muonated radicals, by Heming et al. [43], and, in the  Chapter 3. Diffusive Phenomena and Muon Spin Dynamics  28  Redfield formalism, with the effective interaction 3.16, by Celio and Yen[41, p 57]. The results of these calculations, however, will not be discussed further here.  Chapter 4  Muonium in Copper(I) Chloride  This chapter is a review of experimental studies of muonium as an impurity in CuC1. 4.1  Properties of CuC1  CuC1 is a group I-Vu tetrahedrally co-ordinated binary semiconductor; its ZincBlende crystal structure is illustrated in fig. 4.1.  0  0  0  Figure 4.1: CuC1 crystal structure. The cube has side length 5.42A, and the Cu-Cl distance is 2.35A. The Copper atoms are shown here as small filled circles, the Chlorine as large circles, and the Copper interstitial T site is shown as a point. The properties of CuCl are unusual in many respects, as discussed in detail in the review article by Schwab and Goltzené [44] and references therein. Characteristics of 29  30  Chapter 4. Muonium in Copper(I) Chloride  I form. prob.  1116(4)  Mu 1 66(3)  MuH 9.9(8)  Missing 8(5)  Table 4.1: Formation probabilities in % for the various final states of into CuC1 at temperatures below lOOK, from [48].  +  implanted  CuC1 that may be responsible for the unusual properties of its muonium centres are the atypical electronic and acoustic properties: though it is a semiconductor, it is highly ionic, with ionicity near the critical value separating the tetrahedral compounds, like GaAs, from the octahedral compounds, such as KC1. Also Copper, a transition metal, has d orbitals which influence the band structure of CuCl; for instance, the uppermost 3 hybrid [45,46]. On this valence band is 3d/3p in character rather than the usual sp basis one might expect an unusual electronic structure for impurities. Secondly, the lattice—vibrational properties exhibit anharmonic effects such as a rapid increase in phonon damping with temperature [47]. In light of the success of theoretical models, involving use of the real phonon spectra, in explaining the temperature dependence of the muonium correlation time in KC1 and NaC1 [35], the details of the phonon spectrum may be important in determining the dynamics of muonium in CuCl.  4.2 4.2.1  Muonium in CuC1 High Transverse Field Measurements  Muonium in CuC1 was first observed in high transverse field (HTF) measurements [48]. These studies revealed two separate sets of spectral lines corresponding to two distinct . The 11 1 and Mu forms of muonium with isotropic hyperfine interaction, labelled Mu formation probabilities, determined by the precession amplitudes, are given in table 4.1.  Chapter 4. Muonium in Copper(I) Chloride  31  The hyperfine constants, determined by the frequency differences between two cor responding peaks, were found to be unusually small at roughly 30% of the vacuum value (see table 4.2). Furthermore, significant temperature dependence of these spectra was observed above  ‘-  lOOK. As the temperature was increased, the Mu 1 signal broadened  suddenly, while the Mu” signal increased in amplitude approaching the sum of the Mu’ and Mu 11 amplitudes at low temperature. This behaviour was interpreted as the result of a thermally activated transition from Mu 1 to Mu  The temperature  dependence of the hyperfine parameter, A, , was also strange in that it increased with 1 increasing temperature, opposite to the usual behaviour. Its temperature dependence was fitted using a Debye phonon spectrum, since a linear phonon dispersion is appropri  ate for the acoustic modes that dominate lattice vibration at low temperature. The fit was successful despite anharmonic effects that were expected to play an important role.  The Debye temperatures for both Mu 1 and Mu 11 from these fits (110(7)K for Mu’ and 130(1)K for Mu”) were reasonable compared to those found in heat capacity mea surements ([47] and references therein). Finally, analysis of the line widths and shapes determined by interaction with surrounding nuclear magnetic moments suggested that 1 was static, but reliable characterization of the nuclear hyperfine interaction would Mu have to wait for the LCR experiments (see section 4.2.2). The results mentioned above were then used in an attempt to deduce the locations of the two kinds of muonium in the lattice. The isotropy of A 11 suggested a site of high symmetry or motion fast enough to average out any local anisotropy. The tetrahedral or “T” sites were chosen as the candidate high symmetry locations. There are 4 distinct T sites in the CuC1 lattice: two substitutional sites with 4 nearest neighbour Cu (Cl) atoms at 2.35A and 12 Cl (Cu) next-nearest neighbours at 3.83A, and two interstitial sites with 4 nearest neighbour Cu (Cl) atoms at 2.35A and 10 Cl (Cu) next-nearest neighbours at 2.70A. Though the HTF measurements showed several combinations of  32  Chapter 4. Muonium in Copper(I) Chloride  these locations for Mu 1 and Mu to be unlikely, additional information on the nhf interaction from uLCR experiments would be required to unambiguously determine the sites for muonium in CuC1. It should be noted that, soon after the initial HTF experiments on CuCl, Cox and Symons [49] suggested the possibility that muonium in tetrahedrally co-ordilnated semiconductors could be centred at one of four antibonding sites along the interstitial T site—nearest neighbour nuclear directions and might tunnel locally among these sites (see also [50]). Modelling the muonic spin dynamics in this situation forms the core of this thesis. 4.2.2  LCR Measurements  Although little new information was gleaned from the initial LCR experiment [51] on powdered CuC1, it demonstrated that 1 iLCR on isotropic Mu in CuCl was feasible. The tLCR spectra of both Mu 1 and Mu 11 in single crystal CuC1 was reported soon thereafter [52,53], and became the first experiment to measure the details of the nuclear hyperfine interaction of isotropic muonium in a semiconductor. A reasonable fit to the LCR data could only be obtained if both Mu’ and Mu” were centred about the same interstitial T site surrounded by four Cu atoms. The parameters from these fits are reproduced in table 4.2.  The estimates of the spin  densities were obtained by comparing the measured parameter with the free atom values [54]. A further conclusion of these experiments is that neither form of muonium undergoes long-range motion on the timescale of r because such motion, in a lattice of unpolarized nuclei, would average the static component of the nhf interaction, required to observe an LCR, to zero.  33  Chapter 4. Muonium in Copper(I) Chloride  1 Mu Mu  A 1334.50(3) 1226.69(1)  P 30 28  S  Pc 57 60  Mu 1 CU 63 C1 35  A 1 14.66(3) 27.8(5)  A 1 150.9(1) 50(5)  Mu”j  A 1 -9.15(1) 24.5(5)  1A1 136.3(1) 46(5)  63 CU C1 35  S  Pci  Ptot  29 28  116 116  Q 6.21(3) 1.6(2)  Q 6.75(3) 1.7(2)  Table 4.2: pLCR results for both types of muonium in CuC1, from [53]. Hyperfine and quadrupole parameters are given in MHz. The muonium electron spin density fractions, p , are given in %. 5 The jiLCR measurements settled the debate on the sites of the two forms of muo nium in CuC1, but they did not explain why there should be two such similar forms, located at the same site, yet distinct. There were several hypotheses formulated to solve this problem: • Mu’ is an electronically excited state of Mu”. This, however, seemed unlikely because their hyperfine structures were nearly identical. . This also seemed unlikely since 11 • Mu 1 is a vibrationally excited state of Mu  Mu’ had a lifetime of at least a few microseconds. • Mu’ is accompanied by a metastable lattice distortion. Again the likelihood of such a state being metastable for microseconds seemed small. • Mu’ is a tunnelling excited state of Mu” as suggested in the previously men tioned tunnelling model of Cox and Symons. This hypothesis, though it could  34  Chapter 4. Muonium in Copper(I) Chloride  11 at explain many aspects of the results, required a slow conversion rate to Mu low temperatures to explain the low temperature metastability of Mu. This is discussed in more detail in  §  6.2.  The lack of a satisfactory explanation motivated the subsequent T, experiments that sought insight into the differences between Mu 1 and Mu” by measuring differences in their dynamical behaviour. 4.2.3  Longitudinal Field T 1 Measurements  The most recent experiments were longitudinal field measurements of Mu 1 [9,10] yield ing the spin relaxation rate. The observed T’ spin relaxation was attributed to fluc tuations in the nuclear hyperfine field due to motion of the muonium between sites with varying nhf contribution to the local field. The LCR results together with the longitudinal field quenching curves (F , the non-oscillating part of F(t), vs. B) es 0 tablished that neither form of muonium was undergoing long range motion which is expected to average the uhf field to zero. It was therefore concluded that Mu’ must be moving locally within the tetrahedral cage surrounding the Cu interstitial T site, and that Mu” must be stationary, at least on the  , 1 T  timescale. The results for T 1 and the  inverse correlation time r 1 are shown in fig. 4.2. 1 The T 1 values, obtained by exponential fits to the LF relaxation, exhibit the T minimum (arrows) when r 1 equals  12,  the smallest intra-triplet muonium frequency.  The decrease in the temperatures corresponding to T 1 minima with field indicates an increase in r 1 as the temperature decreases (because (B) 12 increases with B over v this range of fields). These results constitute relatively model independent evidence for local tunnelling motion of Mu in CuC1. The correlation times are extracted from the  35  Chapter 4. Muonium in Copper(I) Chloride  data using, as in equation 3.16, the effective Hamiltonian  6eSe T(t). Because the parameters ea 6  ex 8  1 minima were used to fix are confounded, the T  and  to a value which was assumed temperature independent. Fits using this model  were successful below the Mu’  —*  Mu” transition temperature (fig. 4.3), but they  deteriorated with decreasing temperature below  5K (fig. 4.4). Remarkably, below  1 has similar temperature dependence its minimum the r  (.  27 to that in KC1 T )  where the motion is thought to be long range. The departure of the data from behaviour predicted by Celio’s model was inter preted as a failure of the hopping picture at the lowest temperatures: the assumption of an incoherently varying field (with an exponential correlation function) does not hold at low temperatures where not enough phonons are available to destroy the coherent motion (with a harmonic correlation function), so below  5K it was conjectured that  the motion of Mu 1 was coherently quantum tunnelling. 1 at 3.3K revealed an intriguing peak at Futhermore, a field scan of T  1.5kG (see  fig. 4.5). It was thought that this might be the signature of a level crossing associated with a tunnel splitting in the motional Hamiltonian, but subsequent measurements showed that the apparent peak position varied with T and did not narrow at lower temperatures as expected. Finally, the LE data led to the latest explanation for the existence of two forms of muonium in CuCl. In [10], it is argued that because its low temperature motion 1 inhibits inward is much faster than the rate of lattice relaxation, the motion of Mu relaxation of the Cu atoms surrounding the interstitial T site, and this relaxed state . 11 with muonium at the T site is thought to be the high temperature stable form Mu In conclusion, the iSR experiments in Copper(I) Chloride, to the date of this thesis,  Chapter 4. Muonium in Copper(I) Chloride  36  have discovered some very unusual and interesting properties of muonium, but there remain unanswered questions that can only be addressed by further measurement. The major outstanding problem is to find a consistent explanation of the dynamical properties of the two muonium centres. It is the aim of this thesis, through the model desribed in the next chapter, to provide a theoretical basis on which to test the assertion that Mu 1 is exhibiting local quantum tunnelling below  5K.  37  Chapter 4. Muonium in Copper(I) Chloride  I  I  10  7  I  IIIII,  (°) C  -10  00  6  AC  I  F—  CC 5 io 000 III  I  I  III  I  I  iiliii  I  111111  I  I  iiilil  111111  I  1012 1011  0  E °  10  10  0  E  00  io  (b) °cocxb I  1  2  I  iiliii  I  I  iiilii  4 7 204080200 Temperature (K)  11 calculated assuming 1 and Mu 1 relaxation rates for Mu Figure 4.2: (a) Approximate T exponential depolarization (diamonds, triangles and circles correspond to fields of 1, 2, 1 minima; and (b) fit values for the inverse and 4 kG), the vertical arrows indicate the T , from [9]. 1 correlation time for Mu  38  Chapter 4. Muonium in Copper(I) Chloride  I  I  I  I  —F  I  I  I  I  0.12 0.08 0.04  +  0.00 I  0.0  0.2  I  I  I  0.6 0.4 Time (ps)  I  I  0.8  1.0  Figure 4.3: A successful simultaneous fit to the LF data (at 3 fields) using Celio’s model, from [10].  39  Chapter 4. Muonium in Copper(I) Chloride  .04 .02  0 >N  a-)  E E  > U)  -o  .075 .05 .025  C)  -b  0 C)  0  0  0  .1  B=2kC .06  -  .02  -  I  0  I  2  I  I  4  I  I  6  8  Time (ps) Figure 4.4: Failure in fitting the low temperature LF data to Celio’s model. The calculated relaxation rate is too high at the lowest field and too low at the highest field.  40  Chap ter 4. Muonium in Copper(I) Chloride  CuCI,  <111>IIB, T=3.3 K  7 6 U)  bJ  z  0 0  0.5  1  I  I  I  I  1.5  2  2.5  3  3.5  MAGNETIC FIELD (kG)  1 in Figure 4.5: An unexplained peak in the approximate relaxation rate T’ for M’u CuC1. Subsequent measurements found the position and shape of this feature to be temperature dependent.  Chapter 5  Tunnelling in a Tetrahedron  From the experimental work described in the previous chapter, we are led to a model for Mu 1 at low temperatures in which the muonium tunnels locally between four antibonding sites within a tetrahedral cage defined by four Cu nuclei. In this chapter we develop a model to describe the spin dynamics of the muon in this situation.  5.1  General Considerations  In this section some general assumptions for the model presented in the next section are discussed. Firstly, we will assume that Mu in CuC1 is a deep level impurity, so that only its electronic ground state is involved. For simplicity, we assume an isotropic muon hyperfine interaction term (see  §  2.2), Hhf  =  AS  .  (5.1)  Secondly, we assume that the dominant nuclear hyperfine (nhf) interaction for Mu localized at an anti-bonding (AB) site is with the single closest Cu nucleus, so we neglect interactions with the other nuclei. Clearly this becomes a poor assumption if the AB site is near the T site; furthermore, it should be noted that there are three Chlorine atoms nearer the AB site than the three remaining tetrahedral Copper atoms for most choices of the AB-T site distance. From the trigonal symmetry of the AB site, the nhf interaction is expected to be anisotropic (see table 4.2), in particular the nhf 41  Chapter 5.  Tunnelling in a Tetrahedron  42  term in the Hamiltonian should be axially symmetric, =  S’S’ + A(S’S’ + S’S’), 1 A  (5.2)  where the z’ axis points along the si-Cu axis. Again for simplicity this interaction will be replaced by an isotropic nhf interaction, leaving only AS = 0 transitions. Thirdly, both 63 Cu and 65 Cu have nuclear electric quadrupole (neq) moments which interact with the local electric field gradient caused by the presence of Mu, so, again assuming that only a single nucleus is important, there is a term in the Hamiltonian of the form, =  Q[(ñ.  )(n. )  where ñ is a unit vector directed from the  t  -  STh(S + 1)],  (5.3)  to the Cu. Since the neq term is only a  few percent of the average nhf term (see table 4.2), it will be neglected. Lastly, the motion of the muonium within the tetrahedral cage will be treated analogously to the tight binding method for electronic molecular orbitals. We assume that between any two different sites, the Hamiltonian has a tunnelling matrix element, A; so, the matrix of the spatial part of the Hamiltonian with respect to the site basis is -A-A-A 0 E (Htn)z3  -A  0 E  -A-A .  0 -A-AE  -A  -A-A-A  E  (5.4)  Without any loss of generality, E 0 is set to zero, and the eigenvalues of Htu are easily found to be E 1 = —3A and E 4 = A, the ground state ‘I’ being completely 3 , 2 symmetric, i.e. having equal amplitudes on all four sites. In the site basis (ek), the  Chapter 5. Tunnelling in a Tetrahedron  eigenvectors of Htn, denoted a =  43  k bek,  are (up to a phase) the columns of the  orthogonal matrix 1 1 2/  1 2  =  (5.5) Z7  Since accurate prediction of z requires detailed knowledge of the shape of the potential felt by Mu at the AB site potential minimum, it remains a free parameter. By invoking the tight binding method, we have assumed that the muonium zero point energy in each of the four wells doesn’t exceed the potential barrier between wells. It should be noted that the presence of random strain fields or an externally applied stress will break the tetrahedral symmetry of the tunnelling Hamiltonian and, conse quently, the degeneracy of the triplet. Inclusion of such effects, however, is beyond the scope of this work.  5.2  The Model Hamiltonian  The full Hamiltonian must contain terms for the Zeeman interaction with an external field and the interactions between the spins of the system, and it must incorporate the tunnelling Hamiltonian, so we have, for B applied along the = HhYP  eSB 7  —  SB 711  —  = ASeS+An sites  >  axis (LF),  SB sites i Ei(SeSni),  (5.6) (5.7)  i  and the total Hamiltonian is Htot = Htt + Hzeem + HhYP.  (5.8)  Chapter 5.  44  Tunnelling in a Tetrahedron  Crossing 1 2 3 4 5  6  Mu frequency V3  1134 1112 1114  Levels mem ) 1  ++), ——) +>, ZZ)_  - ——),  ++), +—) ++), —+) +—), —+)  Table 5.1: Muonium frequencies corresponding to the crossings of two Breit—Rabi di agrams. The crossings enumerated in figure 5.1 correspond to matching a muonium frequency, Vjj, to the tunnel splitting, 4zS. E is a spatial projection operator that extracts the probability amplitude for Mu on site i, and we have neglected the small muon—nuclear coupling. We choose the following basis a(i) i) = j1 9  me,m,,mn1,mn2,mn3,mn4>,  i = 1,2.. .d,  (5.9)  where d, the dimension of the Hubert space, is 16(2S + 1). The spectrum of Htot as a function of the applied field is basically two copies of the 11 displaced in energy by 4, with the A Breit-Rabi levels (fig. 2.3), with appropriate , upper copy being triply degenerate (fig. 5.1). Of course, on a finer scale, these levels are split by the nuclear spins. From the figure it is clear that there exist fields B where the levels cross, and the fact that the nuclear hyperfine interaction part of Htot is diagonal in the site basis, implies that it has non—zero matrix elements between the different tunnelling states, so mixing may occur. These crossings shown in the figure correspond, as in table 5.1, to one of the six distinct muonium transition frequencies (numbered as in fig. 2.3) matching the tunnel splitting. For crossings that mix different muon spin states, then, one expects to see level—crossing resonant dips in the integrated muon spin polarization. Since, for  Chapter 5.  45  Tunnelling in a Tetrahedron  Splitting  =  1.1A  9-  0 0)  9-  C 0 >.. .2 C, S.-  ci C  uJ  t’1agnetic Field (Units of A/t)  9-  0 C!) 9-  C  >‘ L.  U)  C  w  Figure 5.1: The six types of intersections of two Breit—Rabi diagrams shifted by various tunnel splittings, 4z\ (given in terms of the muonium hyperfine parameter A), and =  —  7e.  46  Tunnelling in a Tetrahedron  Chapter 5.  any two spins S 1 and 52, [(5z + Sq), 1 S  =  , S] = 0, and S should be ° t 0, [H  conserved. The crossings shown in the figure that appear to disobey this (numbers 2,3,4,5), are in fact mixed, at slightly different field, when the nuclear levels are taken into account. Though Htot does not include direct muon—nuclear flip—flop terms, the mixing is mediated by the electron spin, as in the high field approximate Hamiltonian of  §  2.2.3.  The electron spin remains a good quantum number, if the transition frequencies to a level with different electron spin are large compared to the transition frequency of the avoiding levels. The crossings 3 and 4 are therefore allowed. The crossing 1 is forbidden, however, because H’ supports only /-S  =  0 transitions, and includes  only on—site interaction (with a single nucleus). Crossings that mix only the electron states (2,5), though allowed, have little effect on the muon spin for fields where the electron and muon are effectively decoupled. In the next chapter, results of calculating the muon spin polarization as governed by this Hamiltonian are presented. 5.2.1  High Field Approximation  In high fields, the electron spin can be removed from consideration using the high field effective Hamiltonian of  §  2.2.4. This reduces the dimension d by a factor of 2,  leading to a factor of 8 increase in computing time (since the slowest loop in calculating the muon polarization is d ). For the electron spin down manifold, we have, using 3 [Htumm,  .] 1 H  _  =  [Hem, UThn t H  ]  0,  + man +  (  Hh  )  HhY —  Hh  ),  (5.10)  With respect to the basis 5.9, H 1 has matrix elements  Ii) 1 (iIH  =  u; V +1  (5.11)  Chapter 5.  47  Tunnelling in a Tetrahedron  the diagonal part being = tT’  —  3 + (Z, Ze5  —  3 + Aj(a)  —  Aflt3))(ok)u +  (5.12) and the interaction part being =  AA  (a(l)  ( )ü( )1 +  a(i)a(l)  (u )il (k )1) +  4  (5.13)  zXT is basically the diagonal version of the matrix 5.4, and Z are the appropriate , the Hamiltonian in the electron spin up T Zeeman energies. The matrix elements for H manifold, are of a similar form. This formalism could be used to model high field muon flip transitions within an electron spin manifold such as the high field crossings 3 and 4.  5.3  Calculation of The Muon Polarization Function  The observable of interest is the muon polarization, P(t) =  (5.14)  Tr(p(t)oj,  where p(t)  =  (5.15)  For the initial density matrix po, we assume that the muon is initially 100% spin polarized in the  z  direction and that the other degrees of freedom are equally populated,  i.e. po = N Trnu)(Trnu 0’ N -(I.+o)®I, = -  (5.16)  48  Chapter 5. Tunnelling in a Tetrahedron  where N is a normalization factor and I the identity operator for all degrees of freedom except the muon spin. At temperatures comparable to the tunnel splitting (4), how ever, this initial state is inappropriate. If the local tunnelling system has time to reach thermal equilibrium, a Boltzmann factor for the relative populations of the tunnelling states should be employed. If for some reason one tunnelling state is preferentially populated and does not relax to thermal equilibrium on the timescale of general relative population factor,  ,  , 1 T  then a  should be used. Including the time dependence  explicitly, the polarization is, in general, P(t)  (5.17)  p(u)  = states  where  =  i>j  pt(o)iJ, w are the transition frequencies 2  —  3 and all matrix ), 3 H  elements are calculated in the eigenbasis of H. For the initial state 5.16, this takes an even simpler form: P(t)  cos  [(aj] + 2  = states  i  (5.18)  i>j  Of interest in an integral RSR measurement is the integrated muon polarization, 1 which, using equation 5.17, becomes = I  [  pa +R 1 states  i  i>j  +(W )2].  (5.19)  For the high field effective Hamiltonian, assuming an unpolarized muonium electron, the corresponding observables are just an average of those calculated for the electron spin up and spin down subspaces.  5.4  Finite Lifetime Damping  One expects the energy levels of the Hamiltonian of section 5.2, due to interactions with lattice excitations, to exhibit a finite, temperature dependent lifetime. This may  Chapter 5.  49  Tunnelling in a Tetrahedron  be heuristically included in the model by making the replacement —*  —  il’(T),  which has the following effect on the observables: P(t)e_T’(T)t,  (F)’(t)  (5.20)  and  (Ps)’  =  T  --  =  J°°  e_T(TtP(t)dt  (5.21)  0  _i. £(P(t))I + 3  (5.22)  As an estimate for the effect of the finite lifetime on the muon spin polarization, the linewidth F(T) was calculated as in [21]. This lifetime applies rigorously to the tun nelling levels of a two well system and includes first order phonon interaction rigorously, but with temperature dependent couplings, can include all orders of phonon interaction. I’(T) depends, however, on an unknown dimensionless coupling constant, y, between the system and the phonon bath, which must be left as a free parameter. calculated results presented in the next chapter, this effect has been neglected.  In the  Chapter 6  Results, Discussion and Conclusion  6.1  Results and Discussion  In this section, the results of the calculations laid out in Chapter 5 are presented and discussed. To accurately find the resonances predicted in  §  5.2, a program was written to  calculate F for local quantum tunnelling among four wells (from here on the z will be dropped). To make computing time reasonable, the nuclei were assumed to be spin 1/2 rather than 3/2 (d  =  256 rather than d  =  4096). The main effect of this assumption is  that the iLCR triplet (corresponding to the three /S  =  +1 transitions) is collapsed  into a single resonance. The gyromagnetic ratios used are those given in table A.2, and the hyperfine parameters used were  =  1334.00MHz and A  =  60.07MHz.  1  Unless  otherwise stated, the initial state is 5.16, i.e. unpolarized nuclei and electrons, and equally populated tunnelling states. Some computed resonances are shown in fig. 6.1. For comparison, the crossing field values, from the intersections of two Breit—Rabi diagrams are given in the figure. For values of the splitting where a crossing is very shallow, i.e. the difference in slopes is comparable to the nuclear gyromagnetic ratio, the resonances are broad and the estimated crossing fields are not very good predictions of the resonance positions. The splitting in the resonances 6 (bottom panels of 6.1) is due to the nuclear hyperfine This value is the average A 1  =  +  see table 4.2  50  Chapter 6. Results, Discussion and Conclusion  A=190 MHz, 4 Wells  51  A=140 MHz, 4 Wells  1.00  1.00  =O.398 kG 2 B B = 3 2.999 kG =3.829 kG  B = 2 O.309 kG =l.923 kG 4 B =5.933 kG  0.95  0.95  0.90  0.90 1.  17  1:9  2.1  2.3  2.  O2:22:42:62:8  B (kG)  B (kG)  A=375 MHz, 4 Wells .65 .60  .50 .45 .40  z=lO GHz, 4 Wells  A=25 GHz, 4 Wells  E  T  0.97  0.97  6  0.96 0.95 35.495  =35.508 kG 6 B 35.500  35.505  I  I  35.510  35.515  B (kG)  35.520  0.96  =14.196 kG 6 B  0.95 14.18  14.19  I  14.20  14.21  14.22  B (kG)  Figure 6.1: Predicted tunnelling resonances for muonium locally quantum tunnelling among four wells.  Chapter 6. Results, Discussion and Conclusion  52  splitting. These resonances are sharp (width < 5G) because they involve levels differing in electron spin, so they are approaching at Other resonances, corresponding to the muonium + one nucleus, are also contained in this model. For large tunnel splittings, these resonances are, as in fig. 6.2, indepen dent of A. As A  —  0, though, they bifurcate.  The program was also used to calculate the time dependent muon spin polarization, an example for an on—resonance field is shown in fig. 6.3. This routine could be used to fit LF time differential data in future experiments. As a check, a program was also written for the analogous case of two wells. Some results corresponding to those for four wells are shown in fig. 6.4; note that the two well splitting is 2A rather than 4A. The effects of changing the relative population  i  of the tunnelling states is shown  in fig. 6.5. These results show that tunnelling resonances should exist for the local tunnelling of muonium purported in CuC1. They also confirm that local quantum tunnelling motion does not wash out the resonances originating from interaction with a single nucleus, as, for example, long range motion does. Although Cox’s suggestion [11] that the two forms of muonium correspond the tunnelling singlet and triplet provides no explanation for their different dynamical be haviour,  2  the resonances predicted above may provide a more direct test: if the model  is correct, the predicted tunnelling resonances should be found for both muonium cen tres and should occur at the same fields. Furthermore, the low temperature formation probabilities (table 4.1), should give the correct weighting of these two resonances. If, however, the model of Schneider et al. [10] is right, then the resonances should appear The dynamics could be due to interaction between within the upper tunnelling triplet states, if their 2 degeneracy is broken.  53  Chapter 6. Results, Discussion and Conclusion  t=1O GHz, 4  Wells  =2O  GHz, 2 Wells  O.96  B (kG)  O.96  B (kG)  B (kG)  Figure 6.2: Resonances from interaction with a single nucleus, corresponding to cross 0, these ings within a Breit—Rabi diagram which includes the nuclear levels. As z\ resonances split. Also they are shifted relative to the single nucleus position of 51.118 kG. —  Chapter 6. Results, Discussion and Conclusion  54  t=1O GHz, 4 Wells, B=14.196 kG 1.0  0.8 Z06  0.4 0.2  0.0  1.00  0.90  Time (,us) 11 for local quantum P Figure 6.3: Predicted time dependent muon spin polarization, (t), 1 in CuCl. The top panel also shows the muon decay function, N(t). tunnelling of Mu The field is on the resonance 6 for this tunnel splitting, see first panel of fig. 6.1. The step size for this plot is 5Ons.  Chapter 6. Results, Discussion and Conclusion  55  1.00  0.95  0.90  0.85  0.80  A=750 MHz, 2 Wells  0.95  F61.269Z  68.0  68.1  68.2  68.3  68.4  6 .5  B (kG)  .76  B (kG)  14.20  B (kG)  Figure 6.4: Predicted tunnelling resonances for muonium locally quantum tunnelling in two wells.  Chapter 6. Results, Discussion and Conclusion  L=25 0Hz, 2 Wells, T=oo  56  A=25 GHz, 2 Wells, T=1OK  O.95•  B (kG)  B (kG)  =25 0Hz, 2 Wells, T=5K  A=25 0Hz, 2 Wells, T=1K  O.95  0.90 17.74  (J.90  I  17.75  17.78  17.74  B (kG)  I  17.75  17.76  B (kG)  A=25 GHz, 2 Wells, TO.O2K  =25 0Hz, 2 Wells, T=—1.265K  Inn,’.  -  0.95  0.90  17.74  17.75  B (kG)  17.76  17.74  17.75  17.76  B (kG)  Figure 6.5: The effects of different initial tunnelling state populations on the tunnelling resonances for muonium tunnelling between two levels. The negative temperature in the last panel corresponds to a relative population equal to the ratio of formation 1 and Mu 11 (table 4.1). probabilities for Mu  Chapter 6. Results, Discussion and Conclusion  57  for Mu 1 oniy, and probably oniy below 5K (where Celio’s model fails), and should disappear at higher temperature. The resonances predicted in this thesis, thus, suggest a means to experimentally test these theories, provided the tunnelling matrix element, A, is such that the fields for these resonances are attainable. If the failure of Celio’s model is due to local quantum tunnelling, then its appearance below 5K, suggests that A might be on the order of a few K, corresponding to A  10GHz. For such a A,  the only tunnelling resonance that could be observed would be the exceedingly narrow resonance 6 (bottom of fig. 6.1).  6.2  Conclusion  In conclusion, the results of the calculations of Chapter 5 imply that if muonium is undergoing local quantum tunnelling motion at low temperatures, there should exist one or more resonances corresponding to a spin system energy matching the tunnel splitting.  Such a resonance would be a direct measure of the muonium tunnelling  matrix element, A, and the first direct tLCR measurement of a tunnel splitting in a solid.  Bibliography  [1] G.M. Luke, Ph.D. Thesis, University of British Columbia, 1988 (unpublished). [2] R. Kadono et al., Hyp. Tnt. 31, 205 (1986). [3] 0. Hartmann et al., Phys. Rev. B37, 4425 (1988). [4] S. Fujii, J. Phys. Soc. Jpn. 46, 1843 (1979). [5] V. Storchak et al., forthcoming. [6] R.F. Kiefi et al., Phys. Rev. Letts. 62, 792 (1989). [7] R. Kadono et al., Phys. Rev. Letts. 64, 665 (1990). [8] R. Kadono et al., Hyp. Tnt. 64, 635 (1990). [9] J.W. Schneider et al., Mat. Sci. Forum 83-87, 569 (1992). [10] J.W. Schneider et al., Phys. Rev. Letts. 68, 3196 (1992). [11] S.F.J. Cox, J. Phys. C 20, 3187 (1987). [12] A. Schenck, Muon Spin Roiation Speciroscopy, Bristol: Adam Huger Ltd., 1985. [13] T.D. Lee and C.N. Yang, Phys. Rev. 104, 254 (1956). [14] C.S. Wu et al., Phys. Rev. 105, 1413 (1957). [15] R.L. Garwin et al., Phys. Rev. 105, 1415 (1957). [16] J.H. Brewer, K.M. Crowe, F.N. Gygax, A. Schenck in Muon Physics vol.3. V.W. Hughes, C.S. Wu eds., New York: Academic Press, 1975. [17] R.F. Kiefi and S.R. Kreitzman, forthcoming in Perspectives in Meson Science T. Yamazaki, K. Nakai, and K. Nagamine eds., Amsterdam: Elsevier Science. [18] S.R. Kreitzman and E. Roduner, forthcoming in J. Chem. Phys.. [19] J. Kondo, Hyp. Tnt. 31, 117 (1986). [20] Yu. Kagan and M.I. Klinger, J. Phys. C7, 2791 (1974). 58  Bibliography  59  [21] P.C.E. Stamp and C. Zhang, Phys. Rev. Letts. 66, 1902 (1991). [22] A. Abragam, C.R. Acad. Sci. Paris Ser. 2 299, 95 (1984). [23] A.S. Davydov, Quantum Mechanics, Oxford: Pergamon, 1976. [24] R.F. Kiefi et al., Phys. Rev. A34, 681 (1986). [25] S.R. Kreitzman et al., Phys. Rev. Letts. 56, 181 (1986). [26] R.F. Kiefi et al., Phys. Rev. Letts. 58, 1780 (1987). [27] S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943). [28] K.G. Petzinger, Phys. Rev. B26, 6530 (1982). [29] R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II, Berlin: Springer— Verlag, 1978. [30] A.F. Andreev and I.M. Lifshitz, Soy. Phys. JETP 29, 1107 (1969). [31] A.F. Andreev in Frog. in Low Temp. Phys. vol.8. D.F. Brewer ed., Amsterdam: North-Holland, 1982. [32] K. Yamada, Prog. Theor. Phys. 72, 195 (1984). [33] C.P. Flynn and A.M. Stoneham, Phys. Rev. B1, 3966 (1970). [34] Yu. Kagan and M.I. Klinger, Soy. Phys. JETP 43, 132 (1976). [35] Yu. Kagan and N.V. Prokof’ev, Phys. Letts. A150, 320 (1990). [36] R.S. Hayano, Y.J. Uemura, J. Imazato, N. Nishida, T. Yamazaki, R. Kubo, Phys. Rev. B20, 850 (1979). [37] M. Celio, Hyp. Tnt. 31, 153 (1986). [38] M. Celio and P.F. Meier, Phys. Rev. B27, 1908 (1983). [39] C.P. Slichter, Principles of Magnetic Resonance, Berlin: Springer-Verlag, 1990. [40] M. Celio and P.F. Meier, Phys. Rev. B28, 39 (1983). [41] fl.K. Yen, M.Sc. Thesis, University of British Columbia, 1988 (unpublished). [42] S.R. Kreitzman, Hyp. Tnt. 31, 13 (1986). [43] M. Heming et al., Hyp. Tnt. 32, 727 (1986).  Bibliography  [44] C. Schwab and A. Goltzené, Prog. Cryst. Growth Charact. 5, 233 (1982). [45] A. Zunger and M.L. Cohen, Phys. Rev. B20, 1189 (1979). [46] A. Goldman, Phys. Stat. Sol. B81, 9 (1977). [47] B. Prevot et al., J. Phys. ClO, 3999 (1977). [48] R.F. Kiefi et al., Phys. Rev. B34, 1474 (1986). [49] S.F.J. Cox and M.C.R. Symons, Chem. Phys. Letts. 126, 516 (1986). [50] S.F.J. Cox et al., Hyp. Tnt. 64, 603 (1990). [51] J.W. Schneider et al., Phys. Letts. A134, 137 (1988). [52] J.W. Schneider et al., Phys. Rev. B41, 7254 (1990).  [53] J.W. Schneider et al., Hyp. Tnt. 64, 543 (1990). [54] J.R. Morton and K.F. Preston, J. Magn. Reson. 30, 577 (1978).  60  Appendix A  Note on Signs and Units  In this thesis the following sign convention for the Landé g-factor will be adopted: g  2.002319,  (A.1)  +2.002332.  (A.2)  =  so, =  The magnetic moments are then given by, (A.3)  =  where the magnetons f.t are given in table A.1. The gyromagnetic ratios are (J/T)  Magneton  Bohr Muonic Nuclear  Ieh/2mec Ieh/2mc  9.2741 10_24 4.4906• 10_26  eh/2mc  5.0508 1027  Table A.1: Magnetons (note the muonic magneton occupies an intermediate order of magnitude). Ii  gjLj  is negative for the electron (moment opposite to spin) and positive for the  (A.4) (moment  parallel to spin). Values of the gyromagnetic ratios relevant to this thesis are given in table A.2. 61  Appendix A. Note on Signs and Units  7e 7L  7Cu  62  (MHz/kG) 2802.4706 13.5537 1. 1533  Table A.2: Gyromagnetic ratios (note a factor of 2w). The value for Copper is an isotopic average, see table A.3. Nucleus Cu 63 Cu 65 C1 35  Spin 3/2 3/2 3/2 3/2  Moment () 2.22 2.38 0.82 0.68  I  Abundance 69.1 30.9 75.5 24.5  (%)  Table A.3: Isotope Data for CuC1. With the above sign conventions we have, from equation 2.4, the hyperfine param eter, A  j1O/L1/L2p(0),  =  (A.5)  which has dimensions [A]  so the term AS  .  =  2 T2[o]_1m3h  =  ,  (A.6)  has units of energy.  Commonly in the hyperfine interaction  J  literature (and in this thesis) the spin operators are made unitless, and A is given in energy or frequency units.  

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