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Muonium in C₆₀ fullerites : studies of structural and electronic properties Duty, Timothy Lee 1993

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MUONIUM IN  C60  FULLERITES: STUDIES OF STRUCTURAL AND ELECTRONIC PROPERTIES By Timothy Lee Duty  B. S. Virginia Polytechnic Institute and State University, 1990  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in THE FACULTY OF GRADUATE STUDIES PHYSICS We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA April 1993 © Timothy Lee Duty, 1993  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 Physics The University of British Columbia 1956 Main Mall Vancouver, Canada  Date:  /)gi  //73  Abstract  ,uSR studies of crystalline  C60  reveal the existence of two distinct paramagnetic states.  The experiments described in this thesis confirm that one is an exohedrally bonded muonium radical, while the other is endohedral muonium with a hyperfine parameter  At,  close to that of muonium in vacuum. The signal from the muonium-C60 radical—which is characterized by a small  At,  (10% of A7) and an anisotropic hyperfine interaction—is  sensitive to the molecular dynamics and is used to study the structural phase transition of solid  C60  fullerites  near 260K. Only endohedral muonium is observed in the alkali-metal-doped  K4 C60, K6 C601  and Rb6C60. From its coherent spin precession, we find all three  to be semiconductors with small gaps on the order of 0.5 eV. Our results conflict with the simple band structure model of doped fullerites indicating that electron-electron correlation effects may be important in determining the electronic structure of these solids.  11  ^  Table of Contents  Abstract^  ii  List of Tables^  v  List of Figures^  vi  Acknowledgement^ 1  Introduction  2  C60  vii 1  Fullerene and Fullerites  ^2.1^The  C60  4  Molecule ^  2.2^Crystalline  C60  and AxCso  4 6  ^  2.2.1^Crystal Structure ^ 2.2.2^The Phase Transition and Orientational Dynamics of Solid  3  6 C60  10  2.2.3^Electronic Structure ^  11  2.3^Superconductivity in A3C60 ^  13  ,uSR and Muonium Spectroscopy  15  ^3.1^Introduction ^  15  3.2^Basic Principles of Muon Spin Rotation ^  17  3.2.1^Parity Violation in Weak Decays ^  17  3.2.2^The Experimental Technique ^  21  3.3^Muonium ^  25  111  3.3.1 Hyperfine Levels of Muonium in Vacuum ^  25  3.3.2 Zero-Field and Transverse-Field ,uSR of Isotropic Muonium^27 3.3.3 Anisotropic Muonium ^  31  3.4 Spin Relaxation of Muonium: Spin Exchange with Free Carriers ^34 3.4.1 Spin Exchange Relaxation ^  34  3.4.2 Temperature Dependence of the Spin-Exchange Rate ^ 38 4 Results and Discussion ^ 4.1 Muonium States in Solid  C60 ^  4.2 The First-Order Phase Transition in 4.3  C60  C60 ^  in Zero Field ^  4.4 Endohedral Muonium in the Fullerites ^  41 41 44 50 53  5 Conclusion^  62  Bibliography^  64  iv  List of Tables  4.1 Muonium formation fractions and parameters E9, a, and A0 from fit of the K4C60, K6C60, and Rb6C60 data ^  V  61  List of Figures  molecule ^  2.1  Models of the  2.2  One electron energy levels of the  2.3  Crystal Structure for A3C60, A6C60, and A4C60 ^  3.1  Angular decay distribution of positrons  3.2  Energy spectrum and asymmetry factor of decay positrons ^  20  3.3  Typical Setup for Tranverse Field iLSR ^  22  3.4  Spin precession of the ,u+ in a transverse field ^  24  3.5  Breit-Rabi diagram of vacuum muonium ^  28  3.6  Breit-Rabi Diagrams for isotropic, axially symmetric, and completely  Cal  C60  5  molecule. ^  ^  anisotropic muonium ^ C60  4.2  Frequency spectra near 1/12 of C60Mu"  4.3  Temperature dependence of T2-1,  4.4  Zero field spectrum of C60Mu*  4.5  Temperature dependence of A in zero applied field  4.6  Temperature dependence of the linewidth 7Y1 of the TF-,uSR signal from  in 100G at 5K ^  MuaC60 in pure  C60  9 19  33  4.1  42  ^  A, and 1-1  7  ^  ^ ^  in an applied field of 100 G ^  4.7  ,uSR frequency spectra for K4C60 and K6C60.  4.8  Temperature dependence of the T2-1- linewidths for K4C60, K6C60, and  ^  46 48 52 54  55 57  59  RbsCso . ^  vi  Acknowledgement  I'd like to thank my advisor Jess Brewer for his support and patience, and for giving me the freedom to choose this interesting and timely topic for my thesis. It has been a pleasure to work with Rob Kiefl. I thank him for the many discussions and experimental collaboration from which I have learned a great deal about [tSR, muonium, and condensed matter physics. I would also like to acknowledge the efforts and contributions of the other E658 collaborators, especially Jiirg Schneider, Andrew MacFarlane, and Kim Chow. Special thanks to Curtis Ballard and Keith Hoyle for the technical expertise and advice. The high quality and well characterized samples were provided by Jack Fischer and his colleagues at the University of Pennsylvania.  vii  Chapter 1  Introduction  The recent discovery of the C60 molecule[1] and a simple technique for producing macroscopic quantities of this new form of carbon[2] opened a new chapter in condensed matter physics. Alongside diamond and graphite, crystalline C60 is a third form of solid carbon. Due to the nearly spherical symmetry of the constituent Co molecule, this solid and other C60 fullerites exhibit very interesting physical properties. Exploring the properties of this unusual class of solids has become an exciting endeavour encompassing the whole range of condensed matter techniques, including LtSR. In fact, the richness of the data obtained by iuSR may be attributed to the occurrence of paramagnetic muonium states in these solids.[3] The somewhat inadvertent discovery of the C60 molecule resulted from an interdisciplinary effort to solve a long-standing mystery in astronomy. [4] Spectroscopy of visible light from stars reveals a series of more that 40 absorption features known collectively as diffuse interstellar bands. None of these are unambiguously associated with any particular origin, although carbon clusters have been a suspect since the 1960's. Hoping to observe similar spectra in the lab, researchers set about trying to produce such carbon clusters using laser ablation and arc-burning of graphite in a helium atmosphere. The group led by Richard Smalley of Rice University along with Harold Kroto of the University of Sussex found that by adjusting the timing of the laser pulses and the pressure of the helium atmosphere, they could produce a preponderance of the Cso molecule. Shortly thereafter, they proposed the structure of the molecule to be that  1  Chapter _I. Introduction^  2  of the truncated icosahedron, the symmetry exhibited by the well known soccer ball. Accordingly, the molecule was named buckminsterfullerene, or `Buckyball' for short, after the architect Buckminster Fuller and his geodesic domes which brought him fame. The observation by Kratchmer et al. that the  C60  molecule was readily soluble in  bezene provided the means to separate it from the rest of the carbon soot produced in the arc-burning process which they pioneered.[2] Thus the first macroscopic amounts of  C60  were synthesized as a brand new form of solid carbon. This led to a rapid  growth in research on the properties of the solid. Two interesting phenomena, both of which have been studied using /./SR spectroscopy, motivate the experiments described in this thesis. The first is the existence of a first-order phase transition in the pure solid, and the second is the occurrence of superconductivity at relatively high transition temperatures in alkali-metal-doped  C60  fullerites.[6]  The first order phase transition was initially observed via X-ray diffraction,[5] and subsequently by many other techniques. Formation of a paramagnetic muonium radical, bonded exohedrally to the  Cat  molecule, allows us to study this transition and the  underlying molecular dynamics. The nature of this transition is now well understood. The alkali-metal-doped fullerites, AxC60, whre A refers to an alkali metal, are formed by exposure of pure  Cal  powder to the alkai-metal vapour. When doped to the com-  position A3C60, metallic and superconducting behaviour results. The superconducting transition temperatures of 18K for K-doped  C60  and 29K for Rb-doped  C60  samples are  remarkably high and are surpassed only by the cuprates. These fullerites contrast with the related superconducting graphite intercalation compounds such as KC8, which have transition temperatures below 1 K. The mechanism of superconductivity in these solids is not presently well understood and remains an outstanding issue in the forefront of condensed matter physics. The A3C60 phase is not the only fullerite produced upon exposure to alkali-metal  Chapter 1. Introduction^  3  vapour. Stable phases with the stoichiometry A4 C60 and A6 Co can also be synthesized. We will focus on the electronic properties of these non-superconducting phases. They are studied via iuSR of a paramagnetic muonium state which is determined to be endohedral (inside the buckyball). It is hoped that these results will shed light on the underlying band structure model appropriate for the general description of these fullerites and central to any specific model of superconductivity in A3C60.  Chapter 2  C60  2.1 The  C60  Fullerene and Fullerites  Molecule  The unique properties of this third form of solid carbon stem from the underlying structure of the constituent  C60  molecule (Fig. 2.1a), which is composed of twenty hexagons  and twelve pentagons arranged in a truncated icosahedron with a hollow centre. Three of the four valence electrons of each carbon atom are involved in covalent bonds with the neighbouring carbons. By analogy with graphite these are called a-bonds while the fourth electron is ascribed to a 7r-like orbital. Due to their non-planarity, however, the hybridization of the a-orbital is such that it no longer contains all of the 8-orbital character while the 7r-orbital is no longer of purely p-orbital character.[7] The a-bond hybridization in  C60  is somewhat between the hybridizations of graphite (sp2)  and diamond (89). Within the icosahedral symmetry, the buckyball actually contains two types of C-C bonds (Fig. 2.1b) having different lengths. Comprising the pentagons are the longer (1.45 A) electron-deficient 'single' bonds. Joining two hexagons one finds the shorter (1.40 A) electron-rich 'double' bond. As a result the  C60  molecule aquires a slightly  anisotropic charge distribution having important consequences for the solid, as will be discussed later. The energies of the sixty 7r-like molecular orbitals are of central importance since they form the valence and conduction bands in the solid. The one electron levels formed  4  Chapter 2.  C60  Fullerene and Fullerites^  5  (a)  (b)  Figure 2.1: a: Model of the C60 molecule (from Ref. 8) b: Model of C60 showing the electron-rich 'double' bonds and electron-deficient 'single' bonds (from Ref. 7).  Chapter 2.  Cal and Fullerites^  6  from these radially directed orbitals can be calculated using Hiickel molecular orbital theory [7] or a tight-binding approximation. Figure 2.2 shows the energy levels obtained assuming the hopping amplitudes for both types of C-C bond are the same. These eigenstates reflect the icosahedral point group symmetry exhibited by the potential and are labeled by the letters a, t, g, and h refering to degeneracies of 1, 3, 4 and 5, respectively. The subscripts g and u refer to even (gerade) and odd (ungerade) symmetry, respectively. One recognizes the similarity of these states with the angular momentum eigenstates of an electron confined to a spherical surface. In particular, the degeneracies of the L=0,1,2 and L=4 levels are unchanged. Using distinct hopping amplitudes for each type of bond splits the 9-fold degeneracy of the L=4 level into a 5-fold and a 4-fold degenerate level. The energy levels of the other states shift but their degeneracies remain the same. Ignoring electron-electron interactions, the electronic configuration of the Cal molecule is obtained by filling each of the thirty lowest-lying states with two electrons of opposite spin. The high symmetry of the buckyball greatly simplifies the complexity otherwise expected for a 60-electron molecule. One finds that the lowest lying 30 states are occupied, completely filling the hu shell. Approximately 2 eV above this lie the triply degenerate tiu states.  2.2 Crystalline  C60  and AxCso  2.2.1 Crystal Structure If we ignore the complications of molecular orientation, crystalline C60 can be considered a molecular crystal formed by close packing of spheres. Due to the large diameter of the carbon cage (7.1A), pristine C60 condenses into a face-centered cubic (fcc) lattice with a remarkably large lattice constant of 14.198  A.[9] As a result, the crystal structure  Chapter 2.  C60  7  Fullerene and Fullerites^  tig  t,^1=5  ^ha-H--H- 11^H ^ II^-1+-^II  1=-4  -th^e=3 1= 2  ^-H-^1= 1 1=0  Figure 2.2: One electron energy levels of the C60 molecule calculated using Hiickel MO theory (adapted from Ref. 7). The energy is measured in units of t, the hopping amplitude (assumed to be the same for each C-C bond).The lablels refer to the analogous angular momentum eigenstates of an electron confined to a spherical surface.  Chapter 2.  of  C60  C60  Fullerene and Fullerites^  8  has large interstitial spaces— two with tetrahedral and one with octahedral sym-  metry per  Cal  molecule—availible for deliberate as well as inadvertent doping. (Solvent  molecules left over from the extraction process are a particular nuisance which must be dealt with lest they obscure the interesting physics.) This crystal structure gives  C60  a density of about 1.7 gicm3, considerably lighter than either graphite (2.3g/cm3) or diamond (3.5 gfcm3). When pristine  C60  is exposed to alkali-metal vapour such as potassium or rubidium,  alkali atoms are incorporated into the interstitial spaces forming three stable phases as evidenced by X-ray diffraction.[10,11,12] For arbitrary x, AxCso consists of inhomogeneous mixtures of these three stable phases with x=3, 4, and 6. Attempts to dope in additional alkali atoms beyond the x=6 phase result in formation of regions of pure alkali metal indicating that the x=6 phase is, in some sense, saturated. The x=3 phase, exhibiting superconductivity and also the largest normal state conductivity, retains the original fcc lattice of pure  C60.  (The alkali atoms occupy  the tetrahedral and octahedral sites.) For stoichiometries greater than x=3, some expansion of the lattice is necessary to hold the additional atoms. The crystal structure therefore transforms to body-centered tetragonal (bct) for A4C60 and body-centered cubic (bcc) for A6C60 (see Fig 2.3). It should be noted that not all the aforementioned phases are obtainable for an arbitrary alkali metal. For example, no stable fcc phase has been achieved with only cesium as the dopant. This has been interpreted as due to the large size of the cesium cation which requires a more open  C60  lattice such as bcc or bct. At the other end  of the mass spectrum, neither NasCso nor LirC60 have been found to have a stable fcc phase although binary mixtures (e.g. Na2AC60 A=K,Rb,Cs or Rb2CsC60) utilizing all three are possible.[13] This thesis will be concerned with the non-fcc phases K4C60,  Chapter 2.  A3C60 fcc.  C60  Fullerene and Fullerites^  A6C60 b.c.c.  9  AzIC60  bct.  Figure 2.3: Crystal Structure for A3C60, A6C60, and A4C60 (from Ref. 10).  Chapter 2.  Cal  10  Fullerene and Fullerites^  K6C60, and Rb6C60. Henceforth in the remainder of this paper, the A in AsC60 will be taken to refer solely to K or Rb.  2.2.2 The Phase Transition and Orientational Dynamics of Solid  C60  Due to the strong intra-C60 covalent bonds, the internal molecular structure is basically unchanged in the solid state. This, taken with the experimental fact that the minimum distance between adjacent  C60  molecules in the solid is some 3.1A might lead one to  conclude that the inter-C60 interaction could be adequately described by a Van der Waals interaction. A spectacular demonstration that this is not quite correct is the existence of a first order phase transition in the solid at approximately 250 K.[5] NMR studies using naturally occurring nC nuclei had initially shown dynamical disorder which decreased with temperature.[14] This was interpreted as a consequence of free rotation of the  C60  molecules with a gradual slowing down of reorientations as the  temperature was lowered. Following this, P. Heiney and co-workers used synchrotron X-ray diffraction in an attempt to determine the low temperature structure of the solid. [5] They found that the low temperature phase is orientationally ordered and stable up to 249 K whereupon the solid undergoes a phase transition to the disordered state characterized by quasi-free rotation of the  Cal  molecules. Differential scanning  calorimetry revealed the first-order nature of the transition by indicating a free-energy change of 6.7 J/g at the transition.[5] From the diffraction peaks it was also possible to ascertain that the low temperature lattice can be indexed as simple cubic (sc) with a four-molecule basis. Neutron diffraction measurements clarified the nature of the fcc -- sc transition and the low temperature phase, revealing the ordering configuration of the  C60  molecules.  Chapter 2.  C60  Fullerene and Fullerites^  11  David et a/[15] found that this configuration was the result of an optimized ordering scheme in which the electron-rich, short inter-pentagonal bonds face the electrondeficient pentagon centres of adjacent C60  C60  molecules. The icosahedral symmetry of the  molecules makes this optimization possible for all twelve nearest neighbours and  results in the observed four-molecule basis. Theoretical studies [16,17] including Monte Carlo simulations corroborate these findings upon inclusion of a small anisotropic correction to the previouly assumed Van der Waals interaction. In addition to a Lennard-Jones potential, a small Coloumb potential, parameterized by an effective charge —2q on the short 'double' bonds, is found to reproduce the the transition temperature as well as the orientation of the observed ground state. However, orientational dynamics persist below To, the ordering temperature, as there are many nearly degenerate orientations for each  C60.  These orientations are  related by 7r/3 rotation about a threefold axis and are separated by potential barriers of 300 meV. Resulting from this is a glassy transition characterized by 'frozen in' disorder below a characteristic temperature that depends upon the time scale of the experimental probe.[16] Direct experimental evidence for this transition is found in sound velocity and attenuation measurements[18] and thermal conductivity measurements[19] on single crystal  C60-  2.2.3 Electronic Structure With the molecular energy levels in hand, it is not difficult to arrive at the electronic structure of  C60  solids, at least within the framework of an independent-electron, tight-  binding model. Recall that in the tight-binding approximation, a non-zero amplitude for electrons to hop from one site to the next causes the energy levels of the atom (or molecule as the case may be) to broaden into a band. The validity of such an approach  Chapter 2.  C60  Fullerene and Fullerites^  with the fullerites rests upon the fact that the  C60  12  molecules in the solid are relativly far  apart and consequently have a small hopping amplitude t relative to the on-site energy. Less grounded in certainty is the assumption of non-interacting electrons which is even more of a concern when the energy bands are narrow due to a small t. Charge transfer, the salient feature of doping with alkali-metal atoms, makes the electronic structure of A„C60 no more complicated. One simply uses the alkali-metal atoms to add electrons to the  C60  energy bands. The high electron affinity of the  C60  molecule combined with the low ionization potentials of the alkali metals ensure that the electrons are completely transferred to the formerly neutral  C60  molecule. Structurally  this transfer makes the doped fullerites more ionic and increases the cohesive energy relative to the undoped solid. Electronically, these considerations give the following results, seen simply by inspection of the  Cal  molecular energy levels:  C60  should be insulating with a relativly large  gap of about 2 eV; each added alkali-metal atom per  Cal  adds one electron into the  triply degenerate tit, orbitals so that A3C60 and A4C60 should be conducting; and for six added electrons, the th, orbitals are filled hence A6C60 should be insulating with a gap of a fraction of an eV. This approximate behaviour is observed in conductivity measurements on thin films of AsC60.[20] In these experiments thin films of  C60  are exposed to alkali-metal vapour  while the conductivity is monitored in situ. The films consist of inhomogeneous mixtures of the stable phases with the quoted x an average value of the alkali composition. The conductivity reaches a maximum for x=3, which presumably contains a large fraction of the stable x=3 phase, and decreases beyond this as the doping proceeds to x=6. The lowest resistivity (approximately 2 me-cm) is quite low for a typical metal and implies an unphysically short scattering length — perhaps an indication that this simple model needs modification.  Chapter 2.  C60  Fullerene and Fullerites^  13  More sophisticated calculations predict the actual k-space dependence of the ti other molecular-orbital-derived energy bands, from which one can estimate the density of states.[21,22] These calculations are based on the same ideas, namely complete charge transfer and absence of electron-electron interactions, and therefore provide the same qualitative results. Some authors try to estimate the effect of orientational disorder among the  C60  molecules on the electronic properties [23] but here again they use the  same underlying molecular energy levels.  2.3 Superconductivity in A3C60 The observed transition temperatures for fullerite superconductors (18K for K3C60, 29K for Rb3C60 and 33K for Cs2RbC60) are higher than any other known molecular superconductors and surpassed only by the cuprates. These superconductors are extreme Type II having a coherence length (0) of 26 as measured by magnetization[24] is 2400 4800  A. The penetration depth of K3C60  A, whereas itSR results[25] find a value of  A. Measurements using 13C NMR[26] find an energy gap consistent with weak-  coupling BCS theory while STM measurements[27] favour a larger value supporting strong-coupling models. The mechanism for such high Tc's in these materials is not agreed upon and both electron-phonon and electron-electron[30] pairing mechanisms have been proposed. The classic test of BCS theory is the existence of an isotope effect on the transition temperature. This effect has been seen using isotopically pure K313C60[28,29] however the interpretation that this confirms the validity of BCS theory is not without difficulty. Chakravarty and Kivelson argue that this would also be seen if the pairing resulted from electron-electron interactions. [31,32] Relevant to all models of superconductivity is the measured dependence of Te on the  Chapter 2.  C60  Fullerene and Fullerites^  14  lattice constant [33] which essentially reflects the dependence of T, on N(Ef), the density of states at the Fermi energy . One expects N(Ef) to increase with increasing lattice constant since the bands become more narrow as the hopping amplitude decreases. Virtually all of the conventional theories of superconductivity rely on the underlying band structure model for A,C60 described above. The validity of such an approach should be investigated experimentally. This can be accomplished by studying the nonsuperconducting phases of Ax.C60 in addition to the normal and superconducting states of  A3 C60.  The electronic structures of K4C60, K6C60, and Rb6C60, which are the subject  of part of this thesis, are an important test of any comprehensive theory for the electrical properties of fullerites.  Chapter 3  ,uSR and Muonium Spectroscopy  3.1 Introduction The use of muons for condensed matter studies is possible because of parity violation in the weak interaction and its manifestation in leptonic decays. Spin-polarized beams of low energy muons can be produced; when such muons are stopped in matter, the evolution of their spin polarization may be readily monitored using the experimental techniques of particle physics. Although the method is quite different from that of other spin resonance experiments, the information about the local fields obtained using //SR is similar and often complementary to that obtained from Nuclear Magnetic Resonance (NMR) or Electron Spin Resonance (ESR). The muon is a spin-I lepton with a mass of rs.,- that of the proton and a mean lifetime of , 2.2 its. When negative muons (yr) are stopped in solids, they are quickly captured by an atomic nucleus, cascading down to the lowest muonic orbital which has a radius comparable to the nuclear radius. Positive muons (it+), however, avoid the positively charged nuclei and take up sites in the interstitial regions. The experiments described in this thesis employed positive muons although some condensed matter studies are done with negative muons. Aside from the much smaller mass, the positive muon in a solid is completely analogous to an isolated hydrogen-like impurity. This fact has inspired a whole avenue of research concerned with the dynamics of the muon in the solid. Envisioned as a probe  15  Chapter 3. pSR and Muonium Spectroscopy ^  16  of solids, however, spin relaxation of the p+ is most simply seen as a microscopic probe of the magnetic properties of materials. Within this picture, the dominant interaction of the muon spin is assumed to be its Zeeman interaction with the local magnetic field. In addition to the applied field, the local field includes the dipolar field arising from the electronic and nuclear moments of the host material. Like a bare proton, however, the IL+ is very reactive and does not remain a bare particle in most materials. In good metals it acquires a screening charge which may in turn be polarized by an external field to produce a paramagnetic "Knight shift" of the effective field acting on the ,u+. In other materials, the muon can pick up and bind a single electron to form a hydrogen-like atom (ee) known as muonium or Mu for short. The occurrence of muonium opens up a whole new realm of experimental possibilities as the range of physical properties accessible to //SR techniques is augmented by the presence of the bound electron. This unpaired electron couples much more strongly to the local fields of the host material than the muon. In accordance with the Pauli principle, it can have a rather strong exchange interaction with the other electrons of the host material. The symmetry of the electron wavefunction is indicative of the muonium site and its environment. Additionally, the paramagnetic muonium atom is much more sensitive to local magnetic fields than the p+ simply because of its much larger magnetic moment. For those solids in which muonium forms, electronic structure as well as structural dynamics may be studied via spin relaxation of muonium. Needless to say, knowledge of the dynamics of muonium in solids is also valuable in its own right because of the scientific and technological interest in the behaviour of hydrogen in materials.  Chapter 3. pSR and Muonium Spectroscopy^  17  3.2 Basic Principles of Muon Spin Rotation 3.2.1 Parity Violation in Weak Decays The non-conservation of parity in weak interactions was discovered experimentally by Wu et al. using 'Co /3-decay[34] and Garwin et al. using muon decay[35] in 1957 after a theoretical prediction by Lee and Yang.[36] Parity violation in weak decays is not a small effect. In fact, it is maximal, and forms a cornerstone of the V — A theory of weak interactions. In correspondence with this theory is the experimental observation that neutrinos are exclusively left-handed particles only while antineutrinos are right-handed only (i.e. their spins are antiparallel or parallel, respectively, to their momentum). These facts have important consequences in the decay chain 71 —> /II -.4 e utilized by Garwin et al. in 1957 as well as in current itSR experiments. Spin polarized positive muons are produced from pions from the decay process  +  7r -- it+ +  VA,  a two-body decay. Conservation of momentum requires that the outgoing muon and neutrino be colinear in the rest frame of the pion. The spin of the pion is zero. Since the neutrino must be left-handed (helicity H = —1), conservation of both linear and angular momentum requires that the muon also be left handed. In the rest frame of the pion therefore, the muon is 100% spin-polarized with a helicity H --= —1. If the pions themselves are moving, the muons and neutrinos from the decay—distributed isotropically in the pion rest frame—become Lorentz-boosted into a cone about the initial pion momentum. Hence for a finite acceptance beamline using pion decay in flight, something less than 100% muon polarization is achieved. For positive muons, however, the most common practice is to use a surface muon beam which selects muons from pions decaying at rest near the surface of the production target. In addition to  Chapter 3. pSR and Muonium Spectroscopy^  18  nearly 100% spin-polarization, this type of beam has the advantage of providing low energy muons which are easily stopped in thin samples. It is also amenable to precise beam optics, as it comes from a small, well-defined source. Measurement of the time evolution of the muon ensemble polarization relies upon the asymmetric distribution of positrons from the decay, ,u+ -- e+ + ve + 17,2. As this is a 3-body decay, the kinetic energy of the emerging positron can have a continuous range of energies. Using the V — A theory, the angular probability distribution of the positron can be calculated and is found to depend upon the energy of the decay positron (Fig. 3.1). The angular decay probability has the form WO , x) cx 1 + a(x) cos 0 , where x— (Emax  E  the ratio of the positron energy to the maximum possible energy  f_s_.' 52 MeV), 0 is the angle with respect to the muon spin and a(x) is the energy-  dependent asymmetry factor, a(x) = (2x — 1)/(3 — 2x) (see Fig. 3.2). Averaging over the energy spectrum one finds that the positron is emitted preferentially along the direction of the p+ spin with a net asymmetry of -15.-. Thus by recording the spatial distribution of a large number of decay positrons from muons with the same initial polarization, one is able to determine the direction of that polarization. Although the major result of the work by Garwin et al. was the experimental verification of parity violation, they also noted an interesting observation concerning the nature of the material used to stop the muons. They found that the muon decay asymmetry in nuclear emulsion was only half that of muons stopped in copper or graphite. Friedmann and Telegdi[37) suggested that this loss of asymmetry was due to rapid depolarization on account of muonium formation. This was soon verified in an experiment  Chapter 3. itSR and Muonium Spectroscopy ^  Figure 3.1: Angular Decay Distribution of Positrons from ii+ Decay.  19  Chapter 3. itSR and Muonium Spectroscopy^  20  1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 0  0.2  0.4^0.6  0.8  1  x Figure 3.2: Energy spectrum E(x) and asymmetry factor a(x) of decay positrons.  Chapter 3. iuSR and Muonium Spectroscopy ^  21  performed by Orear et al. [38] in which a large magnetic field was applied to decouple the muon and electron spins and recover the 'missing' asymmetry. 3.2.2 The Experimental Technique There are essentially two spectroscopic methods used in the majority of ,uSR condensed matter experiments: time-integral and time-differential. In this section, a brief discussion of time-differential pSR will be given. Several comprehensive reviews of pSR techniques exist. In particular, the book by Schenck[39] and the review article by Patterson[40] are recommended. The typical time-differential apparatus includes a number of scintillators surrounding a cryostat containing the sample, all of which are placed in a magnetic field. For the transverse-field setup, shown in Fig. 3.3, the applied field is perpendicular to the initial muon polarization. In a longitudinal-field setup, the applied magnetic field is parallel to the muon polarization. On the way to the sample, an incident muon passes through a thin scintillator (detector M in Fig 3.3) before entering the cryostat via a series of Mylar or Kapton windows. A photomultiplier tube amplifies the faint flash of light from the scintillator and generates a pulse which is used to start a digitizing clock.' The decay positron from the stopped muon is detected by another scintillator (detector E in Fig 3.3), and that pulse is used to stop the clock. In this manner, repeated measurements of the time interval between the arrival of a muon in the sample and the detection of its decay positron are made. These time intervals are collected into a histogram which has the form N(t) = B No exp( tjA,) [1 + aelAti)], —  ti jAt (j = 1,N), 'The so-called "clock" is more formally known as a time to digital convertor (TDC).  (3.1)  Chapter 3. ,uSR and Muonium Spectroscopy^  Figure 3.3: Typical Setup for Tranverse Field ,uSR  22  Chapter 3. itISR and Muonium Spectroscopy^  23  where B is a time-independent background, No is the normalization, and ae, the experimental asymmetry. The experimental asymmetry here is not generally equal to the given by the intrinsic angular decay distribution, but instead depends on the experimental setup. This is due to absorption of low energy positrons, which raises the effective average asymmetry, and to the finite solid angle subtended by the scintillator, which lowers it. Fig. 3.4 (top) displays a simple ftSR spectrum of this form, showing clearly the precession and the muon lifetime. The quantity of interest is P(t), the component of the muon polarization in the direction of this particular positron counter.' For the transverse-field geometry depicted in Fig. 3.3, P(t3) is typically the sum of damped sinusoids P(ti) =  E f,Rn(ti) cos(wnt, + on),  (3.2)  where the different frequencies may be due to either inequivalent muon sites or formation of paramagnetic muonium states (to be discussed in the next section). The amplitude fn reflects the relative fraction of muons contributing to the nth signal and R(ti) is a relaxation function which depends on the particulars of the solid and is often taken to be a gaussian or an exponential envelope function. As a simple example, assume that no paramagnetic states exist and that the stopped muons are at equivalent sites having a gaussian distribution of B, the component of the local field along the main applied field Bo =B0 , and perpendicular to the m.uon polarization with width a centered about Bo. In this case, the muon spin precesses at the Larmor frequency so that a2t? P(t3) exp^2 3 ) COS (7 AB Oi j 0) .  (3.3)  During the analysis of many experiments, the histograms from two counters placed symmetrically on opposite sides of the sample are combined to form the asymmetry 2The scintillator/phototube combination is usually called a counter.  Chapter 3. icSR and Muonium Spectroscopy^  24  800  a.)  600  0 400  ° 200 0 0.3 0.2 0.1  0 >, co —0.1 —0.2 —0.3 0 1 2 3 4 5 6 7 8 9 10  TIME (microsec)  Figure 3.4: Spin precession of the it+ in a transverse field. top: raw histogram. bottom: asymmetry.  25  Chapter 3. itSR and Muonium Spectroscopy^  (not to be confused with cte, the experimental decay asymmetry) — aN2(t3) Al2(ti)^ aN2(ti) '  (3.4)  where N1(t3) and N2(t3) are the background-subtracted histograms of counters 1 Sz 2 and a = 14 is the relative normalization. This procedure takes out the muon lifetime. N2 Fig 3.4 (bottom) shows an example of the resulting Al2(ti) which is proportional to  P(t), provided the experimental asymmetries of the two counters are approximately equal.  3.3 Muonium The formation of paramagnetic Mu states in solids is a rather complex phenomenon with many open questions. Nevertheless, if these states do form on a timescale of < 10 ns, they may be readily distinguished by their characteristic precession frequencies in transverse field, by their zero-field oscillations, or indirectly from their decoupling curve (recovery of polarization) in longitudinal field. Although the electronic configuration of Mu in a solid may be quite different from that of the isolated muonium atom, study of the hyperfine levels of muonium in vacuum constitutes an instructive starting point for the discussion of the spin dynamics of the composite muon and electron system. 3.3.1 Hyperfine Levels of Muonium in Vacuum Consider then the isolated muonium atom, which, as mentioned previously, is very similar to an isolated hydrogen atom. The reduced mass m. of muonium, defined by 1/m* = 1/me 1/m„, is only about 0.5% smaller than that of hydrogen. The binding energy,  E —m*e4/2h2,  Chapter 3. ,uSR and Muonium Spectroscopy^  26  is therefore only slightly less than that of hydrogen. Since the first excited state is > 10 ev away, we need only consider the hyperfine structure of the 1S ground state. In this state the electronic configuration is almost perfectly isotropic and the hyperfine Hamiltonian describing the interaction between muon and electrons spins is given by the Fermi contact term 1 1H F = Alia' - 6 4,^ -  -  -  (3.5)  where "cle and 541 are the Pauli spin operators of the electron and muon, respectively. The hyperfine coupling constant, A,„ is given by 87h2 Ail -  3  -Ye-YAK/Mr ,^  (3.6)  with -y,,(-ye), the gyromagnetic ratio of the muon (electron). One notes that A„ is proportional to the probability density of the electron at the muon; for isolated muonium, the hyperfine frequency (also called hyperfine parameter) is determined to be, wo Au, 27r = Ti -  = 4463 MHz.  In a magnetic field the Hamiltonian includes the Zeeman terms for the muon and electron. The spin Hamiltonian for muonium becomes  'H = --yehc7e • B -  —  -y„hc712 • B -  + Apde • d".^(3.7) -  For a uniform B-field, this Hamiltonian is easily diagonalized.[41,42] Taking the field to be along the z-axis and using a basis  xi = I + +)^x2 = I - +)  ^  (3.8)  x3 = I - -)^x4 = I + - ), where the first +(—) refers to the muon spin up (down) along the z-axis, the second similarly to the electron spin, we find the eigenstates  27  Chapter 3. itSR and Muonium Spectroscopy ^  11) = 1+ +  ^ )  13) = 1 - -)  ^  12) = sine) +^+ cos^+)  ^  (3.9)  14) = cos fl —) — sin el — +).  The mixing angle e is given by cos — (1 +  x^ ) ^  V1 -I- X2  i  2  where x is the dimensionless reduced magnetic field, x = B /Bo, with Bo = AJ(-ye We note that in the high-field limit (Paschen-Back regime), the eigenstates are simply the xi's (3.8) we started with as a basis. The corresponding energy eigenvalues are found to be  B„  ^— =^+^- 70) ^4  E2  Ai,^A2, + (7e + 7A )2B? ^2  ^VA2/1 + (7e + -y0)2.13 ^Ai, A4 B„ E3 = — — — (lie — 7 41) E4 4 ^4 ^2 2  (3.10)  and are plotted as a function of the reduced magnetic field (a so-called Breit-Rabi diagram) in Fig. 3.5. An unphysical value of (-ye  —  74)/(7, y) is used in the diagram -  in order to show more clearly the relative behaviour of E1 and E2•3  3.3.2 Zero-Field and Transverse-Field ,uSR of Isotropic Muonium In the preceding section, the spin eigenvalues and eigenvectors of vacuum muonium in a external magnetic field were given. Greatly simplifying the problem was the assumption that the muonium was in the 1S state, thereby reducing the hyperfine Hamiltonian to an isotropic term de • 64 multiplied by a constant, A. We now consider zero or 31.e., their crossing at 19 and respective slopes for large . For the true value of (e )(e+), the crossing occurs at 100 and the slopes of j and 2 in the high-field limit are nearly identical, making these features difficult to portray graphically.  28  Chapter 3. iLSR and Muonium Spectroscopy ^  —110  o  ^^ ^ 1  2  3  X Figure 3.5: Breit-Rabi diagram of vacuum muonium. The eigenstates are labeled by their energies El — RI; X = -1-3Bo with Bo :-_•2 1585G for vacuum muonium.  29  Chapter 3. [ISR and Muonium Spectroscopy^  tranverse-field pSR of muonium characterized by the Hamiltonian of (3.7), leaving Ai, unspecified, and refering to the system simply as isotropic muonium.4 In pSR one observes the time-dependent muon spin polarization, P(t) = (6(t)).^  (3.11)  As an example, the behaviour of P(t) in zero-field will be calculated explicitly. The zero-field case is particularly simple, for in the absence of an external field we can choose the axis of quantization to be parallel to the initial muon polarization, P(0). The electron is assumed to be unpolarized, therefore the initial state is a mixture of + -I-) and I+ —), each with a population of 50%. The eigenstates in zero field are  11 ) = 1 + +)^12) = \-b1 (1 + -) + 1 - +))^(3.12) 13) =^—)^14) =^(I+ —) — — +)). We see that 1-1- +) is an eigenstate, hence 1/2 of the polarization does not change with time. The state 1+ —), however, is a superposition of eigenstates,  1 + - ) =^(12 ) + 14)) , and will therefore change with time. We denote the time-dependent spin state for this half of the ensemble by I W(t))ii. The time dependence is found by applying the time-evolution operator, Iklf(t))ii = /4(t)I^—) = exp^7-it) Using a basis of eigenstates of the Hamiltonian,  /At)  E  4Isotropic muonium states with hyperfine parameters different from that of vacuum are known to exist in semiconductors and insulators e.g. Si, GaAs, diamond, and molecular crystals.[40,39]  Chapter 3. ,uSR and Muonium Spectroscopy^  30  thus IT(t)).11 = 70 1 (e-212)^e- 4t14)) •  The time-dependent z-component of the polarization' is given by  = (GI) = (wmisfoct»  1 =^((210T 12) + (4lai: 14) + eiw0t(41c1 12) + e-nc4(41G1 14)) ,  using wo —  (E2—E4vh—  —hi—L A •  The action of the operator cr,,` on the xi's is given by  from which we have, (210-i: 12) = (410-f. 14) = 0, (210-f 14) =(4I° 12) = 1, therefore,  pzii(t) = eiwot e-iwo ^ = cos(wot). 2  The simple result is that this half of the polarization oscillates between +1 and -1 at the hyperfine frequency, vo = wo/27. We note that this back and forth oscillation of Pz is quite different from a 'precession' in the classical sense of a rotating spin, since 'It can be shown by a similar calculation that the other components, x and y, remain zero at all times.  Chapter 3. ,u,SR and Muonium Spectroscopy^  31  no transverse components develope, and consequently is sometimes referred to as the  muonium heartbeat oscillation. The time dependence of the muon polarization of isotropic muonium in a tranverse field is calculated following a similar procedure, facilitated by use of the density matrix formalism.[42,39,40] With the initial polarization along the x-axis and the field along the z-axis, the polarization is found to oscillate at four frequencies [39], according to  P(t)  1  —4 [(1 + 6) cos cont + (1  —  6)cos wiAt  + (1 + 6) cos w34t + (1 — 6) cos co23t] 1  P(t) = — [—(1 + 6) sin w12t + (6 4 + (1 + 6) sin co34t + (6  —  —  1) sin wiAt  (3.13)  1) sin w23t]  .13,(t) = 0, where wi3 = (Ei — E)/h, corresponding to transitions between energy levels of the Breit-Rabi diagram (Fig. 3.5), and 6 = cos2  —  sin2 = x/V1 + x2.  3.3.3 Anisotropic Muonium The isotropic hyperfine Hamiltonian of (3.5) is inadequate to deal with the phenomenology of m.uonium centres found in many crystalline solids. It is necessary to consider an anisotropic Hamiltonian  7 tHF = cie • A • 6',^ -  -  (3.14)  where A is now a tensor. This generalization of A from a scalar to a second rank tensor reflects the fact that the electron spin density distribution around the muon is in general anisotropic, having non-S-wave components. In an external field, the total Hamiltonian now reads 7-t = --yehrrle • B — y4h5*" • B^e • A •^ -  -  (3.15)  32  Chapter 3. ILSR and Muonium Spectroscopy^  In general, the energy eigenvalues of this Hamiltonian have analytic solutions only if the field is applied along one of the pricipal axes (x', y', z') of A. For the case of B  II  ;',  the energy eigenvalues as a fuction of the field are Ei  .  711 [Azz  E2  =  711  E3 E4  + V (Asx — Ayy)2 + 4.13! (-ye —^  2i  + NI (A,s + Ayy )2 + 4-13! (-ye  -rA)2 I  [—Azz  —V = il = 1 [—Azz — V 4 [Azz  (Ass  —  — Ay y )2 + 4.13! (-ye — -yi,)2 I  (Axx  + Ayy)2 + 4M (-ye —^)2]  (3.16)  These energy levels are show as Breit-Rabi diagrams (Fig 3.6) for 3 types of representative hyperfine tensors: isotropic (Ass =  ric (A Azz,  Ayy =  Az, = htoo), axially symmet-  ^Ayy^Azz, Azz = hwo), and completely anisotropic (As,^Ayy  A„ = hwo). In all cases, the field is directed along the z'-axis. For these dia-  grams, the relative values of the unequal elements of A are taken to be quite different in order to conveniently show all four energy levels. As with isotropic muonium, the muon polarization of anisotropic Mu in a transverse field consists of components oscillating at frequencies 1/12, v23, v14, and v34, given by the Am = +1 transitions between the corresponding energy levels of equations (3.15). However, the amplitudes for the 4 frequency components for anisotropic Mu depend upon the field in a much more complicated way than for isotropic Mu. Of particular interest is the zero-field behaviour evident in the Breit-Rabi diagrams. For isotropic Mu, one observes only one oscillating component of the muon polarization, at the hyperfine frequency. In contrast, axially symmetric Mu exibits three frequencies (1 relatively low and 2 relatively high), while completely anisotropic Mu results in six frequencies (3 low and 3 high). The zero-field Hamiltonian [equation (3.13)] is invariant with respect to a rotation of the coordinate axes; the frequencies, therefore, will not  33  Chapter 3. [SR and Muonium Spectroscopy ^  2/0/2 E2  0-  E,  a  - E4  ^E,  _C 0-  Lil  E,  -v0/2 -  E,  vo/2 -  E, 0-  -  vo/ 2  -  0.0^0.2^0.4^0.6^0.8^1.'0 X  Figure 3.6: Breit-Rabi diagrams for (a) isotropic, (b) axially symmetric, and (c) completely anisotropic muonium.  34  Chapter 3. ,uSR and Muonium Spectroscopy^  depend upon the direction of the initial muon polarization. For the anisotropic cases, the amplitudes, however, will depend quite strongly on the initial direction of P(t), in principle allowing a determination of the orientation of the hyperfine tensor with respect to the crystalline axes.  3.4 Spin Relaxation of Muonium: Spin Exchange with Free Carriers The preceding considerations of the spin dynamics of muonium have neglected relaxation of the muon polarization. In addition to analogous ,u+ relaxation mechanisms' such as (reversible) dephasing due to a distibution of fields or (irreversible) spin transitions due to a fluctuating field, spin relaxation of muonium can occur via several other processes. For example, muonium could form with a distribution of hyperfine parameters, giving rise to dephasing similar to that resulting from a distribution of fields. Another possibility would be transitions between muonium states with different hyperfine parameters, which can be viewed as producing a fluctuating effective field at the muon. Still another mechanism, most relevant to the present studies, is due to the interaction of the muonium electron with the electrons of the medium. 3.4.1 Spin Exchange Relaxation A spin-independent interaction V, such as the Coloumb repulsion between two electrons, combined with the exclusion principle, leads to a so called exchange interaction in the Hamiltonian for the two electrons of the form [43]  V = 2J61 • 62,^ 6  (3.17)  Here we are referring to a + experiencing only the Zeeman interaction with the local magnetic field.  Chapter 3. [tSR and Muonium Spectroscopy^  35  describing a spin-spin interaction between electrons 1 and 2. This term is proportional to J, the overlap integral, which is J = — VabVW bad3xid3x2 = — knaVkliabd3xid3x2, ^(3.18) where Tab( x i, x 2 ) = W a(x 1 )klib( x 2).^  (3.19)  Here Wc, and W b are energy eigenfunctions for a single electron with energies Ea and Eb, respectively.  With one of the electrons the Mu-electron, and the other a Bloch electron of the solid, we can treat 1-1' as a brief, time-dependent perturbation to the muonium spin Hamiltonian of equation (3.7), and find that 7-1' induces spin-exchange transitions between the two electrons:  I + +) mu l —)e <^ I + —> mul+)e I — --) mu l+)e <^ I + ) mul —)e.^(3.20) —  These spin-exchange transitions only involve spin flips of the muonium electron, yet the also affect the muon spin to which it is coupled. A number of authors have evaluated the resulting time evolution of the muon polarization upon the phenomenological inclusion of electron spin relaxation, spin-exchange being one possible mechanism [44,45,46,47,48] We summarize the results for longitudinal and transverse fields in the limiting cases of fast and slow v, the spin exchange rate.  Longitudinal Field: For fast spin-exchange, v > coo(1 x2)1, the time dependence of the it+ polarization[45,46] is  Pr(t),exp(-tfro,^  (3.21)  Chapter 3. [SR and Muonium Spectroscopy ^  36  where ^7  1 = 4v/w02.^  -  (3.22)  That is, the muon polarization is exponentially damped at a rate which is independent of the strength of the applied field, but that decreases with increasing v. Qualitatively, this reflects the weakening of the hyperfine coupling of the electron and muon spin by the large spin-flip rate; the muon behaves more as if it were 'free'. For slow spin-exchange, v < wo(1 + x2), one finds from equation (7.80) from Schenck[39], that ^P(t) =  1 + 2x2 exp ( ^v t) + ^1 ^xp ( (3 + 4x2)vt) 2(1 + x2) ) 2(a + x2)^\^ 1 + x2 ) 4(1 + x2) e  5 + 8x2)v ( x (2 cos wo(1 + x2`)1/2 ^ t + wo(1 + x2)3/2 sin wo(i + x2)1/2t) Averaging over the high frequency oscillating terms, which are unresolved in conventional experiments, we have Pf = Po exP( —07-2)^  (3.23)  with 72 = (1 + x2)/v and Po —  1 -I-- 72  ^ 2(1 + x2)•  (3.24)  We see that for x —> 0, the muon spin relaxes at the electron spin-exchange rate. For increasing x, however, the decoupling of the electron and muon spin reduces the muon spin relaxation rate. This effect provides an enormous practical advantage for longintudinal field measurements of v. One can select the strength of the applied field in order to place the relaxation rate in the range convenient for OR measurements (i. e. 0.1 to 10 s1). For example, if v is strongly dependent on the temperature, one can choose the appropriate field for the temperature range of interest.  Chapter 3. itSR and Muonium Spectroscopy ^  37  Transverse Field: Again we consider the two limits of fast and slow spin exchange.  For fast spin-flipping,[42] v > wo(1 +  1 '15" ..E Ps" + iP" exp[(—iwi, — — ) Y T1.  ti ,  (3.25)  where 7-1 is the same as in equation (3.21). As in longitudinal field, the muon polarization relaxes at a rate inversely proportional to v, and the muon behaves as if it were a 'free' muon precessing at the Larmor frequency, wm. For a slow spin-flip rate, v2 < (wo/2)2x4, Gurevich  et a/.[491 have calculated the x-component of the [t+ polar-  ization. Neglecting the usually unobserved terms with frequencies w14 and w24, their result is  1 52 sin 52.1,t Pf = — exp(—t/7-3) [(cos^ flyt + ^ cos w_t 2^ 3 7-3S24 )  (2w+S22)  + ^ sin S-2-r t sin w_t wof22'y  (3.26)  where CO_ = Lt2+ =  S-2 =  1^1 "i(wi2 + W23) = Owe 1  1  —  iwpi)  (ic.oei + PAD  1 ^\^1 ( 2 w23 — w12) = iwo[(1+ xy _  11,  and the v-dependent beat frequency is Cil, = C2(1 — v2/4522).1. The exponential damping of the precession signal is given by 73 = 2/3v.  (3.27)  For small fields (< 10G), where w12 '-d. co23 E.-: wmu, /3: is given simply by  1 P" = — exp(—t/T3) cos wmut. X^2  (3.28)  Chapter 3. ttSR and Muonium Spectroscopy^  38  3.4.2 Temperature Dependence of the Spin-Exchange Rate We have seen in the last section that spin-exchange interactions give rise to an exponential relaxation of the muonium signal. For a given v, the time evolution of the muon polarization in a longitudinal or transverse field is known. We now consider what factors determine v and in particular we calculate the temperature dependence of v for two simple band structures: (1) a partially-filled band at T=0, i.e. a metal, and (2) a filled band at T=0, above which is a conduction band separated by a gap E9, i.e. a  semiconductor. The spin-exchange rate can be written as  v, nk,,,f(Eki){1- f(Eki)] ,^ (3.29) where Wki,l, the transition probability per unit time for an electron of wavevector k and spin  i  to be scattered to a state with wavevector k' and spin 1, is multiplied  by the probability f (Eki) of finding the state 1k, 1) occupied and by the probability [1 – f (Ek I)] of finding the state Ik',1) empty. From perturbation theory, we have that 2 7r^ A4 Wkt^ i =^-h--I(k, +17-t'iw, -)128(Eki - Ek i – –2–)  A4 = 87 - J2S(k – lc/)6(E/1 – Eki – — 2 )• h  (3.30)  With this, we can convert the sum over k, k' in equation (3.28) to an integral over initial and final state energies of the electron, V OC  Ail 1J2 f i P(Ekt)P(Ek I)f(Eki) [1 - f(Eki)] b(Eki – Eki – — )dEkidEki, 2 h  where p is the density of states (DOS). Ignoring the Zeeman hyperfine energy A4/2, we have 87r 2  V OC i-- j  I p2(E) f (E) [1– f (E)] dE.  (3.31)  Chapter 3. pSR and Muonium Spectroscopy^  39  Thus the entire temperature dependence comes from the Fermi distribution function  f(E) in the above integral which we evaluate for the two cases mentioned above. Temperature Dependence of v in a Metal: In this case, f(E)[1  —  f(E)] is large  only within about kT of Ef. Since kT < Ef, we can pull p(E) out of the integral, evaluating it at Ef. We also notice that  f(E)[1 — f(E)] = —kT dd fE , so that df 2 2 v oc 7 J p (Ef)kT f c° ( -- )^dE  h,^0^dE  and thus v oc 87 J2 p2(Ef)kT.  (3.32)  —  h  Not surprisingly, this result is very similar to Korringa relaxation in NMR where the relaxation rate is found to be proportional to the temperature and the square of p(Ef). [50,51] Temperature Dependence of v in a Semiconductor: In a semiconductor the Fermi level lies somewhere between the valence and conduction bands. The actual value of Ef is determined by the 'law of mass action' which simply states that the number of electrons excited into the conduction band is equal to the number of holes in the valence band.[52] To find the spin-exchange rate, the integral of equation (3.30) is split into two parts representing the contribution of electrons in the conduction band and that of the holes in the valence band. Omitting the details of the calculation, the two parts are found to be equal; for kT < E9, we find that  167- J2 V OC  E9  p(Ec)p(Ei,)kT exp ( ^ h^ 2kT i '  (3.33)  Chapter 3. ,uSR and Muonium Spectroscopy^  40  where p(E) is the density of states (DOS) evaluated at the bottom of the conduction band Etc, and p(E) is the DOS evaluated at the top of the valence band E. In this derivation we assume that the exchange integrals for the conduction electrons and the holes are the same, i.e. Jel = Ade = J, which might be expected if the wavefunctions of the conduction and valence bands are the same. In general, this will not be so, hence the overall constant of proportionality will be different. We have also assumed that the energy level of the bound muonium electron does not lie in the gap. If this were the case, additional relaxation from the process Mu + hole -- Mu + hole would be observed. This type of charge-exchange relaxation—quite similar to spin-exchange relaxation—is negligible if the bound state energy of the electron is far below the Fermi level. Throughout this discussion of spin-exchange relaxation, we have tacitly assumed the existence of muonium states in metals and semiconductors. While muonium is observed in many semiconductors, its formation in metals is usually inhibited by screening of the positive charge by electrons at the fermi surface. The metallic phase A3C60 presents an interesting exception in that screening may not be possible inside the large cavity of the C60  molecule. In fact, muonium in Rb3C60 has recently been observed[53] confirming  this notion. In the next chapter we present results showing spin-exchange relaxation of muonium in other doped fullerites which are found to be semiconducting.  Chapter 4  Results and Discussion  4.1 Muonium States in Solid  C60  Previous pSR experiments show that positive muons stopped in solid  C60  form two  distinct paramagnetic centres.[3] In addition to a small diamagnetic fraction (, 2%), these two centres are easily distinguished in a field of ,--, 100G. The Fourier transform of data taken at 5 K (Fig 4.1) shows a doublet with a comparitively narrow linewidth centered around 150 MHz and two much broader lines near 90 MHz and 240 MHz. The field dependence of the frequencies of these lines allows us to attribute the doublet to the V12  and 1123 transitions of a muonium state with a hyperfine (h f) parameter of 4341+24  MHz—remarkably close to that of vacuum muonium (4463 MHz). Similarly, the much broader lines are ascribed to the v12 and 1134 transitions of a paramagnetic state, in which the isotropic part of the hyperfine interaction Tr[A]/3 is equal to 325 MHz at room temperature (at 5K it is 332 MHz). This is typical of a muonium-substituted radical.{54] From the respective asymmetries of these two signals, we can estimate the relative fractions of implanted muons which form these states: 12±2% for the 4341 MHz signal and 60+10% for the 325 MHz signal. The explanation then, based soley upon these considerations, is that the signal with the vacuum-like h f parameter of 4341 MHz is due to endohedral muonium (i.e. Mu inside the buckyball cage, denoted MuOC60), while the the 325 MHz signal is due to muonium bonded exohedrally to the carbon cage (C60Mu).  41  42  Chapter 4. Results and Discussion^  50  40  1  -  -  -  yr_  -  4  10  0  -^  50^100^150^200 Frequency (MHz)  -  ^  250  Figure 4.1: The Fourier power spectrum of ,uSR in C60 at 100G and 5K. The doublet centred at 150 MHz is from the v12 and v23 transitions of MuOC60, the two broad lines at 90 and 240 MHz are the v12 and v34 transitions of C60Mu'.  Chapter 4. Results and Discussion^  43  As discussed in chapter 3, the hyperfine parameter is proportional to 1 0,(0) 12. With a h f parameter close to that of vacuum, the spin density of the electron of the 4341 MHz centre is not greatly altered with respect to vacuum. This indicates a very small interaction with the buckyball carbons. The 325 MHz centre, on the other hand, indicates a much stronger interaction with the carbon cage. It is typical of a positive muon in a covalent bond, forming what is known as a muonated radical. Since carbon prefers a tetrahedral bonding scheme, we expect that the curvature of the  C60  molecule  makes the radical more likely to form on the outside of the C60 cage. This notion is supported by theoretical calculations of the potential energy surfaces of C6011 [55,54 which show that the most stable configuration has the H attached to one C atom on the outside of the buckyball. We can see that this picture agrees with the observed fractions by the following argument. Suppose all of the implanted muons stopping inside of a buckyball form Mu©  C50  while those stopping outside form C60Mu*. The ratio of the formation fractions  then would follow from volume considerations and be simply the packing fraction of the  Cop  lattice. Due to the large distance between  C60  molecules, the packing fraction  is much smaller than that of the close-packed fcc lattice (ff„=0.74). Using a diameter of 7.1  A for the C60 molecule and a lattice constant of 14.04 A, one finds the packing  fraction to be 0.27. This implies that C60Mu* should form 3.7 times as often as Mu0C60 whereas the observed ratio is 5. The difference is easily explained by supposing that the muon must overcome some potential barrier to get inside the  Cal  cage.  Experimental results discussed in this thesis show that the C60Mu signal is much more sensitive the to the  C60  molecular dynamics than Mu0C60. Along with the  observation of Mu0C60 in the doped fullerites, this supports the basic picture of the two centres presented above. Recently Prassides et al. [57] has reported the observation of Mu0C70 and use its zero-field itSR oscillations to study the molecular dynamics of  44  Chapter 4. Results and Discussion^  the C70 molecule.  4.2 The First-Order Phase Transition in C60 We examined a 500-mg sample of high-purity C60 powder using the M15 beam line at TRIUMF. Conventional transverse-field //SR data were taken in an applied magnetic field of 1.5 T at 22 temperatures ranging from 5 K up to room temperature. The precession frequencies of Mu0C60 are too high to observe at this field, leaving only the signal from the diamagnetic it+ and the C60Mu. radical. In a large field only two frequencies of C60Mu', the v12 and v34 transitions, have an appreciable amplitude. If the spin Hamiltonian of C60Mu* were isotropic, they would depend appoximately upon the field as -y„B Ao  V34 es-,  27r^2h 7t,B At, 27r^ +  Th  (4.1)  where At, is the isotropic hyperfine parameter. If the spin Hamiltonian is anisotropic, however, we can use equation (4.1) (substituting A„ for Am) only if the applied field is along a principal axis of A, the hyperfine tensor. For an arbitrary direction of the applied field, these frequencies are [39] ^ 7r.^2...hz\ 2 + A2 —A] ICYA2B  V12 =^  V34 = [ (7;7" +  )  zhz )  1 2  2 + A2 A2  1  2  (4.2)  where the components of A are now given in the x, y,z system. If the anisotropy is small (i.e. Ass '--'  Ayy'..j.  A„), the off-diagonal terms will be small. Ignoring them  and using the Euler matrices to find Azz in terms of the principal axis components, we  Chapter 4. Results and Discussion ^  45  have V12 Z/34  7ABz  Azz  27r^2h -YABz  Azz  27r^2h  ,  (4.3)  with Azz  = AST sin2 7/, sin2  Ayy COS2  sin2 8 + Azz cos2 0.  (4.4)  The and 0 are two of the standard Euler angles. We see that unless the hyperfine Hamiltonian is isotropic, these frequencies depend on the orientation of the C60Mu* with respect to the applied field. From the difference 1/34 - V12 =  Azz/h,  (4.5)  one could determine Azz if every C60Mu* had the same orientation. More realistically, this difference would give us  A, the average of Azz over all possible orientations.  Fig. 4.2 shows the Fourier transform in the frequency range surrounding v12 for data at selected temperatures. At room temperature, the linewidth is quite narrow and comparable to that measured for muonium substituted radicals in liquids. [54] Below T0=260 K, another signal appears at a slightly lower frequency, having a significantly broader linewidth. Between 242 K and 260 K, both signals are present although the amplitude of the higher frequency signal decreases with the temperature and is matched by a corresponding increase of the lower frequency amplitude. Below 242 K, the linewidth of the remaining signal grows steadily with decreasing temperature. At 5 K the lineshape resembles a broad gaussian. These observations suggest an anisotropic hyperfine interaction for C60Mu. — the broad static linewidth at 5 K is due to the orientational dependence of the transverse field precession frequency. Consider the direction of the applied B-field in the principal  Chapter 4. Results and Discussion^  46  264K -  256K  4141A  5K  10 20 30 40 50 60 70 FREQUENCY (MHz)  Figure 4.2: Frequency spectra near v12 of C60Mu* at selected temperatures. The applied field is 1.5T.  Chapter 4. Results and Discussion^  47  axis system of A. The unoriented nature of the crystallites results in a random distribution of this direction.1 One would expect a powder pattern; however, some additional broadening— probably from a distibution of hyperfine parameters due to the inequivalence of sites with respect to the crystalline axes—effectively produces the gaussian lineshape seen in the Fourier transform at 5 K. At higher temperatures, motional narrowing due to thermally excited rotation of the C60Mu* accounts for the decrease in the linewidth. Using a rotating-reference-frame (RRF) transformation[58], the data were initially fit to an exponential relaxation function. With the RRF, the two frequencies (v12 and v34) are estimated independently, while the exponential decay constant, T2-1, is assumed to be the same for both components of the time signal. We used the estimated frequencies, as prescribed in equation (4.2), to find  A. Fig. 4.3 shows the temperature  dependence of (a) the linewidth, T2-1, and (b) the average hyperfine parameter,  A.  The first-order phase transition is clearly seen as a discontinuity at To, the ordering temperature, in both the linewidth and A. Below To, the correlation times for reorientation of C60Mu (Fig. 4.3c) were estimated by fitting the data at each temperature using the Abragam relaxation function [59] R(t) = exp[—a2r(e-tirc — 1+ thc)]• (4.6) The static linewidth parameter a = 21.2(1.1) its' was obtained from the lowest temperature run and held constant for fits of the other runs. The temperature dependence of the correlation time between 200 and 250 K was fit to an Arrhenius law, 7;1 = A exp(—.Ea/kBT), with A = 6.4(2.4) x 1012 s-1 and Ea = 219(7) meV. At 200 K, = 52(17) ns, and agrees quite well with the NMR result [64] of 64 ns for C60.  'Even for an oriented crystal, the icosahedral symmetry of Co alone would nearly accomplish this.  Chapter 4. Results and Discussion^  10  E:  7  10 6  _ a)^0^0^  0  0  4,  48  cot  E  (a)  i o5 330  326  (b)  (C) F  afb  F E-  r ci  0^100 200 300 TEMPERATURE (K)  Figure 4.3: (a) The linewidth Ti', (b) the average hyperfine parameter A, and (c) the inverse correlation time -1-z-1 extracted from simultaneous fits to the v12 and 113 4 frequencies of the C601V111 radical in fiSR time spectra taken at 1.5T.  Chapter 4. Results and Discussion^  49  Above To, the lineshape increases with temperature, contrary to what is expected from motional narrowing. Very rapid rotation of the C60Mu causes electron-spin relaxation, induced by the coupling of the electron spin to the molecular-rotational-angular momentum. [60] For this mechanism, the linewidth can be approximated as  ,_  ,,.2 r7-7-1 ^'SR ' c 2^— 1 + We27c2  -1-  ___  1  (4. 7)  where o-sR is the electron-spin-molecular-rotation coupling constant and co, = -yeB. The data above To were fitted to an Arrhenius law using equation (4.3) with o-sR = 4.9(1.0) x 108 s-1, A = 5.1(7) x 10' 5-1, and Ea = 98(16) meV. The correlation times above To shown in Fig. 4.3(c) were obtained from the measured linewidth using this fitted value of usR. At 300 K, 7, = 8.5(2.0) ps, also in good agreement with NMR results. [14,65] The discontinuity in A at To shows that the electronic structure of C60Mu also changes abruptly at the phase transition. Most likely, this reflects a shift in the C-Mu bond length. Since the lattice constant decreases only by a small amount (0.044 A)[5] at To, we suggest that the electronic structure depends upon the molecular dynamics of C60Mu*. The shift of  A  could reflect a change in the motion of C60Mu*-from one  characterized by jumping between orientations with nearly equivalent potential minima to one involving quasi-free rotation. Along with the sharp behaviour of T2-1 at To, the discontinuity of  A is suggestive  of a first-order transition. The observation of coexistence of both phases just below To supports this hypothesis. Because of defects, impurities or finite size effects, the individual crystallites may have slightly different transition temperatures, or part of the high-T phase may be pinned upon cooling. If the transition were of a higher order, one would expect an additional broadening of the lineshape just below To, the result of a distribution of A's, rather than two distinct signals.  Chapter 4. Results and Discussion^  50  Additional details concerning this work were reported by Kiefl et. al. [61]  4.3  C60  in Zero Field  Analysis of the transverse-field data, described in the last section, led us to postulate an anisotropic hyperfine interaction for C60Mu*. The details of the anisotropy, however, are not evident from the transverse-field results. Is the hyperfine tensor of C60Mu axially symmetric or completely anisotropic? How large is the anisotropy? (/". e. how large are the relative differences between the principal axis components of A?) Zero-field (ZF) //SR provides the answers to these questions as well as giving us another handle on the molecular dynamics of Co:Mu'. As noted in chapter 3, an axially symmetric spin Hamiltonian will exhibit three transition frequencies in the ZF spectrum—two high (singlet-triplet transitions) and one low (inter-triplet transition) —for a small anisotropy. A completely anisotropic spin Hamiltonian, on the other hand, will show six—three high and three low—for a small anisotropy. Furthermore, these frequencies will not depend upon the orientation of the paramagnetic centre. A distribution of orientations thus will not lead to relaxation due to dephasing, it will affect only the overall amplitude of the signal at each frequency. Fig. 4.4 shows the zero-field iLSR spectrum in orientational motion of the  Cal  C60  at 9 K. At this temperature, the  molecules is frozen. Three low frequency oscillations  characteristic of a completely anisotropic hyperfine interaction are clearly seen in the time and Fourier spectra. They lie on top of a very low frequency (--, 0.07 MHz) background. The unresolved (from one another) high frequency oscillations appear as a single broad component in the first 50 ns of the time spectrum (see inset). The magnitudes of the three low frequencies, 1.2(.1) MHz, 7.4(.1) MHz and 8.6(.1) MHz, are a direct measure of the hyperfine anisotropy. Comparing these to the high frequency  Chapter 4. Results and Discussion^  51  of 332.5(.8) MHz tells us that the anisotropy is small. Using equations (3.16) one finds 7112 = (A„ — A)/2h ^  1113 =  (Ayy  1123 =  (A„ — A)/2h.  Ass)/2h  (4.8)  We denote the measured low frequencies, in increasing magnitude, vi., v2, and 113. Taking Ass < A  ^A„, we can unambiguously assign 713 to 1123; however, we can't  say whether vi = v12 or v13 (similarly for v2). This prevents us from finding the actual components of the hyperfine tensor. The high frequency, P, should be the average of 1114, 1124, and 1134, i.e.  _ VIA + V24 -I- 1/34^Axx + Ayy + v =^=  ^(A) ^ (4.9) 3^3h^— h • Azz  The result is (A)/h = 332.5 (.8) MHz, in good agreement with the high-TF data. In order to unambiguously determine the principal components of A, one would have to measure the field-dependence of the low frequencies for small applied magnetic fields in aligned single crystals of  C60.  We can, however, give the fractional anisotropies  (SARA)) along the principal axes: 0.036(3), 0.223(3), and 0.259(3). Interestingly, the low frequencies are close to those observed in single crystal a-quartz,[62,63] but since (A)/h r.-' 4500 MHz for quartz, these anisotropies are more than 100 times larger than  for quartz. At 9 K the low-frequency signals have an exponential relaxation rate, A_^-2 1.0 us-1, presumably due to slightly inequivalent sites. At higher temperatures, thermallyactivated `ratcheting' of CsoMu* contributes to an addition relaxation. The fitted relaxation rate is plotted as a function of temperature in Fig. 4.5. Assuming that the increase in the relaxation rate is proportional to the 'hop' rate of C60Mu*, we extract  Chapter 4. Results and Discussion^  52  0.12  0.1  0.08 c) F' 0.06  0.04  0.02  0 0^0.5^1^1.5^2^2.5^3 TIME (microsec) 1.E-4  8.E-5  (b  )  E 6.E-5 P.  4.E-5  2.E-5 ^  L„)  A,.  2.5^5^7.5^10^12.5^15 Frequency (MHz)  Figure 4.4: C601VIu* in zero field: (a) time spectrum with the inset showing the first 50 ns; (b) frequency spectrum.  Chapter 4. Results and Discussion^  53  from a fit to an Arrhenius dependence an activation energy Ea = 200(20) meV. This value agrees very well with Ea = 219(7) meV determined from the transverse-field data. As mentioned above, there is a background in the data which appears to be an oscillation at a very low frequency. The frequency is temperature dependent, decreasing from 0.07(.01) MHz at 9 K to 0.02(.01) MHz at 200 K; at room temperature it disappears altogether or is too small to measure. An interesting possibility is that this may come from Mu0C60 as the result of its own weakly anisotropic hyperfine interaction. However, it would be premature to conclude this from the present data. Prassides et al. [57] report zero-field itSR in  C70  and argue that the low-frequency oscillations  seen below 270 K are from MuOC70 with an anisotropic hyperfine interaction.  4.4 Endohedral Muonium in the Fullerites The temperature dependence of the signal characterized by a 4341 MHz hyperfine frequency in pure  C60  is less dramatic than that of the C60Mu* radical (see Fig. 4.6).  Above 230 K and across the phase transtition, its frequency and relaxation rate remain essentially constant. If this signal is from MuOC60 with an isotropic hyperfine interaction, the dominant source of line broadening expected at the lowest temperatures would be the weak dipolar interaction with naturally abundant (1.1%) 13C nuclei. The measured relaxation rate as T --40 is about twice that expected from a simple calculation of the rms width of nuclear dipolar fields at the center of the  C60  molecule.  This may indicate an additional effect due to the zero-point motion of the muonium or the extended nature of the muonium electron. Alternatively, the hyperfine interaction may have a slight anisotropy, as hypothesized in the previous section. Above 100 K, the decreasing linewidth is consistent with with motional averaging of either the anisotropic nuclear dipole interaction or any hyperfine anisotropy, resulting from rapid  Chapter 4. Results and Discussion^  54  10  8  6  4  2  0 0^25 50 75 100 125 150 175 200 225 250  Temperature (K) Figure 4.5: Temperature dependence of A, the relaxation rate of C601VIu* hyperfine oscillations in zero applied field. The solid line is a fit to an Arrhenius dependence giving a activation energy of 200(20) meV.  55  Chapter 4. Results and Discussion^  reorientation of the  C60  cage.  2 1.75  -  1.5  -  -  -^ 1.25 Cl)  -  1 7  cvz  E-I  0.75 0.5 0.25  -  it  0 0^50^100^150^200^250^300  Temperature (K)  Figure 4.6: Temperature dependence of the lin.ewidth 7Y1 of the TF-,uSR signal from MuOC60 in pure C60 in an applied field of 100 G. We also observe muonium centres with vacuum-like hyperfine parameters in K4C60, K6C60 and Rb6C60. Fig 5.7 shows the Fourier transforms of the itSR spectra for K4C60 and  K6 C60  at 5 K in transverse fields of , 100 G. In these doped fullerites, most of the  implanted muons form a diamagnetic centre and the C60Mu radical is conspicuously absent (compare Fig. 4.7 with Fig. 4.1). As in pure  C60,  the two observed frequencies  correspond to transitions between the spin-triplet states of Mu. The sum of the two frequencies v12+ v34 = -ye.13/27 is approximately the Larmor frequency of a free electron.  Chapter 4. Results and Discussion^  56  Their difference provides a measure of Ai, [40]:  h (vi2 + v23+ 2vm)2 + V12 - V23 7 14^ 2^ V23-^Vi2  A=  (4.10)  where the Larmor frequency of the muon = ..4.73 (where = .01355342 MHz/G), is used to determine the field. The estimates of Ai, obtained with this equation are given in Table I. The observation of Mu in the alkali-metal-doped fullerites leaves no doubt to the hypothesis that this muonium is endohedral. The magnitudes of A Mu in the doped fullerites are similar to that of Mu in pure  C60 —  and only a few percent less than that  of Mu in vacuum—indicating a weak interaction with the AxC60 solids are essentially ionic compounds of A+ and  C60  lattice. Given that the  C60-x,  an endohedral site for  Mu is likely to be the most stable. The absence of the C60Mu radical is presumably due to the weakening of the covalent C-Mu bond resulting from the addition of electrons upon forming the  C6-ox  ion. Estreicher et al. argue that the stability of C6011s- (here H  refers to hydrogen or muonium) is maximum for x=0 and decreases with increasing x [55]. The formation fractions for Mu in AxC60 are somewhat less than that in pure  Cal  (see Table I). This might be due to the electrostatic repulsion between the implanted it+ and the charged  C60-x  ion.  While we might have expected to find Mu0C60 in K6C60 and Rb6C60, which are predicted to be semiconductors with gaps of ,0.5 eV,the observation of MuOC60 in K4C60 is somewhat surprising. According to the simple band model described in chapter 2, K4C60 should be metallic. Consequently, we would expect that spin exchange with free carriers would preclude the observation of Mu spin precession—either by extreme line broadening or (for larger carrier concentrations) by effectively decoupling the muon and electron spins, allowing simple ft+ precession in the fast spin-exchange limit. Using equations (3.32) and (3.33), we can get an idea of the spin-exchange enhancement upon  Chapter 4. Results and Discussion ^  .<  (a)  4  57  C 60 -  -  -  -  (b)  -  -  I  -  "ii 60^100^140^180 FREQUENCY (MHz)  Figure 4.7: //SR frequency spectra for (a) K4C60 and (b) K6C60. The line at 150 MHz in the top spectrum is from the TDC.  Chapter 4. Results and Discussion^  58  going from a semiconductor to a metal. Assuming the same density of states, the metal would have a spin-exchange rate which is larger by a factor of roughly exp(E912kT). At T=5 K and for Eg = 0.5 eV, this factor is 6600 (> 10260)! Apparently K4C60, like K6C60 and Rb6C60, is insulating at low temperatures. The temperature dependence of the Mu0C80 linewidths reveals that all have a similar band structure: they are all semiconductors with relatively small bandgaps. Fig. 4.8 shows the T2-1 linewidth parameter obtained by fitting the itSR time spectra to an exponential decay of the muonium precession amplitude R(t) = exp(—t/T2). The data were taken using an applied field of ,--,5 G. Recall that in a low applied field 1112 = 1/23, so the full m.uonium amplitude precesses at the same frequency. The fact that we see muonium spin precession at all tells us that the spin-exchange rate v is slow and hence T2-1 a v [see equation (3.28)]. Indeed, with the addition of a constant background term Ao, the linewidth T2-1 for all three fullerites exhibits the exponential temperature dependence expected for a semiconductor (equation 3.33). We estimate the bandgaps from a fit to the function T2- 1 = akT exp(—Eg/2kT)+ )to.  (4.11)  The estimated bandgap Eg, prefactor a and background relaxation Ao for the three solids are given in Table I. Our results for  K4 C60  conflict with the simple band-structure model of doped ful-  lerites. In this model, the four added electrons partially fill the bands formed from the triply degenerate ti, orbitals. Consequently, K4C60 is predicted to be metallic. We must conclude that some mechanism splits the degeneracy of the th, orbitals resulting in a filled band for x=4. Alternatively, electron-electron interactions may be responsible for the insulating behaviour. Although there is a large statistical uncertainty in the measured gaps, our data suggest that the gap in K4C60 is larger  59  Chapter 4. Results and Discussion^  3.5  K 4C  2.5  60  1.5  0.5  7 6  Cf)  K6  5  C60  4 3 C\t 2  16 14 12 10  50  100  150^200  250  Temperature (K)  Figure 4.8: T2-1 linewidths for MuOC60 in (a) K4C60, (b) K6C60, and (c) Rb6C60•  Chapter 4. Results and Discussion^  60  than that of K6C80. The gap in K6C60 is presumably the difference between the th, and the next higher t19 orbitals. NMR results on Rb4C60 show a non-Korringa-like and strongly temperature dependent TIT, also indicative of a non-metal.[26] The parameters for Rb6C60 are noticeably different than those of K4C60 and K6C60. The energy gap is about that of either K4C60 or  K6 C60.  Also the prefactor a, which is  related to the DOS (equation 3.33), is much smaller and Ao is much larger for RbsCso• The enhanced background relaxation Ao of Rb6C60 may come from a contribution to Ao from the dipolar fields of the alkali ions—in A6C60 each  C60  is surrounded by 24 A  ions. In this case, Ao would scale with the magnetic moment, which for rubidium is times that of potassium. The estimates of a and E9 for Rb6C60 also disagree with the simple band picture. With a larger lattice constant and hence smaller hopping amplitude, Rb6C60 should have a smaller bandwidth. Consequently, we would expect it to have both a higher density of states and a slightly larger bandgap in comparison to K6C60. Our results show both a much smaller bandgap and a reduction in the density of states. The statistical uncertainties for the Rb6C60 may be underestimated, however, as one can fit the data with an a and E9 similar to that of  K4 C60  and  K6 C60  without a significant  increase in the x2/degrees of freedom (0.6 versus 1.2). The results for K4C60 and K6C60 along with additional details concerning these experiments were previously reported by Kiefl et a/166)  Chapter 4. Results and Discussion^  61  Table 4.1: Fraction of injected muons which form a diamagnetic center (fD), muonium (fmu) or radical (fR) at 5K. Ap is the the muon-electron hyperfine parameter of muonium and a, .E9 , and Ao are fitted parameters defined in Eqn. 4.11. A.4 a Eg A0 fp fmu fR (MHz) (eV's') (eV) Cus-1 (%) (%) (%) 2(5) 12(2) 60(10) 4341(24) C60 68(5) 7(2) 4342(66) 3.8(5) x 10 0.40(20) 1.20(6) K4C60 69(5) 6(2) 4230(63) 1.5(2) x 108 0.35(12) 1.40(10) K6C60 81(5) 7(2) 4(20) x 103 0.10(3) 4.30(20) Rb6C60  Chapter 5  Conclusion  The work described in this thesis further characterizes the two muonium centres observed in crystalline  C60.  Zero-field measurements show that the hyperfine tensor of  C60Mu is completely anisotropic and give a measure of the anisotropy. Low-field measurements on single crystals are needed, however, to unambiguously determine the pricipal components of the hyperfine tensor. Apart from its being vacuum-like, less is known about the hyperfine interaction of MuOC60; there is some hint that it may also be slightly anisotropic as in the case of Mu©C70157] Transverse-field ,uSR of C60Mu* provides evidence for the first order nature of the orientational phase transition in solid  C60  and shows a change in the rotational dynamics  at To, the ordering temperature. Below To, the activation energy for reorientation of C60Mu is independently estimated using the zero-field and the transverse-field data. Both results agree within the given uncertainties. The observation of coherent spin precession of muonium in pure  C60  and the several  doped fullerites we examined establishes that all are nonmagnetic at low temperatures. Although this aspect of the experimental results was not discussed in this thesis, we can be sure that any electronic moments in these materials would have to be smaller than a fraction of a nuclear magneton since the positive muon alone is capable of detecting moments on this order. In the presence of electronic moments, one would expect an exchange interaction with the Mu electron which would either substantially broaden the Mu lines or greatly alter them. A detailed analysis with some estimate of the exchange  62  Chapter 5. Conclusion^  63  interaction could in principle set limits on both static and fluctuating moments at low temperatures. Finally, we find ,uSR of Mu@C60 in the doped fullerites to be sensitive to the electronic band structure. 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