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Theoretical spin dynamics on muonium level-crossing resonance Yen, Hon Kit 1988

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T H E O R E T I C A L S P I N D Y N A M I C S O N M U O N I U M L E V E L - C R O S S I N G R E S O N A N C E By H O N K I T Y E N B . E . S c , University of Western Ontario, 1986 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF M A S T E R OF A P P L I E D S C I E N C E in T H E F A C U L T Y OF G R A D U A T E STUDIES Department of Physics We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A September 1988 © Hon Ki t Yen, 1988 . ' 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 Redfield's theory and the theory of master equations have been reviewed and their applications to muonium spin dynamics discussed. It was found that both theo-ries are equivalent in the Markov limit. In some cases, analytical expressions for relaxation rates are found. In addition, Redfield's theory was applied to describe spin relaxation of muonium-substituted free radicals near level-crossing resonances. Theoretical predictions were compared with experimental data for the CQFQ-MU radical and the results suggest there are several relaxation mechanisms involved. n Table of Contents Abstract ii List of Figures v Acknowledgements vi I Introduction 1 1.1 Introduction to Muon Spin Rotation 1 1.2 Introduction to L C R 2 II Basics of Muonium Spin Dynamics and L C R Spectroscopy 5 II. 1 Theory 5 II. 1.1 Muonium Spin Dynamics 5 II. 1.2 Muon polarization in longitudinal field; Treatment in Hilbert Space 11 II.1.3 Muon polarization in longitudinal field; Treatment in Liou-ville Space 14 II. 1.4 Muonium Interacting With Nuclear Spins: the idea of Level Crossing 19 III Theory of Spin Dynamics 22 III. l Master Equations 22 III. 1.1 Formal Derivation 22 III. 1.2 Born Approximation 26 III. 1.3 Applications to Muonium 29 III.2 Redfield Theory of Relaxation Processes 35 iii 111.2.1 Formal Derivation 35 111.2.2 Application to Muonium Relaxation 39 IV Explicit Calculations and Results 43 IV. 1 Introduction 43 IV.2 Muonium Relaxation 43 IV.2.1 Application of Redfield's Theory . 43 IV.2.2 Relaxation in Transverse Fields 44 IV.2.3 Muonium Relaxation in Longitudinal Fields 46 IV.3 Muonium Relaxation in L C R 51 IV.3.1 L C R without relaxation 51 IV.3.2 Spin Relaxation in Electron Tt Model 54 IV.3.3 Spin Relaxation in Anisotropic Randomly Fluctuating Local Field 57 IV.3.4 Explicit Application to C 6 H 6 - M u 60 V Conclusions 62 Bibliography 64 IV List of Figures 1 Breit-Rabi diagram for muonium in Ge 8 2 Fourier transforms of /iSR spectra in quartz and in slightly p-doped Si. 10 3 Residual muon polarization versus longitudinal magnetic field strength. 12 4 Parallel(A||) and transverse(Ai2 and A 2 3 ) relaxation rates vs field for Sex = 3.16 x 1 0 85 _ 1 and r c = 10 _ 1 15. The hyperfine parameters used are normal muonium in Si 48 5 Parallel and transverse relaxation rates vs field for 6ex = 10 8 s _ 1 and r c = 10- 1 0s 49 6 Parallel and transverse relaxation rates vs field for 8SX = 3.16 x 10 7 6 _ 1 and rc = 10~9s 50 7 Time evolution of the muon decay asymmetry in L F on F(6) level-crossing resonance 61 v Acknowledgements It has been my pleasure to work with Dr. M . Celio whose creativity and hospitality greatly benefits me both as a student and a friend. I would like to thank Prof. J . Brewer for providing my fundings during the past two years and for introducing me to the friendly and inspiring /.iSR group at T R I U M F . Finally, I would like to express my gratitude to Dr. R. Kiefl for carefully reading the manuscript and my appreciation to R. Balden for his patience in teaching me to handle the Symbolic Manipulation Program on the V A X system. VI Chapter I Introduction Muon spin rotation ( fj,SR ) experiments involving the formation and description of paramagnetic muon states (muonium) have reached a considerable level of maturity in recent years. Still, a simple theoretical description of depolarization processes involving muonium is missing. This constitutes the main motivation for this study which tries to provide a simple solution to the problem of muonium spin relaxation in connection with standard fiSR and level-crossing- resonance (LCR) experiments. 1.1 Introduction to Muon Spin Rotation Muon spin rotation (fj,SR) spectroscopy date back to the year 1957, when Garwin et. al. detected the parity violation in weak interaction. In the experiment, it was observed that the muons were subjected to depolarization processes inside the stopping target which depended on the target material used. This observation lead to the to use muons in solid state research. The muons used in a (J.SR experiment are produced from the decay of pions decay. After stopping in the target, the muon spin precesses around some effective field before decaying after a mean lifetime of about 2.2 fis . The muon spin polar-ization can be observed via its weak decay, since the latter implies the emission of a positron in the direction parallel to the muon spin polarization. In many solids, liquids, and gases a positive muon can bind with an electron and form an hydrogen-like atom called "muonium" (Mu). The properties of muo-nium are very similar to those of a hydrogen since their reduced masses are almost 1 equal(m* = .995m#). In a solid, the electron wave-function is usually distorted due to the interaction of electron wave function with its environment. This change , which is not completely understood yet (at least quantitatively) leads to a change in the hyperfme interaction which is directly measured by (J.SR technique. In the present work we deal with the behavior of the muon spin polarization in a muonium atom in cases where the latter is interacting with its environment. We will introduce describe a theoretical framework with the aim to describe muonium depolarization processes, especially in cases where muonium is used in connection with the recently developed level-crossing-resonance (LCR) technique. 1.2 Introduction to L C R The investigation of the properties of free radicals is of considerable importance for both environmental studies and industrial applications. The study of the structure of the different species of free radicals, their reaction mechanisms and reaction rates is therefore of fundamental interest in order to get some understanding about the relevant chemical reactions and process optimizations. The technique of fj,SR compliments other more conventional methods such as optical spectroscopy and electron spin resonance(ESR), which have been applied to study free radicals for more than twenty years. Muonium-substituted free radicals are formed when unsaturated organic molecules like olefins, dienes, aromatic compounds or carbonyls are irradiated with positive muons. How the electronic environment is moclefied by the presence of a postive muon can be inferred from the measurement of the electron-muon hyperfine cou-pling constant which is the aim of standard /J.SR experiments. The latter are usually carried out using the transverse field technique at very high fields where the Zeeman terms in the spin Hamitonian are dominant. A^, which is proportional to the Fermi contact interaction between the electron and the muon, provides infor-2 mation on electronic structure of the radical near the muon. For a more complete review of free radical fiSR prior to the development of the L C R technique, the readers should refer to the review article written by Fischer [1]. The possibilty of using L C R in f.iSR was first suggested by Abragam [2] in 1984. Subsequently Kreitzman et. al. [3] applied this technique to determine the nuclear quadrupolar interaction of the nearest neighbor nuclei to muons in cop-per [3]. The possibility of using L C R to study free radicals was also developed at T R I U M F [4] and stimulated considerable theoretical and experimental work in the whole fj,SR community. The principles of L C R will be explored in details in the next chapters. Here, we only give a rough sketch of the basic principles of this method. Usually in muon L C R , one is faced with a system composed by a muon spin, an unpaired electron spin and surrounding nuclear spins. Roughly speaking, the muon L C R phenomenon occurs when a muon spin transition frequency is tuned, by varying the applied magnetic field, to some other spin transitions in the coupled spin system. From the position of the detected resonances, information about the electron-nuclear hyperfine interaction (nhf) can be obtained, which is important for understanding the electronic structure of the radical. The most remarkable feature of L C R is that in high field, the position and magnitude of each resonance are insensitive to the number of nuclei off resonance. Therefore each nuclear resonance can be observed independently and this makes L C R a powerful spectroscopic technique in resolving nuclear hyperfine(hf) structure in complicated paramagnetic spin systems involving the muon. Although in this work we will only treat muonium states in free radicals, it is worth mentioning that L C R has already been successfully used to provide infor-mation about muonium centers in semiconductors. The resolution of the nuclear hf of the muonium centers in several semiconductors has placed severe constraints 3 on the various models which were originally proposed to describe hydrogen-like defects. This was particular interesting since until very recently no electron param-agnetic resonance(EPR) measurement on hydrogen defects in semiconductors could be reported. Although one of the basic aims of L C R is to determine nuclear hf couplings, the relaxation effects associated with the muon spin polarization function contain information as well, in particular about possible dynamical processes in which the free radical may be involved, such as diffusion, averaging of the anisotropic part of the nuclear hf interaction through tumbling of the radical etc. . In this thesis a theoretical framework will be derived which should allow one to extract information about these relaxation processes. In chapter two, the spin dynamics of muonium and that of muonium plus one nuclear spin will be discussed. The theory of master equations and Redfield's theory for spin relaxation will be treated in chapter three. The explicit application of the concepts presented in chapter three to the problem of muonium relaxation and of muonium relaxation in connection with L C R experiments will be found in chapter four, where the results will be compared with previous theoretical predictions and experimental results. Chapter five contains the conclusions. 4 Chapter II Basics of M u o n i u m Spin Dynamics and L C R Spectroscopy II.l Theory II.1.1 Muonium Spin Dynamics In /J.SR experiments where a paramagnetic state is formed, one primarily studies the dynamics of the muonium spin system in an external magnetic field. Two types of interactions are involved: the Zeeman and the hyperfine interaction. The Zeeman interaction arises from the fact that both the positive muon and the electron spins couple to the external magnetic field. In addition, the two spins are coupled together through the hyperfine interaction. To simplify the discussion, we assume that the electron wave function around the muon is spherically symmetric, i . e. , the hyperfine interaction contains only the Fermi contact term: Hhf = -^-\yiS(0)\2^-?e (1) where | ^ 1^(0) | 2 is the electron density at the muon and the operators for the mag-netic moments are AV = -QvH^Jh (2) fxe = -ge/.iBSe/h- (3) ge = +2 and = —2 are the g-factors of the electron and the muon respectively in the muonium state; \x% is the Bohr magneton for electron and is the Bohr magneton for muon which is 1/207 /^(because / i M = 207m e). Hence, the Zeeman energy splitting is more or less dominated by the spin eigenstates of the electron. 5 The spin-Hamiltonian of a muonium atom is in its lS-ground state in an external field B is given by „ hi/,,- ^ geftB- , D m i i = a • r H r • i? H a • B (4) 4 2 2 ( J where a and r correspond to the Pauli spin operators for the muon and the electron respectively, and hva = Ku0 = ——\^{0)\2gepiB9^tJ, (5) is the hyperfine energy splitting which, in vacuum, is about 4463.3 MHz . The spin Hamiltonian 4 can be solved analytically. Assuming the magnetic field B along the z-axis, the Hamiltonian becomes, „ hu0 _ _ gefiB g^n^ H = —o • T + ~Y~rzB + —^-°zB (6) The state space of the muonium spin system is the tensor product of the state spaces of the muon and the electron spin, both in the z-representation \ap > = \a <g>|/? >e a,/3 = ±l (7) We adopt the notation 1 v l = -Mn = 13.55MHz/kG (8) and r = ^ B 9 ^ = 2.§2MHz/G (9) and introducing the four states: I x i > = | + + > |X2 > = I - + > | X 3 > = | + - > |X4 > = ! - - > the spin Hamiltonian is represented by a 4 x 4 matrix with elements: (10) H ( ^ + " o o o \ n —i^- 4- ^ n 4 2 2 P R 2 4 2 w ^ „ V o o o y~t-v-f) (11) The eigenvalues of the Hamiltonian can easily be found to be: (12) ^2,4 And the corresponding eigenvectors are: + + > E2 > cos/?| - + > +sin/?| + - > E3 > > \E4 > - s in /3 | - -f > +cos^| + - > (13) where cot 2/9 = x = TB/u0. From equations 13, it can be seen that \E\ >,\E2 > and \E3 > form a triplet state which is degenerate in zero fields. The levels E\ and E^(M = 1) diverge linearly with the magnetic field, while E2 and E±(M = 0) have a.non-linear depen-dence on 5 . A plot of the energy levels of the Hamiltonian 6 versus field strength in the case of Germanium (y0 = 2361MHz) is shown in figure 1. The muon spin polarization is the quantity which is measured in p,SR spec-troscopy. The spin polarization in an arbitrary direction is defined as the average value of the muon Pauli spin matrix, First, we shall consider the case where the external field is applied in a direction perpendicular to the initial muon polarization. Assuming that the electron captured by the muon is initially not polarized, one finds for the time dependence of the muon polarization [5]: p =< a > (14) pf(t) = -{cos2 f3[cosu2\t + costc^i] + sin 2 /?[cos uj_t + cosu^]} (15) 7 Figure 1: Breit-Rabi diagram for muonium in Ge 8 From Eq. 15 one sees that only the | A M | = 1 transitions are allowed. The fre-quencies corresponding to the transitions 1 *-> 4 and 3 <-> 4 are usually higher than the experimental resolution (about 600 MHz) and averaged to zero, leaving the experimental transverse field polarization to be Pii(t) = 2 ^ c o s 2 @ cosu2it + sin 2 /3 co s t ^ i } (16) Because of cot 2/5 = x = TB/u0, and using 1 i -cos2/? = -(1 + 2 v y r + ^ 2 ; = 5<1 + 7 T ^ ) ( 1 7 ) Equation 16 can be written as 1 1 x p / ( i ) = - cos Q+t cos Q~t + - , = s i n f i + t sinft~i (18) 2 2 A/I + x2 w i ith 0± = i ( u ; 2 3 ± u , 1 2 ) (19) which indicates that the recorded polarization should show beats. Note also that u23 + vl2 = ( r - 27JB (20) and i P 2 R 2 F7? *23 - V12 = ^ o 2 + T 2 5 2 - = [1 + 0 ( —)] (21)' A Fourier transform of the fiSR spectrum should exhibit two lines with the centre depending linearly and the separation depending quadratically on B. The hyperfine frequency can then be extracted from the splitting of the lines. In the case of silicon, figure 2 shows two additional peaks around 43 M H z that are not predicted by our spin Hamiltonian. These frequencies are associated with the "anomalous" muonium state Mu* [5], whose description is beyond the scope of this work. ... , QUARTZ 293 K 101 G 1 • . SILICON ! 20 K if 101 G • 1 'j • v II' .rj : • . ••• i i 0 50 100 150 200 FREQUENCY (MHz) Figure 2: Fourier transforms of pSR spectra in quartz and in slightly p-doped Si. 10 In the second typical experimental setting, the muons are initially polarized in a direction parallel to the applied magnetic. For the time dependence of muon spin polarization one then finds [5] Pl?(t) = hi + cos22/? + sin22/?cosu>2 4i) (22) Note that x 2 c o s 2 2 / J = r ^ ( 2 3 ) Equation 22 can be rewritten as = 2 ( l + x 2 ) ( 1 + + C ° S C J 2 4 f ) ( 2 4 ) Again, the frequency W24 is very high and effectively averaged to zero. Hence one obtains an expression for p£(t) which is essentially time-independent, 1 + 2x2 . "» = 2T2? ( 2 6 ) The longitudinal polarization, according to equation 25, increases from 1/2 at zero field and approaches 1 asymptotically at high fields where the p,+ and e~ spins are decoupled (Paschen Back region). At the critical field, i . e. , x — 1 or p* = 3/4, the hyperfme frequency is obtained using x = TB/u0. The results of a longitudinal field experiment in Si at room temperature is shown in figure 3. II.1.2 M u o n pola r iza t ion i n long i tud ina l field; Treatment i n H i l b e r t Space In this section, the time dependence of muon spin polarization in a longitudinal field is worked out in detail using the Hilbert space formalism. Using equation 14, the polarization of the muon along the magnetic field is given by PM) =< az(t) > (26) 11 W H,(kG) 1 5 Figure 3: Residual muon polarization versus the longitudinal magnetic field strength. 12 where < . . . > represents the trace operation, i.e. an average over a complete set of states and where In the Hilbert space a basis is given by a complete set of states, and any state can therefore be represented by a linear combination of the basis state vectors. As the state vector evolves with time, one can imagine that the state vector under-goes a unitary in the Hilbert space, which corresponds to a change in the linear combination of the basis state vectors. Assuming the muon is initially polarized in the z-direction, i . e., at t — 0 Since the spin of the bound electron is initially unpolarized, we assume that half of the electron spins are up and the other half are down, a_(t) = eiHtlho-z($)e-iml% (27) (28) (29) The initial state of muonium is therefore : - > (30) First of all, we consider the time evolution of | + + > (which is an eigenvector): + + >= e-iwt\E_ > (31) For the | - |— > state, using 13 we get + - >= sin f3\E2 > + cos j3\EA > (32) Hence aze-wt/n\ + _ > (cos2 f3e~iuJ2i + sin 2 (5e~iuJii)\ + - > sin /3 cos /3(e ) ! - + > 13 and < + - \ e i H t / h a ; e - , H t / n \ + -> = cos2 f3e~iuJ2t + sin 2 (3e sin 2 (3 cos2 j3\e cos2 2/? + sin 2 2(3 cos u>24t (33) Combining equations 26, 30, 31, and 33 the longitudinal field polarization of the muon is A similar treatment can of course be used in the transverse field case. II.1.3 M u o n po la r i za t ion i n longi tud ina l field; Treatment i n L i o u v i l l e At this point we introduce a new formalism, the Liouville space formalism, which offers substantial advantages over the more familiar Hilbert space formalism when dissipation effects have to be considered. After laying out the basics of the Liouville's space, the previous calculation of the polarization of the muon in muonium in longitudinal field will be repeated. The Liouville space is a linear vector space over the complex number field C . While in the Hilbert space the elements are given by the quantum states, the elements in the corresponding Liouville space are given by the set of all Hilbert-Schmidt operators in the old Hilbert space. This means that for a n-dimensional Hilbert space, the dimension of the correspond-ing Liouville space is n2 and the basis in C will consist of n 2 linearly independent operators of 7i. The scalar product in C is defined as p*(t) = -(1 + cos2 2(3 + sin2 2 (3 cos u2At) (34) Space Hilbert-Schmidt operators* > elements of C (35) 14 The operators in C are called 'superoperators' so as to distinguish from ordi-nary operators in Hilbert space. One of most important superoperators in quantum statistical mechanics is called the Liouville operator which is defined as L A = [H, A] = HA - AH (36) In the muonium problem, given by two coupled spins 1/2, one deals with a 4-D state space which means a 16-D Liouville's space. Explicitly, the basis in the Liouville space for this problem is given by: {1 j &x i ®y i Oz i fx i^~yi^"zi &x 7~x i &y Ty) & z Tz, &xT~y; axTz , 0~yTx, 0~yTz , 0~ZTX, OzTy j-where the tensor product operation between the muon Pauli spin matrix cr,- and the electron Pauli spin matrix Tj is implied. The dynamics of the density matrix is governed by the von-Neumann's equa-tion: p = -^[H,p] = -^Lp (37) which has the formal solution p(t) = p(0)e-lLt/n (38) The density matrix of the muonium is given by the tensor product of the density matrices of the muon and the electron: pMu(t) = (^1 + pjt) • a + pe(t) • r + pa0aaTp) . . (39) The spin Hamiltonian 4 that accounts for the hyperfine and Zeeman interac-tion can be rewritten into the form HMU W 0 _ _ <SE • f cJ^  • a . . —-— = —a • T H (4U) 15 where the following substitutions have been made hu0 TlLU0 Tiioe Now, the explicit'forms for the Hamiltonian and the muonium density matrix 39 can be inserted into the von Neumann equation 37 . The solution implies the evaluation of the commutators appearing the right hand side of the von-Neumann equation The calculation is fairly complicated, grouping all the terms together we finally get The time dependence of a particular polarization component can then be obtained by multiplying equation 41 with the relevant Pauli spin matrix and then by taking the trace. For instance, the time development of any muon polarization In the 4 x 4 tensor product space that we are considering the trace of the unit matrix is equal to 4, therefore the final form of the equation for the muon polarization components is: (41) p^(t) is found by multiplying <7j with equation 41, which leads to: (42) (43) Similarly one finds: (44) dt 16 and dp3 dt (45) Equations 43, 44 and 45 define a system of fifteen coupled ordinary differen-tial equations with "constant coefficients". The latter can be rewritten in a more compact form where — iL is a 15 x 15 matrix that contains all the information on the polarization of the muonium. L is called the Liouvillian. In our specific case, the matrix — iL is actually of dimensions 14 X 14 since pzz(t) does not couple to the other polarization components. Furthermore, — iL is block diagonal and can be reduced into one 6 x 6 block, which contains pffi) and is therefore relevant in the longitudinal field case, and one 8 x 8 block. There are two methods to solve the problem. The first involves the diagonalization of — iL and a decomposition of the initial density matrix in a linear combination of eigenvectors of — iL. The second method is based on the Laplace transform, which is defined as p(t) = -iLp(t) (46) 5 > 0 (47) Applying it to equation 46, we get (48) which is solved as p(s) = (si + iL)-lp(0) (49) where p(0) is the initial muon polarization. 17 The explicit calculation is again quite lengthy and implies the evaluation of a 6 x 6 determinant. For p*(s) one finds: which can be rewritten using the method of partial fractions as ,2 j 2 2 The final result for the time dependence of the muon polarization component parallel to the applied field is finally obtained by performing the inverse Laplace transform Pl(t) = 1 - , . , a . s a ( l + cos Jul + K + uJH (51) which, using the identities: r = 2ITB Wn . 1 o; 2 4 = u2 — U 4 = yjoji + (27rF 5 ) 2 s = = ———- = cot 2/5 is of course identical with expression obtained in the previous section using the Hilbert space formalism. From the previous calculations it is not clear to identify the advantages of the Liouville space formalism compared to the Hilbert space one; rather the former seems more complicated. However, one should notice that this mathematical tool becomes really useful when one has to describe dissipation phenomena, i.e. in cases where the system of interested is no longer isolated but, by means of some 18 interaction with its surroundings, evolves in time in an irreversible way. In such cases, as we will show in the next chapter, the Liouville space formalism can be extended to include dissipation effects which explicitly show up in a modification of the Liouville operator —iL. II.1.4 M u o n i u m Interact ing W i t h Nuc lea r Spins: the idea of Leve l Cross ing Unti l now we have only discussed the time evolution of an isolated muonium spin system in an external magnetic field. Here we extend the previous considerations to the case where the spin of the electron which forms muonium is also coupled to a nuclear spin, a situation which is commonly found in fiSR experiments in free radicals. In the following we will restrict ourselves to the case of a 1/2 nuclear spin, which e.g. represents the situation where muonium interacts with a proton spin. This system is very useful to introduce the level-crossing technique, which was first suggested by Abragam [2] and was further developed at T R I U M F [6]. We write the spin Hamiltonian as H = Aj-T+ANS-J + g6iiBS-B +gliH„I-B+gPHPJ-B (52) where S, I and J are the spin operators for the electron, muon and the proton and where A ^ , A N are the muon (nuclear) hyperfine parameters respectively. One should notice that the interaction between the muon and the nuclear spins is neglected due to the comparatively small muon magnetic moment. If we employ frequency units, the Hamiltonian takes the form: ? = vu— + vv— + ^rzB^azB - ^AZB (53) h " 4 P 4 2 2 2 ( J where <T,T and A are the Pauli spin operators for the muon, electron and the proton respectively. We choose the quantization axis along the z-direction, parallel to 19 applied magnetic field. Futhermore, defining the three positive gyromagnetic ratios as: 9ef^B 7e = h 7^ = h 7 P = h and using a-f = ^(r+a~ + r~a+) + r z a z (54) A - f = ^ ( r + A ~ + r - A + ) + TZAZ (55) we rewrite the Hamiltonian as: f = % ^ + | ( r ^ - + r-a+) + ^ A z + ^ ( r + A - + r - A + ) + f T ^ " "sf^ ~ ~2AzB ( 5 6 ) It should be noted that this Hamiltonian reduces to two l x l blocks and two 3 x 3 blocks, each corresponding to a specific total z-spin. We now concentrate on the spectrum of the Hamiltonian at high magnetic field. B ecause of the fact that >^ 'y^'Tp, the effect of the direction of the electron spin will dominate the energy spectrum. The spectrum will effectively be split into two regions: the higher energy part composed by states with electron spin up and the lower energy part with states with electron spin down. Also, at high field, the Zeeman effect is so dominant that, in general, it is sufficient to approximate the eigenstates of the 3-spin system by the eigenstates of the z-component of the spin operators \SZIZJZ >. However, this is not true when accidental degeneracies occur, so that when two muon-nuclear hf levels are energetically close to each other and an effective mixing occurs. In such a case , all eigenstates that are not directly involved are still good whereas the two energetically close states must be replaced by appropriate linear combinations in the spirit of second order perturbation theory. Looking at a level crossing within the context of /J,SR, where the spin polar-ization of the muon is the quantity which is measured, the most interesting feature is that the muon polarization develops amplitude oscillations at the magnetic fields where two energy levels with different muon spin states get close. Consequently, in an integral fiSR experiment [6] a resonant like effect on the muon decay asymmetry is expected as the magnetic field is scanned. From the position of the resonances, and knowing the value of the muon hyperfine interaction, information about the nuclear hyperfine interaction (nhf) can then be inferred from the experiment. This is known as level-crossing resonance (LCR) technique, a method which has already found wide applications in the study of defects in semiconductors and of muonated free radicals [6]. 21 Chapter III Theory of Spin Dynamics III.l Master Equations III.1.1 Formal Derivation The theory of Master Equations and the projection operator technique were devel-oped by Zwanzig [7] and Nakajima [8] in the late 1950's. The aim of the theory is to investigate the dynamics of an open system(A) interacting irreversibly under the influence of the surroundings(S) [9]. In general, both A and B can be com-posed of many subsystemsjhowever, for the sake of simplicity, we shall restrict our analysis to a total of two subsystems, namely, the subsystem of interest(S) and the reservoir(i?). In describing dissipation phenomena(e. g. spin relaxation), it is assumed that the reservoir is macroscopically large in order to bring in the irreversible nature of dissipation. To obtain the equation of motion for S, we further assume that S and R are initially separated and that at t = 0 and only the macroscopic properties of R are known. The equation of motion for S can be obtained by averaging over the degrees of freedom that are not required, namely, those of R. A very elegant way to perform this operation is given by the Zwanzig's projection operator technique [7]. In quantum statistical mechanics, the states of a system are described by the density operator p, which dynamics obey the von-Neumann equation, .dp(t) i- dt 22 = [H,p(t)] (57) where H is the Hamiltonian of the total system, and where we have set U = 1. Introducing the Laplace transform p(z)= / dte-ztp(t) (58) Jo , with z £ C, Equation 57 becomes: izp(z) - ip(0) = Lp(z) (59) where L is the Liouville operator introduced in Chapter 2. Since S and are assumed to be uncorrelated at t = 0, the system density matrix can be written as p(0) = p°spR (60) where p% is a stationary distribution. We write the total Hamiltonian as: H = HS+J1R + (61) H0 interaction and we define the reduced density matrix as Ps = TTRP (62) In order to calculate the equation of motion for p$, we introduce a linear operator P which acts on elements X of C in the following manner: PX = PRTrRX (63) and P2X = p%TrRPRTrRX - PX (64) It is evident that the introduction of p°R in the definition 63 is to ensure proper renormalization. We denote the complement of P by Q. 23 If we operate with P and Q on the von-Neumann equation 57 separately, we get two coupled equations for Pp and Qp respectively: izPp(z)-ip(0) = PLPp(z) + PLQp(z) (65) izQp(z) = QLPp(z) + QLQp(z) (66) (note that Pp(0) = p(0) and as a consequence Qp(0) = 0). Solving equation 66 for Qp(z), we obtain, Qp(z) = (iz - QL)-1 QLPp(z) (67) Now we can substitute equation 67 into equation 65 to obtain the equation of motion for Pp(z): izPp(z) - z>(0) = PLPp(z) + PL(iz - QL)-1QLPp(z) (68) It is further assumed that TrR(Vp°R) = 0 (69) Actually the last equation imposes no restrictions since one can always include the diagonal elements of V into H0 by an appropriate redefinition. By assuming TrR(VP°R) = 0, we get a group of useful relationships: PLyP = 0 (70) P(LS + LR) = LSP (71) LRP = 0 (72) (PL)Q = (PLv)Q (73) QLP = LVP (74) (QL)Q = (LS + LR + QLV)Q (75) Using equation 71, we can write PLPp(z) = LsPp(z) (76) 24 Combining equation 71 - equation 76 and also equation 69, we get izPp(z) - ip(Q) = LsPp(z) + PLv(iz - L S - L R - QLvyxLvPp(z) izpRTrRp(z) - ip0Rp°s = LsP°RTrRp(z) + p°RTrR(Lv(iz - Ls - LR - QLv)~1)Lvp°RTrRp(z) Since by definition TrRp(z) = ps{z), the equation of motion of the reduced density matrix Ps(z) i s [iz-Ls-M{z)]ps(z) = ips (77) where the memory kernel is given by M{z) = TrR{Lv(iz - L S - L R - QLy)-1 Lv P°R] (78) Rewriting equation 77 as izps(z) - ip°s = Lsps(z) + M{z)ps(z) (79) and performing the inverse Laplace transform, the evolution of pg in the time do-main is given by .dpg) = ^ + ft ^ , M ( t , ) p ( t _ t l ) ( 8 0 ) at Jo with M(t) = -iTrR(Lve-^Ls+LR+QLvt^ LvP°R (81) The first term on the right of equation 80 describes the unperturbed motion of ps while the second term, which describes dissipation, may be viewed as a generalized collision operator correct to all orders of V. The above formalism provides a foundation for obtaining phenomenological expressions that describe dissipation. Although equation 77 - equation 81 are microscopically exact, they are usually too complicated to be used to solve practical problems. Obviously, the difficulties lie in the complexity of M. Practically, the application of the Master equation formal-ism to physical problems requires an approximate treatment of M. Conventionally this can be done by performing perturbation expansion on the exponential operator exp[—i(Ls + LR + QLy)t] in powers of Ly: -| CO M(z) = TrR[Lv, — Y\QLv(iz - Ls - LRx]nLvpR (82) iz - L s - L R n = 0 The convergence of equation 82 is guaranteed only if the interaction between R and S is not too strong, i.e. only if the order parameter 0{Cy)/0(Cs + £n) is small enough for the series to converge. If the interaction is small enough, then the Master Equation approach offers substantial advantages over the ordinary perturbation solution of equation 57, which simply consists in an expansion of the time evolution operator in ps(t) = TrR{exp[-i(Ls + LR + Lv)t}}p(0) (83) in terms of Ly. The reason for this is that an infinite number of terms of the expansion 83 has to be summed up in order to recover a finite order approximation to the generalized Master Equation [10]. III .1.2 B o r n A p p r o x i m a t i o n In the case of a system interacting weakly with the reservoir, the Born approxima-tion can be used, which consists in replacing the exponential operator exp[—i(Ls + LR + QLy)t] by exp[—i(Ls + LR)t], and in treating Ly as the perturbation param-eter. In the lowest order approximation, the memory kernel M(t) is therefore of order V2. 26 To simplify the analysis, it is convenient to expand V in terms of a complete set of system and reservoir operators: V = X>^ (84) k where uk and vk operate in the space of the reservoir and the subsystem of interest, respectively. Using e-iL°TV{t) = e-iH°rV(t)eiH°T = V(t - r ) (85) Equation 67 can then be written as {dp^) =Lsp_(t) + j* drM(T)ps(t _ T) (86) In the Born approximation, the second term on the right side of equation 86 becomes = -iTrR f dTLve-zL°TLvp°Rps(t - r) Jo = -iTrR f dT[J2ukVk,e-iL°TlJ_u\V\, P°Rps(t - T)}} Hence ips(t) = Lspsit) -i__f dr{ckX(r)[vk, e-iL^vxPs(t - r)] - cXk(-r) k\ J o [vk,e-tLsTps(t-r)vx]} (87) Note that all references to the reservoir are now concentrated in the reservoir ther-modynamic correlation functions ckX(r) or cXk(—r) which are defined by ck\(j) = TrR[pRuk(r)ux) The correlation functions can be rewritten as ctx(r) = \[ckX(r) ± cXk(-r)} = ± c ^ ( - r ) (89) 27 Therefore, Ckx{r) = 4x(r) + ca( r ) (90) cxk(-r) = Ck\(r) - Ckx(r) (91) The Laplace transform of equation 87 is given by (iz - Ls)ps(z) = iPs(Q) + M{z)~ps(z) (92) with M(z)ps{z) = -iY_[vkJUz + iLs)[vx, ps(z)} + jkX(z + iLs)[vx, ps(z)}+] (93) kX where [p, q]+ = pq + qp, and jk\ = / dre-^cUr) (94) Jo If the temperature of the reservoir is much higher than the typical system energies, the influence of the reservoir can then be approximated by a random field of force acting on the electron spin [11]. In this case Ck\(r) —+ CfcA( | r | ) and jk~x = 0. This result considerably simplifies equation 93, M(z)~ps(z) = -i_Z[vkJkx(z + iLs)[vx,ps(z)]] (95) it A So far we have considered memory effects in obtaining the equation of motion. How-ever, if the relaxation time of the reservoir, r c , is much shorter than the relaxation time of the system of interest, the memory effect becomes insignificant and the formulism reduces to the Markovian approximation. This can be mathematically characterized by the condition that all thermodynamic correlation functions Ckxij) satisfy Ckx(r) == 0 for |r | > r c (96) The energy spectrum of the reservoir is then continuous unless the C^A'S are quasi-periodic functions of r as implied by equation 87. 28 Under this restriction, in the long time limit(i >> r c ) , the equation of motion for the subsystem of interest simplifies to: ips(t) = Lsps{t) -iJ2[vk,jjt\(iLs)vx,ps(t)] + jk\(iLs)vx,Ps(t)]+] (97) k\ The Laplace transform of this Markovian equation of motion is [iz - L s - M(z)]ps(z) = ip°s (98) and M{z)ps{z) = -iYXvk,jHiLs)vx,ps(t)] + jkx(iLs)vx, Ps(t)]+] (99) kX Futhermore, if the influence of the reservoir on the subsystem of interest can be described by a random field of force, the memory kernel of the Markovian equation of motion further reduces to the following form: M(z)ps(z) = -iY,[vkJkx(iLs)vx,ps(z)]] (100) III.1.3 Applications to Muonium In this section, the theory discussed in section III. 1.1 will be applied to describe the behavior of muon spin polarization of a muonium atom in a magnetic field which is interacting with the surroundings. In this model, the bound electron is responsible for the interactions with the reservoir. In dealing with the spin dynamics of a muonium system, it is convenient to represent the observables by the direct product of the Pauli spin matrices T1 and a 3 (hJ —1>2,3) of the electron and the muon respectively, and r° = a° = 1, where 1 is the unit matrix. Of course, the direct products of the Pauli spin matrices will serve as the basis in Liouville's space as we have discussed in section II. 1.3. The basis vectors are denoted symbolically by: fAP = A A ®Tq a,(3 = 0,1,2,3 (101) 29 The scalar product is naturally defined as (fQ(SJ^) = Tr{aarpa'itv) = 4£ a„fy„ (102) In this representation, the Hamiltonian 4 for the muonium takes the form HMu = \f + \f°l + f f° (103) where bi = gefiBBi and c,- = g^p^Bi. (the reader should compare the results with those obtained in section II.1.3). Since we are interested only in the dynamics of the system(the muonium atom), all we need to know is the action of the linear transformation Ls on the 16 basis vector. Borrowing the results from section II. 1.3 and rewriting them in the present notation, we get Lsf°° = 0 (104) Lsf3 = iejki^flk + blfok) (105) LsP° = * ' e i « ( | / w + c//*°) (106) Lsfij = ieijk^(fok - f^) + iejkicifki + iejklb,fik (107) These results can be neatly summarized by representing them in a matrix equation: LsfaP = {LsU/cpr (108) where ( Z ^ ) ^ / ^ i s the 16 x 16 Liouvillian that we discussed in section II.1.3. The initial value of the reduced density matrix p°$ is most conveniently repre-sented by using polarization matrices, P°s = \n°aPfa(i (109) 30 where the initial polarization matrix is defined as Kip = Trs(psfaP) (110) and its Laplace transform reads Tlap(z) = Trs(pS(z)rp) ( H I ) From equation 77, the resolvent operator can be defined as: R(z) = [z + iLs + iM(z)]-1 (112) and is given by the 16 x 16 matrix with elements R{z)rp = {R),ulaP{Z)r (H3) In terms of the resolvent operator, the initial polarization matrix can be expressed alternatively as nQj3(*) = Trs[R(z)psf^} (114) and its matrix elements, in the final form, are given by I M z ) = Rap^z)Trs{psD = RaP/lw{z)Illv (115) Before we can advance with our calculations, the interaction between the muonium atom(S) and its surroundings(R) has to be specified. Due to the domi-nant nature of the electron spin in the spin system, to a first approximation only the interaction between the electron spin and the reservoir is considered. Instead of investigating a particular microscopic interaction in which Se couples to some magnetic moments in the reservoir, we choose a less specific model where all relax-ation mechanisms are approximated by a randomly fluctuating field and discuss the general features of that result. This field is assumed to jump between the values 31 ±Ta(a = x,y,z) with a probability per unit time equals to ( 2 r c ) _ 1 , where r c is the relaxation time of the reservoir. The advantages of this model are that it is capable, in no loss of generality, to describe several possible relaxation phenomena, such as the diffusion of muonium atom in a lattice, the collision of the atom with charge carriers or the flipping of localized spins due to exchange or spin-lattice in-teractions. This simple semi-classical random field relaxation model is discussed in detail in Appendix C of Slichter's book [12]. The main consequence of this model is that the correlation function of the reservoir can be described by a simple decaying exponential function with a correlation time (2 r c ) _ 1 . We write the interaction Hamiltonian V as: V = Y^t)^ (116) where T2's are the cartesian components of the fluctuating field representing the reservoir and 8ex is the interaction strength between the electron spin and the mag-netic field. This model bears some of the essential characteristics of dipolar coupling except that it is much more easier to treat. In fact, one of the main purposes of introducing this model is to reduce the complexity of the formalism discussed in section III.1.1. The correlation functions for the reservoir are defined as ckx - TrR[uK(T)uxp°R] = 8-fTrR[Tk(r)Tx(0)pR] (117) However, taking the trace is nothing more than computing the quantum mechanical average of the correlation Tk(r)Tx(0). Classically, this just represents an ensemble average over the random process. Using the result from Appendix C in Slichter [12], the correlation function is given by, ca ( r ) = S-f6kX < Tl >av e-M/T<, k,\ = x,y,z (118) 32 If we further assume the fluctuating field is isotropic, < T2X >AV=< T] >AV=< T] >AV= 1 (119) we can rewrite Equation 95 as M{z)ps{z) = -iJ2[vk,Jk(z + iLs)[vk,ps(z)}} (120) k\ where /•oo jk(z) = / dre-ZTck(r) (121) Jo or, expressed in our representation, M{z)f°e = -i J2 {fk,Jk(z + iLs)[fokJaP}} (122) fc=l,2,3 The action of jk(z + iLs) on a general basis vector is: Jk(z + iLs)fal3 =8-f[[z + -]1 + ^Ls]- 1 r / 3 (123) Finally, this leads to the final form of the relaxation matrix M(z), M(z) = -i^f Fok{(z + -)l + iLs]-1Fok (124) 4 k=l,2,3 T c where the 16 x 16 matrix Fok represents the action of the commutator [rk,...] on all the elements of the basis. The results we have obtained so far is valid for arbitrary long correlation time r c under the Born approximation. As a consequence of the interaction between the muonium spin and the reser-voir, it is natural to expect either a broadening or a splitting of the different transi-tion lines between the four energy levels of the muonium spin system. The features of the line shape are basically determined by the two parameters 6ex and r c . Dif-ferent combinations and choices of 6ex and r c will therefore lead to a variety of relaxation effects which will be discussed in more detail in the next Chapter. If the correlation time of the reservoir is very small, for example, smaller than the spin relaxation time of our spin system (which is found experimentally to be of 33 the order of 1 pis), memory effects can then be neglected. Consequently, the Markov approximation, (see equation 100) becomes valid. The Laplace transform of the von-Neumann equation can then be written as: (z + iLs + M)ps(z) = p°s (125) where M is now frequency independent(because from equation 100, we know that M does not depend on z in the Markov limit) and its matrix elements are given, in our basis representation, by M=^fE K " 1 + iLsYXp/okF^F^ (126) Although the evaluation of M is still very tedious, in the Markov limit both M and iLs are at least matrices with constant coefficients. Equation 125 can be rewritten as (z + iLeJJ)ps(z) = p°s (127) where Leff=Ls-iM (128) At this stage, in the markov limit, it is clear that solving the equation of motion is nothing more than diagonalizing the Liouvillian Leff. The complex eigenvalues of the matrix —iLeff, which are c numbers repre-senting the complex frequencies governing the time evolution of the polarization components HQp(t), fully determine the spin dynamics of the muonium spin sys-tem. In its most general form, the polarization components are given by i W W = E V A j i (129) j where Aj's are the complex amplitudes found by decomposing the initial polariza-tion in a linear combination of eigenvectors. 34 In the case of free muonium (Lejf = Ls ), no dissipation will occur, and all the eigenvalues A are purely imaginary. The introduction of irreversible processes results in first order relaxation ef-fects which "damp" the transition frequencies. If 8ex is small enough, the free muonium frequencies are barely changed , and the polarization functions 15 and 22 are modified according to: p\{t) = ( l - a ) e - A l l * (130) p\(t) = ^(cos 2 /?e- A 2 l i cos u21t + sin 2 pe~A23t cos u23t) ' (131) where a = | sin 2 2/? and where we have omitted all undetectable terms correspond-ing to high frequencies. A | | ( A 2 1 and A 2 3 ) are called the parallel(transverse) relax-ation rates, and their dependence on the applied field, respectively on 6ex and r c , will be discussed in the Chapter 4. III.2 Redfield Theory of Relaxation Processes III.2.1 F o r m a l De r iva t i on The derivation of the Redfield theory starts from the time evolution equation for the density matrix in the interaction picture[12]: ~ = ~[p*(0)^m] + (^) j\P\Q),H\{t%m{t)]dt' (132) where < n\p*(t)\m >= e^"-^1 < n\p(t)\m > and p(t) = e~hH^p*{t)e^Hot. The same type of relations holds between 7ii(t) and 7i\(t). The aa' matrix elements of equation 132 can be written as: <a| [p*(0) ,« i (<) ] |a '> = Y,i<<ApXW><P\Wt)\<*'> P - < a\H\(t)\(3 >< / V ( 0 ) K > (133) 35 To solve equation 133, we must know the form of the the interaction 7i\(t), es-pecially with respect to the properties of its ensemble average. We assume that we are dealing with a collection of ensembles where each one starts with the same density matrix at t = 0 but evolve differently according to different perturbations T~t\{t) [12]. Futhermore the ensemble average of rii vanishes so that 7ii(t) has vanishing diagonal elements. We define H1{t) = YJHq{t)K* (134) where Kq,s are functions of the spin coordinates and Hq{t) represents a classical field. Since we have assumed that Ti-i(t) is stationary, in this semi-classical approx-imation, the ensemble average of the perturbation is equal to its time average. We can further assume that the time average of Hq(t) is equal to zero.Therefore < a\H\(t)\/3 > = 0 (135) We are now left with the second term on the right of equation 132. To simplify the expression we use the fact that < 0\Hi(t)\0' >= j^p-"?)* < > and T — t — t', and we carry out an expansion of the right hand side of equation 133. After this an ensemble average has to be performed over different 7ii(t), using assumption that the average < a.\H\(t — T)\J3 >< (3'\7ii(t)\a' > is independent of time and vanishes when r > r c . If we consider the range t > r c , the limit of integration can be set to oo. The correlation function GapaiQi{r) is defined as GaPa.p,{T) = < alTi^t)^ >< P'lH^t + r)\a'> (136) where the a's are the row indices and the f3's are the column indices. Substituting equation 134 into the definition of the correlation function, we get Gcwv =J2< < * I # ' I 0 > < P'\K"'\<x' > Hq(t)Hq,(t + r) (137) 36 We define the spectral density Lqq>(uj) of the interaction as Lqq,(u) = / Hq(t)Hql(t + r)e-luJTdr (138) Jo and we decompose the spectral function into its real and imaginary parts 1 f°° Re{Lqq>(uj)} = - H,(t)Hql(t + r) cos urdr = &„# (139) Z J —oo (•OO Jm{L m/(a;)} = - / Hq(t)Hq>(t + T)smcordT (140) Jo The imaginary part of the spectral function can be shown to give rise to a second order frequency shift which is usually omitted [12]. Now, ^p*aa' c a n be written as dp*aa, 1 ^ {< _\K«\/3 X 0'\K«'\a' > [kqql(a - /?) + kqq,(a' - /?') n WW x < a|A' 9 ' | /? > kqql{(5 - /3')e'(<"<»-<'w)'} (141) Using the result of equation 137 and equation 140, the real part of the spectral density(hereafter it will be referred to as the spectral density, neglecting the name 'real') results in the form: Jaa>(}f3>(w) = J2 < <*\Kq\a' >< /3'\Kq \P > / Hq(t)Hq,(t + r) cosurdr qq> J-°° = 2j_ < a\Kq\a' >< P'\K9'\P > kqq, (142) Substituting equation 142 into equation 141 and renaming the variables, the time evolution of the density matrix elements can be cast into the following simple matrix form: ^ = £ R ^ ' w e ^ - ^ ' - ^ p ^ O ) (143) 37 w here I Raa'W = ^ 2 {Jaf3a'f3>(®' ~ P') + Ja/3a'f3'(o! — P)— 8a'0' ^ J-yPiotij ~ P) -<W£*W(7-/? ' )} (144) 7 At this point,we have to ensure that the perturbation expansion is justified, i.e. we must check whether a time range t r c exists such that pfo>(t) = ph>{0) (145) It implies that the evolution of the system concerned must be very slow compared with the scale of time of the surroundings, so that We can now replace p*pp>(0) by p*pp'(t) and obtain a differential equation for p* which will enable us to solve the equation of motion of p* by integration at times much later than t — 0. The result is the Redfield's equation: = Y,R^M^a~^-^']tPh'{t) (147) at The information for the system over the time interval r c is never asked and physically it means that during this period of time the density matrix does not change too much and as a consequence it implies that T i ,T 2 rc. In the long time limit, Equation 147 can be written as o* ' ~ ~ ~%2Raa'PP'P*pp' (148) dt where J2' represents the sum over all states ct's and /3's for which the relation ua — k-V = ^>p — wpi holds. The equation of motion of a physical observable, A,of the subsystem of inter-est can be obtained using the density matrix resulted from solving the Redfield's 38 equation 147 From Paa' >)p*aa> + el(' dt 'a and after substituting these results into the Redfield's equation, we obtain the equation of motion of the physical observable A, III.2.2 Application to Muonium Relaxation Basically, the dynamics of the density matrix in Redfield theory is determined by the form of the matrix R in equation 148. The explicit form of R can be obtained using various methods and it obviously depends on the representation which is used. If we choose the eigenvectors of HMU as the basis representation, we can rewrite the equation of motion for the matrix elements of p as d < A > dt = Yl {j[P,'K0]aa> + Raa'M'PPP1} < Ct\A\a > (149) < m|/j|n >= —i < m\Lsp\n > — < m\Mp\n > (150) First of all, we concentrate on the first term on the right side: < m\LsP\n > = < m\HMup\n > — < m\pHMu\n > = uim < m\p\n > - w „ < m\p\n > Wmn < m\p\n > (151) 39 where, of course, HMU is given by equation 4. Thus, for any operator A, Ls < m\A\n > = ] T ( L s ) m n A , < k\A\l > ki = <-Omn8km8\n < k\A\l > Ls < m\A\n > = u>mn < m\A\n > (152) To evaluate the term < m\Mp\n >, we first consider the matrix elements of Jk(iLs), 82 1 < m\jk(iLs)A\n > = - f ( - + iLs)'1 < m\A\n > 4 rc < m\A\n > mn rc{1 l U - T c 2 ) < m \ A \ n > 1 mn c = r c D m n < m\A\n > (153) where x lLOmnTc /-, r A \ ^ = 1 + 0 , * T 2 ( 1 5 4 ^ 1 mn c and 82 <m\Mp\n> = -^rc < m\[Tk,\jk(iLs)Tk,p]]\n > 82 = -fTcJ2{DJi < mW\j >< j\rk\l >< l\p\n > jkl - D l n < m\rk\j >< j\p\l >< l\rk\n > - D m j < m\rk\j >< j\p\l >< l\rk\n > +D3l < m\p\j >< j\rk\l >< l\rk\n >} (155) After substituting < j\p\n >= E; < j\p\l >< l\n > 40 in the first term on the right hand side of equation 155: and j -» /, / -» p, < m\p\l >= _]j < m\J >< J\PV > in the second term of the same equation, 150 can now be written in a form similar to that of equation 148: < m\p\n >= -iu)mn < m\p\n > -_ZRmn/ji < i\p\l > (156) with S2 Rmn/jl = ~YTC < m\Tk\P >< P\n\j >< l\n > Dpj 4 k V + _Z < l\Tk\P >< P\Tk\n >< m\j > Dip p - <m\Tk\j >< l\rk\n > (Dln + Dmj)} (157) Equation 156 and 157 are the Redfield equations that describe the time evolution of the muonium density matrix. The time dependence of a component of the po-larization vector for the muon or the electron spin obeys: p°P = Tr{p{t)aar0) = ] P < m\p\n >< n\aaTp\m > mn — -i_>2umn < m\p\n >< n\aaTp\m > mn - _Z Rmn/ji < j\p\l >< n\crairp\m > (158) mnjl Futhermore, if we use the fact that P = \EPh(i>^s 7,8 = 0,1,2,3 (159) £7 we can write PaP = _ZQ^h^5 (i6o) •yS 41 where Qafihs = -- Yl^rnn < m\a^TS\n >< n\aaT0\m > ^ mn Rmn/ji < JW^rs\l >< n\aaT/3\m > (161) mnjl Equations 156, 157, 160 and 161, enable us to calculate the effect of different relaxation processes on muonium depolarization. A couple of examples will be presented in Chapter 4, and the results will be compared with those obtained by Celio and Meier [13] using the Master Equations approach. 42 Chapter I V Expl ic i t Calculations and Results IV. 1 Introduction In chapter three, the basic formulation of the problem of muonium relaxation was discussed within two different frameworks. In section III.1.3 the problem was in-vestigated using the master equations approach following the paper of Celio and Meier [13], while the Redfield's theory of relaxation was introduced in section III.2.2. We also discussed that, as the relaxation time of the reservoir gets much shorter than that of muonium (Markov limit), the two formalisms describe exactly the same physical process but in two different representations. This chapter is divided into two parts: in the first part we will focus on the model for muonium relaxation described in chapter three. We will derive analytical expressions for the muonium relaxation rates using the Redfield's theory and we will compare the results with those obtained by Celio and Meier [13]. In the second part we will discuss muonium relaxation processes in a L C R experiment; where analytical expressions for the muonium relaxation rates will be obtained for two different models. IV.2 Muonium Relaxation I V . 2 . 1 A p p l i c a t i o n of Redfield 's Theo ry In section III.1.3, we have discussed the application of Redfield's theory to muonium in the Markov limit and we have obtained the time evolution equation 156 for the muonium spin system. A closer look at the right side of equation 156 shows that 43 it contains two parts: an unperturbed part corresponding to the pure transition frequencies between the muonium eigenstates , and a second term describing the relaxation process. At high enough fields, one can assume that the non-zero matrix elements of the unperturbed part are very large compared to matrix elements of R. Therefore, as a first approximation, it is allowed to neglect the off diagonal elements of R and consider the corrections to the twelve eigenfrequencies are solely due to the diagonal elements of R < m\p\n > (162) for m ^ n. This is true for m ^ n; for m = n instead the diagonal part of the unperturbed term disappears and the full 4 x 4 submatrix of Rmn/mn becomes important and has to be diagonalized. If we denote the diagonal relaxation matrix elements Rmn/mn by A m „ , the solution to equation 162 is given by < m\p(t)\n >= e~iUmnt~Xmnt < m\p(0)\n > (163) where the initial density matrix can be written as K O J ^ E A . r / , (164) * a(3 The polarization vector of the muonium can be expressed as pa(3 = < m\p(t)\n X n\aaTp\m > (165) mn I V . 2 . 2 R e l a x a t i o n i n Transverse Fie lds In the transverse field case, the muon is prepared with a polarization perpendicular to the applied magnetic field, let say in the x direction (while B || z) and the electron 44 is not polarized. The initial density matrix for the muonium can then be written as: p(0) = -(l + ax) (166) Expressed in the basis of the eigenvectors of the muonium, p(0) takes the form MO = j / 1 cos/5 0 -s in /5 \ cos (3 1 sin (3 0 0 sin (3 1 cos (3 V - sin/5 0 cos,3 1 J (167) Substituting these results into equation 165, the transverse muon spin polarization is given by p* - ^ e ( " , W m " " A m " ) ( < m\p(0)\n >< n\ax\m > mn = - ^ e ( - ! ' w " > n - A m „ ) i < m | 1 + a ^ n > < c n^^-n > (168) In the representation of the eigenvectors of the muonium, the non-zero matrix ele-ments of ax are: 12, 14, 21, 23, 32, 34, 41, 43. The sum contains only eight terms and the muon polarization function is then pi = | [ cos2 /5(cos co2lte'X2lt + cos w 3 4 te~ A s 4 i ) + sin 2 /5(cos u>41ie"A"* + cos u23te'X2it)} (169) where we have made use of the fact that A,-j = A ; t- (since only the real parts of A tj's are considered) and LO^J = —u>ji. A n explicit form for the muonium relaxation rates can then be obtained: A21 — 61xTC[ 4 sin2/5 4 L l + a , 2 1 r 2 sin 2 2(3 + 2 cos2 p( + l + ^ V c 2 1 + ^ V 2 ) 23 1 c 1 + "iVc* £ e Vc r 4cos2/5 + 4 sin 4 P] 2^3 4 l l + w 2 3 r 2 sin 2 2/5 + 2 s h V / 5 ( r 1 + + u)\2 r 2 1 + a > 3 2 T 2 ) 1 + ^ V 2 + 4 cos4 /5] (170) (171) 45 A41 — 3^4 ^eVcr 4 C O S 2 / ? • . 2 1 1 1 + u f e 2 • l + ^ 2 4 r c 2 ) sin 2 2ft  + l +cu f 4 r 2 6Lrcr 4sin 2/? + 4 cos4 /?] 4 L l + ^ 4 r 2 sin 2 2/5 + 2 cos2 /?( l + cu 2V 2 + l + w 2 4 r c 2 ' 1 + ^ 2 V 2 + 4 sin4/?] (172) (173) (174) IV.2.3 Muonium Relaxation in Longitudinal Fields In the longitudinal field case, the muon is initially polarized in a direction parallel to the applied field and the electron is unpolarized. The density matrix at t = 0 is then given by P(0) ; ( ! + **) / 2 0 0 0 \ 0 l - c o s 2 / ? 0 sin 2/5 0 0 0 0 \ 0 sin 2/3 0 1 + cos 2/9 / (175) The longitudinal polarization of the muon, which is given by equation 165, takes the form = 7 £ e ( _ ' u ' m n _ A m " ) t < m | l + a2\n >< n\az\m > ^ mn Pl(t) = p(t)u-p(t)33 +cos 2(3(p(t)4A-p(t)22) sin 2 28 . . \ ^ j coscj 2 4(t)e"A 2 4 < (176) where A o 4 — 1 ^ + T T ^ ) + 2 c o s 2 / J ( I 1 1 1 + ^ 2 1 ^ 2 sin 2 2(3 + ^43 ' c 2 , + w|3rc 2 ^ - 2 1 2—J-TrV^ + 4 c o s 2fl (177) 46 In most cases, the frequency U24 is too high a frequency to be detected experimen-tally; therefore, all the information about muonium relaxation in longitudinal fields is contained in the terms p(t)a . Unlike the transverse case, where no diagonal elements of the transition fre-quencies are involved, we now have to diagonalize the self transition elements of the relaxation matrix Rmn/mn- Considering the submatrix alone, we can write down the following equation of motion: Psub — RsubRsub (178) where psub = (pn,P22, P33, PAA) and Rsub is given by / - A . 1 B_ A_ Q B A A R — _ X 2 T •H'Sub — ex c A'12 . A'14 A'12 A'14 B , 2AB B 2AB A'12 X12 A'21 , A'23 „ 0 — 2 - -A_ + _B_ 4-X22, Xn X23 . A 3 4 \ A"l4 A'24 A 3 4 A 3 4 A'14 A 2 4 ' (179) where A = sin 2 /3;B = cos2 /?; X , j = 1 + o;?-rc2. The problem now reduces to the determination of the eigenvalues and eigen-vectors of Rsub- As Tr(psub) = 0, the matrix Rsub is singular and one of its four eigenvalues is equal to zero. Also, Rsub is real symmetric and this implies that all the eigenvalues are real. In general, the eigenvalues and eigenvectors of Rsub depends on the applied magnetic field, and also, of course, on 8ex and rc. In figures 4- 6 the results of the numerical solutions of the master equations approach [13] are shown as discrete points while the results obtained by using the Redfield's theory are displayed as continuous lines for a representative set of pa-rameters 8ex and r c . The hyperfine parameters used are those of normal muonium in silicon (2012 MHz) . As expected, the agreement between Redfield's theory and master equations in the Markov limit is excellent and this supports the fact that both approaches are actually one theory but presented in different representations in the Markov limit. 47 2.0 T i i 1 r 0 100 200 300 4 0 0 Ffeld(G) Figure 4: Parallel(A||) and transverse(A1 2 and A 2 3 ) relaxation rates vs field for 6ex = 3.16 x 1 0 8 5 _ 1 and r c = 1 0 _ 1 1 5 . The hyperfine parameters used are normal muonium in Si. 48 2.0 0 ' 100 200 300 400 Field(G) Figure 5: Parallel and transverse relaxation rates vs field for 8ex = 10 8 .s - 1 and 49 r c =10" 9 s S =3.15x10 7s F f e l d ( G ) Figure 6: Parallel and transverse relaxation rates vs field for 8ex = 3.16 x 1 0 7 5 _ 1 and rc — 1 0 _ 9 3 . 50 It is important to notice that in this work all expressions for the muonium re-laxation rates have been obtained either as analytically expressions(in the transverse field case) or as the roots of a third order algebraic equation (in the longitudinal case). IV.3 Muonium Relaxation in L C R In chapter two, the idea of a system composed by the muon, electron and nuclear spin was introduced and the the corresponding spin Hamiltonian was briefly dis-cussed. In this section we will investigate in more detail the dynamics of this spin system, first in absence of any relaxation process and second by describing two different relaxation mechanisms. In the following we will make use again of the Redfield's formalism . Unlike the master equation approach, which relies on the numerical diagonalization of very large Liouville operators, the Redfield's theory will lead again to analytical expressions for the muonium relaxation function. IV.3.1 L C R without relaxation In this section we will study in detail the mixing between two energy levels involved in a level crossing. Near a level crossing one has to use degenerate perturbation theory, in which in order to estimate the eigenstates and eigenvalues of the nearly degenerate manifold, one treats the off diagonal elements of the Hamiltonian as the perturbation. In such a situation , where the degeneracy involves two levels, say |1 > and |2 >, we have to diagonalize the 2 x 2 matrix whose elements are given up to second order by: 51 £n = e 2 i (180) = Bn + Ep— (181) w. here H' = H u + H 2 2 (182) The reason to use H1 instead of H is to make the 2 x 2 matrix e hermitean. The L C R effect occurs at those fields where the two diagonal elements of the e matrix become equal. If we define tan 26 = 2 ^ (183) e l l — ^22 d2 = |e 1 2 |e-^ (184) the eigenvalues of e can be expressed as E± = ^ ( e i i + e 2 2) ± ^ ( e n - e 2 2 ) 2 + 4|e 1 2 | 2 (185) Similarly, for the eigenvectors one finds |£+ > = c o s 0 e - , W 2 | l > + sinfle i , / ' / 2 |2 > (186) | £_ > = - s i n 0 e - ^ / 2 | l > + cos0e^ / 2 |2 > (187) The value of the phase ip is equal to zero because the matrix elements of e are real. For the upper resonance(electron spin up), the two levels involved are |i > = !+.-+> | 2> = | + + - > and for the lower resonance(electron spin down), the levels are |6> = | - - + > |7 > = | - + - > 52 The evaluation of the eigenvalue problem is straightforward and the results can be found in Ref. [14]. Here we will simply outline the basic ideas of the calculation and quote the results of Heming et al. [14] . To obtain the e matrix, we pick out the appropriate elements from the Hamilto-nian for the muonated free radical shown in chapter two and apply equations 181,185 and 184. The results are expressed in a form which is independent of the domains of resonance(i. e. true for both electron spin up and spin down). The transition frequency between the two energy levels close to the region of resonance is given by: m = { [ ( 7 , - 7 p ) B - ^ ^ + ^ ^ ] 2 A2 A2 , and the mixing angle of the states is tan 29 = A,Ap/27eB ( l i l - l p ) B - \ \ A , - A p \ + ^ The field of closest approach is found by equating the diagonal elements of e and by solving the obtained equation with respect to B The result is given by; A A A2 — A2 ° ' 2 ( 7 , - 7 P ) 27e(A,~Apl U y U j This automatically leads to a transition frequency at the resonance field: • < B . ) = ( 1 9 1 ) From equation 191 it is evident that, once the value of the muon hf parameter is known , one can obtain the value of the nuclear hf parameter of (Ap) either by measuring either the field at which the L C R occurs or the value of the transition frequency at resonance. 53 IV.3.2 Spin Relaxation in Electron Tt Model In this section we are going to discuss two different models which can cause re-laxation effects as detected in pSR experiments in free radicals. In the present situation it should be noticed that the muonated free radical (the three-spin sys-tem) plays the role of the system of interest S while the environment represents as usual the reservoir R. We start with an electron T i relaxation model, which is actually very similar to the one described previously in the context of muonium relaxation. The only difference here is that in muonated free radicals one more nuclear spin is involved. The interaction looks as before: V = '£SfTi(t)Ti (192) i ~^ where i = x,y,z and T;(i) is the general form of the random field. As discussed in chapter three, the fluctuating field is assumed to jump stochastically between the values ± T a with a probability per unit time l / 2 r c . The correlation functions are expressed as ckx = 6-fSkxe~^ (193) As the in the L C R technique the field is applied parallel to the initial muon polarization, we have to diagonalize the full 8 x 8 submatrix representing all the self transitions. The polarization tensor of the system is a straightforward generalization of equation 160 and is given by p a ^ ( t ) = J2< m\p(t)\n >< n | a a r / 3 A 7 | m > (194) mn The initial density matrix is given by the tensor product of the initial density matrices of muon, electron and nuclear spins. We assume that the muon is fully 54 polarized in the z-direction (B || z) while the electron and the nucleus are both unpolarized, i.e. • P(0) = g(l + ^ ) (195) or, in an explicit matrix form P(0) / 2 0 0 0 0 0 0 0 \ 0 1 - cos 29 sin 29 0 0 0 0 0 0 sin26> 1 + cos 29 0 0 0 0 0 0 0 0 0 0 0 0 0 0' 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 \ o 0 0 0 0 0 0 o ) (196) The longitudinal polarization of the muon is pl = < m\p(t)\n >< n\az\m > mn With our intial conditions P£(*) = Pn(t) ~ Putt) + Pss(t) + P6e(t) - p77(t) - pss(t) + cos29(p33(t) - p22(t)) + sm29(p23(t) + p32{t)) (197) (198) Again, we assume that, at high enough field, the transition frequencies (except the self transitions) are much larger than the elements of the relaxation matrix, so we can write Pmn(t) = e ( - ' ' W m "- A m n ) Vmn(0) for 171 / Tl (199) Hence, the term p23(t) becomes sin 29 P23(t) 5 ( - i w 2 3 - A 2 3 ) t (200) Therefore + cos 29{p33{t) - p22(t)) + cosu23(t)e-x^ ( 2 0 1 ; 55 One can check that, without relaxation, the last expression reduces to m = 4 If we use the following representation 3 cos2 29 sin 2 29 COS U>23< (202) IXi >= | H >; |X2 >= |55 >; Ixa = |44 >; |x< >= |88 > | X 5 > = | 2 2 > ; | X 6 > = | 3 3 > ; | X 7 = |66 >; \ X s >= \77 > the submatrix that is going to be diagonalized is I xn -xn 0 0 0 0 0 0 -xtl 0 0 0 0 0 0 0 0 —Xu 0 0 0 0 0 0 —A'g4 XB* 0 0 0 0 0 0 0 0 j4.Ae: + BX7j 0 — .4A"6; -BX72 0 0 0 0 0 AX73 + B A ' M -BX63 —AX-3 0 0 0 0 -AX63 -BXa AXa + BXa 0 \ 0 0 0 0 -BXn -AX73 0 AX73 + B.y (203) where A — sin 2 8, B = cos2 6 and A',-j = . \ 7 . The structure of Rsub is compara-tively simple, it is composed by two 2 x 2 blocks and one 4 x 4 block which can be diagonalized separately. A further approximation can further simplify the calculation of the eigenvalues and eigenvectors of the 4 x 4 block. Since all the X-jS in the block involves levels with opposite electron spins and because of 7 e 7/J,7 p, it is resonable to assume that, at high field, = xe (204) where u>e = %B . With this approximation, the final form of the muon polarization function in a longitudinal field and subjected to an electron 7\ relaxation process turns out to 56 be 1 sin 4 0 _ A i i cos^e X2t e 2 2 sin 2 20 , , - — - — cos u23te~X23t (205) wnere and Ai = (1 + cos2fl)X e (206) A 2 = (1 - cos2(9)Xe (207) A 2 3 = ^[sin 2 6(D62 + D37) + cos2 6(D36 + D72)} (208) and where all relaxation rates are expressed in units of o\xrc . The time integrated muon polarization function is given by the Laplace trans-form of the polarization function evaluated at s = l / r M , i.e. 1 , sin 4 6 cos4 6 1 -t- A2T^ I 4-"} (209) 2 1 l + A 1 r M  + A r„sin 2 26 1 + A 2 3 r M 2 (1 + A 2 3 r M ) 2 + (u23r^y IV.3.3 Spin Relaxation in Anisotropic Randomly Fluctuating Local Field In the previous section, we have discussed a model in which the electron spin-lattice interaction is the only relaxation process. However, very often in the studies of free radicals, this model is not sufficient to describe all the involved processes. One possibility is represented by a model in which the anisotropic part of the nuclear hf interaction varies stochastically in time, as a result of the tumbling motion of the radical in its environment. In this model, the interaction looks slightly more complicated than before V = 6fTPl{t)rpK, (210) 57 where (3,y = x,y,z and (5 ^ 7. T is assumed to be a randomly fluctuating tensor, whose components have a transition probability of l / 2 r c per unit time to switch between two values. The correlation functions are therefore 81 4 c/?7,/?'y(T) = ~SPl^ll < T | 7 >av e M / T c (211) For simplicity, we assume that <• T 2 > = < T 2 > —< T 2 > = <r T 2 > = < T2 > — <r T 2 > — 1 (212) To study the relaxation of the spin system in a longitudinal field, we simply have to repeat the procedure carried out in treating the electronic T\ model, i.e., we have to solve the eigenvalue problem for the submatrix Rsub which leads to the equations of motions for pa's and then use equation 197 to obtain the polarization of the muon. The initial conditions are assumed to be the same as the ones used in the previous model but here we focus on a L C R involving two states with electron spin down. As a consequence, the initial density matrix p(0) assumes a slightly different form compared to equation 196: / 2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1 - cos 26 0 0 0 0 0 sin 25 V 0 0 0 0 0 0 The polarization function for the muon is given by 0 0 \ 0 0 0 0 0 0 0 0 sin 26> 0 1 + cos 26 0 0 0 } (213) Pl(t) = (Pll(t) + P22(t) + P55(t) - P33{t) - P44(t) - p88(t)) + cos26(p77(t) - p66(t)) + S m . 2 9 cosco67(t)e-Xe7t (214) Again, we make the assumption that the imaginary part of the elements of R contribute only to minor corrections of the transition frequencies at high field, and 58 therefore can be neglected. Futhermore, we approximate the terms X), j ' s in high field by _ 1 - JUjjTc ' J ~ 1+U2-T2  1 t j c 1 + LV2T2 1 = xP if U{j involves the transitions between states with opposite electron spins, and by 1 for transitions between states with same electron spins. Then, one can obtain for example A6 7 = i?67/67 = sin 2 6(D6i + D62 + D65 + D47 + D37 + DS7) + cos2 0 ( D 6 4 + D63 + D68 + D71 + D72 + D75) w 2Xe + 1 where the relaxation rate is again expressed in units of 62xTC . A l l elements of the submatrix R can be calculated, leading to (215) (216) R J U6 — / 2Xe + 1 - 1 0 0 -Ac -AXe -BXC - 1 2 A \ + 1 0 0 -xc -AXe -BXC 0 0 2A\ + 1 - 1 0 —BXe —AXe 0 0 - 1 2 A \ + 1 0 -BXe -AXC -A% -xc 0 0 2 A E + 1 -A -B -AXt -AXt —BXt -BXC -A 2Xe +1 0 -BXt —BXt -AXe —AXe -B 0 2A'« + 1 \ o 0 -xt -xe 0 -B -A 0 0 -xe -xc 0 -B -A 2A% + 1 / (217) where A = s in 2 8 and B = cos2 8. After the submatrix has been diagonalized, the initial conditions pu(0) are used to express the polarization component of interest as a linear combination of eigenvectors. 59 IV.3.4 Explicit Application to C 6 H 6 - M u We have applied the model presented in the previous section to the case of a muonated free radicalformed by muonium addition to hexafluorobenzene(CQFQ), a system which which was recently investigated earlier by Kiefl et. al. [15]. In this system there are six equivalent sites on the carbon ring where muonium can add. After addition there are four L C R ' s expected from the four inequivalent spin 1 /2 flourine nuclei. The muon hf parameter is known from previous experiments in transverse field [16]. Here we are interested in the time dependence of the muon decay asymmetry, which is proportional to the muon polarization function and which is shown in figure 7 for the F(6) L C R , corresponding to a miclear hf coupling of 200.68 MHz . From figure 7 one can see that the entire muon polarization relaxes to zero within a few /is , indicating that both electron and fluorine spins are involved in the relaxation process. Such a behavior can not be described by a Ti electron model because the latter can make only half of the muon polarization is expected to relax. Therefore a process in which the fluorine spin is also involved has to be taken into account, as e.g. the model describing the stochastic fluctuation of the anisotropic part of the nuclear hf interaction. Of course it can not be excluded that some electron Ti process also plays a role in the present situation. This can be accounted quite easily in our description since one can obtain the total relaxation matrix R simply by summing the single contributions. However, the resulting 8 x 8 matrix can not be solved analytically any more. So, for simplicity, in the present investigation, we neglect any electron T i process. A number of simulations were produced by varying the two parameters SEX and TC . We observed that the fit is only sensitive to the product S2XTC but not sensitive to their individual values: Our best fit was obtained with 8ex = 408 /J5 - 1 60 0 2 4 6 8 10 TIMEQusec) Figure 7: Time evolution of the muon decay asymmetry in L F on F(6) level-crossing resonance. and T c = 6.0 x 10~eps , which corresponds to S\xrc of the order of 1.0/is - 1. The fit of the oscillating part of the polarization function can be improved if a nonlinear fit is performed, putting more weight on data at earlier times. The anisotropic fluctuating local field model fits quite well to the C&F6 data except for the oscillating part, which appears to be damped more rapidly than predicted. This further reinforces our previous suggestion that the some electron Ti spin relaxation process is also involved. 61 Chapter V Conclusions In this work we have presented two theories, the master equation formalism and the Redfield's theory, which can applied to describe relaxation processes associated with a muonium spin system. Although there is no doubt that the master equa-tion approach is a very powerful tool when applied to irreversible phenomena, the mathematical complexity associated with it makes it impossible to obtain analyt-ical results and one must rely on tedious numerical calculations. Fortunately, in the Markov limit the master equation approach is fully equivalent to the Redfield's theory, which has the substantial advantage to have the Liouvillain represented in its diagonal form. This makes the whole calculation more transparent and allows one, in many cases, to obtain analytical expression for the observables of interest. The comparison of the theoretical predictions based on this formalism with experimental data from a muon L C R experiment on the C6F6-Mu radical, although not completely sucessful, shows that the present approach deserves further atten-tion. At present we can only say that neither the electron spin-lattice relaxation model nor the anisotropic fluctuating local field model alone are sufficient to describe the observed relaxation. We believe that a more complete description could be achieved by considering a linear combination of the two dissipation models. In that case, one more paramter representing the relative weight of each mechanism would be needed. In conclusion we believe that, in connection with L C R experiments, the ex-tension of the present work to more sophisticated relaxation models could be very useful in describing dynamical processes associated with muonated free radicals. 63 Bibliography [1] H. Fischer. Hyperfine Interactions, 17-19:751, 1984. [2] A . Abragam. C. R. Acad. Sc. Paris Series II, 299:95, 1984. [3] S. R. Kreitzman, J . H . Brewer, D. R. Harshman, R. Keitel, D. L . Williams, K . M . Crowe, and E. J . Ansaldo. Phy. Rev. Lett., 56:181, 1986. [4] R. F . Kiefl, R. Keitel, S. R. Kreitzman, G. Luke, J . H . Brewer, D. R. Noakes, P. W. Percival, T. Matsuzaki, and K . Nishiyama. JJ,SR Newsletter, 31:1794, 1985. [5] A . Schenck. Mxion Spin Rotation Spectroscopy. Hilger, 1985. [6] R. F . Kiefl. Hyperfine Interactions, 32:707, 1986. [7] R . J . Zwanzig. Led. Theor. Phys.(Boulder), 3:106, 1960. [8] S. Nakajima. Progr. Theor. Phys., 20:948, 1958. [9] For a review of the applications of the Nakajima-Zwanzig theory, see F. Haake, in Quantum Statistics in Optics and Solid State Physics, Vol. 66 of Springer Tracts in Modern Physics, edited by G. Hohler and E . A . Niekish(Springer, New York, 1973), p. 98. [10] Moreno Celio. Master Equations and Applications to Spin Dynamics. PhD thesis, Universitat Zurich, 1985. unpublished. [11] P. N . Argyres and P. L . Kelley. Phys. Rev., 134:A98, 1964. [12] C. P. Slichter. Principles of Magnetic Resonance. Volume 1 of Springer Series in Solid-State Sciences, Springer-Verlag, second edition, 1978. [13] Moreno Celio and P. F . Meier. Phys. Rev. B, 28:39, 1983. [14] M . Heming, E.Roduner, D. Patterson, W. Odermatt, J . Schneider, and I. Savic. Chemical Physics Letters, 128(1):100, 1986. [15] R. F . Kiefl, S. Kreitzman, M . Celio, R. Keitel, G. M . Luke, J . H . Brewer, D. R. Noakes, P. W. Percival, T. Matsuzaki, and K . Nishiyama. Phys. Rev. A, 34:681, 1986. [16] E . Roduner, G. A . Brinkman, and P. W. Louwrier. Chem. Phys., 73:117, 1982. 64 


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