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THEORY OF NUCLEAR MAGNETISM OF SOLID HYDROGEN AT LOW TEMPERATURES By YING LIN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1988 ACKNOWLEDGEMENTS I am greatly indebted to Professor Neil S. Sullivan for his clear physical understanding and guidance during this research. His spirit of devotion to work always affected me. I would also like to thank Professor E. Raymond Andrew for his clear course lectures that gave me basic insight into nuclear magnetic resonance. I am grateful to Professors Neil S. Sullivan, E. Raymond Andrew, James W. Dufty, Charles F. Hooper, Pradeep Kumar, David A. Micha, David B. Tanner and William Weltner for their guidance, help, and concern and willingness to serve on my supervisory committee. It is my pleasure to thank Dr. Carl M. Edwards, Dr. Shinll Cho and Daiwei Zhou for their helpful suggestions and discussions. The help from my friends Laddawan Ruamsuwan, James K. Blackburn, Qun Feng and Stephan Schiller with the computer work is greatly appreciated. The cooperation and friendship of my fellow graduate students, as well as that of the staff and faculty of this department, has made my stay at U.F a pleasant and rewarding experience. This research was supported by the National Science Foundation through Low Temperature Physics grants DMR8304322 and DMR86111620 and the Division of Sponsored Research at the University of Florida. TABLE OF CONTENTS page ACKNOWLEDGEMENTS .............................................ii ABSTRACT ............................................... .............. v CHAPTER 1 INTRODUCTION ................... ............................. 1 2 THEORY OF NMR RELAXATION............................... 6 BloembergenPurcellPound Theory .............................. 6 Theory for Liquids in terms of Mori's Formalism .................. 7 Kubo and Tomita Theory.......................................11 Nuclear SpinLattice Relaxation in NonMetallic Solids ........... 16 Nuclear SpinLattice Relaxation for Solid Hydrogen ............... 18 3 NUCLEAR SPINLATTICE RELAXATION ...................... 22 Formulation of Longitudinal Relaxition Time T ................. 22 Temperature Dependence of T1 ................................... 27 Spectral Inhomogeneity of T1 ..................................... 33 NonExponential Relaxation of Nuclear Magnetization............. 38 4 ORIENTATIONAL ORDER PARAMETERS .................... 45 Density Matrix Formalism....................................... 45 Application to Solid Hydrogen .................................... 50 A Proposition for a Zero Field Experiment....................... 55 A Model for The Distribution Function of a....................... 58 5 NMR PULSE STUDIES OF SOLID HYDROGEN.................64 Solid Echoes...................................................... 64 Stimulated Echoes ..............................................70 iii LowFrequency Dynamics of Orientational Glasses...............73 6 SUMMARY AND CONCLUSIONS..............................85 APPENDIX FluctuationDissipation Theory ..................................89 REFERENCES ........................................................ 91 BIOGRAPHICAL SKETCH.............................................96 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THEORY OF NUCLEAR MAGNETISM OF SOLID HYDROGEN AT LOW TEMPERATURES By Ying Lin April 1988 Chairman: Neil Samuel Charles Sullivan Major Department: Physics Systematic studies of the nuclear magnetism of solid orthopara hydrogen mixtures at low temperatures are presented. The formulation of the nuclear spinlattice relaxation time T1 for the case of the local ordering in the orientational "glass" phase of orthopara hydrogen mixtures is given. The temperature dependence of T1 is discussed. A strong dependence on the position of the NMR isochromat in the line shape is found and this is in good agreement with the experimental results of a group of physicists at Duke University, provided that the crossrelaxation is taken into account. The relaxation is found to deviate considerably from an exponential recovery. The orientational degrees of freedom of ortho hydrogen molecules in terms of density matrices and the irreducible tensorial operators associated with unit angular momentum are described. The range of allowed values for the orienta tional order parameters is determined from the positivity conditions imposed on the density matrix. A theory of solid echoes and nuclear spin stimulated echoes following a two pulse RF sequence and a threepulse sequence in the quadrupolar glass phase v of solid hydrogen is developed. The stimulated echoes can be used to compare a "fingerprint" of the local molecular orientations at a given time with those at some later time (less than T1 ) and thereby used to detect ultraslow molecular reorientations. CHAPTER 1 INTRODUCTION The orientational ordering of the rotational degrees of freedom of hydrogen molecules at low temperatures has been carried out by a number of groups in recent years (1 19). The principal reason for this interest is that the ortho hydrogen (or para deuterium) molecules with unit angular momentum, J = 1, represent an almost ideal example of interacting "spin1" quautum rotators, and therefore a valuable testing ground for theoretical models of cooperative behavior. A suggestion that a random distribution of ortho molecules in a para hydrogen matrix may behave as a quadrupolar glass at low temperatures for ortho concentration X < 55% was proposed. It was based on the observa tion that orthopara hydrogen mixtures provide a striking physical realization of the combined effects of frustration and disorder on collective phenomena. These effects play a determining role in spin glasses, and the behavior of solid hydrogen mixtures is analogous to that of a spinglass such as EuxSrl_zS in a random field(20,21). The electrostatic quadrupolequadrupole interaction is the dominant inter action which determines the relative orientations of the ortho molecules, and there is a fundamental topological incompatibility between the configuration for the lowest energy for a pair of molecules (a Tee Configuration for EQQ) and the crystal lattice structure: one cannot arrange all molecules so that they are mutually perpendicular on any 3D lattice. 2 At high temperatures the molecules are free to rotate, but on cooling, the ortho molecules tend to orient preferentially along local axes to mini mize their anisotropic EQQ interactions. As the temperature is reduced, this leads to a continuous but relatively rapid growth of local order parameters, (a = (3J2 2)) which measure the degree of alignment along the local sym metry axes oz. There is a broad distribution p(a) of local order parameters at low temperatures (12,22,23), but no clear phase transition has been detected (18,24). The degree of cooperativity in the slowing down of the orientational fluctua tions of the molecules as the samples are cooled is of special interest in these sys tems and this has motivated several independent experimental studies(2528) of NMR relaxation times T1, which are determined by the fluctuations of the molecular orientations. One of the most striking results reported by S. Wash burn et al. (28) is the observation of a very strong dependence of TI on the spectral position within the NMR absorption spectrum. While a qualitative in terpretation of this behavior in terms of "sloppy" librons has been offered(1,28), a more detailed treatment has been lacking. One of the aims of my research is to extend earlier work (29) by using a straightforward theory and to compare the results with the experimental data. In the low concentration regime, although it is agreed that (i) the low temperature NMR lineshapes indicate a random distribution of molecular ori entations (for both the local axes and and the alignment a =< 3J2 2 >, and (ii) that there is apparently no abrupt transition in the thermodynamic sense; there has been disagreement (1,3) over the interpretation of the behavior of the molecular orientational fluctuations on cooling from the high temperature (free rotator) phase to very low temperatures (T < 0.1K). While some early 3 work reported a very rapid, but smooth variation (25,27,30) with tempera ture, corresponding to a collective freezing of the orientational fluctuations, subsequent studies (14,26,28) indicated a slow, smooth dependence with no evidence of any strong collective behavior. In order to resolve this problem it is important to understand two unusual properties of the nuclear spin relax ation rates in the glass regime. These two properties which will be discussed in Chapter 3 are (i) the spectral inhomogeneity (14,29,30) of the relaxation rate T'1 across the NMR absorption spectrum, and (ii) the nonexponential decay (14,31) of the magnetization of a given isochromat; both result from the broad distribution of local axes and alignments for the molecular orientations. Both properties alone lead to variations by more than an order of magtitude and need to be understood theoretically before attempting to deduce charac teristic molecular fluctuation rates from the relaxation times. It will be shown that the strong departure from exponential decay for the magnetization can be understood provided that the broad distribution of local order parameters is correctly accounted for. In the socalled "quadrupolar glass" the quantum rotors cannot in gen eral be described by pure states and a density matrix formalism is needed to describe the orientational degrees of freedom. It is needed to determine the precise limitations on the local order parameters (molecular alignement, etc.) from the quantum mechanical conditions imposed on the density matrix and to discuss the implications for the analysis of NMR experiments. Some of the considerations for the spin1 density matrix description have been given else where (29,32 37) but solid H2 is a special case because the orbital angular momentum is quenched. The case will be discussed in chapter 4. 4 In molecular solids, one has to consider two kinds of degrees of freedom: translational degrees of freedom for the centreofmass motion and orientational degrees of freedom for the rotational motion of the molecules. One of the most fascinating problems encountered in the molecular solids is the existence of glasslike phases in which the molecular orientations become frozen without any significant periodic correlation from one site to another throughout the crystal. The most striking example of these "orientational glasses" is probably that observed in the solid hydrogen when the quadrupole bearing molecules are replaced by a sufficient number of inert diluants. If the quadrupoles in this frustrated system (20) are replaced by "inert" molecules, it leads to large reorientations of the quadrupolebearing molecules in the neighbourhood of the inactive diluants and the disappearance of long range order when the quadrupole concentrations is reduced below 55%. The HCP lattice is apparently stable down to very low temperatures (7) and the NMR experiments indicate that the molecular orientations vary in a random fashion from one site to another, both the directions of the local equilibrium axes and the degree of orientation with respect to these axes vary randomly throughout a given crystal. The important questions are (i) whether or not the freezing of the molecular motion persists for time scales much longer than those previously established for the glass phase and (ii) how the freezing occurs on cooling. In order to answer these questions a new type of experiment was clearly needed. The echo techniques are of great practical importance in NMR measure ments on the orientationally ordered hydrogen, because they allow one to ex tract information which is not easily or unambiguously determinable from ei 5 their steadystate line shape or FID analyses. Conventional continuous wave (CW) and free induction decay (FID) NMR techniques have only been able to show that the orientational degrees of freedom appear to be fixed for times up to 104 105 S. A considerable improvement may be achieved by the analy sis of solid echoes for which the observation times during which one can follow molecular reorientations are extended to an effective relaxation time (T2)eff, which may (by a suitable choice of pulses) be much longer than the transverse relaxation time T2 that limits the conventional techniques. It will be shown that spin echoes and stimulated echoes following a two pulse sequence and a threepulse sequence, respectively, provide a more pow erful means of investigating the orientational states and particularly the dy namics of the molecules bearing the resonant nuclei, than the conventional continuouswave technique. Spin echoes were observed in solid H2 a long time ago (38) and have been used to study the problem of orientational ordering (11, 14, 27, 28, 39, 40). In order to gain a deeper insight into this problem, a series of questions will be disscused in chapter 5. These are the formation of spin echoes (including solid echoes and stimulated echoes), explanation and comparison with experiments, and the motional damping of echoes. 'CHAPTER 2 THEORY OF NMR RELAXATION BloembergenPurcellPound Theory The theory of spin relaxation in liquids (or gases) is based upon time dependent perturbation theory. In liquids where the spinspin coupling is weak and comparable to the coupling of the spins with the lattice, it is legitimate to consider individual spins, or at most groups of spins inside a molecule, as separate systems coupled independently to a thermal bath, the lattice. It is well known that the expression for the interaction between two mag netic dipoles of nuclear spin I and I can be expanded into 6 terms: HDD = I" *[ Ii2[ j 3(1i n)(Ij n)] = (A +B+C+D+E+F) (2.1) r3 where A + B = (3cos20 ij 31izz) C = j(IIz + lIiz3)sinOcosOei0 D = (Irliz + lizi7)sinOcosOe"i E = I+I+sin2Oe2io 4 z 3 F = IT1:8in26e22o 4 3 7 If we carry out firstorder timedependent perturbation calculations to ob tain the transition probability W between the megnetic energy levels, we find for a nuclear spin I = 1. W = 4h2I(I 1)[J (vo) + J2(2vo)] (2.2) 4 2 and the longitudinal relaxation time 1 T = 2W = 142I(I + 1)[Jl(o) + J2(210)] (2.3) S2 2 This is the BPP expression(41,42); where +00 J(w) = G(r)ei'dr oo00 G(r) = F(t)F*(t + r) 1 Fo = (1 3os20) r3 1 F = sinOcosOei r3 F2 = sin20e2io Theory for Liquids in Terms of Mori's Formalism Daniel Kivelson and Kenneth Ogan reformulated the study of spin relax ation in liquids in terms of Mori's statistical mechanical theory of transport phenomena(43,44,45,46). They started with some wellknown phenomenologi cal magnetic relaxation relations and formulated them in a manner most suit able for comparison with Mori's theory. They obtained simple Bloch equations by the Mori method and extend the treatment to a time domain not adequately described by the Bloch equations. For this dissertation I will just demonstrate 8 the theory for the simple case in which the Bloch equations can describe the relaxation phenomena. We know for a Brownian particle that the Langevin equation is a valid equation of motion for times much longer than the characteristic molecular times: d7 md = v+ F(t) (2.4) Where v is the velocity of the Brownian particle, 7 is the force on it due to an externally applied field, v represents the slow, frictional force where is the friction constant, and F(t) is a force which is a rapidly varying random function which averages to zero. Mori developed a "generalized Langevin equation" which provides a divi sion of time scales into a slow time scale associated with the motion of the Brownian particle and a fast time scale associated with collective motion. Mori chose a set of dynamical variables, for example, A(t), which describe the relevant slow variations in the system. A(t) describes a displacement from equilibrium, i.e. (A(t)) = 0. Let & represent component of the time derivative of A that are orthogonal to A, and are rapidly varying = (1 P)A (2.5) where P is a projection operator. The time dependence of A(t) can be expressed in terms of the superoper ator, N of the Hamiltonian as A=im A (2.6) (2.7) A(t) = e iA The time evolution of a(tp) is determined by a propagator composed only of those components of the Hamiltonian Nx, which lie outside the subspace determined by the slow variables, i.e. a(tp) = exp[t(1 P)i)z]ea (2.8) We now define a memory function matrix k(t), an effective memory func tion matrix K(t) and a relaxation matrix K as follows: Memory function matrix: k(t) = ((tp)). (AA)1 (2.9) Effective memory function matrix: K(t) = k(t)eint (2.10) where ilt =A (A(). (AAt)1 (2.11) Relaxation matrix K: = K(t)dt (2.12) The relaxation matrix may be complex: S= KRE iKIM (2.13) The real part is associated with relaxation times and the imaginary part with frequency shifts. Substituting (2.13) into generalized Langevin equation, we obtain the fun damental relation Mori's equation: A(t) = [i(l + KIM) + KRE] A(t) (2.14) and its Fourier transform: iwA(w) A(0) = in A(w) K(w n) A(w) (2.15) We can now derive Bloch's equation and relaxation time formulae in a simple case (one variable). We express the IIamiltonian as l = )t + ML + ISL (2.16) where MS depends only on the spin variable, ML represents the molecular mo tions and interactions which are independent of the spins, and XSL involves those interactions which involve both spins and nuclear spatial coordinates. The slow variables A can be selected as: A= A'Sz (2.17) S_ We assume that k(t), the memory function matrix, decays rapidly so that K(w n) can be replaced by KRE iKIM. Equation (2.15), transformed back to the time domain, then becomes: Sz (t) =T[Sz(t) (Sz) (2.18) d Si(t) = [i(wo + a) T1]S(t) (2.19) where T = ReKzz = 4 ([)1SL(tp), Sz(tp)][Sz, M ]) dt (2.20) T2 = (kRE) = Re(N) f ()sL(tp), S(tp)][S, )IsLDeTiwot dt (2.21) o = (AIM) = Im( j) ([ ( SL(tp) S(tp)][Sy, SLI)e:Fiw t dt (2.22) Equations (2.18) and (2.19) are the simple Bloch equations except that the a term which represents the socalled nonsecular or dynamic frequency shift is given explicitly. 11 Kubo and Tomita Theory The theory originally introduced by Kubo and Tomita emphasizes the simi larity of magnetic relaxation to other nonequlibrium phenomena (46,47,48,49). Description We know that the Bloch equation can describe the relaxation of the mag netization Mi. It is a linear equation. Sy(t) H(t) mx(t)i + my(t)j mz(t) mZk (2.23) dt T2 T1 In matrix notation the Bloch equation (2.23) can be written as AMi(t) = L AM(t) (2.24) where Amk(t)= mk(t) m~ (k = i,j,k) (2.25) mO is the thermal equilibrium value S0 0 L= 0 iwo 0 (2.26) \0 0 T + iwo If we transform from the laboratory frame to a frame rotating with the Larmor frequency around the z axis, we will have AMi(t) = L'AMR(t) (2.27) The formal solution of Eq. (2.24) is the following: Ama(t) = (eLt)aAmP(t) (2.28) The Fourier transform of the formal solution is therefore a(w) = )a,)Amfi(O) (2.29) 12 + means a positive Fourier transform and in the rotating frame: 1 Amj(w)+ = ( )Ami(O) (2.30) 33 Response Function The linear response formalism begins by calculating the linear response of a dynamic variable to a disturbance created by a timedependent external force. Consider the Hamiltonian: Htotal(t) = H MIHi(t) (2.31) where (i)H describes all the interactions responsible for the motions of the spins, including the effect of the large, static Zeeman field. (ii)Hi(t) is a small, timevarying field which is responsible for the nonequlib rium behavior of the system. The response function fkl(t) is defined as Amk(t) = fkt(t r)Hi(T)dT (2.32) The response function fkl(t) gives the effect of the disturbance at time t. The Fourier transform of eq. (2.32) is Amk(w) Xkl(w)Hi(w) (2.33) where Xkl(W) = eitfkl(t)dt (2.34) Equation (2.34) defines the susceptibility Xkl(w). It has a real and imaginary part that are connceted by the KramersKronig relations. The imaginary part 13 of the susceptibility is called the absorptive part which is related to the power the sample absorbs. The real part of the susceptibility is called the dispersive part and is related to the measured line shape. Relaxation Function The relationship between the relaxation function Fkl(t) and the response function fkl(t) is given by Fkl(t)= fkl()dr (2.35) The relaxation function describes the time change of the response after the external disturbance is cut down to zero. Assume a step disturbance: HI(t) = HleEtO(t) (2.36) where (t) = t 0 (2.37) and e is a small positive constant that will be taken to zero at the end of the calculation. In principle this disturbance corresponds to having a field in addition to the Zeeman field. The response to the step disturbance in the limit E , 0 is: Amk(t) = Fkl(t)H t > 0 (2.38) and Amk(0) = Fkl(0)Hi (2.39) Here Fkl(t) is the relaxation function which describes how the response to a step function disturbance decays in time. 14 If we regard equations (2.38) and (2.39) as matrix equations and formally eliminate the external force in these two equations, we obtain Ama(t) = Fa(t) F"(0)Am (0) t > 0 (2.40) The central assumption of the linear response theory is that Eqs. (2.40) and (2.28) can be combined to yield a molecular expression for L as [eLt]ap = Fa,(t)F73(0) (2.41) We define akl(w) as the following k() Xkl() Xkl(O) = dteitFk(t (2.42) W Jo From equations (2.38) and (2.39): Ama(w) = oaa(w)Hp (2.43) Ama(0) = Xac(O)Hp (2.44) ma(w) = aa'(w)X7(0)Amp (0) (2.45) Comparing equation (2.28) with equation (2.45) yields ( )af = cw)X (0) = ( (2.46) L iw uaJX' (0) in the rotating frame 1 akk(w + Kwo) LTi k = 0, 1 (2.47) Lkk Xkk(O) i.e. Lkk Reakk(w + Kwo) L=kk + 2 (2.48) L'k + Xkkk (0) Using a symmetric form of Okk, equation (2.48) becomes L'kk Reakk(w) L1 Xkk) (2.49) Lkk + Xkk(O) 15 Time correlation Function Formulas for Transport Coefficients The transport coefficient is expressed as a time integral of a correlation function of magnetization. These formulas will be derived in the limit of weak coupling between the spin and lattice degrees of freedom. We assume that the Hamiltonian of the system H(A) can be split into two parts: a part Io that contains the Zeeman Hamiltonian and a part H' that couples the spins to the lattice and is responsible for the relaxation: H(A) = Ho + AH' (2.50) We also assume the transport coefficient L'(A) may be developed in a power series in A with a leading term in A2: L'(A) = A2 E AnL'(n) (2.51) n Since we are assuming weak coupling, we identify the measured transport coefficient with the first term of the sum in Eq. (2.45), i.e. A2L'(0) = L'(A). The results of transport coefficients are 1 ( )2 00 L T1 2(AM2 O dt ([H',Mz[H'(t), Mz)o + C.C. (2.52) 1 L11 = L11 = T2 ( (1)2 0 0  dt {([H', M+ [H' (t), M_ ]) 0 4XT2 +([H'(t),M][H',M+])o + C.C.} (2.53) where XT2 = ([AM, AM+]+)o (2.54) H'(t) = e H' (2.55) H'(t) = ekffotH, kH0() (2.55) 16 The subscript zero on the braket indicates that the trace is taken over the equilibrium density matrix exp(fHo) STr(exp(Ho)) ) which does not involve the spinlattice coupling. Nuclear SpinLattice Relaxation in NonMetallic Solids The problem here is essentially the same as that for liquids and gases, namely to calculate the probability of a flip of a nuclear spin caused by its coupling with the thermal motion of a "lattice." There are, however, some significant differences. The internal motion in solids will often have much smaller amplitudes and/or much longer correlation times than in liquids. In rigid solids because of the tight coupling between nuclear spins exemplified by frequent flipflops between neighbours, the correct approach to nuclear mag netism is a collective one, where single large spinsystem with many degrees of freedom are to be considered, rather than a collection of individual spins. The assumption is usually made that the strong coupling of the nuclei simply establishes a common temperature called a "spin temperature," and that the lattice coupling causes this temperature to change(50,51,52). A quite general equation can be derived. d# 1 dt = ( ) (2.57) where = T8 Spin temperature 0o = T Temperature of the lattice 17 Under certain conditions (practically all experimental situations), high spin and lattice temperature, Abragam and Slichter (51,52) give the following result 1 1 n,m Wmn(En Em)2 . T, 2 ()/0 (2.58) T1 2 (N2) Where I m) and I n) are the eigenstates of No. Wmn = Wnm is the transition probability from the state I m) to the state In). There is another way to calculate T1. It is a density matrix method, which is quite general and especially suited to discussing cases in which motional narrowing takes place. Let the density matrix p describe the behavior of the combined quantum mechanical system, spins + lattice. In the interaction representation p* = e peXot (2.59) i dt [)1(t),p*] (2.60) Where )1 is a perturbation, ((t) = e ockleot Equation(2.60), integrated by successive approximation, gives dp* i 1 to dt h hlLt 2 foi +higher order terms (2.61) or dp* i 1 t d Nh (t),p*(0)]1 dr[ I(t), LI(t r),p*(O)]1 +higher order terms (2.62) 18 Since all the observations are performed on the spin system, all the relevant information is contained in the reduced density matrix a* a* = trf{p*} (2.63) with matrix elements (a I a* a') = f(fa I p*  fa'). By making some assumptions Abragam (51) gives a general master equa tion dt [j (t), [N(t r),a* o]]dr (2.64) where the bar represents an average of many particles. If there is a spin temperature, then d# 1 *+00 odt 2 ( , ) [)t(t), [l (t r), o]]d (2.65) i.e. dp 1 (,o P) [ c dt 2 ()102) (),o[( r), No])dr (2.66) 1 11 O2)0 T1 = 2 7)2 o [(t))o][ tr),4])dr (2.67) For the relaxation of like spins by dipolar coupling the result for T1 is the same as the result of Kubo and Tomita theory. Nuclear SpinLattice Relaxation for Solid Hydrogen The molecular hydrogens (H2,D2,HD,etc.) form the simplest molecular solids. The properties of solid mixtures of ortho (angular momentum J = 1, nuclear spin I = 1) and para (J = 0, I = 0) hydrogen molecules have been extensively studied both theoretically and experimentally in the past decade (53,54,55,56). A popular method of experimentally probing this system has been through nuclear magnetic relaxation studies (57,58,59,60). The relax ation is determined by the orientational fluctuations of the molecules which 19 is in turn determined by the EQQ interaction between the ortho molecules. This relaxation rate, which is a consequence of the intramolecular nuclear spin interactions, is given by (56,61) 1 16 162 S= r {c J (wo) + d [ J (w) (2wo)]} (2.68) Where c denotes the constant of spinrotational coupling and d that of the intramolecular dipolar coupling, with respective values of 113.9 and 57.7 khz. The spectral density functions J(mw) are taken at w0 and 2wo where w0 is the Larmor frequency(51,p278). We consider two regimes. The High Concentration Regime A. B. Harris (56) calculated the spectral functions for the correlation func tions for both infinite and finite temperatures. He used a high temperature expansion method to calculate the second moment and obtained good agree ment with the high concentration experiments of Amstutz et al. (62), with regard to both the temperature and concentration dependence of the relax ation time. Myles and Ebner (63) used a high temperature diagrammatic technique, combined with a simple method of impurity averaging over the distribution of OH2 molecules. The averaged equations were then solved numerically to obtain the spectral functions for solid H2 selfconsistently for the first time. The resulting spectral functions were used to compute the TI as a function of the orthomolecule concentration and this was shown to agree well with experiments at 10K and over concentraiton range of 0.5 < X < 1. They obtained a VX concentration dependence for T1, which was in agreement with the data of Amstutz and colleagues (62) for X > 0.5. The Low Concentration Regime The low concentration regime (X < 0.5) had been explored 'by Sung(64), A.B. Harris(56), Hama et al. (65),Ebner and Sung (66), and Ebner and Myles (67) at an earlier date. Recently, the work has been concentrated on X < 0.5 and very low temperatures (T < 400mk), which details we will discuss in Chapter 3. Sung (64) applied the high temperature statistical theory, developed for paramagnetic resonance with a small concentration of spins, to the calculation of the angular momemetum correlation functions and Harris used an improved version of the same theory. The TI resulting from these calculations had a concentration dependence of X3, which was in agreement with the data of Weinhaus and Meyer (61), but the magnitude of TI obtained in this way was in disagreement with that data. Hama et al. (65) developed a theory which was capable of treating the T = oo correlation functions at all concentrations and which gave a concen tration dependence and magnitude for TI which were in fair agreement with experiment for all X (61,62). Both methods had the defect that the impurity averaged correlation func tions were obtained by statistically averaging assumed functional forms and no attempt was made to determine the shape of the spectral function. The first attempt in the small X region to calculate the high temperature correlation functions selfconsistently and thus to overcome the above defect was made by Ebner and Sung (66). They used the Sung and Arnold (68) method of impurity averaging the Blume and Hubbard (69) correlation func tion theory and obtained a T which had the experimentally observed X tion theory and obtained a T1 which had the experimentally observed XX 21 concentration dependence at small X. Since they made no attempt to prop erly account for the anisotropy of the intermolecular interactions, they did not obtain quantitative agreement with the experimental magnitude of T1. Ebner and Myles (67) improved the calculation of Ebner and Sung by properly treating the anisotropy of the electric quadrupolequadrupole (EQQ) interaction, which is the dominant orientationally dependent interaction be tween two O HI2 molecules in solid H2 and which therefore, almost totally determined the shape of the angular momentum spectral functions. Sung and Arnold's method of impurity averaging the infinite temperature Blume and Hubbard (69) correlation function equations was employed, but the equations were obtained using the full EQQ interaction rather than an isotropic approximation to it. The spinlattice relaxation time was computed as a function of the O H2 concentration using a formula for l derived by applying the Blume and Hubbard theory (69) to the nuclear spin correlation functions in this system. The resulting T1 was compared to the data of Wein haus et al. (61) at a temperature of T = 10K and agreement was generally good with regard to both its magnitude and concentration dependence. CHAPTER 3 NUCLEAR SPIN LATTICE RELAXATION Formulation of Longitudinal Relaxation Time T1 There is a striking resemblance between the phase diagram for the magnetic alloys such as CuMn, AuFe ... and orientationally ordered orthohydrogen parahydrogen alloys(Fig.l2). The nuclear spinlattice relaxation of ortho molecules at low temperatures is determined by the modulation of the intramolecular nuclear dipoledipole interactions HDD and the spinrotational coupling HSR by the fluctuations of the molecular orientations(55, 56). The calculations are particularly transpar ent if we use orthonormal irreducible tensorial operators 02m and N2m for the orientational (J = 1) and nuclear spin (I = 1) degrees of freedom, respectively. The 02M are given by (3J2 2) 020 = z 021 = T (JJ + JzJ) 2 022 = 1 (J)2 (3.1) 2 and similar expressions hold for the N2m in the manifold I = 1. The intramolecular nuclear dipoledipole interaction HDD and the spin rotation interaction HSR can be written in the above notation as HDD = hD Nt(i)O2m(i) i and SR = hC (1)mN m(i)Olm(i) (3.2) i respectively. D=173.1 khz and C=113.9 khz. The index i labels the ith molecule. The 01 and Nim are the operator equivalents of the spherical harmonics Ylm in the manifolds J = 1 and I = 1, respectively. The relaxation rate due to HDD can be shown to be 1T TD I2) ([Iz,WDDI [JLD(t),Iz])Tdt 1 0DD (Ir) Jo'] = (f(Iz, [DD eeHt OMDDe' Iz]dt (3.3) where Ho = hwolz. It is the Hamiltonian responsible for the molecular dy namics. Using the commutators [Iz, N2m] = mN2m we find T1 1=D2 m22m(mwo) (3.4) m=1,2 where the spectral density at the Larmor frequency J2m (w) = (O2m(t)O tm)) eic"otdt (3.5) oo The expectation value < ... >T must be calculated with respect to the fixed Z axis given by the direction of the external magnetic field. This is the quantization axis for the nuclear Zeeman Hamiltonian, which is perturbed by the weaker HDD and HSR terms. The orientational order parameters, however, are evaluated with respect to the local molecular symmetry axes. We must therefore consider the rotations 02m = d2 (x)02O (3.6) where the dmi are the rotation matrix elements for polar angles X = (a, /) defining the orientation of OZ in the local (x,y,z) reference frame. %e3.0 / rW Hexagonal / <2.0 / Cubic wL / 1.0 / Disorder Long Range Diore Order //// Glass / I a I :1 I I ' 0 0.2 0.4 0.6 0.8 1.0 ORTHO CONCENTRATION Figure 1. Phase Diagram of OrthoPara H2 Mixtures (a) T ferromognet 0 ~tate ). spn glass (equiibriu phase) 15 EuxSr.xS T(K) & Fea*/. (b) 10 so PM PM 60 FM 5 FM 1.2 1.6 2.O 00 o.5 X Il T'(103 K1) TIAJ PM (C) 1[ FM SG 0 1 Figure 2. (a) Theoretical Phase Diagram for A Short Range System (b) Experimental Phase Diagram for EuxSrlxS And AuFe (c) Phase Diagram of The Ising Spin Glass (A) LONG RANGE ORDER PARA" (B) GLASS Figure 3. Picture of Long Range Order And Quadrupolar Glass 27 We assume the simplest possible case (02m(t)O ()) =(02m(0)02m(0)))2m(t) (3.7) In the following, we consider only this case and further assume that the relax ation is dominated by the fluctuations of the u = 0 component. We find 1 2 T D = 21 a 2)D2 m2idmo(a)lg2o(mo) (3.8) m=1,2 where the g20(mwo) are the Fourier transforms of the reduced correlation func tions g20(t) and a =(3J2 2)T. The prefactor (2 a a2) is the mean square deviation of the operator 020 evaluated in the local symmetry axis frame. The contribution from the spinrotational interaction HSR is SR 2= C2Jl1(wo) = C2(2 + )dlo1(a)2glO(WO) (3.9) the total rate Tfl = T' + TR Temperature Dependence of T1 Minimum Values of T1 From the previous result (3.8 and 3.9) 1 1 1 ( () = (a,) + (a,O) T1 T1DD T1SR where 1 (o, ) = (2aa2)D2 m21dmo(0)12g2o(mwo) TDD m=1,2 1(oa, ) = 1 c2(2 + cr)dlo(0)12gO(O) TISR 6 If we take a powder sample average, the T1SR is small. T1DD (a) = 12 (2 a a2)D2 9g20(wo) + 920(2wo) (3.10) To a good approximation we have 1 1 1 S(o) (2 a a2)D2 520(W) + g20o(2wo) (3.11) For the simplest case we can restrict a to negative values with appropriate definitions of principal axes (22 and Chapter 4) i 02 o _2 d 1 1 4 S 2 (2,  x 12 i 20(wo) + ,20(2wo) 1 f_2 da 12 5 5 = D2(g20(wo) + 4g20(2w0)) (3.12) Assume g20(wo) and g20(2wo) can be taken to be in the Lorentzian form: 920(wO) = 2 (3.13) 1 + wOc2 g20(2w") = Tc (3.14) 1 + 4wor (3.14) when woTc a 0.6156, T1 = Timin The eqs. (3.12) (3.14) result in the following 1) wo = 2I x 100 x 106 Timin = 13.4172 msec 2) w = 2r x 25 x 106 Timin = 3.3542 msec Tiexp = 1.03 msec Fig.4 (ref.25) 3) wo = 27r x 9 x 106 Timin = 1.20574msec Tlexp = 2.25 msec Fig.5 (ref.70) 103 S102 I 10 1500 500 T(mK) Figure 4. Experimental Curve of TI 5, (msec) Figure 5. Experimental Curve of T1 Temperature Dependence of T1 It has been observed that for the solid, a Gaussian Free Induction Decay is a good approximation for small t and so we should also consider a Gaussian form for g20(mwo) for high frequencies. From eqs. (3.7) and (3.8): t2 (02m(0)02m(0))g(t) = 02m(0)02m(O))e' (3.15) g(W) t2 a 2ae 2 g(w) = e 'wtdt = ave 4 (3.16) Eq. (3.12) becomes 1 D2VW  (ae 4 + 4aea) (3.17) T, 36 36 1 T1 = (3.18) D2 w.02,2a2 ae 4 + 4ae T1 passes through a minimum when d= 0, i.e for a = 5.082 x 109, and the minimum value is Timin = 1.1384msec, at = 25MHz. The result of the theoretical value Timin = 1.1384msec is in very good agreement with the experimental value Tlmin = 1.03 msec(25). Now the question is how T1 varies at fixed w0 over a wide range of tem peratures which causes rc to vary. Similar to the discussion of the dynamics of spin glasses (71), we would like to try an Activation Law of the VogelFulcher type: a = aoe TTo (3.19) where A is the activation energy in temperature unit a0 is a time factor which is the value of a as T , oo, and To is some characteristic temperature (transition temperature), such that as T To, long relaxation times become important. Figure 6. T1T Curve (solidTiexp,dashedTitheo) 33 By using two experimental values (T1 = 2.550msec at T = 0.450K and Tl = 2.514msec at T = 0.380K for X = 38%) we find the following formula: 0.1880 a = 5.4128 x 108 x eTo.5o15 (3.20) The Fig.6 shows curves of results of calculation and experiment. The solid line is experimental curve(25) and the dashed line is theoretical curve. Spectral Inhomogeneity of T1 Results of Calculations The dependence of TI on Av was evaluated by considering the line shape to be a sum of Pake doublets. Each Pake doublet consists of a positive branch given by Av = + DP2(a)u and a negative branch given by Av = DP2(a)a (Fig.7). As previously demonstrated we restrict a to negative values. That is Av = D 1(3cos2 a 1) (positive branch) 22 A = DOl 1(3cos2a 1) (negative branch) 22 In order to test the theory against the experimental data we consider only the low temperature limit for which the fluctuations are slow compared to the Larmor frequency w0. In this case the spectral density functions are given by g(wo) = 4g(2wo) (low temperature limit of Lorentz form), where r1 is the characteristic fluctuation rate, which was taken to be a unique value for simplifying calculations. For the dipolar contribution at fixed a and Av we obtain S2(2 a)2 ( (3.21)] TD (3.21) W01 I(AV) ve branch *D lr/2 P,(a)I1/2  + ve branch , DP ( a ) 2, Figure 7. Allowed Range of Values of a for A Given Frequency Av 35 and for the spinrotational contribution: 1 18C (2 Ial)(1 ) SR (3.22) wo The frequency dependence should be obtained by summing over all allowed a for a given frequency Av. The calculations were straightforward but quite tedious. Table I and Table II show the numerical results. Table I Spectral Dependence of DipoleDipole Relaxation Rate Ti~ D Frequency range Av T17D x (2 1) 0 < 10D212vD+4v2 2v2 2 2Vv S3D2 D(D2v) log H +D(D log 'D < Av < D 5D'7vD+8v' 22 v 2 3D D (Dv) Table II Spectral Dependence of SpinRotational Relaxation Rate T1R Frequency range Av TIIR x (18 2) 2D3v 2L 2v 2u log v o0 < v D log + log D Discussion Fig.8 shows the results of the calculations for the dipolar contribution T1 D and the spinrotational contribution T1 as function of frequency. All curves L1SR in this figure have been normalized to unity at Av = 0 in order to facilitate the comparison with the experimental data. The net relaxation rate T1 1(calc) = TIDD + T1R. The curves show discontinuities in slope at AI = D, but this was not seen experimentally. If we take the finite crossrelaxation rate T1 into 36 account, which will bring the very slowly relaxing isochromats at At/I = 2D into communication with the rapidly relaxing components, T_1 (3.23) T1 T1(alc) + T (3.23) when the individual spinlattice relaxation times are much longer than the crossrelaxation times the spins come to a common spin temperature via the crossrelaxation mechanism before relaxing to the lattice via the rapidly relax ing components. On the other hand, when the direct spinlattice relaxation is fast and TI < T12, the magnetization of each molecule relaxes directly to the lattice and the spins do not achieve a common spin temperature. In this latter case the nuclear magnetization will be spatially inhomogeneous. Yu et al. (14) defined T12 by the probability of a spin flipflop transition via the intermolecular dipoledipole interaction for two isochromats VI and /2 given by T2 = (T2flip fl)op 2( v2)2 2r MArnte J = (T2flipflP) l1 ep I ntra M ntra) (3.24) where the exponential factor is the overlap given by Abragam (51) for two lines centred at vi and /2 and their individual widths are determined by the intermolecular dipoledipole interaction. M ra tra and AMinter are the moments resulting from the intra and intermolecular dipoledipole interac tions, respectively. Yu et al. (14) observed a much weaker dependence than the exponential variation with M'ntra given by the overlap factor. Due to the discrepancy between theory and experiment we chose the empirical values reported by Yu et al. (14) as the most reliable estimate of T12. 500 100 T, (Z/ T, (o) 0 0.25 0.5 0.75 Z/ D Figure 8. Frequency Dependence of TI 38 The most complete studies of the spectral inhomogeneity of T1 in the quadrupolar glass phase has been carried out for an ortho concentration X = 0.45 at T = 0.15K(14), and the data of ref.14 would place the crossrelaxation time T12 in the range 7.5 10.5 msec. While the crossrelaxation is faster than the direct relaxation to the lattice, the values of T12 reduce the spectral inhomogeneity of the relaxation of the NMR line shape. In this case we expect to observe T1 = T1 + T1 for 1(obs) 1(cal) 12 the relaxation of the magnetization of a given isochromat. This is shown in Fig.8. In view of the uncertainties in the crossrelaxation and the simplifying assumptions that have been made, the overall agreement with the experimental results is good. The correct overall behavior is predicted as well as the subtle change in spectral dependence at the half width points which has already been seen in the experimental data of Yu et al. (14). NonExponential Relaxation of Nuclear Magnetization Although it is agreed that (i) the low temperature NMR lineshapes indi cate a random distribution of molecular orientations (for both the local axes and the alignment a =(3J2 2)), and (ii) that there is apparently no abrupt transition in the thermodynamic sense; there has been disagreement (1,3) over the interpretation of the behavior of the molecular orientational fluctuations on cooling from the high temperature (free rotator) phase to very low tem peratures (T < 0.1K). Is it a very rapid, but smooth variation (25,27,70) with temperature, corresponding to a collective freezing of the orientational fluctuations or a slow, smooth dependence (9,14,28) with no evidence of any strong collective behavior? In order to resolve this problem it is important 39 to understand two unusual propertiesthe spectral inhomogeneity of the re laxation rate T.1 across the NMR absorption spectrum (14,29,30,72) and the nonexponential decay of the magnetization of a given isochromat (14). The spectral inhomogeneity has been discussed previously and the purpose of this part is to show that the strong departure from exponential decay for the mag netization can be understood provided that the broad distribution of local order parameters is correctly accounted for. Variation of Relaxation Rates within Given Isochromats As previously proved (3.21,3.22) 1 S (22 a )2 (2Av2 1DD 2 1 (2I a)C2(1 ) 1SR  The frequency of a particular component of the NMR absorption line is given by 1 Atv = DP2(a)a (3.25) 2 and this can be satisfied by very different values of a and a; e.g. Av = D occurs for a = 1,P2 = 1; a = ,P2 =1; = ,P2 = ....; the only constraints being that a and P2 lie within their limits; 2 < a < 0, and  in very different values of T1DD and T1SR. Molecules which contribute to the same isochromat of the NMR line but which have different values of a and P2 will therefore relax at different rates. This is illustrated in table III,IV and 1D, 1D and 0, respectively. The V, which give the variation of T1 for Av D, D and 0, respectively. The relative contribution of these rates can be determined from the probabilities 40 II(a),HI(P2) of finding a and P2. At low temperatures, the analysis of the line shape indicates that a good approximation for II(a) is a triangular distribution 11(a) oc a (12,23). For P2 we assume a powder distribution of local axes (i.e. very glassy) which requires that H(P2) oc 1 The calculated rates (2P2+1) Tl(ac, P2) have a relative weight P = H(a) x II(P2) for the relaxation of the component Av = TDiolP2. Magnetization ratio M(t) Pe Ti (3.26) M(0) P The weighted relaxations using the indicated probabilities are given in Fig.9 and Fig.10. The experimental results reported by Yu et al. (14) for different Av are indicated by the symbols. Table III Variation of Relaxation Rates within A Given Isochromat (Av = 'D) parame. >arame. prob. prob. rates rates rates(a P2(a) a n() ) r(P T ( T1 1 1.022 1.067 0.058 0.234 0.267 S1.044 1.143 0.107 0.431 0.492 1 ~ 1.069 1.231 0.144 0.583 0.666 ~ 4 1.095 1.333 0.167 0.681 0.777 1% 1.124 1.455 0.170 0.706 0.804 SS 1.155 1.600 0.150 0.634 0.720 9 16 1.188 1.778 0.097 0.422 0.478 2 1.225 2.000 0.0 0.0 0.0 (a)Rates given in units of D2 24wor The Determination of The Molecular Correlation Time r and TI at A v = 0 The only unknown parameter for eqs. (3.21) and (3.22) is the molecular correlation time r and the best fit represented by the solid lines and the broken 41 Table IV Variation of Relaxation Rates within A Given Isochromat (Av = 1D) parame. parame. prob. prob. rates rates rates(a) P2(a) a II(P2) I1(a) T T,1 T1 1.022 0.533 0.092 0.272 0.325 a 1.044 0.571 0.179 0.526 0.629 16 1 1.069 0.615 0.260 0.760 0.910 4 1.095 0.667 0.333 0.972 1.165 1 1 1.119 0.727 0.398 1.159 1.389 j 1.155 0.800 0.450 1.316 1.576 S1.188 0.889 0.486 1.435 1.715 1 1 1.225 1.000 0.500 1.500 1.789 1.265 1.143 0.482 1.485 1.763 1.309 1.333 0.417 1.337 1.577 T 5 1.359 1.600 0.275 0.938 1.097 j 2 1.414 2.000 0.0 0.0 0.0 (a)Rates given in units of D2 24wor line at Au = 0 in Fig.9, is obtained r = 1.51 x 107S. For S= 1.51 x I7S 1 103 1 Av=o= 1.49597 x 10 T, 10.0446 T1 IAv=o= 6.7144(msec) (3.27) T1 at Av = 0 is 6.71 ms which is in excellent agreement with the experimental value of 6.6 0.5ms (14,70). Since D=173.1 khz, in Fig.9 the Av for the calculated are 87,43.5 and M(o, are 87,43.5 and 0 khz which are below the values chosen by the Duke group (14). The same theroy to calculate for Au = 98 and 58 khz and the results are depicted M(O) in Fig.10. The overall agreement is very satisfactory. Comparison of the calculated decays MA,(t) with the experimental results shows that not only is the correct overall deviation from exponential decay predicted, but that there is also a significant long tail to the decay which ought to be tested for experimentally. This long time behavior is unique to the glassy regime of the hydrogen mixtures. Table V Variation of Relaxation Rates within A Given Isochromat (Av = 0) parame. prob. rates rates rates(a) Jao 11(a) 7,1 T1 T~11 1 0.533 1.467 2.249 3.096 0.571 1.429 2.245 3.070 8 0.615 1.385 2.237 3.036 0.667 1.333 2.222 2.992 0.727 1.273 2.198 2.933 4 0.800 1.200 2.160 2.853 0.889 1.111 2.099 2.740 1 1.000 1.000 2.000 2.577 8 1.143 0.857 1.837 2.331 1.333 0.667 1.556 1.940 8 1.600 0.400 1.040 1.271 2 2.000 0.0 0.0 0.0 We consider P2(a) = 0 for the range of Ja\ considered in Table IV in order to facilitate the comparison. (a) Rates given in units of D2 24w7r 1.0 0.8 0.6 0.4 0.3 0.2 0.1 15 30 45 t (msec) Figure 9. Time Dependence of M(t) for Different Av 1.0 08 0.6 0.4 0.3 02 0.1 15 30 45 t (msc) Figure 10. Time Dependence of M(t) for Av = 98 and Av = 58 khz CHAPTER 4 ORIENTATIONAL ORDER PARAMETERS Density Matrix Formalism A particle (e.g., an atom, molecule or nucleus) isolated in space and with nonzero angular momentum in it3 rest frame has a manifold of states with equal energy. The problem is how to specify the orientational degrees of freedom of an individual molecule. We describe the degrees of freedom of the quantum rotors with angular momentum J = 1 in terms of single particle 3 x 3 density matrices pi (for each site i). The pi are completely described by (1) the molecular dipole moments (Jx)i, (Jy)i,(Jz)i and (2) the quadrupole moments (J2)i, (JxJy)i, (JyJ)i.... In the absence of interactions which break time reversal symmetry, the dipole moments (Jx), (Jy), and (Jz) vanish and we need only consider 5 independent variables. Instead of Cartesian components, it is more convenient to use a set of irreducible tensorial operations IILM with 0 < L < 2J for general J and the associated multiple moments tim = Tr(pIllm) (4.1) For simplicity the site index has been dropped. The expansion of the single particle density operator in terms of the multiple moments is given by 2J L S(2J 1) (2L + 1)tLMILM (4.2) L=0 M=L There are three conditions imposed on p: Hermiticity and both weak and strong positivity conditions(34). p is a Hermitian operator and tLM = (1) tL_M (4.3) loo is a unit matrix operator and too = 1 = Trp The weak positivity conditions are given by 2J < Tr(p2) 1 (4.4) 2J +1  When one eigenvalue is equal to 1, the others being null; then the matrix p describes a pure state and Tr(p2) = 1. The minimum of Tr(p2) is reached when all the eigenvalues are equal to (2J + 1)1; then the matrix describes a completely unpolarized state and Tr(p2) = (2J + 1)1. For density matrices of spin2 particles, condition (4.4) is the only con dition imposed by the positivity condition. But for J > 1, further conditions are imposed on the density matrix and on the multiple parameters by the positivity property. , The eigenvalues of p must be positive definite because they represent the probabilities of realizing some given state and this leads to the strong positivity conditions 0 < An < 1 (4.5) where An is the nth eigenvalue. These conditions place the strongest limitations on the allowed values for the multiple moments and thus on the allowed values of the local order parameters for orthoH2 molecules in the solid mixtures. It is useful to construct orthonormal matrix representations of the irre ducible operators IILM in the representation (J2, J). For J=1 these are given 47 by the following 3 x 4,4natrix operators with rows (and columns) labelled by the eigenvalues 1,0,1 of Jz. 1 1 1 0 oO IIlo = Jz= 0 0 0 JZ0 0 1 1 1 10 ii1 = J+ = V2 0 0 1 ,. J 0 0 1 1 1 I 0 0\ 1120= (3J j2)= 0 2 0 (4.6) 0 0 1 1 1 ( 1 0 n2 2' (JzJ+ J+Jz) ( 0 0  n,22 = J 0 0 0 (4.7) 2 0 0 1L,M = (1)MIM (4.8) and Tr(IILMIL,M',) = 6LL'6MM' (4.9) The quadrupole operators 112M transform analogously to the spherical har monics Y2m(a, 3) with respect to rotations of the coordinate axes. The reference axes have remained arbitrary in the discussion and we are therefore free to choose local references axes that correspond to the local sym metry for each molecule. The natural choice for the zaxis is along the net com ponent of the angular momentum at a given site, i.e. such that (Jx) = (Jx) = 0. The general form for pj=I can be identified by the mean values of its magnetic dipole and electric quadrupole moments in the above notation: P = II3 + pnln + Qrnl2m (4.10) n m where An =(Iln) and Qm =(nIIm) 48 The expression (4.10) is identical with that was derived by ref.35 and ref.73. We still can choose x and y axes such that Q2 is real. i.e. Q2 = 2 YJ ) (4.11) and (JXJy + JyJX) = 0 (4.12) Q2 measures the departure from axial symmetry about the z axis and is, some times called the eccentricity (29, 30). It can be shown that with this choice of local reference frame Qi and Q1 also vanish and the density matrix may be written as / =0 0 Q2 1 1 f1 0 0 QV p = 3 + 0 0 0 + 0 Q 0 (4.13) O 1 Q2 0 1 Q where 1 AtO (JZ) Qo = ((3J2 2)) (4.14) Qo is the alignment (29, 30) along the zaxis. Sometimes it is convenient to define the alignment and the eccentricity as o' =(1 J) = /Qo (4.15) S=(2 j2y) = 2Q2 (4.16) Both a' and Y] have maximum amplitude of unity. In terms of these parameters p becomes 1 1 1 0 0 0 (_o 0 2 ) p = A + 0 0 1 + 0 32 0 (4.17) 3 2 0 0 1 1 0  ( 277 3 ) (,0,I) P+ = Figure 11. The Allowed Values oftt =(Jz), a And r7 for Spin1 Particle S, where j =(Jz). The three eigenvalues of p are 1 2, 3 3 A2,(3) = a 2 +2 (4.18) The strong positivity conditions, An > 0, are therefore seen to restrict the allowed values of the local order parameters a',r and / to the interior of a cone (Fig.11) in the 3D parameter space. The vertex of the cone is located at a' = 1, A = t = 0, corresponding to the pure state 1 ) =1 Jz = 0), and the base of the cone is defined by a' = and p2 + r12 = 1 which corresponds to the pure state  I = cosy I Jz = 1) + sin I\ Jz = 1) with (Jz) = cos27 and (J2) = sin2' (The polar angle y generates the points on the circle of the cone's baseplate). The positivity domain shown in Fig.ll is the same as those obtained by Minnaert (34) using the EbhardGood theorem and similar to those given by W.Lakin (32) in his analysis of the states of polarization of the deuteron. Having established the physical considerations which determine the limited range of allowed values for the order parameters, we now turn to the special case of solid hydrogen. Applications to Solid Hydrogen In the absence of interactions which break time reversal symmetry, the expectation value (Ja) must vanish for all a in solid hydrogen. This is the socalled "quenching" of the orbital angular momentum (74). The reason for this is that in the solid the electronic distribution of a given molecule may (to a first approximation) be regarded as being in an inhomogeneous electric 51 field which represents the effect of the other molecules. This inhomogeneity removes the spatial degeneracy of the molecular wave function which must be real and the expectation value of the orbital angular momentum (ih4) must accordingly vanish. The separation of the rotational energy levels is given by Ej = BJ(J +1) with B=85.37K, and in the solid at low temperatures only the lowest J values, J = 0 for paraH2 and J = 1 for orthoH2, need be considered. At low temperatures the anisotropic forces between the ortho molecules lift the rotational degeneracy and the molecules align themselves with respect to one another to minimize their interaction energy. For high ortho concentrations one observes a periodic alignment in a Pa3 configuration with four interpene trating simple cubic sublattices, the molecules being aligned parallel to a given body diagonal in each sublattice and the order parameters, oa =(1 J ), are the same at each site au = 1. The long range periodic order for the molecular alignments is lost below a critical concentration of approximately 55% and the NMR studies indicate that there is only short range orientational ordering with a broad distribution of local symmetry axes and local order parameters throughout the sample. This purely local ordering has been referred to as a quadrupolar glass in analogy with the spin glasses, but unlike the dipolar spin glasses, there is no welldefined transition from the disordered state to the glass regime. In order to describe the ortho molecules in the glass regime, where there is a large number of sites with various values of au, a density matrix formalism must be used. The quenching of the angular momentum in solid H2 has two consequences for the limits on the allowed values for the order paramters: 52 (1) From the previous discussion,: the allowed values of a' and rq lie within a triangle bounded by the three lines (Fig.12 AABC). 1 2 + o' > 3 3 1 1 1 a la' r > 0 (4.19) 3 3 2 which represent the strong positivity conditions for the eigenvalues of p when the angular momentum is quenched. (2) Since (.,) 0 for all a, we're free to choose the zaxis which was previously fixed by the net component of the angular momentum. The natural choice for the local reference axes is now the set of principal axes for the quadrupolar tenisor Qa =( (J Jp + JpJa) 23J26,) (4.20) The choice of principal axes is not unique, however, because after finding one set we can always find another five by relabeling axes. We can prove that not all of the points in the "allowed" portion of the (a', rl) plane are inequivalent (every point represents one state), and we only need to consider the hatched region (triangle CFE) in Fig.12. The states of all other areas of the triangle of allowed values can be obtained from the ACFE by suitable rotations (i.e. relabel the axes). The important point is that the pure states bc =1 Jz = 0) (4.21) and OA (= z 1+ Jz = 1)) (4.22) 2Z 53 are not inequivalent (The labels A through F refer to the special points on the triangle of allowed values shown in Fig.12), and the corresponding wave functions are listed in Table VI. The state OA can be obtained from Pc by a rotation of the axes by it about x axis. The rotation operator Rx(3 )'E( rJ) J ( 2 + i ) + 1 (4.23) 2 h 2 i + and 3r Rx( 2 ) eC = e A (4.24) The rotation Rx() leaves the state F invariant and maps D onto the point G in parameter space. We can furthermore show that the rotation Rx(3) maps the following triangular regions of parameter space onto one another ACDF AAGF ACEF  AAEF and ABDF ABGF (4.25) It is also seen by considering the transformation (",) = R()p(o',rl)Rx( (4.26) 2 2 using the matrix representation Rx( 2 2 (4.27) ~2 = 2 54 Table VI Special Point,; in Orientational Order Parameter Space. (see Fig.12) Label parame. Wave Function A (1,1) [j 1)+ )]/ B ( 1) [ 1) 1)/ C 1,010) 0) D (t, 2) 1) /20)]/2 E t(,", [i( 1)+ 1)) V2 0)/2 F (0,0) (1 + i)1) + (1 + i) 1) V2 0)1//6 G (1,0). (1+ ) 1)+(1+i) 11)]/2 R(~) : c + C A, D GC; B, F,i E fixed. Rz(1) : A + D[, *k E ID; DIC, F,G fixed. The Rz(I)) transforms the points (a', r) into new points (o", r') given by r = (1 (4.28) which corresponds to the mapping given by eqs.(4.25). There is some confusion in the literature concerning the values of a'. In Van Kranendonk's(75) book the negative value of a' had been ruled out as it was unphysical. But this is not strictly correct. Van Kranendonk's remarks refer to considerations of the pure states I Jz = 0) and I Jz = 1) only. He does not discuss either the density matrix approach or the formulation of the intrinsic quadrupolar order parameters a' and 7t needed to describe the orientational degrees of freedom of the orthoH2 molecules. The results of the relabeling transformation are particularly easy to under stand if we consider the pair (S = ao, N = /r3) which transform orthonor mally when the axes are rotated. The parameter space (S, N) is shown in 55 Fig:13. The rotation R(~) 'in (S, N) parameter space through the line BZ with the transformed points given by 2=2 N (4.29) 2 2 Similarly, the relabeling (x, y, z) * (y, x, z) corresponds to a reflection through the line CG. The entire area of allowed values in parameter space can therefore be mapped out starting only with the triangle ACFE by simply relabelling the principal axes. We only need to consider the hatched region of parameter space shown in Fig.12 in order to describe all physically distinguishable orientational states for orthoH2 molecules in the solid state. This is not the only choice that can be made for a "primitive" area of inequivalent values of (a', rI). We may also choose the sum of ACFK and AGFJ in Fig.13. In that region the states have the minimum values of the eccentricity N. Both choices are equivalent. A Proposition for A Zero Field Experiment An equivalent expression for the spectrum for each molecule is A = D(r20(,))oz (4.30) As previously discussed, considering the transformation to the local frame we obtained Ati = D[o'p2(cosOi) + 3risin2i0cos2ki] (4.31) where (O8, ,i) are the polar angles defining the orientation of the applied mag netic field with respect to the local molecular symmetry axes. (4,') G (,o)I (4,0) B Sm(1 0) (I,0) Figure 12. The Allowed Values of a And rl for Spin1 Particle AN I I,0) Figure 13. Diagram of Allowed S And N 58 We propose that the assumption of axial symmetry can be tested exper imentally by examining the zero field NMR absorption spectrum. Reif and Purcell (76) have carried out zero field studies for the long range ordered phase where it is known that or is constant and tr = 0, but it has not previously been considered for the glass phase. In zero applied magnetic field the degeneracy of the nuclear spin levels is lifted only by the intramolecular spinspin interactions. HDD(i)= hD[_a (3I I2) + .(I + 2I)] (4.32) The tensorial operators for the rotational degrees of freedom have been re placed by the expectation values a' and r1. An applied radiofrequency field can induce magnetic dipole transitions between the nuclear spin levels in anal ogy with the socalled pure quadrupole resonance absorption(51 Chapter VIII). The eigenstates of HDD are I I = 0) and I ) = (1 Iz = 1) Iz = 1). For each molecule i, three resonance lines can be expected corresponding to the transitions +) + 0),I ) + 0) and +) > ) with frequencies ,1 = D(a[ + ), i.2 = D(ua 1)i) and = Dr, respectively. If axial symmetry is a good approximation, there is only one line at vi = Dao and the detailed shape of the NMR absorption spectrum in zero field will be identical with that observed at high fields. Otherwise the high and zero field spectra will not be identical. A Model for The Distribution Function of a X. Li, H. Meyer and A. J. Berlinsky (23) have proposed a model for the distribution function of ao. They assume that for a single crystal the orientation 59 of the principal axes are uniformly distributed. They considered a Cartesian basis for the description of the single particle density matrices p= J2 I J2 + J2) 2 2 1 / 2 2 Pi =(1 i) (J, J Jz) The pure states pa) = 1 correspond to lawave functions for a = y, or z. This leads to a natural parameterization of the p ) given by pM,)_ ePE^ ( e Ea ePEai where the "energies" Ea, are in general temperature dependent. Li et al (23) then made the further assumption that the effective site energies are normally distributed about zero with a width A(T) subject to the constraint Ea Ea, = 0. i.e. P(E,Ey,E) (E +Er +E A 6(Ex + Ey + Ez) These assumptions lead to definite predictions concerning the variation of the NMR line shape parameters M2 and M4, and in particular of the variation of Sas a function of the degree of local orientational order parameter measured by (a ff)rms Instead of Cartesian symmetry for the effective site energies, we explored the same trends for a cylindrical symmetry for the effective site energies because of the axial symmetry of the quadrupolar interactions. This also leads to a natural description of the energy states in terms of local "two level system," corresponding to Jz = 1 and Jz, = 0 separated by an energy gap Ai. 60 The probability distribution of A is Gaussian as following 2 A2 P(A)= D2 e 2" (4.33) The order parameter a =(3J2 2) will be 2e KT 2 (o(A)) (4.34) 2eKT + 1 We can prove that M4 15 (a4) (4.35) M2 7 ((o2/)2 where c2 2 (2eKT 2) 2 _A2 (a ) =  e dA (4.36) o (2eKT + 1)2 V 7D2 let = X 2) = (2ep 2)2 2 x2 () (2e X e 2 dX (4.37) o (2eKT 4 1)2 4f 0 (2e X 2) 2 x2 (4) = (2e 1)4 2 dX (4.38) o (2e KTX +1)4 7r By using numerical integral method the curve in Fig.14 shows the ratio Sversus /(a2)(order). (a2) Considering the intermolecular and the intramolecular dipolar broadening, the fourth moment M4 and the second mement M2 should be (51) M2= Tr{I[ }]2 Tr { i2 (4.39) Tr{I2)} where m = mntra + mnter Nintra 11 h 1 3cs2Okl (3Ikz 1) kl kl k kI1 61 enter = y'sh (1 3cos20ij)S Fioo wn ai qut omi rce r3. s i.j ri3j Following a quite complicated but straightforward calculation we can prove that M2 = Mintra nter (4.41) M4 = Mntra + M ter + 4MantraMs2nter (4.42) For a Gaussian shape one has Minter = 3(2nter)2 and M4 = Mntra + 3(A4 nter)2 + 4MintraMnter (4.43) Taking the same value of Minter = 20(khz)2 as in ref.23 we obtain the curve in Fig.15. The upper one is ,Mt versus V(2) and the lower one is 2 (Atintra)2 JA versus (2). The experimental lineshapes (23,24,27) are below the calculated lineshape, but the shape and slope are almost the same. The value of maximum of M2 experimental lineshape is 3.15 (18) and the value of maximum M of calculated lineshape is 5.60. lineshape is 5.60. MODEL (INTRA) 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 ORDER Figure 14. Diagram of versus /(2) 3.5 3.0 2.5 2.0 1.5 1.0 L 0.0 MODEL 7 6 I 4 3 2 1 I I I I I I I I 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 ORDER Figure 15. Diagram of versus order (M2)e CHAPTER 5 NMR PULSE STUDIES OF SOLID HYDROGEN The experimental technique of nuclear magnetic resonance with spin echoes has been used widely in recent years as a tool to investigate the dynamical properties of crystals. In particular, much attention has been paid to the study of relaxation rates of dynamical processes in molecular solids by means of the spin echoes occurring after application of resonant rf 900 r (o) pulse sequences and rf 900 t tw 1'0) pulse sequences. The purpose of this chapter is to develop the theory of spin echoes of solid H12 in orientationally ordered phase and discuss the lowfrequency dynamics of orientational glasses. Solid Echoes There are several publications which have presented calculations for the amplitude of solid echo responses to 90 r 0 pulse sequences (39, 78, 79). We would like to examine the time dependence of the nuclear operators and the conditions for the focussing of solid echoes. The Time Dependence of The Nuclear Operators A general twopulse sequence denoted by (I r ) is sketched in Fig.16. In NMR experiments one observes only the orthomolecules (I = 1). In a first approximation we will treat the system as a set of independent molecules. The signal we measured is proportional to Tr[p(t)I+] = Tr[p(t)(Iz + ily)], where p is the density operator at time t and I the nuclear spin operator. R (.P) y ii' //Nj / MRP x x Figure 16. Solid Echo (twopulse sequence) ReW R'(W) tw *T tw 2T Figure 17. Stimulated Echo (threepulse sequence) Ro( ) R(~) 67 In equilibrium the expression of density matrix is given by eB o Pieq Treo) (5.1) e = Tr(ePNo) Let Z be the direction of applied magnetic field. o. = wohzi where wo is the Larmor frequency. Since < 1, e lKT 1 + ,zi (1 +1 )wo Pie (1 + 1(1 + zi) (5.2) Tr(1 + Izi) 3 KT We know that the radio frequency field is responsible for nonequilibrium behavior of the system. In the rotating frame after the first 900 pulse, the density matrix p(0+) would be p(O+) = ef Hdtpi(0)e f Hidt (5.3) If the radio frequency field is in the y direction of the rotation frame, we obtained p(O+)= eitl (1+ z)e + 3 KT 1 hw0 (54) p(O+) = (1 + I) (5.4) 3 KT This represents that the first pulse Ro puts the magnetization along the x axis (for simplicity we have dropped the site index i). We shall consider the time evolution of the nuclear spin operators under the effect of the intramolecular Hamiltonian in the rotating frame. Hs = hAwI+ eDhPi(O)aolo (5.5) i i where 2 1 2 aIo = (3Ji 2) iOk z and p(r) = e H (t')dt' (1 + hw x)e f H (t')dt (5.6) 3 KT It is assumed that the second pulse is a 0 pulse and the phase of that is (the angle between radio frequency field and ^ axis in rotating frame). After the second pulse the density matrix is given by p(r+) = ei (IIsinb+Iycos) p(r_)ei(Isin+lycos4) (5.7) The nuclear spin density matrix at time t after second pulse is therefore p(t + r) = u(t)R(O)u(r) (1 + h I )ut(r)Rt(q)ut(t) (5.8) KT where u(t) = e[iAwIzi!DP2(0)a(312)lt (59) R( ) = eiO(I.,inO+IrcosO) (5.10) u(r) = e[iAwlzi DP2(0)a(3I2)}] (5.11) and the signal S(t + r) oc Tr{p(t + r)I+} Tr{p(t + T)I+} =1 hwoTr{u(t)R(O)u(r)Ixut(r)Rt (O)ut(t)I+} 3 KT = E (ml I u(t) I m2z(m2 R(4) m3)(m3 I u(r) I m4) m4 1 mg(m u ) 8(6 (m4  lx  ms)(m5 I u(r)  m6)(m6  R(]) ma7) 69 (m7 I ut(t) I m8)(m8g +  mi) (5.12) Formation of Solid Echoes After a lengthy calculation the results of eq.(5.12) are still quite compli cated (for general 4 and ). For some special values of 4 and 4 the results are the same as that in ref.39. We would like to give the results in some special conditions. (1)0 = 0 (apply one pulse) Tr{p(t)I+} = e cosDP2(0)at] (5.13) 3 KT 2 This is called the Free Induction Decay. (2) ) = 0, = r This is a 90' r 90' pulse sequence Tr{p(t + 7)I+} = h eiAwtcos[ DP2(o)a(t r)I(1 eiAw2r) (5.14) 3 KT 2 Obviously, Tr{p(t + r)I+} has a maximum value at t = r. This is what is the meant by a solid echo. Another question is that according to eq.(5.14) the inphase echo (Aw = 0) does not exist. This is in disagreement with the experimental results. The reason lies in neglecting intermolecular dipoledipole interaction and the difficulty of getting exact Aw = 0 in experiments. (3)0 = 1, > = This is a 90' r 90' pulse sequence Tr{p(t + r)I+} 1= hwe iwtcos[ DP2()(t r)]( + eiw2) (5.15) 3 KT 2 From eq.(5.15), same as the second case, Tr{p(t+r)I+} obtaines maximum value at t = r, forming the solid echo. 70 Stimulated Echoes It is already shown that in certain cases a sequence of three 900 pulses may be advantageous (80,81). We first describe the formation of nuclear spin stimulated echoes. The stimulated echoes can be used to compare a "fingerprint" of the local molecular orientations at a given time with those at some later time (less than Ti) and thereby used to detect ultraslow molecular reorientations. Applications to the study of the molecular dynamics on cooling into the quadrupolar glass phase of solid hydrogen will be discussed. Formation of Stimulated Echoes A threepulse sequence is sketched in Fig.17. As the analysis of solid echoes, the nuclear spin density matrix at different time are given by 1 hw0 p(O) = Peq 11 + oIz) 3 KT p(O+) = (1+ hIx) 3 KT p() = e d H(ti)dt 'p(o+)e f Hec(t')dt' p(r+) = ei(lx in +Iycos ) p(r_) i(I ,sinf+Iycos~) ,i `_pec ( i H'eci p[(t + r)] = e i Htp(r+)eH ,t, p[(tw + T)+] = eit'(Isin~'+cos')p[(t, + )_]ei(l2si ) p(tw + r + t) = eH1f tp[(tw + r)+]er) Hj t The signal S(tw + r + t) o Tr{p(tw + r + t)I+} (5.16) The first preparatory pulse creates a transverse magnetization in the rotat ing frame. Under the influence of the Hamiltonian given by equation(5.15), the 71 evolution during the short time r(< T2) leads to the formation of nuclear spin states described by p(r_) which contains both transverse magnetization and transverse alignment. The transverse components are transferred by a second pulse R(tk) into longitudinal components corresponding to spin polarization and spin alignment. They will be stored and evolved during long waiting time tw (chosen short compared with the longitudinal relaxation times). Therefore we can obtain a "fingerprint" of the local alignments a. After the waiting period tw < T2 the stored components can now be 'read' by means of a third pulse of rotation angle 1'. The spin polarization and spin alignment 'stored' during tw are transferred by the third pulse into coherent transverse states, which then evolve in a reverse manner to that occurring during the first evo lutionary period r and the transverse signal focuses to stimulated echo after a delay time r following the third pulse. Tr{p(tw + r + t)I+} 1 hwoc 3K= T (m I eH t m2)2 R(42)  m3)(m3 Ie iH t, I M4) m1 m12 (m4 I R(O) I ms)(m5 eH cr Ime I) Ix m7) i slect I msee (m7 1 eiH' r I m)(mg I R( ) I mg)(mg  eH~t"  mo0) (mlo I R+(O') I n11)(mn1 I e Hct I m12)(ml12 I+ I mi) (5.17) Since the calculation is extremely tedious and the results for general 0,4' and general i,0' are very complicated, we only investigate the results of cal culations by taking two special cases. (1) =0, =2 = j, = j 72 This is a 90' r 90' tw 90' pulse sequence Tr{p(tw + 7 + t)I+} = 3 Teiwt icos(Awr)cos2(Awt.)cos[ DP2(8)o(t r)] + 3 eiticos(Awr)sin2 (Awt)cos[ DP2(0)a(t + r)] 2 KT 2 2 e wtsin(Awr)sin(Awt,)cos[1DP2(0)a(t tw + r)] (5.18) 3 KT 2 (2) S= Of = This is a 90 T 90' tw 90' pulse sequence Tr{p(tw + r + t)I+} = wo ewtisn2 (Awt,)sin(Awr)cos[ 1 DP2(0)(t r)] 3 KT 2 +2 hwo eiwtisin(Awr) cos2(Awt)cos[1 DP2(O)a(t + r)] 3 KT 2 +2hwo eiwto(AWt)cos(AWt,)co5[1DP2(0)o(t t + r)] (5.19) 3 KT 2 Examining expressions of eqs.(5.18) and (5.19), besides the echo at t = r after the third pulse, there is a additional echo at t = tw r. The results are sketched in Fig.18. The appearance of multiple echoes has been seen in experiments (Fig.19) for orthopara hydrogen mixtures at low temperatures. Engelsberg et al. (81) have presented results for nuclear spin stimulated echoes in glasses. The curve of 11B echoes in borosilicate glass at 4.2 K (Fig.20) shows that in addition to the solid echo at t = 2r(after first pulse) and a stimulated echo at t = T + 2r, other echoes at t = 2T (image echo) and t = 2T + 27 (primary echo) were clearly observed. For longer waiting times, the solid echo(which they called spontaneous echo) decays rapidly and only the stimulated echo remains detectable. 73 Fig.21 shows the temperature dependence of the ( r r tw () stimulated in the quadrupolar glass phase of solid hydrogen (r = 25/s, tw = 2ms). The theoretical results (eqs 5.18 and 5.19) and the experimental results clearly show that the amplitude of stimulated echo is proportional to 1. For experimental curve only a very slight modification (indicated by the arrow) was observed. We will discuss this phenomenon later. The experimental curve in Fig.22 gives the relation of stimulated echo versus waiting time tw of solid hydrogen (orthoconcentration x = 0.54, T = 220mK). The time scale is logarithmic. The logarithmic decay behavior can be understood in terms of the motional damping of stimulated echoes. LowFrequency Dynamics of Orientational Glasses The orientational glasses (20, solid orthopara H2 mixtures (6), N2/A mix tures (83) and the KBrlxK(CN)x mixed crystals (82, 84, 85)) form a sub group of the general family of spinglasses which continue to generate intense interest because of the apparently universal low temperature properties ob served for a very diverse range of examples (dilute magnetic alloys, mixed crys tals, dilute mixtures of rotors, partially doped semiconductors (86), Josephson junction arrays (87) and others). The most apparent striking universal fea tures (20) are an apparent freezing of the local degrees of freedom on long time scales without any average periodic long range order, characteristic slow relaxations and historydependence following external field (magnetic, electric, elasticstrain....) perturbations, and a very large number of stable low energy states. Soli; echo o 0 (t4r+ t (tv Ae. S~im. \ echo g90 t0wt Figure 18. Sketch of The Results of Calculations Figure 19. Experimental Curve (x = 23%,T = 38mK) Nuclear spin stimulated echoes in glasses T 2t St r T TrT T*2t 2T 2T.2T + 1, 472 +I 2T Figure 20. 11B Spin Echoes in Borosilicate Glass at 4.2K 3635 25 cn =3 d L20 0 CL 0 015 0,o Li O IO 300 T (mK) Figure 21. Temperature Dependece of Sequence ( 25s,t = 2ms) Sequence (r = 25ils,t, = 2ms) o.e 0 0 0.2 r II II II..I \, s .I \ lI,,, 0.1 0.5 1 2 5 10 20 SO 10 iWaiting te Is) Figure 22. The Observed Decay of Stimulated Echo (x = 0.54, T = 220mK) (squares:r = 12.5As; circles and triangles: r = 251s) 79 The echo calculation mentioned above was based on the static case. If the ortho molecules are in slow motion, the stimulated echoes will be damped. At low temperatures the random occupation of lattice sites for solid H2 mixtures (for X < 55%) leads automatically to the existence of local electric field gra dients, the field conjugate to the local order parameter, which plays the same role as the magnetic field for the dipolar spin glasses. This random local field therefore makes the problem of local orientational ordering in random mix tures equivalent to the local dipolar ordering in spin glasses in the presence of random magnetic fields. In analogy with the analysis for spin glasses we assume that the existence of local electric field gradients leads to clusters (or droplets) of spins (88). Based on Fisher and Huse's recent picture (89), we provide a explanation of the low frequency relaxation and the lowtemperature specific heat of solid ortho para hydrogen mixtures. In the scaling model of Fisher and Huse the lowenergy excitations which dominate the longdistance and longtime correlations are given by clusters of coherently reoriented spins. Their basic assumptions are: (1) Density of states at zero energy for droplets (d dimension) length scale L as L0, where 0 < 0 < d. (2) Free energy barriers EB for cluster formation scale as Eg L with 0 <, With these assumptions, Fisher and Huse show that the autocorrelation function Ci(t) =((Si(0)Si(t))t(Si)b)c (5.20) 0d decays as (log t) 3 for t + oo. 80 For our system, assuming axial symmetry, the quasistatic local orienta tional order parameters are the alignments ri =(3J2 2)i and the correspond ing autocorrelation functions Ci(t) = ai(0)ai(t) can be studied directly by NMR. For a 900y r 900y t, 90' pulse sequence we assume at t = r, order parameter a = a(r) and at t = r + tw,a = a(r + tw) for each ortho molecule Tr{p(tw + r + t)I+} = 2 woe_iAticos(Awr)cos2(Awtw)cos[1DP (0)a(r + tw)t DP2(0)a(r)r] 3 KT 2 2 +2 o iticos(Awr)sin2(Atw)cos[lDP2(0)a(r + tw)t + 1DP2(0)a(Tr)r 3 KT 2 2 2 hw ,it (a,t,) 2 A e wtsin(Awr)sin(Awtw) 3 KT 1 1 1 cos[ DP2(0)a(r + tw)t 1DP2(0)a(r + tw)tw + DP2(O)a(r)r] (5.21) 2 2 2 Considering t = r, the stimulated echo amplitude A oc((cos[Dirai(r)]cos[Dir7i(r + tw)])) (5.22) where the double brackets refer to an average of configuration and Di = DP2(co0si). The important point is that if the local order parameters ai remain fixed during tw, there is no damping of the stimulated echo, while ai changes due to local reorientations, then the contribution to the echo is severely attenuated. The product Dr can in practice be made very large and this method can therefore be used to study ultraslow motions in solids. We believe that in Fig.21 the departure portion from T, (indicated by an arrow) is due to slow motion. A barrier will have a characteristic life time given by an Arrhenius Law 1 1 S= eKBT r To or tunneling rate F EB r(EB) = Foe KBT (5.23) where To is the characteristic attempt frequency for clusters of this size. In the long time limit ro is reasonably welldefined because it is associated with a characteristic cluster size. In a time t the only barriers crossed will be those sat isfying 0 < EB < Emax(t) where Emax(t) = KBTlog 1. Any barriers crossed lead to significant changes in the local order parameters and the amplitude of the stimulated echo is then simply A(t) = P(EB)dEB (5.24) Emax(t) At low temperatures, assuming a constant density of barrier heights P(EB), we find A(t) = 1 KBTPo log() (5.25) to The prefactor Po can be determined from the low temperature behavior of the heat capacity. For the orthoH2 molecules with angular momentum J = 1, we can asso ciate a simple two level system with the energy states for a given molecules; the states Jo = 1 being separated from the state J = 0 by a gap 3Aj (The states Ji = +1 are degenerate if there are no interactions which break time reversal symmetry). At low energies we can, following the above arguments, identify the low energy excitations (which determine Cv at low T) with a broad quasiconstant distribution P(A) for 0 < A < Ao for the spins in a cluster. Identifying P(A = 0) with Po, the density of low energy barriers, we have Cr Ao 18 A2 N18 = PoAo 3 3A dA (5.26) NR K22 4e KBT+ eKBT +4 where x is the orthoH2 concentration. let u = 3p C, 2KBT _KB u2 _= Po KT U du (5.27) NxR 3 Jo 4eu + e + 4 set t = Poot u2du (5.28) NxR 3 o4eu + eu + 4 let ,2 vu2du C1 = tf t (5.29) V 3 4eu + eU + 4 The resulting C' (Figure 23) has a linear temperature dependence at low T and a peak at Tpk = 0.70A in close resemblance to the temperature behavior observed by Haase et al. (90). From the peak position in the experimental data, o = 1.27, PoK = 0.86 and for the stimulated echo decay this value gives Acac.(t) = 1 0.43 loglo() (5.30) to and the observed decay Aob.(t) = 1 0.55 loglo( ) (5.31) at T = 0.22K for x = 54%. The agreement is remarkably good. The experimental curves indicate to 104s. It should be noted that the argument relating the logarithmic decay to maximum barrier height crossed in time t can also apply (over a short time scale) to the case of orientational ordering in pure N2 studied in reference (11,91) because one also observes a relatively large distribution of order pa rameters centered on a' = 0.86 and with width 0.12 in this case. The essential point is that the time scale of the slow relaxations in the glass phase is simply related to the low temperature behavior of the heat capacity. 83 Another important point is that the characteristic times to are much shorter than the spectraldiffusion time scale (~ sec) seen by the recovery of holes burnt in the NMR lineshape. We therefore find it difficult to attribute the logarithmic decay seen in H2 to spectral diffusion across the NMR linshape. Cv' t 0.5 I'' I 0.4 0.3 0.2 0.1 0.0 IIII 0.0 0.4 0.8 1.2 1.6 2.0 Figure 23. Calculated Curve of C' t CHAPTER 6 SUMMARY AND CONCLUSIONS The orthopara mixtures of solid H2 are studied theoretically for fcc lattices (X > 0.55) of finite site by Akira Mishima and Hiroshi Miyagi (92). The sys tematic theoretical studies of nuclear magnetism for quadrupolar glass regime are carried out in this dissertation. We have developed a theory of the nuclear spinlattice relaxation of ori entationally ordered ortho hydrogen molecules for the case of local ordering in the quadrupolar glass phase of solid hydrogen. We have investigated the temperature dependence of T1. It shows that Gaussian Free Induction Decay is a quite good approximation. The calculations indicate a strong spectral in homogeneity of the relaxation rate T (Av) throughout the NMR absorption line. The detailed dependence is much stronger than the simple dependence T1y1(Av) oc (2 +o) given by earlier estimates by A. B. Harris et al. (1). There is also a strong variation with the orientation of the local symmetry axis with respect to the external field. The variation is in agreement with that found by Hardy and Berlinsky (93) for the longrange ordered Pa3 phase. If the crosssection relaxation between different isochromats is taken into account, the results of calculations of spectral inhomogeneity is in good agreement with the experimental results. In addition to the spectral inhomogeneity, the relaxation is also found to be strongly nonexponential. This can be understood easily. As a result of the glassy nature of the system, there is a broad distribution of both local orientational order parameters(a) and the orientation (a) of the local symmetry 86 axes for the molecular alignments. The existence of these distributions at low temperatures means that a given frequency Av in the spectrum comes from all the different allowed combinations of a and a that satisfy Av = 1DP2(a)a. The dependence of the relaxation rates on a and P2 then leads to a distribution of local relaxation rates for a fixed Av. It is this distribution in rates that leads to the observed nonexponential behavior of M(t). Calculations based on the expected probability distributions for a and P2 yield results in good agreement with the results reported in the publications (14). The most important conclusion of the study of T1 is that the spectral inho mogeneity and nonexponential recovery both results in an order of magnitude variation of the nuclear spin relaxation and must therefore be correctly under stood and accounted for before attempting to analyze the experimental results in terms of the fundamental molecular motions. We have determined that in general the orientational degrees of ortho H2 molecules in the solid need to be described in terms of density matrices. The ortho molecules have momentum J = 1 and the single particle density matrices are completely determined by five independent parameters (if the angular momentum is quenched). These parameters are (1) the three principal axes (x,y,z) for the second order tensor. (2) the alignment ao =(1 jJ), and (3) the eccentricity =(J2 J2). The positivity conditions for the density matrix show that the only allowed values of (a', r1) are those enclosed in a triangle in (a', ??) space whose vertices are the pure states I Jz = 0) and I Jz = 1). Not all of these allowed values are physically inequivalent because one may relabel the principal axes and we have shown that one can determine a simple primitive set of order parameters 87 which are inequivalent by the choice 2a' > rl > 0. Orientational states with negative ao are not excluded on theoretical grounds. Studies of molecular dynamics have shown the existence of paralibrons, i.e. collective excitations that exist in large clusters with welldeveloped short range order. Nuclear spin stimulated echoes have proved to be very effective for the study of ultraslow molecular motions both in the molecular orientational glasses and in ordinary glasses (81). The existence of multiple echoes which is given by theoretical calculations is also in good agreement with experimental results of solid hydrogen and ordinary glass at low temperature. We have offered a unified explanation of the slow relaxational behavior and low temperature heat capacity of the quadrupolar glass phase of solid hydrogen in terms of the density of lowenergy excitations in the system. Since the stimulated echoes are damped by any change in the orientational states during tw, they can be used to detect very slow molecular motions. The phenomenon of logarithmic decay of the stimulated echo, which is similar to decay of magnetization in metal spin glass, can be explained by using Fisher and Huse's recent picture of the short range spinglasses and the domination of the longterm relaxation by low energy largescalecluster excitations. In analogy with spin glasses (e.g. alloys like Cu Mn where a random configuration of spins condenses at low temperature) the linear specific heat with temperature at low T can be understood in terms of the suggestion (94) that the dominant contribution to the specific heat will be from clusters of particles for which the energy barrier is sufficiently great so that resonant tunneling between the two local minima does not occur, but sufficiently small so that tunneling between the two levels can take place and thermal equilibration 88 can occur during the time span of the specific heat experiment. We obtained a very good linear dependence curve at low T by using a numerical integral method. Having combined with the results of logarithmic decay of stimulated echoes and linear lowtemperature specific heat, a very good numerical expression of the amplitude of the stimulated echo as a function of waiting time tw for x = 54%, T = 0.22K is obtained. Most people agree that molecular orientations of random orthopara hydro gen mixtures become frozen at low temperature and there is no evidence for any welldefined phase transition on cooling from the completely disordered high temperature phase. However, the gradual transition to the glass state in these systems involves strong cooperativity as evidenced by Monte Carlo calculations. Studies of the nuclear relaxation and molecular dynamics have shown quantitatively how important the cooperativity is. Further theoretical study of the slow motions needs to be carried out to improve the understanding of the nature of the quadrupole ordering in random mixtures at low tempera tures and their relation to the family of glasses. Further experiments at very low temperatures will also provide a deeper understanding of the properties of the quadrupolar glass state. APPENDIX FLUCTUATIONDISSIPATION THEOREM The fluctuationdissipation theorem is extremely important because it re lates the response matrix to the correlation matrix for equilibrium fluctuations. In Chapter 2 we mensioned the definitions of the response function(eq.2.32) and the relaxation function(eq.2.35). The response function my be written in a number of equivalent ways: fkl(t)= W Tr{[peq,Mj(t)]Mk} (A.1) 1 = ([Ml(t),Mk]) (A.2) 1 fkl(t)= [MI,Mk(t)]) (A.3) Using an identity due to Kubo the response function may also be written as Ik() = ds(MI(ihs)Mk(t)) (A.4) fkl(t) = ds(ML(ihs)Mk(t)) (A.5) Here ihs plays the role of time. Such formulas appear in quantum statistical mechanics because of the formal similarity between the way time and temper ature must be treated. In the calculation of a canonical partition function one must take into account the factor e~H. In calculating the time evolution of operators one must consider operators of the form e(iHt. By writing t = ihs one can see that s will play a role identical to # in all formal developments. 90 From eq.(2.35) and eq.(A.5) it follows that Fkl(t)= ds(M(ihs)Mk(t)) lim ds(M(ihs)Mk(T)) (A.6) Jo T ooJo As T apparoches infinity in the second term of Eq.(A.6) we can assume that the correlation is lost between various components of the magnetization so that the correlation function factors into (M1)(Mk). 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