Theory of nuclear magnetism of solid hydrogen at low temperaures

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Theory of nuclear magnetism of solid hydrogen at low temperaures
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Lin, Ying, 1946-
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Nuclear magnetism   ( lcsh )
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Thesis (Ph. D.)--University of Florida, 1988.
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Includes bibliographical references.
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by Ying Lin.
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Typescript.
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Vita.

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Full Text







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. Shin-ll 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 co-operation 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 DMR-8304322 and DMR-86111620 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

Bloembergen-Purcell-Pound Theory .............................. 6

Theory for Liquids in terms of Mori's Formalism .................. 7

Kubo and Tomita Theory.......................................11

Nuclear Spin-Lattice Relaxation in Non-Metallic Solids ........... 16

Nuclear Spin-Lattice Relaxation for Solid Hydrogen ............... 18

3 NUCLEAR SPIN-LATTICE RELAXATION ...................... 22

Formulation of Longitudinal Relaxition Time T ................. 22

Temperature Dependence of T1 ................................... 27

Spectral Inhomogeneity of T1 ..................................... 33

Non-Exponential 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









Low-Frequency Dynamics of Orientational Glasses...............73

6 SUMMARY AND CONCLUSIONS..............................85

APPENDIX

Fluctuation-Dissipation 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 ortho-para hydrogen

mixtures at low temperatures are presented.

The formulation of the nuclear spin-lattice relaxation time T1 for the case

of the local ordering in the orientational "glass" phase of ortho-para 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 cross-relaxation 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 three-pulse 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 ultra-slow molecular

re-orientations.












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 "spin-1" quautum rotators,

and therefore a valuable testing ground for theoretical models of co-operative

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 ortho-para 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 spin-glass such as EuxSrl_zS in

a random field(20,21).

The electrostatic quadrupole-quadrupole 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(25-28)

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 so-called "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 spin-1 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 centre-of-mass 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 glass-like 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 quadrupole-bearing 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 steady-state 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 10-4 10-5 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 three-pulse 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

continuous-wave 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


Bloembergen-Purcell-Pound Theory

The theory of spin relaxation in liquids (or gases) is based upon time-

dependent perturbation theory.

In liquids where the spin-spin 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)sinOcosOe-i0

D = (Irliz + lizi7)sinOcosOe"i

E = -I+I+sin2Oe-2io
4 z 3
F = IT1:8in26e22o
4 3






7

If we carry out first-order time-dependent 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 well-known 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)e-int (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 so-called 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 non-equlibrium 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) = (e-Lt)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 time-dependent 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, time-varying 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 Kramers-Kronig 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

[e-Lt]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 ak-k(w + Kwo)
LTi k = 0, 1 (2.47)
Lkk- Xk-k(O)
i.e.
Lkk Reak-k(w + Kwo)
L=kk + 2 (2.48)
L'k + Xk-kk (0)
Using a symmetric form of Ok-k, equation (2.48) becomes

L'kk Reak-k(w)
L1 Xk-k) (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 = L-1-1 =
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 spin-lattice coupling.


Nuclear Spin-Lattice Relaxation in Non-Metallic 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 flip-flops between neighbours, the correct approach to nuclear mag-

netism is a collective one, where single large spin-system 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][ t-r),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 Spin-Lattice 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 spin-rotational 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 O-H2 molecules. The averaged equations were then solved numerically to

obtain the spectral functions for solid H2 self-consistently for the first time.

The resulting spectral functions were used to compute the TI as a function

of the ortho-molecule 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 self-consistently 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 quadrupole-quadrupole (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 spin-lattice 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 ortho-hydrogen-

para-hydrogen alloys(Fig.l-2).

The nuclear spin-lattice relaxation of ortho molecules at low temperatures

is determined by the modulation of the intramolecular nuclear dipole-dipole

interactions HDD and the spin-rotational 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 dipole-dipole 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

T-1 1=D2 m22m(mwo) (3.4)
|m|=1,2
where the spectral density at the Larmor frequency

J2m (w) = (O2m(t)O tm)) e-ic"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 Ortho-Para 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 K-1)

TIAJ PM

(C) 1[ FM
SG

0 1



Figure 2. (a) Theoretical Phase
Diagram for A Short Range System
(b) Experimental Phase Diagram for
EuxSrl-xS 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 spin-rotational interaction HSR is


-SR 2= C2Jl1(wo) = C2(2 + )|dlo1(a)2glO(WO) (3.9)

the total rate Tfl = T' + T-R


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, ) = (2-a-a-2)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 + 4ae-a) (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 10-9, 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 Vogel-Fulcher

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. T1-T Curve (solid-Tiexp,dashed-Titheo)







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 10-8 x eT-o.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 r-1 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 spin-rotational 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 Dipole-Dipole Relaxation Rate Ti~ D

Frequency range Av T17D x (2 1)
0 < 10D2-12vD+4v2 2v2 2 2Vv
S3D2 D(D-2v) log H +D(D log
'D < Av < D 5D'-7vD+8v' 22 v
2 3D D (Dv)




Table II Spectral Dependence of Spin-Rotational Relaxation Rate T-1R

Frequency range Av TIIR x (18 2)
2D-3v 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 spin-rotational 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 + T-1R. The curves show discontinuities in slope at |AI = D, but this
was not seen experimentally. If we take the finite cross-relaxation 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 spin-lattice relaxation times are much longer than the

cross-relaxation times the spins come to a common spin temperature via the

cross-relaxation mechanism before relaxing to the lattice via the rapidly relax-

ing components. On the other hand, when the direct spin-lattice 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 flip-flop transition

via the intermolecular dipole-dipole interaction for two isochromats VI and /2

given by

T2 = (T2flip- fl)op -2( v2)2
2r MArnte J

= (T2flip-flP) -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 dipole-dipole interaction. M ra tra and AMinter are the

moments resulting from the intra- and inter-molecular dipole-dipole 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 cross-relaxation

time T12 in the range 7.5 10.5 msec.

While the cross-relaxation 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 T-1 = T1 + T-1 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 cross-relaxation 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).


Non-Exponential 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 properties-the spectral inhomogeneity of the re-

laxation rate T.1 across the NMR absorption spectrum (14,29,30,72) and the

non-exponential 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 = TDio|lP2.

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 ( T-1
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 T-1
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 10-7S. For


S= 1.51 x I-7S

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 T-1 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 spin-2 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 ortho-H2 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 z-axis 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 Q-1 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 z-axis. 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 Spin-1 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 Ebhard-Good 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

so-called "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 para-H2 and J = 1 for ortho-H2, 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 sub-lattices, the molecules being aligned parallel to a given

body diagonal in each sub-lattice 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 well-defined

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 z-axis 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) 1-1)]/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 ortho-H2 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 ortho-H2 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) + 3-risin2i0cos2ki] (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 Spin-1 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 spin-spin 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 radio-frequency field

can induce magnetic dipole transitions between the nuclear spin levels in anal-

ogy with the so-called 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 la-wave functions for a = y, or z.

This leads to a natural parameterization of the p ) given by

pM,)_ e-PE^-
( e
Ea e-PEai
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)
2e-KT + 1
We can prove that
M4 15 (a4)
(4.35)
M2 7 ((o2/)2
where
c2 2 (2e-KT 2) 2 _A2
(a ) = -- e-- dA (4.36)
o (2e-KT + 1)2 V 7D2
let = X
2) = (2e-p 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 + 4MintraM-nter (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 low-frequency 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 two-pulse sequence denoted by (I r ) is sketched in Fig.16.

In NMR experiments one observes only the ortho-molecules (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 (two-pulse sequence)


ReW




















R'(W)


tw *T tw 2T


Figure 17. Stimulated Echo (three-pulse sequence)


Ro( )


R(~)






67

In equilibrium the expression of density matrix is given by

e-B o
Pieq Treo) (5.1)
e = Tr(e-PNo)

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+) = e-f 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_)e--i(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[iAwIz-i!DP2(0)a(31-2)lt (59)

R( -) = eiO(I.,inO+IrcosO) (5.10)

u(r) = e[iAwlz-i DP2(0)a(3I-2)}] (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 e-iAwtcos[ 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 in-phase echo (Aw = 0) does not exist. This is in disagreement with the

experimental results. The reason lies in neglecting intermolecular dipole-dipole

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 ultra-slow molecular re-orientations.

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 three-pulse 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, + )_]e-i(l2si )

p(tw + r + t) = e--H1f 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 e-H t m2)2 R(42) | m3)(m3 Ie- iH t, I M4)
m1 -m12

(m4 I R(O) I ms)(m5 e-H 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 Te-iwt icos(Awr)cos2(Awt.)cos[ DP2(8)o(t r)]


+ 3 e-iticos(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 e-wtisn2 (Awt,)sin(Awr)cos[ 1 DP2(0)(t r)]
3 KT 2
+2 hwo e-iwtisin(Awr) cos2(Awt)cos[1 DP2(O)a(t + r)]
3 KT 2
+2hwo e-iwto(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 ortho-para 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 (ortho-concentration 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.


Low-Frequency Dynamics of Orientational Glasses

The orientational glasses (20, solid ortho-para H2 mixtures (6), N2/A mix-

tures (83) and the KBrl-xK(CN)x mixed crystals (82, 84, 85)) form a sub-

group of the general family of spin-glasses 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 history-dependence following external field (magnetic, electric,

elastic-strain....) 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, 47-2- -+-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 low-temperature specific heat of solid ortho- para

hydrogen mixtures.

In the scaling model of Fisher and Huse the low-energy excitations which

dominate the long-distance and long-time 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 L-0, 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 quasi-static 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 re-orientations, 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 ultra-slow 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 well-defined 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 ortho-H2 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

quasi-constant 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
N-18 =- PoAo 3 3A dA (5.26)
NR K22 4e KBT+ eKBT +4


where x is the ortho-H2 concentration.









let u = 3p

C, 2KBT _KB u2
_= Po KT U du (5.27)
NxR 3 Jo 4e-u + e + 4

set t =

Poo-t u2du (5.28)
NxR 3 o4e-u + eu + 4

let
,2 vu2du
C1 = -tf t (5.29)
V 3 4e-u + eU + 4

The resulting C' (Figure 23) has a linear temperature dependence at low T

and a peak at Tpk = 0.70-A 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 10-4s.

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 spectral-diffusion 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 ortho-para 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 spin-lattice 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 long-range ordered Pa3 phase. If the

cross-section 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 non-exponential 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 para-librons,

i.e. collective excitations that exist in large clusters with well-developed short

range order.

Nuclear spin stimulated echoes have proved to be very effective for the

study of ultra-slow 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 low-energy 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 spin-glasses and the domination

of the long-term relaxation by low energy large-scale-cluster 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 low-temperature 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 ortho-para hydro-

gen mixtures become frozen at low temperature and there is no evidence for

any well-defined phase transition on cooling from the completely disordered

high temperature phase. However, the gradual transition to the glass state

in these systems involves strong co-operativity as evidenced by Monte Carlo

calculations. Studies of the nuclear relaxation and molecular dynamics have

shown quantitatively how important the co-operativity 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


FLUCTUATION-DISSIPATION THEOREM

The fluctuation-dissipation 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). Hence the relaxation

function may be written in its more usual form

Fk (t) = ds(AM(-ihs)AMk(t)) (A.7)

In this form there exists a mathematical identity between the relaxation func-

tion and the correlation function gkl(t), given by

Fkl(t) = B(t r)gkl(r)dr (A.8)
-oo
where
1
gk(t) = ([AM(t), AM(0)]+) (A.9)
2
the + subscript indicates the anticommutator; or in terms of Fourier transforms

SW) = ^ ) (A.10)
B(w)
where
3 7 ltl
B(t) = log[coth( )] (A.11)
hir 2#h
and
1 hw hw
= -_coth (A.12)
B(w) 2 2
The derivation of these identities is called the quantum-mechanical fluctuation

dissipation theorem. This theorem relates the response of step function distur-

bance to the correlation function of fluctuations in the equilibrium ensemble.

In the classical limit (h -+ 0) B(t) P36(t), and one obtains the classical

fluctuation-dissipation theorem


Fkl(t) = 9kl(t)


(A.13)












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