
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00004815/00001
Material Information
 Title:
 Theory of nuclear magnetism of solid hydrogen at low temperaures
 Creator:
 Lin, Ying, 1946
 Publication Date:
 1988
 Language:
 English
 Physical Description:
 vi, 96 leaves : ill. ; 28 cm.
Subjects
 Subjects / Keywords:
 Acoustic echoes ( jstor )
Degrees of freedom ( jstor ) Hydrogen ( jstor ) Low temperature ( jstor ) Magnetic fields ( jstor ) Molecules ( jstor ) Nuclear spin ( jstor ) Relaxation time ( jstor ) Symmetry ( jstor ) Temperature dependence ( jstor ) Low temperature research ( lcsh ) Nuclear magnetism ( lcsh ) Solid hydrogen ( lcsh )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Florida, 1988.
 Bibliography:
 Includes bibliographical references.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Ying Lin.
Record Information
 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 024514383 ( ALEPH )
AFL1242 ( NOTIS ) 19908577 ( OCLC ) AA00004815_00001 ( sobekcm )

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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
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. Shin11 Cho and
Daiwei Zhou for their helpful suggestions and discussions.
The help from my friends Laddawan Ruamsuwan, James K. Blackburn,
Qun Feng and Stephan Schiller with the computer work is greatly appreciated.
The cooperation and friendship of my fellow graduate students, as well as
that of the staff and faculty of this department, has made my stay at U.F a
pleasant and rewarding experience.
This research was supported by the National Science Foundation through
Low Temperature Physics grants DMR8304322 and DMR86111620 and the
Division of Sponsored Research at the University of Florida.
11
TABLE OF CONTENTS
page
ACKNOWLEDGEMENTS ii
ABSTRACT v
CHAPTER
1 INTRODUCTION 1
2 THEORY OF NMR RELAXATION 6
BloembergenPurcellPound Theory 6
Theory for Liquids in terms of Moris Formalism 7
Kubo and Tomita Theory 11
Nuclear SpinLattice Relaxation in NonMetallic Solids 16
Nuclear SpinLattice Relaxation for Solid Hydrogen 18
3 NUCLEAR SPINLATTICE RELAXATION 22
Formulation of Longitudinal Relaxition Time T\ 22
Temperature Dependence of T\ 27
Spectral Inhomogeneity of T\ 33
NonExponential Relaxation of Nuclear Magnetization 38
4 ORIENTATIONAL ORDER PARAMETERS 45
Density Matrix Formalism 45
Application to Solid Hydrogen 50
A Proposition for a Zero Field Experiment 55
A Model for The Distribution Function of o 58
5 NMR PULSE STUDIES OF SOLID HYDROGEN 64
Solid Echoes 64
Stimulated Echoes 70
iii
LowFrequency Dynamics of Orientational Glasses 73
6 SUMMARY AND CONCLUSIONS 85
APPENDIX
FluctuationDissipation Theory 89
REFERENCES 91
BIOGRAPHICAL SKETCH 96
iv
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
THEORY OF NUCLEAR MAGNETISM OF SOLID
HYDROGEN AT LOW TEMPERATURES
By
Ying Lin
April 1988
Chairman: Neil Samuel Charles Sullivan
Major Department: Physics
Systematic studies of the nuclear magnetism of solid orthopara hydrogen
mixtures at low temperatures are presented.
The formulation of the nuclear spinlattice relaxation time T\ for the case
of the local ordering in the orientational glass phase of orthopara hydrogen
mixtures is given. The temperature dependence of T\ is discussed. A strong
dependence on the position of the NMR isochromat in the line shape is found
and this is in good agreement with the experimental results of a group of
physicists at Duke University, provided that the crossrelaxation is taken into
account. The relaxation is found to deviate considerably from an exponential
recovery.
The orientational degrees of freedom of ortho hydrogen molecules in terms
of density matrices and the irreducible tensorial operators associated with unit
angular momentum are described. The range of allowed values for the orienta
tional order parameters is determined from the positivity conditions imposed
on the density matrix.
A theory of solid echoes and nuclear spin stimulated echoes following a two
pulse RF sequence and a threepulse sequence in the quadrupolar glass phase
v
of solid hydrogen is developd. 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 T\ ) and thereby used to detect ultraslow molecular
reorientations.
vi
CHAPTER 1
INTRODUCTION
The orientational ordering of the rotational degrees of freedom of hydrogen
molecules at low temperatures has been carried out by a number of groups in
recent years (1 19). The principal reason for this interest is that the ortho
hydrogen (or para deuterium) molecules with unit angular momentum, J = 1,
represent an almost ideal example of interacting spin1 quautum rotators,
and therefore a valuable testing ground for theoretical models of cooperative
behavior.
A suggestion that a random distribution of ortho molecules in a para
hydrogen matrix may behave as a quadrupolar glass at low temperatures for
ortho concentration X < 55% was proposed. It was based on the observa
tion that orthopara hydrogen mixtures provide a striking physical realization
of the combined effects of frustration and disorder on collective phenomena.
These effects play a determining role in spin glasses, and the behavior of solid
hydrogen mixtures is analogous to that of a spinglass such as EuxSri_xS in
a random field(20,21).
The electrostatic quadrupolequadrupole interaction is the dominant inter
action which determines the relative orientations of the ortho molecules, and
there is a fundamental topological incompatibility between the configuration
for the lowest energy for a pair of molecules (a Tee Configuration for EQQ)
and the crystal lattice structure: one cannot arrange all molecules so that they
are mutually perpendicular on any 3D lattice.
1
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,
(er = (3J% 2)) which measure the degree of alignment along the local sym
metry axes oz. There is a broad distribution p(cr) 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 experiemental studies(25 28)
of NMR relaxation times T\, 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 T\ 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(l,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 experiemental 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 o = , 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 < 0.1/f). 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 Tj1 across the NMR absorption spectrum, and (ii) the nonexponential
decay (14,31) of the magnetization of a given isochromat; both result from the
broad distribution of local axes and alignments for the molecular orientations.
Both properties alone lead to variations by more than an order of magtitude
and need to be understood theoretically before attempting to deduce charac
teristic molecular fluctuation rates from the relaxation times. It will be shown
that the strong departure from exponential decay for the magnetization can
be understood provided that the broad distribution of local order parameters
is correctly accounted for.
In the socalled quadrupolar glass the quantum rotors cannot in gen
eral be described by pure states and a density matrix formalism is needed to
describe the orientational degrees of freedom. It is needed to determine the
precise limitations on the local order parameters (molecular alignement, etc.)
from the quantum mechanical conditions imposed on the density matrix and
to discuss the implications for the analysis of NMR experiments. Some of the
considerations for the spin1 density matrix description have been given else
where (29,32 37) but solid #2 is a special case because the orbital angular
momentum is quenched. The case will be discussed in chapter 4.
4
In molecular solids, one has to consider two kinds of degrees of freedom:
translational degrees of freedom for the centreofmass motion and orientational
degrees of freedom for the rotational motion of the molecules.
One of the most fascinating problems encountered in the molecular solids
is the existence of glasslike phases in which the molecular orientations become
frozen without any significant periodic correlation from one site to another
throughout the crystal. The most striking example of these orientational
glasses is probably that observed in the solid hydrogen when the quadrupole
bearing molecules are replaced by a sufficient number of inert diluants.
If the quadrupoles in this frustrated system (20) are replaced by inert
molecules, it leads to large reorientations of the quadrupolebearing molecules
in the neighbourhood of the inactive diluants and the dissappearance 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
ther steadystate line shape or FID analyses. Conventional continuous wave
(CW) and free induction decay (FID) NMR techniques have only been able to
show that the orientational degrees of freedom appear to be fixed for times up
to 1CF4 105 S. A considerable improvement may be achieved by the analy
sis of solid echoes for which the observation times during which one can follow
molecular reorientations are extended to an effective relaxation time (?2)e//>
which may (by a suitable choice of pulses) be much longer than the transverse
relaxation time I2 that limits the conventional techniques.
It will be shown that spin echoes and stimulated echoes following a two
pulse sequence and a threepulse sequence, respectively, provide a more pow
erful means of investigating the orientational states and particularly the dy
namics of the molecules bearing the resonant nuclei, than the conventional
continuouswave technique.
Spin echoes were observed in solid #2 a long time ago (38) and have been
used to study the problem of orientational ordering (11, 14, 27, 28, 39, 40). In
order to gain a deeper insight into this problem, a series of questions will be
disscused in chapter 5. These are the formation of spin echoes (including solid
echoes and stimulated echoes), explanation and comparison with experiments,
and the motional damping of echoes.
CHAPTER 2
THEORY OF NMR RELAXATION
BloembergenPurcellPound Theory
The theory of spin relaxation in liquids (or gases) is based upon time
dependent perturbation theory.
In liquids where the spinspin coupling is weak and comparable to the
coupling of the spins with the lattice, it is legitimate to consider individual
spins, or at most groups of spins inside a molecule, as separate systems coupled
independently to a thermal bath, the lattice.
It is well known that the expression for the interaction between two mag
netic dipoles of nuclear spin /t and Ij can be expanded into 6 terms:
HDD = Ij 3(f, S)(/y n)]
rij
= {A + B + C + D + E + F) (2.1)
r
ij
where
A + B = (3cos26 1)(/, Ij 3IizIjz)
C = + IlzI+)sin0cosOe~i(l>
D = (//y, + IizIpsinOcosOe'*
E = 1^ sirfer2'*
4 ]
F = /r r sin2 0e2l
4 3
6
7
If we carry out firstorder timedependent perturbation calculations to ob
tain the transition probability W between the megnetic energy levels, we find
for a nuclear spin I
W = ?nf4ft2I(/ + 1MM + ^2(2^o)] (2.2)
and the longitudinal relaxation time
Ti =
1
2W
i = 4h2I{l + l)[Ji(^) + ^J2( 2//0)]
This is the BPP expression(41,42); where
G{r)eiu)TdT
G(t) = F(t)F*(t + r)
Fq = ^(l 3 cos20)
F\ = zsinQcosOe1^
r6
F2 = \sin20e2i
r6
(2.3)
Theory for Liquids in Terms of Moris Formalism
Daniel Kivelson and Kenneth Ogan reformulated the study of spin relax
ation in liquids in terms of Moris statistical mechanical theory of transport
phenomena(43,44,45,46). They started with some wellknown phenomenologi
cal magnetic relaxation relations and formulated them in a manner most suit
able for comparison with Moris 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:
= f $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 f 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 ^4(t) can be expressed in terms of the superoper
ator, )ix, of the Hamiltonian as
A = i)ixA (2.6)
A(t) = e'X'^A
(2.7)
9
The time evolution of ci[tp) is determined by a propagator composed only
of those components of the Hamiltonian Hx, which lie outside the subspace
determined by the slow variables, i.e.
a(tp) = exp[t( 1 P)i)ix\a (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) =(d(tp)a)'() 1
Effective memory function matrix:
K{t) = k(t)e~iilt
where
in =(^)(]yl
Relaxation matrix K:
/
Jo
The relaxation matrix may be complex:
R = Y'HE K1M
(2.9)
(2.10)
(2.11)
(2.12)
(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 Moris equation:
i(i) = [(n +
(2.14)
10
and its Fourier transform:
iu(oj) A(0) = il A(u) K(u 2) A(w) (215)
We can now derive Blochs equation and relaxation time formulae in a
simple case (one variable).
We express the Hamiltonian as
H = *S+ XL + Usl (216)
where Mg depends only on the spin variable, Hr represents the molecular mo
tions and interactions which are independent of the spins, and "Hgi involves
those interactions which involve both spins and nuclear spatial coordinates.
The slow variables A can be selected as:
=^A52j (2.17)
We assume that k(t), the memory function matrix, decays rapidly so that
K(cj f) can be replaced by Kre Kim Equation (2.15), transformed back
to the time domain, then becomes:
^$*(0 1[^(0 (*$*)] (218)
^S(0 = [*(w0 + ff) ^"^(0 (2.19)
where
4 f
Ti = ReKzz = {[MsL{tp)Sz{tp)][SziMsL]) dt (2.20)
T2l =
2 f
= Re(~) J (2.21)
= [KlAi) = Jq ([HsL{tp)S(tp)][STHsL])eTlUJt dt (222)
Equations (2.18) and (2.19) are the simple Bloch equations except that the o
term which represents the socalled nonsecular or dynamic frequency shift is
given explicitly.
11
Kubo and Tomita Theory
The theory originally introduced by Kubo and Tomita emphasizes the simi
larity of magnetic relaxation to other nonequlibrium phenomena (46,47,48,49).
Description
We know that the Bloch equation can describe the relaxation of the mag
netization m. It is a linear equation.
= 7m(i) x Hit) mAt)U ^
dt v K T2 Ti
In matrix notation the Bloch equation (2.23) can be written as
(2.23)
Am(t) L Am{t)
(2.24)
where
Amk(t) = mk(t) mk (k = i,j,k)
is the thermal equilibrium value
/_ 1
TT
L =
0
0 ~ % ia,o
0 \
0
V
(2.25)
(2.26)
0 0 f iojQ j
If we transform from the laboratory frame to a frame rotating with the
Larmor frequency around the z axis, we will have
Am(i) = L'Am^(t)
(2.27)
The formal solution of Eq. (2.24) is the following:
Ama(t) = (e Lt)a/3Amp[t)
(2.28)
The Fourier transform of the formal solution is therefore
Ama(cu)+ (..JapAmp(0)
(2.29)
12
+ means a positive Fourier transform and in the rotating frame:
Amy(w)+ = ( 1 )Am,(0) (2.30)
Ln ~ lbJ
Response Function
The linear response formalism begins by calculating the linear response
of a dynamic variable to a disturbance created by a timedependent external
force.
Consider the Hamiltonian:
= H (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)H(t) is a small, timevarying field which is responsible for the nonequlib
rium behavior of the system.
The response function fki(t) is defined as
= / fklit T)Hi{T)dr (2.32)
J oo
The response function fki(t) gives the effect of the disturbance at time t.
The Fourier transform of eq. (2.32) is
Amfc(u>) = XkMHM (233)
where
roo
XklM = / eMtfkl(t)dt (2.34)
Jo
Equation (2.34) defines the susceptibility Xfc/(w) ^ h35 a real and imaginary
part that are connected by the KramersKronig relations. The imaginary part
13
of the susceptibility is called the absorptive part which is related to the power
the sample absorbs. The real part of the susceptibility is called the dispersive
part and is related to the measured line shape.
Relaxation Function
The relationship between the relaxation function and the response
function fki{t) is given by
/oo
fkliT)dT
The relaxation function describes the time change of the response after the
external disturbance is cut down to zero.
Assume a step disturbance:
Ht{t) = Hietd{t) (2.36)
where
(<)={; 5J (237)
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:
A mk(t) = Fkl{t)Hl t>0 (2.38)
and
Amfc(0) = Fkl{0)Hi (2.39)
Here Fki(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) Fai{t)F^l{0) Arribo) 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
() (2.41)
We define okÂ¡{u>) as the following
okl{u) = ~ Xkl^ = f dteiwtFki(t) (2.42)
JO
From equations (2.38) and (2.39):
AmQ(w) = oap{u)Hp
(2.43)
AmQ(0) = Xap{Q)Hp
(2.44)
Ama(w) = aa7(w)x/31(0) A mp(0)
(2.45)
Comparing equation (2.28) with equation (2.45) yields
= ^ (">* = So}
(2.46)
in the rotating frame
1 = Vkk(u + Ku0) k =
Lkk tuJ Xjfcfc(O)
(2.47)
i.e.
L'kk Reck_k{u + Ku0)
Lkk + oj2 Xfc(O)
Using a symmetric form of ok_k, equation (2.48) becomes
(2.48)
L'kk RekkH
Lkk+u2 Xkk{)
(2.49)
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 //(A) can be split into two
parts: a part IIo that contains the Zeeman Hamiltonian and a part H' that
couples the spins to the lattice and is responsible for the relaxation:
H{ A) = H0 + A H' (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) = A2J^A nL\n)
n
(2.51)
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. A2Z/(0) = L'(A).
The results of transport coefficients are
Loo = t1
i \2
(*)
2(A Ml)
roo
/ dt{[H',Mz][H'{t),Mz})0 + C.C. (2.52)
Jo
Ln = Lli =
12
i \2
(L
4XTi
+pr'(<),M_l[ff',M+l>0 + C.C.}
roo
/ d{<[tf',M+][tf'(),M_])0
Jo
(2.53)
where
xr2 = (254)
H'(t) = ehHot H'
(2.55)
16
The subscript zero on the braket indicates that the trace is taken over the
equilibrium density matrix
_ exp((3H0)
P Tr(exp((3H0))
which does not involve the spinlattice coupling.
Nuclear SpinLattice Relaxation in NonMetallic Solids
The problem here is essentially the same as that for liquids and gases,
namely to calculate the probability of a flip of a nuclear spin caused by its
coupling with the thermal motion of a lattice. There are, however, some
significant differences. The internal motion in solids will often have much
smaller amplitudes and/or much longer correlation times than in liquids. In
rigid solids because of the tight coupling between nuclear spins exemplified by
frequent flipflops between neighbours, the correct approach to nuclear mag
netism is a collective one, where single large spinsystem with many degrees
of freedom are to be considered, rather than a collection of individual spins.
The assumption is usually made that the strong coupling of the nuclei simply
establishes a common temperature called a spin temperature, and that the
lattice coupling causes this temperature to change(50,51,52).
A quite general equation can be derived.
d(3 1
fjrVM (257)
where
0 = jflr Ts Spin temperature
00 = 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 Wmn{En Em)2
(SfT
(2.58)
Where  m) and  n) are the eigenstates of Hq.
Wmn = Wnm is the transition probability from the state  m) to the state
I n)
There is another way to calculate T\. 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~h^tpek^t (2.59)
hdp*
i dt
(2.60)
Where "Hi is a perturbation,
X{{t) = e_Â£*%eS*ot
Equation(2.60), integrated by successive approximation, gives
t
dt
+higher order terms (261)
or
^ f drK(t), K(t r),/(0)
+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 o*
a* = tr f{p*} (2.63)
with matrix elements (o: I * I a') =Â£/(/ I P* I /')
By making some assumptions Abragam (51) gives a general master equa
tion
do* f
~M~ = ~ JQ WO.^* T)>* ao]]dr (2.64)
where the bar represents an average of many particles.
If there is a spin temperature, then
dd l f+
= 2{Po ~ 0) J m WW. [ ( 0. (2.65)
i.e.
f = Â£7
~ = ~Â£Â£ (W(0.o![,*(<r).%l>* (2.67)
For the relaxation of like spins by dipolar coupling the result for T\ is the
same as the result of Kubo and Tomita theory.
Nuclear SpinLattice Relaxation for Solid Hydrogen
The molecular hydrogens (H2,D2,HD,etc.) form the simplest molecular
solids. The properties of solid mixtures of ortho (angular momentum J 1,
nuclear spin 7 = 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,Gl)
r = y*r3{c2JiVo) + d2[9j}(u 0) + yj(2u,0)]} (2.68)
Where c denotes the constant of spinrotational coupling and d that of the
intramolecular dipolar coupling, with respective values of 113.9 and 57.7 khz.
The spectral density functions J(moj) are taken at uq and 2wq where u>o is the
Larmor frequency(5l,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 simlple method of impurity averaging over the distribution
of OH2 molecules. The averaged equations were then solved numerically to
obtain the spectral functions for solid H2 selfconsistently for the first time.
The resulting spectral functions were used to compute the T\ as a function
of the orthomolecule concentration and this was shown to agree well with
experiments at 10K and over concentraiton range of 0.5 < X < 1. They
obtained a y/X concentration dependence for T\, which was in agreement with
the data of Amstutz and colleagues (62) for X > 0.5.
20
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 < 400mfc), 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 rnomemetum correlation functions and Harris used an improved
version of the same theory. The T\ resulting from these calculations had a
5
concentration dependence of X3, which was in agreement with the data of
Weinhaus and Meyer (61), but the magnitude of T\ 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 T\ which were in fair agreement with
experiment for all X (61,62).
Both methods had the defect that the impurity averaged correlation func
tions were obtained by statistically averaging assumed functional forms and no
attempt was made to determine the shape of the spectral function.
The first attempt in the small X region to calculate the high temperature
correlation functions selfconsistently and thus to overcome the above defect
was made by Ebner and Sung (66). They used the Sung and Arnold (68)
method of impurity averaging the Blume and Hubbard (69) correlation func
5
tion theory and obtained a T\ which had the experimentally observed J3
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 T\.
Ebner and Myles (67) improved the calculation of Ebner and Sung by
properly treating the anisotropy of the electric quadrupolequadrupole (EQQ)
interaction, which is the dominant orientationally dependent interaction be
tween two O #2 molecules in solid H and which therefore, almost totally
determined the shape of the angular momentum spectral functions.
Sung and Arnolds method of impurity averaging the infinite temperature
Blume and Hubbard (69) correlation function equations was employed, but
the equations were obtained using the full EQQ interaction rather than an
isotropic approximation to it. The spinlattice relaxation time was computed
as a function of the O H2 concentration using a formula for derived by
applying the Blume and Hubbard theory (69) to the nuclear spin correlation
functions in this system. The resulting T\ 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 T]
There is a striking resemblance between the phase diagram for the magnetic
alloys such as CuMn, AuFe ... and orientationally ordered orthohydrogen
parahydrogen alloys(Fig.l2).
The nuclear spinlattice relaxation of ortho molecules at low temperatures
is determined by the modulation of the intramolecular nuclear dipoledipole
interactions Hqd and the spinrotational coupling HgR by the fluctuations of
the molecular orientations(55, 56). The calculations are particularly transpar
ent if we use orthonormal irreducible tensorial operators O2m and N2m for the
orientational (J = 1) and nuclear spin (I = 1) degrees of freedom, respectively.
The O2M are given by
20 =
@2l = T~{JJz T JzJ)
022=2(J)2 l31)
and similar expressions hold for the N2m in the manifold 1=1.
The intramolecular nuclear dipoledipole interaction Hjjr) and the spin
rotation interaction II SR can be written in the above notation as
HDD = ^2m(02m(*)
22
23
and
Hsr = he Â£(lP<,(*')lm(0 (3.2)
i
respectively. D=173.1 khz and C=113.9 khz. The index i labels the ith
molecule. The 0/m and Nm are the operator equivalents of the spherical
harmonics YÂ¡m in the manifolds J = 1 and / = 1, respectively.
The relaxation rate due to Hjjd can be shown to be
1 f
T1DD = Jfzj y0
= JJrj J0 (Ii*>Hdd\ Â£';J"1^co'_5,",.4])
(3.3)
where H0 hoj0Iz. It is the Hamiltonian responsible for the molecular dy
namics. Using the commutators [Iz,N2m\ mN2m we find
TIDD = \2 Y m2j2m{u) q) (3.4)
M=l>2
where the spectral density at the Larmor frequency
/0 .
T
oo
The expectation value < ... >y 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 Hjjd and terms. The orientational order parameters,
however, are evaluated with respect to the local molecular symmetry axes. We
must therefore consider the rotations
^2w (36)
where the dmix are the rotation matrix elements for polar angles x = (a,/?)
defining the orientation of OZ in the local (x,y,z) reference frame.
TEMPERATURE (K)
24
ORTHO CONCENTRATION
Figure 1. Phase Diagram of OrthoPara i/2 Mixtures
25
Figure 2. (a) Theoretical Phase
Diagram for A Short Range System
(b) Experimental Phase Diagram for
EuxSrixS And AuFe (c) Phase
Diagram of The Ising Spin Glass
26
Figure 3. Picture of Long Range
Order And Quadrupolar Glass
27
We assume the simplest possible case
< ={O2m(0)Om(0))g2m(Â¡) (3.7)
In the following, we consider only this case and further assume that the relax
ation is dominated by the fluctuations of the p = 0 component. We find
T1DD = (2 o2)D2 Y m2Mmo()2i?2o(^o) (3.8)
m=l,2
where the <720(mwo) are the Fourier transforms of the reduced correlation func
tions <72o(0 and a = {3J2 2)T. The prefactor (2 a a2) is the mean square
deviation of the operator O20 evaluated in the local symmetry axis frame.
The contribution from the spinrotational interaction Hrr is
SR = 2C2jn(U,0) 0C'2(2 + a)Mlo(a)2<7lo(wo) (39)
the total rate 1 = T^D + T^gR
Temperature Dependence of T]
Minimum Values of T1
From the previous result (3.8 and 3.9)
=r(M) = (M) + rfaO)
DD SR
where
jT~(a 0) = (2 ~ 0 ~ 2)d2 Y mVmo(0)202o("^o)
1DD m= 1,2
= l2(2 + ff)ldioWI2io(w0)
SR 6
If we take a powder sample average, the Tirr is small.
,2\ n2
7?2o(wo) + 7ff2o(2wo)
(3.10)
28
To a good approximation we have
jri) ~ (2cra2)L>2
r^2o(wo) + 2(2wo)
(3.11)
For the simplest case we can restrict o to negative values with appropriate
definitions of principal axes (22 and Chapter 4)
1 f2(2 a a2)da l
X u
Ti
f2d(J
12
riteoi^o) + rff2o(2wo)
o o
 (ff2o(wo) + 402o(2wo))
(3.12)
Assume <720(^0) and
92o(uo) = (3'13)
92o(2wo) = TTtfl (314)
when ojqtc 0.6156, T\ = rlmin
The eqs. (3.12) (3.14) result in the following
1)
uq = 2n x 100 x 106
T\min = 13.4172 msec
2)
u>q = 2it x 25 x 106
Tlmin ~ 3.3542 msec
T\eXp 1.03 msec Fig A (ref.25)
3)
u0 = 27T x 9 x 106
Tlmin = 1.20574 msec
T\exp = 2.25 msec Fig.5 (ref.70)
(08S) J.
29
T(mK)
Figure 4. Experimental Curve of T\
T, (msec)
30
Figure 5. Experimental Curve of T\
31
Temperature Dependence of T j
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 ff2o(ma;o) fr high frequencies. From eqs. (3.7) and (3.8):
(O2m(0)OÂ¡m(0))9(f) =(02m(0)0'2m(0))e J
t
t
/OO t2 w2a
e o2 e~l(i}tdt = a\e T
OO
(3.15)
(3.16)
Eq. (3.12) becomes
1
Dy/ _w2 2 \
(ae 4 + 4ae 0 '
T\ 36 v
Ti =
36
ae 4 + 4ae woa
(3.17)
(3.18)
irp _
T\ passes through a minimum when = 0, i.e for a = 5.082 X 10 and
the minimum value is Tlmin = 1.1384msec, at = 25 MHz. The result of
the theoretical value 7\Win = 1.1384msec is in very good agreement with the
experimental value Tlmn = 1.03msec(25).
Now the question is how T\ varies at fixed cjq over a wide range of tem
peratures which causes rc to vary. Similar to the discussion of the dynamics of
spin glasses (71), we would like to try an Activation Law of the VogelFulcher
type:
a = a0e 'T_7o.
(3.19)
where
A is the activation energy in temperature unit
q is a time factor which is the value of a as T > oo, and
Tq is some characteristic temperature (transition temperature), such that
as T Tq, long relaxation times become important.
32
0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48
T(K)
Figure 6. TrT Curve (solidriezp,dashedrlt^eo)
33
By using two experimental values (T\ = 2.550msec at T = 0.450/ and
T\ 2.514msec at T = 0.380 for X = 38%) we find the following formula:
o 0.1880
a = 5.4128 x 10 8 X ero.6on> (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 Inhomogeneitv of Ti
Results of Calculations
The dependence of 7\ 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(ol)o and a negative branch given by Av = \DP2(a)o
(Fig.7). As previously demonstrated we restrict o to negative values. That is
lal 1. 9
Av = D (3cosot
2 2V
Av Z) (3cos2a
2 2v
 1) (positive branch)
1) (negative branch)
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 wq In this case the spectral density functions are given by
ff(wo) = 4<7(2u>q) = 5 (low temperature limit of Lorentz form), where r1 is
UJqT
the characteristic fluctuation rate, which was taken to be a unique value for
simplifying calculations.
For the dipolar contribution at fixed o and Av we obtain
iiV 2)D2 \l (2&)2
1DD
w0r
(3.21)
34
Figure 7. Allowed Range of Values of a for
A Given Frequency Aw
35
and for the spinrotational contribution:
wor
The frequency dependence should be obtained by summing over all allowed
o for a given frequency At/. The calculations were straightforward but quite
tedious. Table I and Table II show the numerical results.
(3.22)
7l _
SR ~
1/^2
18 U
(2
Table I Spectral Dependence of DipoleDipole Relaxation Rate
Frequency range Au
rp\ 24
1 lDD x \ m )
0 < Au < \D
lOD'i l2uD+4u2 21/2 i 2i/ 2i/2 i u
3D2 + D(D2u) log D + D[Dv) log V
2D < A u < D
5D27uD+3u2 2i/2 i_
3DP + D(Du) log D
Table II Spectral Dependence of SpinRotational Relaxation Rate TiRR
Frequency range Au
T1 x f18 r 1
J1 SR x 1 r?2 )
0 < Au <\D
2U3u 2i/ i 2/ 2u i v
D D2v loÂ§ U + TJ=U io8 77
\D < Au < D
77+i/ 2i/ i i/
~n~ + Dv log V
Discussion
Fig.8 shows the results of the calculations for the dipolar contribution
and the spinrotational contribution as function of frequency. All curves
in this figure have been normalized to unity at Au = 0 in order to facilitate
the comparison with the experimental data. The net relaxation rate T^aic\ =
^1DD "* SR The curves show discontinuities in slope at \Au\ = y, but this
was not seen experimentally. If we take the finite crossrelaxation rate T^ into
36
account, which will bring the very slowly relaxing isochromats at Ai/ = 2D
into communication with the rapidly relaxing components,
= T.
1
l(calc)
+ T.
l
12
(3.23)
when the individual spinlattice relaxation times are much longer than the
crossrelaxation times the spins come to a common spin temperature via the
crossrelaxation mechanism before relaxing to the lattice via the rapidly relax
ing components. On the other hand, when the direct spinlattice relaxation is
fast and
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 Ti2 by the probability of a spin flipflop transition
via the intermolecular dipoledipole interaction for two isochromats and u2
given by
{TÂ¡lip~flopylexp
7T2(//i t/2)2
Mmter
= (T^f^r'exp
jyrintra ^^jintra^2
2 Ml2nter
(3.24)
where the exponential factor is the overlap given by Abragam (51) for two
lines centred at i/j and 1/2 and their individual widths are determined by the
intermolecular dipoledipole interaction. Mytra, Mjnira and M^nier are the
moments resulting from the intra and intermolecular dipoledipole interac
tions, respectively. Yu et al. (14) observed a much weaker dependence than
the exponential variation with Mytra 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 T\2
37
O 0.25 0.5 0.75
I//D
Figure 8. Frequency Dependence of T\
38
The most complete studies of the spectral inhomogeneity of T\ in the
quadrupolar glass phase has been carried out for an ortho concentration X
0.45 at T 0.15/f(14), and the data of ref.14 would place the crossrelaxation
time T12 in the range 7.5 10.5 msec.
While the crossrelaxation is faster than the direct relaxation to the lattice,
the values of 1\2 reduce the spectral inhomogeneity of the relaxation of the
NMR line shape. In this case we expect to observe Tl(lbs) ~ Tl(lal) + T12 for
the relaxation of the magnetization of a given isochromat. This is shown in
Fig.8. In view of the uncertainties in the crossrelaxation and the simplifying
assumptions that have been made, the overall agreement with the experimental
results is good. The correct overall behavior is predicted as well as the subtle
change in spectral dependence at the half width points which has already been
seen in the experimental data of Yu et al. (14).
NonExponential Relaxation of Nuclear Magnetization
Although it is agreed that (i) the low temperature NMR lineshapes indi
cate a random distribution of molecular orientations (for both the local axes
and the alignment a =(3J 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 < O.llf). Is it a very rapid, but smooth variation (25,27,70)
with temperature, corresponding to a collective freezing of the orientational
fluctuations or a slow, smooth dependence (9,14,28) with no evidence of any
strong collective behavior? In order to resolve this problem it is important
39
to understand two unusual propertiesthe spectral inhomogeneity of the re
laxation rate 2^ 1 across the NMR absorption spectrum (14,29,30,72) and the
nonexponential decay of the magnetization of a given isochromat (14). The
spectral inhomogeneity has been discussed previously and the purpose of this
part is to show that the strong departure from exponential decay for the mag
netization can be understood provided that the broad distribution of local
order parameters is correctly accounted for.
Variation of Relaxation Rates within Given Isochromats
As previously proved (3.21,3.22)
*)!>
1SR OJqT
The frequency of a particular component of the NMR absorption line is
given by
A u = DP2{cc)o (3.25)
2
and this can be satisfied by very different values of o and a; e.g. Av = \D
occurs for o l,P2 = = ~ l;o = Â§> f*2 = Â§ the only
constraints being that a and P2 lie within their limits; 2 < o < 0, and
i < P2 < 1. Obviously, different pairs of a and P2 (for fixed Av) can result
in very different values of Tujd and Tig ft. 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
V, which give the variation of T\ for Au = \D, \D and 0, respectively. The
relative contribution of these rates can be determined from the probabilities
40
n(a),n(P2) of finding o and P2. At low temperatures, the analysis of the line
shape indicates that a good approximation for IT(cr) is a triangular distribution
I1(
very glassy) which requires that II(P2) oc , 1 The calculated rates
P2) have a relative weight P = II(
component Au = :f.D<7P2.
Magnetization ratio
M(t) Â£Pe Tl
M(0) ~ Â£P
(3.26)
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
/\u are indicated by the symbols.
Table III Variation of Relaxation Rates within A Given Isochromat (Av = \D)
parame.
M)
larame.
kl
prob.
n^)
prob.
i%)
rates
j.l
1\SR
rates
yl
1\nn
rates(a)
rr1
13
16
1,5
1.022
1.067
0.058
0.234
0.267
7
8
7
1.044
1.143
0.107
0.431
0.492
13
16
13
1.069
1.231
0.144
0.583
0.666
3
4
i
3
1.095
1.333
0.167
0.681
0.777
n
6
V
1.124
1.455
0.170
0.706
0.804
5
8
8
5
1.155
1.600
0.150
0.634
0.720
9
1fi
6
9
1.188
1.778
0.097
0.422
0.478
1
2
2
1.225
2.000
0.0
0.0
0.0
Rates given in units of
The Determination of The Molecular Correlation Time t and T^ at Av 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 (At^
parame.
P2H
Darame.
cr
prob.
n (p2)
prob.
nw
rates
j.l
1\ SR
rates
T~ 1
J1 on
rates
rr1
15
lf
~S~
15
1.022
0.533
0.092
0.272
0.325
8
1.044
0.571
0.179
0.526
0.629
13
T6
13
1.069
0.615
0.260
0.760
0.910
3
4
i
3
1.095
0.667
0.333
0.972
1.165
11
16
8
11
1.119
0.727
0.398
1.159
1.389
5
8
4
1.155
0.800
0.450
1.316
1.576
^I
1G
5
1.188
0.889
0.486
1.435
1.715
1
2
1
1.225
1.000
0.500
1.500
1.789
7^
16
8~
7
1.265
1.143
0.482
1.485
1.763
J
A
3
1.309
1.333
0.417
1.337
1.577
16
8
5
1.359
1.600
0.275
0.938
1.097
jl
4
2
1.414
2.000
0.0
0.0
0.0
Rates given in units of 24^2 T
line at Au = 0 in Fig.9, is obtained t = 1.51 x 10_75. For
T = 1.51 x 10_75
1
TÂ¡
I At/=0 1.49597 X
103
10.0446
T\ At/=0= 6.7144(msec)
(3.27)
T\ 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 Aiu for the calculated are 87,43.5 and
0 khz which are below the values chosen by the Duke group (14). The same
theroy to calculate for Av = 98 and 58 khz and the results are depicted
in Fig. 10. The overall agreement is very satisfactory.
Comparison of the calculated decays M^u{t) with the experimental results
shows that not only is the correct overall deviation from exponential decay
42
predicted, but that thei'e 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.
M
prob.
n(
rates
J.1
1\ SR
rates
T~ 1
11DD
rates(a)
rr1
r~
15
0.533
1.467
2.249
3.096
4
1
0.571
1.429
2.245
3.070
8
13
0.615
1.385
2.237
3.036
2^
3
0.667
1.333
2.222
2.992
11
0.727
1.273
2.198
2.933
5
0.800
1.200
2.160
2.853
8
9
0.889
1.111
2.099
2.740
1
1.000
1.000
2.000
2.577
S~~
7
1.143
0.857
1.837
2.331
4
3
1.333
0.667
1.556
1.940
8
5
1.600
0.400
1.040
1.271
2
2.000
0.0
0.0
0.0
We consider = 0 for the range of cr considered in Table IV in order
to facilitate the comparison.
Rates given in units of np~
24wr
M(t)/M(o)
43
O 15 30 45
t (msec)
Figure 9. Time Dependence of M(t) for Different Au
M(t)/M(o)
44
Figure 10. Time Dependence of M(t) for
Av  98 and Aw = 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 its 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 3x3 density
matrices (for each site i). The p are completely described by
(1) the molecular dipole mements (Jx)i, {Jy){,{Jz)i and
(2) the quadrupole moments (JxJy)i, (JyJz)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 operatiors with 0 < L < 2J for general J and the
associated multipole moments
tlm = Tr{PTllm) (41)
For simplicity the site index has been dropped. The expansion of the single
particle density operator in terms of the multipole moments is given by
2 J L
0 = (2 > T 1) E E (2i+1)iLWnÂ£M (42)
1 1 L=0M=L
There are three conditions imposed on p: Hermiticity and both weak and
strong positivity conditions(34).
45
46
p is a Hermitian operator aiul
hjM = ((43)
Tl00 is a unit matrix operator and t00 = 1 = Trp
! '' ) \ I
Tlie weak positivity conditions are given by
27TT Tr(/) 1 ^
. 1 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 + l)1; then the matrix describes a
completely unpolarized state and Tr(p2) = (2J + l)1.
For density matrices of spin^ particles, condition (4,4) is the only con
dition imposed by the positivity condition. But for J > further conditions
are imposed on the density matrix and on the multipole 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 multipole moments and thus on the allowed values
of the local order parameters for orthoH2 molecules in the solid mixtures.
It is useful to construct orthonormal matrix representations of the irre
ducible operators Hlm Â¡n the representation (J2,JZ). For J=1 these are given
47
by the following 3x3, matrix operators with rows (and columns) labelled by
the eigenvalues 1,0, 1 of Jz.
and
1 1 Z1 0 0
Hio = fiJz = 7= 0 0 0
v ...... V? \ o o l
i i ( 1
1111 = 0 0 1
V2 v2\0 0 0
1 /_ ,2 .9> f
n20 = ^^) = ^
1 0 0
02 0
0 0 1
1 1 (0 1 o
n2i z(JzJ+ + J+Jz) = j= ooi
2 V2 V0 0 0
,n22
1 2 ( 1
J+ =10 0 0
0 0 0,
Mirt
n,M = (i) nl
M
LL'MM'
(4.6)
(4.7)
(4.8)
(4.9)
The quadrupole operators transform analogously to the spherical har
monics Y2m(a,/3) with respect to rotations of the coordinate axes.
The reference axes have remained arbitrary in the discussion and we are
therefore free to choose local references axes that correspond to the local sym
metry for each molecule. The natural choice for the zaxis is along the net com
l
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 rnements in the above notation:
P ^ ^ Mnffn + ^ ^ QmJ^2r
m
where nn = {fl{n) and Qm = (n2m)
/rrt
(4.10)
48
The expression (4.1U) is identical with that was derived by ref.35 and ref.73.
We still can choose x and y axes such that Qi is real. i.e.
Q2 = \(4~J2y) (4.11)
(JxJy 4* JyJx) = 0 (412)
Q2 measures the departure from axial symmetry about the z axis and is some
times called the eccentricity (29, 30). It can be shown that with this choice of
local reference frame Qi and also vanish and the density matrix may be
written as
i"3 + TT0
/I
0
0 \
0
0
0 +
Vo
0
1
Ve'
Q 2
0
o
o
(4.13)
where
Qo = 4j<(3J% 2)) (4.14)
Qo is the alignment (29, 30) along the zaxis. Sometimes it is convenient to
define the alignment and the eccentricity as
=(1 j?) = ~\JIqo (4.15)
V ={4 Jy) = 2Qi (4.16)
Both o' and ?/ have maximum amplitude of unity. In terms of these parameters
p becomes
I l 'i \ ji
3 2
/I
0
0 >
0
0
1
+
Vo
0
1)
3U
0
h
V
3
\
0
Â¥)
(4.17)
49
V
tr'
Figure 11. The Allowed Values of/r =(JZ),
o And rj for Spin1 Particle
50
where p =(JZ).
The three eigenvalues of p are
x 12,
A i T o
1 3 3
= {j1 \/ p2 + i/2
3 3 9 V ** '
(4.18)
The strong positivity conditions, An > 0, are therefore seen to restrict the
allowed values of the local order parameters o',r\ and p to the interior of a
cone (Fig. 11) in the 3D parameter space. The vertex of the cone is located at
o' = 1, p = t] 0, corresponding to the pure state \ ip) =\ Jz = 0), and the
base of the cone is defined by o' \ and p2 + rj2 = 1 which corresponds
to the pure state  rp) = cos7  Jz = 1) + sin'y  Jz 1) with (Jz) = cos2^Â¡
and (J\) = sin2') (The polar angle 7 generates the points on the circle of the
cones baseplate).
The positivity domain shown in Fig. 11 is the same as those obtained by
Minnaert (34) using the EbhardCood theorem and similar to those given by
W.Lakin (32) in his analysis of the states of polarization of the deuteron.
Having established the physical considerations which determine the limited
range of allowed values for the order parameters, we now turn to the special
case of solid hydrogen.
Applications to Solid Hydrogen
In the absence of interactions which break time reversal symmetry, the
expectation value (Ja) must vanish for all a in solid hydrogen. This is the
socalled quenching of the orbital angular momentum (74). The reason for
this is that in the solid the electronic distribution of a given molecule may
(to a first approximation) be regarded as being in an inhomogeneous electric
51
field which represents the eiFect 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 must
accordingly vanish. The separation of the rotational energy levels is given by
Ej = 737(7 f 1) with 13=85.37K, and in the solid at low temperatures only the
lowest J values, 7 = 0 for par a7/2 and 7 = 1 for ortho772, 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 Pa% configuration with four interpene
trating simple cubic sublattices, the molecules being aligned parallel to a given
body diagonal in each sublattice and the order parameters, c'a =(l Â§7a),
are the same at each site a'a = 1.
The long range periodic order for the molecular alignments is lost below
a critical concentration of approximately 55% and the NMR studies indicate
that there is only short range orientational ordering with a broad distribution of
local symmetry axes and local order parameters throughout the sample. This
purely local ordering has been referred to as a quadrupolar glass in analogy
with the spin glasses, but unlike the dipolar spin glasses, there is no welldefined
transition from the disordered state to the glass regime. In order to describe
the ortho molecules in the glass regime, where there is a large number of sites
with various values of a', a density matrix formalism must be used.
The quenching of the angular momentum in solid 772 has two consequences
for the limits on the allowed values for the order parainters:
52
(1) From the previous discussion, the allowed values of o' and rÂ¡ lie within
a triangle bounded by the three lines (Fig.12 AABC).
J L ,
 + o' > 0
3 3
'r a T] > 0
3 3 2
(4.19)
which represent the strong positivity conditions for the eigenvalues of p when
the angular momentum is quenched.
(2) Since (Ja) = 0 for all a, we are free to choose the zaxis which was
previously fixed by the net component of the angular momentum. The natural
choice for the local reference axes is now the set of principal axes for the
quadrupolar tensor
Qap ~(2 (J
(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 (o', rj) 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 A CFE by suitable rotations (i.e.
relabel the axes).
The important point is that the pure states
VC =1 z = )
(4.21)
and
A = 7/(l Jz 1)+ I Jz = _1))
(4.22)
53
are not inequivalenl (Â¡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 can be obtained from i/>c by a
rotation of the axes by 4^ about x axis.
The rotation operator
rfr=^(if^)=(f)2+'(t)+i )
and
(4.24)
The rotation Rx(4r) leaves the state F invariant and maps D onto the point
G in parameter space.
We can furthermore show that the rotation Rx(j) maps the following
triangular regions of parameter space onto one another
A CDF > A AGF
ACER  AAEF
and
ABDF > ABGF
(4.25)
It is also seen by considering the transformation
= i(y)p(a,,7?)i2a:(y)
,3tt,
(4.26)
using the matrix representation
/ I _L
2 ^
_l_ n
V2 V2
1 i 1
V 2 ^2 2 J
(4.27)
54
Table VI Special Points in Orientational Order Parameter Space, (see Fig. 12)
Label
A
parame.
(In)
Wave Function
(1 1>+ 1)1/V2
B
(VV
I!1 DA/2
C
(Ml)
DS
D
(Li)L
111) DV2I 0)1/2
E
M:
l!'(l !)+ 1 1 V'i 1 0)1/2
F
(0.0)
FT + 0U) + (i + 0li)V2o)l/v/6
G
H.O).
K1+ 011)+ (1 + 01 1)1/2
Rx{^Â£) : i'Ai'l V;Gi VtB> tJF> flxed
#z(f) : 1>A * 0B> 0D5 VC VF. VG fixed
The i?x(4r) transforms the points (cr',77) into new points (o",rÂ¡') given by
(4.28)
which corresponds to the mapping given by eqs.(4.25).
There is some confusion in the literature concerning the values of o'. In Van
Kranendonks(75) book the negative value of o' had been ruled out as it was
unphysical. Hut this is not strictly correct. Van Kranendonks remarks refer
to considerations of the pure states  Jz = 0) and  Jz = 1) only. He does not
discuss either the density matrix approach or the formulation of the intrinsic
quadrupolar order parameters o' and rÂ¡ needed to describe the orientational
degrees of freedom of the orthoi/2 molecules.
The results of the relabeling transformation are particularly easy to under
stand if we consider the pair (5 = o', N = ^77) which transform orthonor
inally when the axes are rotated. The parameter space (S, N) is shown in
55
Fig.13. The relation i2i(^f) 'in (>S, N) parameter space through the line BZ
with the transformed points given by
S'
N'
(4.29)
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 orthoH molecules in the solid state. This is not the only choice that
can be made for a primitive area of inequivalent values of We may
also choose the sum of ACFK and AGF J 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
D(n20 (i))oZ
(4.30)
As previously discussed, considering the transformation to the local frame
we obtained
Ai/ = D\o[p2{cosOi) + ^r]isin20icos2i\ (4.31)
where (0, <^>) are the polar angles defining the orientation of the applied mag
netic field with respect to the local molecular symmetry axes.
56
("i'1)
(T
B
Figure 12. The Allowed Values of o And 77
for Spin1 Particle
57
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 o' is constant and 77 = 0, but it has not previously been
considered for the glass phase.
In zero applied magnetic field the degeneracy of the nuclear spin levels is
lifted only by the intramolecular spinspin interactions.
UddM = if) + + /Â£,)] (4.32)
The tensorial operators for the rotational degrees of freedom have been re
placed by the expectation values o' and rj. An applied radiofrequency field
can induce magnetic dipole transitions between the nuclear spin levels in anal
ogy with the socalled pure quadrupole resonance absorption(51 Chapter VIII).
The eigenstates of Hdq are  Iz 0) and  ) = ^( Iz l)  h = 1).
For each molecule i, three resonance lines can be expected corresponding to
the transitions  +) > 0),  ) > 0) and  +) > ) with frequencies
= D{o[ + J?7Z), uf = D[o[ Jr?) and isf = DrÂ¡l, respectively. If
axial symmetry is a good approximation, there is only one line at = Do
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 o
X. Li, H. Meyer and A. J. Berlinsky (23) have proposed a model for the
distribution function of o'. 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
Pi ~ 2^~ 4 + 4 + 4)
p=( 44 + 4)
p\ =(i 4> = \(Jl + 4 4)
The pure states p= 1 correspond to rpawave functions for a = x,y,or z.
(a)
This leads to a natural parameterization of the p1 given by
where the energies Eai 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 ]T)a Eai
0. i.e.
P(EX, Ey, Ez)
\[El+El+E?M A2
2ttA2
fi{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
as a function of the degree of local orientational order parameter measured
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 JZi = 1 and JZi 0 separeted by an energy gap A.
60
The probability distribution of A is Gaussian as following
P(A) =
nD2
e 2o'2
(4.33)
The order parameter a =(3 J2 2) will be
We can prove that
2e kt 2
= _a_
2e kt + 1
M4 15 (tr4)
M2
7 (<22
where
roo
(o2) = /
Jo
(2e at 2)2 / 2 _
(2e kt + l)2 V ttD2
A2
dA
(4.34)
(4.35)
(4.36)
let % = X
, f00 (2e~KfX 2)2 /y ^
(<7)/ __d_v wV~e 2 (437)
/o (2e at"* f 1)2 V 7T
D X
<4>
/ (2e at
A)
2)4 12 x2
e~ dX
(4.38)
(2e ktx + l)4 V X
By using numerical integral method the curve in Fig.14 shows the ratio
versus y/ (ct2) (order).
{a2)
Considering the intermolecular and the intramolecular dipolar broadening,
the fourth moment M4 and the second mement M2 should be (51)
where
,, rrp;,/*]2}
"lhi}
3V{M,X',A]2}
Mi= tTW)
^intra __ ^inter
MÂ¡n,ra = {3jkÂ¡I j* f)
4 > > r L.1
(4.39)
(4.40)
k,l
61
*rr=r,^E(1 3T2e',]nsi
r3.
i]
Following a quite complicated but straightforward calculation we can prove
that
M2 = M\ntra + Mrjnier
(4.41)
M4 = M\ntra + M\nter + 4 MlntraMl2nter
(4.42)
For a Gaussian shape one has Mnter = 3(M2 er) and
= M'tntra + + 4MflraM)n"r
(4.43)
Taking the same value of Mnter = 20[khz)2 as in ref.23 we obtain the curve
in Fig. 15. The upper one is ^Mintry2 versus y (cr2) and the lower one is p^2
versus
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
experimental lineshape is 3.15 (18) and the value of maximum of calculated
lineshape is 5.60.
62
MODEL (INTRA)
ORDER
Figure 14. Diagram of
{^)
P?
versus
vV>
63
MODEL
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)
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 occuring after application of resonant rf 90 r ip^ pulse
sequences and rf 90 r ip^ tw pulse sequences.
The purpose of this chapter is to develop the theory of spin echoes of solid
#2 in orientationally ordered phase and discuss the lowfrequency dynamics of
orientational glasses.
Solid Echoes
There are several publications which have presented calculations for the
amplitude of solid echo responses to 90 r xp^ pulse sequences (39, 78, 79).
We would like to examine the time dependence of the nuclear operators
and the conditions for the focussing of solid echoes.
The Time Dependence of The Nuclear Operators
A general twopulse sequence denoted by (j t ipis sketched in Fig. 16.
In NMR experiments one observes only the orthomolecules (7 = 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)(Ix + Hy)],
where p is the density operator at time t and 1 the nuclear spin operator.
64
65
R (.
/V
Figure 16. Solid Echo (twopulse sequence)
66
Rot?) R()
0 T 2T
Figure 17. Stimulated Echo (threepulse sequence)
67
In equilibrium the expression of density matrix is given by
eWo
Pie
ieq Tr(eWo)
Let Z be the direction of applied magnetic field.
\ = ~uohIzl
(5.1)
where ui0 is the Larmor frequency.
Since
Pie
(! + Jet hi)
(5.2)
Tr(l + ^4,) *3' KT
We know that the radio frequency field is responsible for nonequilibrium
behavior of the system.
In the rotating frame after the first 90 pulse, the density matrix p(0+)
would be
p(0+) = e~*f HLdtpi{0)ehf Hldt (5.3)
If the radio frequency field is in the y direction of the rotation frame, we
obtained
r J 1 HoJq
/,(0+) = S54i(i + ^4)]eH4
(0+) = I(1 + t'1
(5.4)
This represents that the first pulse R0 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.
Hdd = Â£ Â£ \\jlDhP2(et)atlÂ¡
2
0
(5.5)
68
where
and
Â¡ =(34 2)
t2o ~ v^3^1 ~~
, \ 4 rHcu')dt'iH ,hujoT\
p(t) = e hJ0 dd (1 + iz;/z)eft Jo
r,ec
V<
(')dt'
KT
(5.6)
It is assumed that the second pulse is a pulse and the phase of that is
(the angle between radio frequency field and y axis in rotating frame). After
the second pulse the density matrix is given by
p(T+) eirl)(Ixsin+IyC08)p^T_}eitl>[Ixsin+Iyco8)
(5.7)
The nuclear spin density matrix at time t after second pulse is therefore
p{t + t) = tt()i2(^)u(0^(l + ^4)t(r)!t(i)ut() (5.8)
where
_ e\iAwIzi^DP2(9)<7(3I^2)]t
R((f>) = eilP(Ixsin+IyCos)
tt(r) = e[tAW4iiÂ£>P2(i)a(3/?2)]r
and the signal
S(t + r) a Tr{p(t + r)/+}
Tr{p(t + r)/+}
(5.9)
(5.10)
(5.11)
= lj^Tr{u(t)R()u{r)Ixu\T)rt{)u\t)I+}
= \^T ^ (mi  u(i)  m2)(m2  R{4>)  m3)(m3  u(t)  m4)
mxm8
(m4  Ix  m5)(m5  u^r)  m6)(m6  ^(0)  m7)
69
(m7  ut(i)  m8)(m8  1+  mi) (5.12)
Formation of Solid Echoes
After a lengthy calculation the results of eq.(5.12) are still quite compli
cated (for general
the same as that in ref.39. We would like to give the results in some special
conditions.
(1)xp = 0 (apply one pulse)
Tr{p(t)I+} = e~lAujtcos[^DP2{e)at} (5.13)
This is called the Free Induction Decay.
(2) cp = 0,xp  This is a 90 r 90 pulse sequence
Tr{p(t +t)I+}
= c'^cos^DP^oit r)](l e*A"2') (5.14)
Obviously, Tr{p{t + r)/+} has a maximum value at t = t. This is what
is the meant by a solid echo. Another question is that according to eq.(5.14)
the inphase echo (Aw = 0) does not exist. This is in disagreement with the
experimental results. The reason lies in neglecting intermolecular dipoledipole
interaction and the difficulty of getting exact Aw = 0 in experiments.
(3)<^> = xp = j This is a 90^ r 90Â£ pulse sequence
Tr{p(t +t)I+}
= ~iAucos^DP2(0)a(t t)](1 + eA"2") (5.15)
From eq.(5.15), same as the second case, Tr{p(t + r)I\} obtaines maximum
value at t = t, forming the solid echo.
70
Stimulated Echoes
It is already shown that in certain cases a sequence of three 90 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 2\) and
thereby used to detect ultraslow molecular reorientations.
Applications to the study of the molecular dynamics on cooling into the
quadrupolar glass phase of solid hydrogen will be discussed.
Formation of Stimulated Echoes
A threepulse sequence is sketched in Fig. 17. As the analysis of solid echoes,
the nuclear spin density matrix at different time are given by
/>(<>) = Peq j(l +
/>(<>+) = (1 + "ix)
(r_) = e~if HÂ£Wp(0+)eifoT WH'
p(T+) = e^(h^n+Iycos(Ixsin+IyCos)
p[{tw + t)~] = e hndd twp(r+)ehndd
p[{tw +r)+] = p[{tw + r)_]e^'(7^*^W')
p{tw + t + t) = e~hHddt p\[tw + r) + ] ehHdd l
The signal
S{tw + r + t) cx Tr{p(tw + t + 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(
states described by p(r_) which contains both transverse magnetization and
transverse alignment. The transverse components are transferred by a second
pulse R (xp) 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 o. After the waiting
period tw
pulse of rotation angle xp'. 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 + t + f)/+}
1 hojQ
3 ~KT
y] (mi I e hHdd I m2)(m2 \ R(') I m3)(m3 I e hHdd tw  m4)
mimi2
(m4 I R(
(m7 I e'idd I m8)(mg  R(4>) \ mg)(mg \ e* dd w  mio)
(mw I B+(0') I mu)(mn   m12){m12  1+  m,)
(5.17)
Since the calculation is extremely tedious and the results for general
and general xp^xp' are very complicated, we only investigate the results of cal
culations by taking two special cases.
(1)
= 0,
* = f, xP' = f
72
This is a 90 r 90 tw 90 pulse sequence
Tr{p(tw + t + Â£)/+}
= ^e~^ujticos(A
2 KT 2
H ^e~l^'u}ticos(Au>T)sin2(Aijjtw)cos[DP2(6)o(t + r)]
2 KT 2
^^e_Awsn(Au;f)sm(Au;ly)cos[^Z)P2(^)0'( tw + 01 (5.18)
3 KT 2
(2)
=Â§,<' = Â§
0 = 1,^ = !
This is a 90 r 90Â£ tw 90 pulse sequence
Tr{p(K; + r + 0^+}
= ^e_^wsm2(AcuU!)sn(Ac<;r)cos[Z)P2(^)cr(^ ~ r)]
3 ii T 2
+ ^^e_lAwism(Ao;r)cos2(Au;ly)cos[^)P2(0)cr( + r)]
3 ii T 2
+ ^e*^wco.s(Aw)co.s(Awty)co.s[.D.P2(0)cr( ~ ^ + 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 t. The results
are sketched in Fig. 18. The appearance of multipole echoes has been seen in
experiments (Fig.19) for orthopara hydrogen mixtures at low temperatures.
Engelsberg et al. (81) have presented results for nuclear spin stimulated
echoes in glasses. The curve of 11P 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 + 2t (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 j tw ^)
stimulated in the quadrupolar glass phase of solid hydrogen (r = 25ps, 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 ^ For
experimental curve only a very slight modificaion (indicated by the arrow) was
observed. We will discuss this phenomenon later.
The experimental curve in Fig.22 gives the relation of stimulated echo
versus waiting time tw of solid hydrogen (orthoconcentration x = 0.54, T =
220mK). The time scale is logarithmic. The logarithmic decay behavior can
be understood in terms of the motional damping of stimulated echoes.
LowFrequency Dynamics of Orientational Glasses
The orientational glasses (20, solid orthopara H2 mixtures (6), N2/A mix
tures (83) and the KBri_xK(CN)x mixed crystals (82, 84, 85)) form a sub
group of the general family of spinglasses which continue to generate intense
interest because of the apparently universal low temperature properties ob
served for a very diverse range of examples (dilute magnetic alloys, mixed crys
tals, dilute mixtures of rotors, partially doped semiconductors (86), Josephson
junction arrays (87) and others). The most apparent striking universal fea
tures (20) are an apparent freezing of the local degrees of freedom on long
time scales without any average periodic long range order, characteristic slow
relaxations and historydependence following external field (magnetic, electric,
elasticstrain....) perturbations, and a very large number of stable low energy
states.
74
Figure 18. Sketch of The Results of Calculations
75
Figure 19. Experimental Curve (x = 23%,T 38mK)
76
Nuclear spin stimulated echoes in glasses
3635
Figure 20. Spin Echoes in Borosilicate Glass at 4.2K
77
78
Waiting tmt,tm (at)
Figure 22. The Observed Decay of Stimulated Echo
(x = 0.54, T = 220mK)
(squaresir = 12.5Â¡xs\ circles and triangles:
t = 25 fxs)
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 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 Huses recent picture (89), we provide a explanation of the low
frequency relaxation and the lowtemperature specific heat of solid ortho para
hydrogen mixtures.
In the scaling model of Fisher and Huse the lowenergy excitations which
dominate the longdistance and longtime correlations are given by clusters of
coherently reoriented spins. Their basic assumptions are:
(1) Density of states at zero energy for droplets (d dimension) length scale
L as L~6, where 0 < 6 <
(2) Free energy barriers Eb for cluster formation scale as Eg ~ with
0 <4>
With these assumptions, Fisher and Huse show that the autocorrelation
function
CM =((s,(o)sM)t(s>)?)c
_0_
decays as (logi) for f > oo.
(5.20)
80
For our system, assuming axial symmetry, the quasistatic local orienta
tional order parameters are the alignments cr =(3 J2 2)t and the correspond
ing autocorrelation functions Ct() = <7j(0)cTj(i) can be studied directly by
NMR.
For a 90y t 90y tw 90 pulse sequence we assume at t r, order
parameter a =
Tr{p(tw + r + t)I+}
^^^e~lAulticos(AuT)cos2(Autw)cos[\DP2(0)g(t + tw)t \dP2{6)o{t)t\
o K1 2 2
+ ^^^e~lAuticos(AujT)sin2(Autw)cos[^DP2{0)a(T + tw)t + ^Z>P2(^)cr(r)r]
o K1 2 2
^e~lAu,tsin(A
3 KT
cos[^DP2{6)o(t + tw)t ^DP2{9)o(t + tw)tw + ^DP2(0)a(r)T] (5.21)
Considering t r, the stimulated echo amplitude
A ({(cosID^o.^cosID^o^t + tu)])) (5.22)
where the double brackets refer to an average of configuration and Dt =
7jDP2(cos0i). The important point is that if the local order parameters <7j
remain fixed during tw, there is no damping of the stimulated echo, while
(Tj changes due to local reorientations, then the contribution to the echo is
severely attenuated. The product Dr can in practice be made very large and
this method can therefore be used to study ultraslow motions in solids. We
believe that in Fig.21 the departure portion from ^ (indicated by an arrow) is
due to slow motion.
A barrier will have a characteristic life time given by an Arrhenius Law
81
or tunneling rate T
E
T(Eb) = r0e"^
(5.23)
where Tq is the characteristic attempt frequency for clusters of this size. In
the long time limit Tq is resonably welldefined because it is associated with a
characteristic cluster size. In a time t the only barriers crossed will be those sat
isfying 0 < EB < Emax[t) where Emax{t) = KBTlog Any barriers crossed
lead to significant changes in the local order parameters and the amplitude of
the stimulated echo is then simply
fc
A(t) = /
Je,
P(EB)dEB
max(t)
(5.24)
At low temperatures, assuming a constant density of barrier heights P(EB),
we find
A(t) = 1 KBTP0 log() (5.25)
The prefactor Pq can be determined from the low temperature behavior of the
heat capacity.
For the orthomolecules with angular momentum J 1, we can asso
ciate a simple two level system with the energy states for a given molecules;
the states J = 1 being separated from the state J^ = 0 by a gap 3At (The
states J^ = 1 are degenerate if there are no interactions which break time
reversal symmetry). At low energies we can, following the above arguments,
identify the low energy excitations (which determine Cv at low T) with a broad
quasiconstant distribution P(A) for 0 < A < Aq for the spins in a cluster.
Identifying P(A = 0) with Pq, the density of low energy barriers, we have
NxR
18 A2
ts2t2 3A 3A
^ 1 4e kbT + eKBT
dA
+ 4
(5.26)
where x is the ortho#2 concentration.
let u = 3^^
Cv
NxR
2 KBT
82
o in
f3t<Â£
Jo
u
4e~u + eu + 4
du
(5.27)
set t =
let
^=^1*
 = it
V 3 Jo
f t v?du
Jo 4e~u + eu
+ 4
u2du
(5.28)
(5.29)
4e_u + eu + 4
The resulting C'v (Figure 23) has a linear temperature dependence at low T
and a peak at Tp. = 0.70^ in close resemblance to the temperature behavior
observed by Haase et al. (90). From the peak position in the experimental
data, = 1.27, PqK 0.86 and for the stimulated echo decay this value
gives
(5.30)
Acalc) = 1 O.43log10()
to
and the observed decay
4>&a.(0 = 1 O.55log10()
to
(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 o' = 0.86 and with width 0.12 in this case. The essential
point is that the time scale of the slow relaxations in the glass phase is simply
related to the low temperature behavior of the heat capacity.
83
Another important point is that the characteristic times to are much shorter
than the spectraldiffusion time scale (~ sec) seen by the recovery of holes
burnt in the NMR lineshape. We therefore find it difficult to attribute the
logarithmic decay seen in H2 to spectral diffusion across the NMR linshape.
84
Cv' t
t
Figure 23. Calculated Curve of C'v t
CHAPTER 6
SUMMARY AND CONCLUSIONS
The orthopara mixtures of solid #2 are studied theoretically for fee lattices
{X > 0.55) of finite site by Akira Mishima and Hiroshi Miyagi (92). The sys
tematic theoretical studies of nuclear magnetism for quadrupolar glass regime
are carried out in this dissertation.
We have developed a theory of the nuclear spinlattice relaxation of ori
entationally ordered ortho hydrogen molecules for the case of local ordering
in the quadrupolar glass phase of solid hydrogen. We have investigated the
temperature dependence of T\. 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^(Au) throughout the NMR absorption
line. The detailed dependence is much stronger than the simple dependence
T~l(Av) oc (2 + c) given by earlier estimates by A. B. Harris et al. (l). There
is also a strong variation with the orientation of the local symmetry axis with
respect to the external field. The variation is in agreement with that found
by Hardy and Berlinsky (93) for the longrange ordered Pa% phase. If the
crosssection relaxation between different isochromats is taken into account,
the results of calculations of spectral inhomogeneity is in good agreement with
the experimental results.
In addition to the spectral inhomogeneity, the relaxation is also found
to be strongly nonexponential. This can be understood easily. As a result
of the glassy nature of the system, there is a broad distribution of both local
orientational order parameters(cr) and the orientation (a) of the local symmetry
85
86
axes for the molecular alignments. The existence of these distributions at
low temperatures means that a given frequency Au in the spectrum comes from
all the different allowed combinations of o and a that satisfy Au = DP2{ol)o.
The dependence of the relaxation rates on a and P2 then leads to a distribution
of local relaxation rates for a fixed Au. It is this distribution in rates that leads
to the observed nonexponential behavior of M(t). Calculations based on the
expected probability distributions for o and P2 yield results in good agreement
with the results reported in the publications (14).
The most important conclusion of the study of T\ 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 o' ={ 1 and
(3) the eccentricity r] ={J% Jy).
The positivity conditions for the density matrix show that the only allowed
values of [o',r]) are those enclosed in a triangle in (cr,,r/) space whose vertices
are the pure states  Jz = 0) and  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 2o' > rÂ¡ > 0. Orientational states with
negative o' are not excluded on theoretical grounds.
Studies of molecular dynamics have shown the existence of paralibrons,
i.e. collective excitations that exist in large clusters with welldeveloped short
range order.
Nuclear spin stimulated echoes have proved to be very effective for the
study of ultraslow molecular motions both in the molecular orientational
glasses and in ordinary glasses (81). The existence of multiple echoes which is
given by theoretical calculations is also in good agreement with experimental
results of solid hydrogen and ordinary glass at low temperature.
We have offered a unified explanation of the slow relaxational behavior and
low temperature heat capacity of the quadrupolar glass phase of solid hydrogen
in terms of the density of lowenergy excitations in the system.
Since the stimulated echoes are damped by any change in the orientational
states during tw, they can be used to detect very slow molecular motions. The
phenomeno 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 Huses recent picture of the short range spinglasses and the domination
of the longterm relaxation by low energy largescalecluster excitations.
In analogy with spin glasses (e.g. alloys like Cu Mn where a random
configuration of spins condenses at low temperature) the linear specific heat
with temperature at low T can be understood in terms of the suggestion (94)
that the dominant contribution to the specific heat will be from clusters of
particles for which the energy barrier is sufficiently great so that resonant
tunneling between the two local minima does not occur, but sufficiently small so
that tunneling between the two levels can take place and thermal equilibration
88
can occur during the time span of the specific heat experiment. We obtained
a very good linear dependence curve at low T by using a numerical integral
method.
Having combined with the results of logarithmic decay of stimulated echoes
and linear lowtemperature specific heat, a very good numerical expression of
the amplitude of the stimulated echo as a function of waiting time tw for
x = 54%, T = 0.22K is obtained.
Most people agree that molecular orientations of random orthopara hydro
gen mixtures become frozen at low temperature and there is no evidence for
any welldefined phase transition on cooling from the completely disordered
high temperature phase. However, the gradual transition to the glass state
in these systems involves strong cooperativity as evidenced by Monte Carlo
calculations. Studies of the nuclear relaxation and molecular dynamics have
shown quantitatively how important the cooperativity is. Further theoretical
study of the slow motions needs to be carried out to improve the understanding
of the nature of the quadrupole ordering in random mixtures at low tempera
tures and their relation to the family of glasses. Further experiments at very
low temperatures will also provide a deeper understanding of the properties of
the quadrupolar glass state.
APPENDIX
FLUCTUATIONDISSIPATION THEOREM
The fluctuationdissipation theorem is extremely important because it re
lates the response matrix to the correlation matrix for equilibrium fluctuations.
In Chapter 2 we mensioned the definitions of the response function(eq.2.32)
and the relaxation function(eq.2.35). The response function my be written in
a number of equivalent ways:
(A.l)
= I) (A.2)
M*) = 4 (A.3)
in
Using an identity due to Kubo the response function may also be written
as
fP
fM = / ds{Ml(ihs)Mk(t)) (A.4)
Jo
fkM = f (A.5)
Jo
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~P^. in calculating the time evolution of
operators one must consider operators of the form By writing t = ihs
one can see that s will play a role identical to (3 in all formal developments.
89
90
From eq.(2.35) and eq.(A.5) it follows that
rP rP
Fki[t)= ds(Mi(ihs)Mk(t)) lim / ds(Mi(ihs)Mk(T))
Jo TyooJo
(A.6)
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 (MÂ¡}(Mk). Hence the relaxation
function may be written in its more usual form
Fki[t) = ds(AMi(ihs)AMk{t)}
Jo
(A.7)
In this form there exists a mathematical identity between the relaxation func
tion and the correlation function gki{t), given by
r+oo
Fkl{t) = B(t r)gkl(T)dT
J oo
(A.8)
where
(A.9)
the + subscript indicates the anticommutator; or in terms of Fourier transforms
II
<^55
(A.10)
where
B(t) = Ajr log[coifc( 2]0h )]
(A.ll)
and
1 hu> (/3hu>\
A coth
B[u) 2 V 2 J
(A.12)
The derivation of these identities is called the quantummechanical 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) Â¡36(i), and one obtains the classical
fluctuationdissipation theorem
Fkl (0 = PSkM
(A.13)
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15. M. A. Klenin, Phys. Rev., B28, 5199 (1983)
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20. Debashish Chowdhury, Spin Glasses and Other Frustrated Systems, Prince
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21. H. Maletta and W. Felsch, phys. Rev. B20, 1245 (1979)
22. Y. Lin and N. S. Sullivan, Mol. Cryst. Liq. Cryst. 142, 141 (1987)
23. X. Li, H. Meyer and A. J. Berlinsky, Submitted for publication (1987)
24. N. S. Sullivan, C. M. Edwards, Y. Lin and D. Zhou, Can. J. Phys.(in press)
(1987)
25. N. S. Sullivan and D. Esteve, Physica B + C 107, 189 (1981)
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Full Text 
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AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
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. Shin11 Cho and
Daiwei Zhou for their helpful suggestions and discussions.
The help from my friends Laddawan Ruamsuwan, James K. Blackburn,
Qun Feng and Stephan Schiller with the computer work is greatly appreciated.
The cooperation and friendship of my fellow graduate students, as well as
that of the staff and faculty of this department, has made my stay at U.F a
pleasant and rewarding experience.
This research was supported by the National Science Foundation through
Low Temperature Physics grants DMR8304322 and DMR86111620 and the
Division of Sponsored Research at the University of Florida.
11
TABLE OF CONTENTS
page
ACKNOWLEDGEMENTS ii
ABSTRACT v
CHAPTER
1 INTRODUCTION 1
2 THEORY OF NMR RELAXATION 6
BloembergenPurcellPound Theory 6
Theory for Liquids in terms of Moriâ€™s Formalism 7
Kubo and Tomita Theory 11
Nuclear SpinLattice Relaxation in NonMetallic Solids 16
Nuclear SpinLattice Relaxation for Solid Hydrogen 18
3 NUCLEAR SPINLATTICE RELAXATION 22
Formulation of Longitudinal Relaxition Time T\ 22
Temperature Dependence of T\ 27
Spectral Inhomogeneity of T\ 33
NonExponential Relaxation of Nuclear Magnetization 38
4 ORIENTATIONAL ORDER PARAMETERS 45
Density Matrix Formalism 45
Application to Solid Hydrogen 50
A Proposition for a Zero Field Experiment 55
A Model for The Distribution Function of o 58
5 NMR PULSE STUDIES OF SOLID HYDROGEN 64
Solid Echoes 64
Stimulated Echoes 70
iii
LowFrequency Dynamics of Orientational Glasses 73
6 SUMMARY AND CONCLUSIONS 85
APPENDIX
FluctuationDissipation Theory 89
REFERENCES 91
BIOGRAPHICAL SKETCH 96
iv
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
THEORY OF NUCLEAR MAGNETISM OF SOLID
HYDROGEN AT LOW TEMPERATURES
By
Ying Lin
April 1988
Chairman: Neil Samuel Charles Sullivan
Major Department: Physics
Systematic studies of the nuclear magnetism of solid orthopara hydrogen
mixtures at low temperatures are presented.
The formulation of the nuclear spinlattice relaxation time T\ for the case
of the local ordering in the orientational â€œglassâ€ phase of orthopara hydrogen
mixtures is given. The temperature dependence of T\ is discussed. A strong
dependence on the position of the NMR isochromat in the line shape is found
and this is in good agreement with the experimental results of a group of
physicists at Duke University, provided that the crossrelaxation is taken into
account. The relaxation is found to deviate considerably from an exponential
recovery.
The orientational degrees of freedom of ortho hydrogen molecules in terms
of density matrices and the irreducible tensorial operators associated with unit
angular momentum are described. The range of allowed values for the orientaÂ¬
tional order parameters is determined from the positivity conditions imposed
on the density matrix.
A theory of solid echoes and nuclear spin stimulated echoes following a two
pulse RF sequence and a threepulse sequence in the quadrupolar glass phase
v
of solid hydrogen is developd. The stimulated echoes can be used to compare a
â€œfingerprintâ€ of the local molecular orientations at a given time with those at
some later time (less than ) and thereby used to detect ultraslow molecular
reorientations.
vi
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 â€” 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 â€œspin1â€ quautum rotators,
and therefore a valuable testing ground for theoretical models of cooperative
behavior.
A suggestion that a random distribution of ortho molecules in a para
hydrogen matrix may behave as a quadrupolar glass at low temperatures for
ortho concentration X < 55% was proposed. It was based on the observaÂ¬
tion that orthopara hydrogen mixtures provide a striking physical realization
of the combined effects of frustration and disorder on collective phenomena.
These effects play a determining role in spin glasses, and the behavior of solid
hydrogen mixtures is analogous to that of a spinglass such as EuxSri_xS in
a random field(20,21).
The electrostatic quadrupolequadrupole interaction is the dominant interÂ¬
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.
1
2
At high temperatures the molecules are free to rotate, but on cooling,
the ortho molecules tend to orient preferentially along local axes to miniÂ¬
mize their anisotropic EQQ interactions. As the temperature is reduced, this
leads to a continuous but relatively rapid growth of local order parameters,
(er = (3J% â€” 2)) which measure the degree of alignment along the local symÂ¬
metry axes oz. There is a broad distribution p(<7) 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 fluctuaÂ¬
tions of the molecules as the samples are cooled is of special interest in these sysÂ¬
tems and this has motivated several independent experiemental studies(25 â€”28)
of NMR relaxation times T\, which are determined by the fluctuations of the
molecular orientations. One of the most striking results reported by S. WashÂ¬
burn et al. (28) is the observation of a very strong dependence of T\ on the
spectral position within the NMR absorption spectrum. While a qualitative inÂ¬
terpretation of this behavior in terms of â€œsloppyâ€ librons has been offered(l,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 experiemental data.
In the low concentration regime, although it is agreed that (i) the low
temperature NMR lineshapes indicate a random distribution of molecular oriÂ¬
entations (for both the local axes and and the alignment o = , 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 < 0.1/f). While some early
3
work reported a very rapid, but smooth variation (25,27,30) with temperaÂ¬
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 relaxÂ¬
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 Tj1 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 characÂ¬
teristic molecular fluctuation rates from the relaxation times. It will be shown
that the strong departure from exponential decay for the magnetization can
be understood provided that the broad distribution of local order parameters
is correctly accounted for.
In the socalled â€œquadrupolar glassâ€ the quantum rotors cannot in genÂ¬
eral be described by pure states and a density matrix formalism is needed to
describe the orientational degrees of freedom. It is needed to determine the
precise limitations on the local order parameters (molecular alignement, etc.)
from the quantum mechanical conditions imposed on the density matrix and
to discuss the implications for the analysis of NMR experiments. Some of the
considerations for the spin1 density matrix description have been given elseÂ¬
where (29,32 â€” 37) but solid H2 is a special case because the orbital angular
momentum is quenched. The case will be discussed in chapter 4.
4
In molecular solids, one has to consider two kinds of degrees of freedom:
translational degrees of freedom for the centreofmass motion and orientational
degrees of freedom for the rotational motion of the molecules.
One of the most fascinating problems encountered in the molecular solids
is the existence of glasslike phases in which the molecular orientations become
frozen without any significant periodic correlation from one site to another
throughout the crystal. The most striking example of these â€œorientational
glassesâ€ 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 â€œinertâ€
molecules, it leads to large reorientations of the quadrupolebearing molecules
in the neighbourhood of the inactive diluants and the dissappearance 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 measureÂ¬
ments on the orientationally ordered hydrogen, because they allow one to exÂ¬
tract information which is not easily or unambiguously determinable from ei
5
ther steadystate line shape or FID analyses. Conventional continuous wave
(CW) and free induction decay (FID) NMR techniques have only been able to
show that the orientational degrees of freedom appear to be fixed for times up
to 1CF4 â€” 105 S. A considerable improvement may be achieved by the analyÂ¬
sis of solid echoes for which the observation times during which one can follow
molecular reorientations are extended to an effective relaxation time (?2)e//>
which may (by a suitable choice of pulses) be much longer than the transverse
relaxation time I2 that limits the conventional techniques.
It will be shown that spin echoes and stimulated echoes following a two
pulse sequence and a threepulse sequence, respectively, provide a more powÂ¬
erful means of investigating the orientational states and particularly the dyÂ¬
namics of the molecules bearing the resonant nuclei, than the conventional
continuouswave technique.
Spin echoes were observed in solid #2 a long time ago (38) and have been
used to study the problem of orientational ordering (11, 14, 27, 28, 39, 40). In
order to gain a deeper insight into this problem, a series of questions will be
disscused in chapter 5. These are the formation of spin echoes (including solid
echoes and stimulated echoes), explanation and comparison with experiments,
and the motional damping of echoes.
CHAPTER 2
THEORY OF NMR RELAXATION
BloembergenPurcellPound Theory
The theory of spin relaxation in liquids (or gases) is based upon time
dependent perturbation theory.
In liquids where the spinspin coupling is weak and comparable to the
coupling of the spins with the lattice, it is legitimate to consider individual
spins, or at most groups of spins inside a molecule, as separate systems coupled
independently to a thermal bath, the lattice.
It is well known that the expression for the interaction between two magÂ¬
netic dipoles of nuclear spin /t and Ij can be expanded into 6 terms:
HDD = â– Ij  3(f, â– r>)(/y â€¢ n)]
rij
= â€” (A + B + C + D + E + F) (2.1)
r
ij
where
A + B = ^(3cos26  1)(/, â€¢ Ij  3IizIjz)
C = \{I+Ijz + Iizl+)sin0cos0e~i(l>
2 1 3
D = + IiJ^sMcoste*
E =
4 * 3
F = r r sin2 0e2i
4 * 3
6
7
If we carry out firstorder timedependent perturbation calculations to obÂ¬
tain the transition probability W between the megnetic energy levels, we find
for a nuclear spin I â€”
W = ?l4fc2/(/ + lMK) + ij2( 2wo)} (2.2)
and the longitudinal relaxation time
*1 =
1
2W
+ lMM + o)]
This is the BPP expression(41,42); where
G{r)eiu)TdT
G(t) = F(t)F*(t + r)
Fq = ^(l â€” 3 cos20)
F\ = â€”zsinQcosOe1^
r6
F2 = \sin20e2i
r6
(2.3)
Theory for Liquids in Terms of Moriâ€™s Formalism
Daniel Kivelson and Kenneth Ogan reformulated the study of spin relaxÂ¬
ation in liquids in terms of Moriâ€™s statistical mechanical theory of transport
phenomena(43,44,45,46). They started with some wellknown phenomenologiÂ¬
cal magnetic relaxation relations and formulated them in a manner most suitÂ¬
able for comparison with Moriâ€™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:
 f  tv + 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 f is
the friction constant, and F(t) is a force which is a rapidly varying random
function which averages to zero.
Mori developed a â€œgeneralized Langevin equationâ€ which provides a diviÂ¬
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 Ã¡ represent component of the time derivative of A that are orthogonal
to A, and are rapidly varying
Â¿=(1 P)A (2.5)
where P is a projection operator.
The time dependence of ^4(t) can be expressed in terms of the superoperÂ¬
ator, )ix, of the Hamiltonian as
A = iHxA (2.6)
A(t) = Ã©*''A
(2.7)
9
The time evolution of ci[tp) is determined by a propagator composed only
of those components of the Hamiltonian , which lie outside the subspace
determined by the slow variables, i.e.
a(tp) = exp[t( 1 â€” P)i)ix\a (2.8)
We now define a memory function matrix k(t), an effective memory funcÂ¬
tion matrix K(t) and a relaxation matrix K as follows:
Memory function matrix:
k(t) =(d(tp)a)'(ÃÃ) 1
Effective memory function matrix:
K{t) = k(t)e~iilt
where
in =(ÃÃ^)(ÃÃ]yl
Relaxation matrix K:
/
Jo
The relaxation matrix may be complex:
R = Y'HE  ÃK1M
(2.9)
(2.10)
(2.11)
(2.12)
(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 funÂ¬
damental relation  Moriâ€™s equation:
â€”iâ€™(i) = [Â¿(n +
(2.14)
10
and its Fourier transform:
ÃuÃ(oj) â€” A(0) = tfl â€¢ Ã(u) â€” K(u â€” H) â€¢ A(w) (215)
We can now derive Blochâ€™s equation and relaxation time formulae in a
simple case (one variable).
We express the Hamiltonian as
H = MS + HL + usl (2.16)
where Mg depends only on the spin variable, Hr represents the molecular moÂ¬
tions and interactions which are independent of the spins, and "Hgi involves
those interactions which involve both spins and nuclear spatial coordinates.
The slow variables A can be selected as:
Ã=^A52j (2.17)
We assume that k(t), the memory function matrix, decays rapidly so that
K(cj â€” fÃ) can be replaced by Kre â€” ÃKjm. Equation (2.15), transformed back
to the time domain, then becomes:
â€” Tj 1[52(t) â€”(5'2)] (218)
^SÂ±(t) = [*(Â±w0 + oÂ±)  r2_1]5Â±(i) (2.19)
where
4
Ti = ReKzz = Jq {[HsL{tp)â€™Sz{tp)]{SziMsL}) dt (2.20)
T2l = (^Ã¼Â£?)Â±Â±
2 fÂ°Â°
= Re(~) J {[Â«st(
2 rÂ°Â°
Â°Â± = [KlAi)Â±Â± = Jq ([HsL{tp)â€™SÂ±(tp)][STâ€™HsL])eTlUJÂ°t dt (222)
Equations (2.18) and (2.19) are the simple Bloch equations except that the oÂ±
term which represents the socalled nonsecular or dynamic frequency shift is
given explicitly.
11
Kubo and Tomita Theory
The theory originally introduced by Kubo and Tomita emphasizes the simiÂ¬
larity of magnetic relaxation to other nonequlibrium phenomena (46,47,48,49).
Description
We know that the Bloch equation can describe the relaxation of the magÂ¬
netization m. It is a linear equation.
= 7m(t) x Hit)  mAt)U gÃ¼M _ â„¢Â»W ~
dt ' v â€™ â– v â€™ T2 Ti
In matrix notation the Bloch equation (2.23) can be written as
(2.23)
Am(t)  â€”L â€¢ Am{t)
(2.24)
where
Amk(t) = mk(t)  mÂ°k (k = i,j,k)
is the thermal equilibrium value
/ 1
TT
L =
0
0   ÃU 0
0 \
0
V
(2.25)
(2.26)
0 0 â€” f iojq j
If we transform from the laboratory frame to a frame rotating with the
Larmor frequency around the z axis, we will have
Am(i) = â€”L'Am^(t)
(2.27)
The formal solution of Eq. (2.24) is the following:
Ama(t) = (e Lt)apAmp(t)
(2.28)
The Fourier transform of the formal solution is therefore
Ama(cu)+  {L_.Ja/3Amp{0}
(2.29)
12
+ means a positive Fourier transform and in the rotating frame:
Am;(w)+ = ( , 1 . )Am,(0) (2.30)
Ln ~ lbJ
Response Function
The linear response formalism begins by calculating the linear response
of a dynamic variable to a disturbance created by a timedependent external
force.
Consider the Hamiltonian:
Htotal(0 = H  (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)HÂ¿(t) is a small, timevarying field which is responsible for the nonequlib
rium behavior of the system.
The response function fki(t) is defined as
= Ã fklit T)Hi{T)dr (2.32)
J â€” oo
The response function fki(t) gives the effect of the disturbance at time t.
The Fourier transform of eq. (2.32) is
A mk[u) = Xjfc/H#/(w) (2.33)
where
roo
XldH = / (2.34)
Jo
Equation (2.34) defines the susceptibility Xfc/(w) ^ h35 a real and imaginary
part that are connected by the KramersKronig relations. The imaginary part
13
of the susceptibility is called the absorptive part which is related to the power
the sample absorbs. The real part of the susceptibility is called the dispersive
part and is related to the measured line shape.
Relaxation Function
The relationship between the relaxation function and the response
function fki{t) is given by
/oo
fkliT)dT
The relaxation function describes the time change of the response after the
external disturbance is cut down to zero.
Assume a step disturbance:
Ht{t) = Hieâ‚¬t0{t) (2.36)
where
Â«(<)={; SfÃ¼ (237)
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 â€”> 0 is:
A mk{t) = Fkl{t)Hl t>0 (2.38)
and
Amfc(0) = Ffcz(0)tfz (2.39)
Here FÂ¿.z(Â£) 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
A ma(t) â€” Fai{t)Flf}{0)Amp{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
= riWVW (*â€¢Â«)
We define okÂ¡{u>) as the following
okl{u) = â€”Xkl^ = f dteiwtFki(t) (2.42)
JO
From equations (2.38) and (2.39):
A ma(u) = oafi(u)Hp
(2.43)
AmQ(0) = Xafi[Â°)Hp
(2.44)
A ma(w) = A mp{0)
(2.45)
Comparing equation (2.28) with equation (2.45) yields
(2.46)
in the rotating frame
1 =ak_k^ + Kuo) k =
Lkk  tuJ Xjfcfc(O)
(2.47)
i.e.
L'kk Reokk{u + Kuo)
Lkk + u2 Xfcâ€”Â¿(O)
Using a symmetric form of ok_k, equation (2.48) becomes
(2.48)
L'kk ReÂ°kkH
Lkk+u2 Xkk{Â°)
(2.49)
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 IIo that contains the Zeeman Hamiltonian and a part H' that
couples the spins to the lattice and is responsible for the relaxation:
H{ A) = H0 + A H' (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]Ta nL\n)
n
(2.51)
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. A2Z/(0) = L'(A).
The results of transport coefficients are
Loo = t1
i \2
(*)
2(A Ml)
roo
/ dt{[H',Mz][H'{t),Mz})0 + C.C. (2.52)
Jo
Ln = Lli = Tfr
12
i \2
(sL
4XTi
roo
Jo
(2.53)
where
XT2 = AM+]+)o (254)
H'(t) = e*HotH'e*HÂ°W
(2.55)
16
The subscript zero on the braket indicates that the trace is taken over the
equilibrium density matrix
_ exp((3H0)
PÂ° Tr(exp((3H0))
which does not involve the spinlattice coupling.
Nuclear SpinLattice Relaxation in NonMetallic Solids
The problem here is essentially the same as that for liquids and gases,
namely to calculate the probability of a flip of a nuclear spin caused by its
coupling with the thermal motion of a â€œlattice.â€ There are, however, some
significant differences. The internal motion in solids will often have much
smaller amplitudes and/or much longer correlation times than in liquids. In
rigid solids because of the tight coupling between nuclear spins exemplified by
frequent flipflops between neighbours, the correct approach to nuclear magÂ¬
netism is a collective one, where single large spinsystem with many degrees
of freedom are to be considered, rather than a collection of individual spins.
The assumption is usually made that the strong coupling of the nuclei simply
establishes a common temperature called a â€œspin temperature,â€ and that the
lattice coupling causes this temperature to change(50,51,52).
A quite general equation can be derived.
d(3 1 â–
(2^)
where
0 = jflr Ts  Spin temperature
00 = fiY 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 Yln,m Wmn{En Em)2
(SfT
(2.58)
Where  m) and  n) are the eigenstates of Hq.
Wmn = Wnm is the transition probability from the state  m) to the state
I n)
There is another way to calculate T\. 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~h^tpeh^t (2.59)
hdp*
i dt
(2.60)
Where "Hi is a perturbation,
#Ã(t) = eâ€œ^0%e^ot
Equation(2.60), integrated by successive approximation, gives
â€¢t
df_
dt
+higher order terms (261)
or
^ Â¿ f [Vi*{i  r),/(0)
+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 o*
a* = tr f{p*} (2.63)
with matrix elements (o: I Â°* I a') =â– Â£/(/Â« I P* I /Â«')â€¢
By making some assumptions Abragam (51) gives a general master equaÂ¬
tion
do* fÂ°Â°
Â¿ = yo [#Ã(0Â»[tfÃ(*  T)>Â°* Â°o\\dT (264)
where the bar represents an average of many particles.
If there is a spin temperature, then
dd l f+Â°Â°
= 2{Po W(<  r),%]]dr (2.65)
i.e.
f =  Â£7
Â£ = 5 * (2.67)
For the relaxation of like spins by dipolar coupling the result for T\ is the
same as the result of Kubo and Tomita theory.
Nuclear SpinLattice Relaxation for Solid Hydrogen
The molecular hydrogens (H2,D2,HD,etc.) form the simplest molecular
solids. The properties of solid mixtures of ortho (angular momentum J â€” 1,
nuclear spin 7 = 1) and para (J = 0, 7 = 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 relaxÂ¬
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,Gl)
r = y* +
Where c denotes the constant of spinrotational coupling and d that of the
intramolecular dipolar coupling, with respective values of 113.9 and 57.7 khz.
The spectral density functions J(moj) are taken at uq and 2wq where u>o is the
Larmor frequency(5l,p278).
We consider two regimes.
The High Concentration Regime
A. B. Harris (56) calculated the spectral functions for the correlation funcÂ¬
tions for both infinite and finite temperatures. He used a high temperature
expansion method to calculate the second moment and obtained good agreeÂ¬
ment with the high concentration experiments of Amstutz et al. (62), with
regard to both the temperature and concentration dependence of the relaxÂ¬
ation time.
Myles and Ebner (63) used a high temperature diagrammatic technique,
combined with a simlple method of impurity averaging over the distribution
of OH2 molecules. The averaged equations were then solved numerically to
obtain the spectral functions for solid H2 selfconsistently for the first time.
The resulting spectral functions were used to compute the T\ as a function
of the orthomolecule concentration and this was shown to agree well with
experiments at 10K and over concentraiton range of 0.5 < X < 1. They
obtained a y/X concentration dependence for T\, which was in agreement with
the data of Amstutz and colleagues (62) for X > 0.5.
20
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 < 400mfc), 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 rnomemetum correlation functions and Harris used an improved
version of the same theory. The T\ resulting from these calculations had a
5
concentration dependence of X3, which was in agreement with the data of
Weinhaus and Meyer (61), but the magnitude of T\ obtained in this way was
in disagreement with that data.
Hama et al. (65) developed a theory which was capable of treating the
T â€” oo correlation functions at all concentrations and which gave a concenÂ¬
tration dependence and magnitude for T\ which were in fair agreement with
experiment for all X (61,62).
Both methods had the defect that the impurity averaged correlation funcÂ¬
tions were obtained by statistically averaging assumed functional forms and no
attempt was made to determine the shape of the spectral function.
The first attempt in the small X region to calculate the high temperature
correlation functions selfconsistently and thus to overcome the above defect
was made by Ebner and Sung (66). They used the Sung and Arnold (68)
method of impurity averaging the Blume and Hubbard (69) correlation func
5
tion theory and obtained a T\ which had the experimentally observed J3
21
concentration dependence at small X. Since they made no attempt to propÂ¬
erly account for the anisotropy of the intermolecular interactions, they did not
obtain quantitative agreement with the experimental magnitude of T\.
Ebner and Myles (67) improved the calculation of Ebner and Sung by
properly treating the anisotropy of the electric quadrupolequadrupole (EQQ)
interaction, which is the dominant orientationally dependent interaction beÂ¬
tween two O â€” #2 molecules in solid HÂ¿ and which therefore, almost totally
determined the shape of the angular momentum spectral functions.
Sung and Arnoldâ€™s method of impurity averaging the infinite temperature
Blume and Hubbard (69) correlation function equations was employed, but
the equations were obtained using the full EQQ interaction rather than an
isotropic approximation to it. The spinlattice relaxation time was computed
as a function of the O â€” H2 concentration using a formula for derived by
applying the Blume and Hubbard theory (69) to the nuclear spin correlation
functions in this system. The resulting T\ 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 T]
There is a striking resemblance between the phase diagram for the magnetic
alloys such as CuMn, AuFe ... and orientationally ordered orthohydrogen
parahydrogen alloys(Fig.l2).
The nuclear spinlattice relaxation of ortho molecules at low temperatures
is determined by the modulation of the intramolecular nuclear dipoledipole
interactions Hqd and the spinrotational coupling HgR by the fluctuations of
the molecular orientations(55, 56). The calculations are particularly transparÂ¬
ent if we use orthonormal irreducible tensorial operators O2m and N2m for the
orientational (J = 1) and nuclear spin (I = 1) degrees of freedom, respectively.
The O2M are given by
Â°20 =
@2Â±l = TT JzJÂ±)
^2Â±2 = 2 (^i)2 Ã3'1)
and similar expressions hold for the N2m in the manifold 1=1.
The intramolecular nuclear dipoledipole interaction Hj)Â£) and the spin
rotation interaction II SR can be written in the above notation as
HDD = hD^Z ^2m(*)Â°2m(0
22
23
and
Â¡Â¡SR = he Â£(lP<,(*')Â°lm(0 (3.2)
i
respectively. D=173.1 khz and C=113.9 khz. The index i labels the ith
molecule. The 0/m and NÂ¿m are the operator equivalents of the spherical
harmonics YÂ¡m in the manifolds J = 1 and 1=1, respectively.
The relaxation rate due to Hjjd can be shown to be
1 fÂ°Â°
T1DD = JQ H^^Dd]
= fQ ^Iz'^Dd] ehHotHDDe~hH^\l^)dt (3.3)
where H0 = hu>0h It is the Hamiltonian responsible for the molecular dyÂ¬
namics. Using the commutators [Iz,N2m] â€” m^2m we find
TIDD = \D2 Y m2j2m(mw0) (3.4)
m=l,2
where the spectral density at the Larmor frequency
/0Â° ,
T
oo
The expectation value < ... >y 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 and Hgn terms. The orientational order parameters,
however, are evaluated with respect to the local molecular symmetry axes. We
must therefore consider the rotations
^2w â€” ^ ^ (36)
where the dmix are the rotation matrix elements for polar angles x = (a,/?)
defining the orientation of OZ in the local (x,y,z) reference frame.
TEMPERATURE (K)
24
ORTHO CONCENTRATION
Figure 1. Phase Diagram of OrthoPara i/2 Mixtures
25
Figure 2. (a) Theoretical Phase
Diagram for A Short Range System
(b) Experimental Phase Diagram for
EuxSrixS And AuFe (c) Phase
Diagram of The Ising Spin Glass
26
Figure 3. Picture of Long Range
Order And Quadrupolar Glass
27
We assume the simplest possible case
=(O2m(0)OÃm(0))g2m(Â¡) (3.7)
In the following, we consider only this case and further assume that the relaxÂ¬
ation is dominated by the fluctuations of the n = 0 component. We find
TIDD = Â¿(2 â€ Â°  o2)D2 Y m2Mmo(Â«)2i?2o(^o) (3.8)
m=l,2
where the <720(mwo) are the Fourier transforms of the reduced correlation funcÂ¬
tions <72o(0 and a = {3J2 â€” 2)T. The prefactor (2 â€” a â€” a2) is the mean square
deviation of the operator O20 evaluated in the local symmetry axis frame.
The contribution from the spinrotational interaction Hrr is
SR = 2C'2jn(U,0)  0Câ€™2(2 + a)Mlo(a)2<7lo(wo) (39)
the total rate 1 = T^D + T^gR
Temperature Dependence of T]
Minimum Values of T1
From the previous result (3.8 and 3.9)
=r(M) = (M) + 7jÂ¡râ€”{p,e)
DD SR
where
jT~(a' 0) = Â¿(2 ~ 0 ~ Â°2)d2 Y mVmo(0)202o("^o)
1DD m= 1,2
7fTâ€”{
SR 6
If we take a powder sample average, the Tirr is small.
,2\ n2
7?2o(wo) + 7ff2o(2wo)
(3.10)
28
To a good approximation we have
^(CT) Â« Â¿(2  o  o2)D2
r^2o(wo) + 02u(2wo)
(3.11)
For the simplest case we can restrict o to negative values with appropriate
definitions of principal axes (22 and Chapter 4)
1 _ f%{2  a  a2)da l
X u
Ti
f2d(J
12
c^oi^o) + 02o(2wo)
o o
 (ff2o(wo) + 4^20(2^0))
(3.12)
Assume <720(^0) and
92oW = r(313)
92o(2wo) = Trtfi (314)
when uqtc Â« 0.6156, T\ = rlmin
The eqs. (3.12) â€” (3.14) result in the following
1)
wq = 2n x 100 x 106
Tlrmn = 13.4172 msec
2)
wq = 2tt x 25 x 106
~ 3.3542 msec
T\ exp = 1.03 msec â€” Fig A (ref.25)
3)
w0 = 27T x 9 x 106
Tlmin â€” 120574 msec
T\exp = 2.25 msec â€” Fig.5 (ref.70)
(08S) J.
29
T(mK)
Figure 4. Experimental Curve of T\
T, (msec)
30
Figure 5. Experimental Curve of T\
31
Temperature Dependence of T j
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 ff2o(ma;o) fÂ°r high frequencies. From eqs. (3.7) and (3.8):
(02m(0)0Â¡m(0))9(i) =(02m(0)0'2tn(0))e j
t
tâ€˜
/OO t2 w2u
e <Â«2 e~l(i}tdt = a\ÃÃ±e T
OO
(3.15)
(3.16)
Eq. (3.12) becomes
1
DÂ¿y/Ã± _w2 2 \
â€” â€” (ae 4 + 4ae 0 '
T\ 36 v
Ti =
36
ae 4 + 4ae woa
(3.17)
(3.18)
irp _
T\ passes through a minimum when = 0, i.e for a = 5.082 X 10â€œ , and
the minimum value is Tlmin = 1.1384msec, at = 25 MHz. The result of
the theoretical value 7\win = 1.1384msec is in very good agreement with the
experimental value TlmÂ¿n = 1.03msec(25).
Now the question is how T\ varies at fixed wq over a wide range of temÂ¬
peratures which causes rc to vary. Similar to the discussion of the dynamics of
spin glasses (71), we would like to try an Activation Law of the VogelFulcher
type:
a = a0e 'T_7o
(319)
where
A is the activation energy in temperature unit
Ã¼q is a time factor which is the value of a as T â€”> oo, and
Tq is some characteristic temperature (transition temperature), such that
as T â€”*â€¢ Tq, long relaxation times become important.
32
0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48
T(K)
Figure 6. TrT Curve (solidriezp,dashedrlt^eo)
33
By using two experimental values (T\ = 2.550msec at T = 0A50K and
T\ â€” 2.514msec at T = 0.380ÃÃ for X = 38%) we find the following formula:
O 0.1880
a = 5.4128 x 10 8 X gro.6oi6 (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 Inhomogeneitv of Ti
Results of Calculations
The dependence of 7\ 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(ol)o and a negative branch given by Av = â€” \DP2(a)o
(Fig.7). As previously demonstrated we restrict o to negative values. That is
Av = â€” D^  (Seos2a â€” 1) (positive branch)
Av â€” ~(3cos2a â€” 1) (negativebranch)
2 2
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 wq In this case the spectral density functions are given by
ff(wo) = 4<7(2u>q) = 5 (low temperature limit of Lorentz form), where r1 is
UJqT
the characteristic fluctuation rate, which was taken to be a unique value for
simplifying calculations.
For the dipolar contribution at fixed o and Av we obtain
M2  Â°  (tj?)2
ADD
w0r
(3.21)
34
Figure 7. Allowed Range of Values of a for
A Given Frequency Av
35
and for the spinrotational contribution:
wor
The frequency dependence should be obtained by summing over all allowed
o for a given frequency At/. The calculations were straightforward but quite
tedious. Table I and Table II show the numerical results.
(3.22)
7â€™l _
SR ~
!_ r<2
18 U
(2
Table I Spectral Dependence of DipoleDipole Relaxation Rate
Frequency range Au
rp\ , 24Ã¼jÂ¿t i
1 lDD x \ m )
0 < Au < \D
10772â€”12i/7?+4i/2 , 21/2 i 2i/ , 21/2 i u
3Â£>2 + 77(772u) log 7? + D[Dv) log 77
D < A u < D
5772 â€” 7i/77+8i/2 , 2i/2 iâ€ž_ i/
3i?* + 77(77*) l0g D
Table II Spectral Dependence of SpinRotational Relaxation Rate
Frequency range Au
T1 x f18 Â°r 1
SR x 1 r?2 )
0 < Au <\D
2773* 2* i 2Ã/ , 2* i v
D 772* loÂ§ 77 + TJ=U io8 77
\D < Au < D
77+i/ i 2* intr *
~U~ + 77* Â°8 77
Discussion
Fig.8 shows the results of the calculations for the dipolar contribution
and the spinrotational contribution as function of frequency. All curves
in this figure have been normalized to unity at Au = 0 in order to facilitate
the comparison with the experimental data. The net relaxation rate T^aic\ =
^1DD "*â– sR The curves show discontinuities in slope at \Au\ = y, but this
was not seen experimentally. If we take the finite crossrelaxation rate T^1 into
36
account, which will bring the very slowly relaxing isochromats at Ai/ = 2D
into communication with the rapidly relaxing components,
= T.
1
l(calc)
+ T.
l
12
(3.23)
when the individual spinlattice relaxation times are much longer than the
crossrelaxation times the spins come to a common spin temperature via the
crossrelaxation mechanism before relaxing to the lattice via the rapidly relaxÂ¬
ing components. On the other hand, when the direct spinlattice relaxation is
fast and 7\
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 Ti2 by the probability of a spin flipflop transition
via the intermolecular dipoledipole interaction for two isochromats and U2
given by
(TflipflÂ°P)lexp
7T2(//i  t/2)2
Mmter
= {Tllipflop)lexP
jyrintra _ ^^jintra^2
2 Aqnter
(3.24)
where the exponential factor is the overlap given by Abragam (51) for two
lines centred at and 1/2 and their individual widths are determined by the
intermolecular dipoledipole interaction. Mâ„¢tra, Mjnira and M^nier are the
moments resulting from the intra and intermolecular dipoledipole interacÂ¬
tions, respectively. Yu et al. (14) observed a much weaker dependence than
the exponential variation with 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 T\2
37
O 0.25 0.5 0.75
I//D
Figure 8. Frequency Dependence of T\
38
The most complete studies of the spectral inhomogeneity of T\ in the
quadrupolar glass phase has been carried out for an ortho concentration X â€”
0.45 at T â€” 0.15/f(14), and the data of ref.14 would place the crossrelaxation
time TÂ±2 in the range 7.5 â€” 10.5 msec.
While the crossrelaxation is faster than the direct relaxation to the lattice,
the values of 1\2 reduce the spectral inhomogeneity of the relaxation of the
NMR line shape. In this case we expect to observe Tl(lbs) ~ Tl(lal) + T12 for
the relaxation of the magnetization of a given isochromat. This is shown in
Fig.8. In view of the uncertainties in the crossrelaxation and the simplifying
assumptions that have been made, the overall agreement with the experimental
results is good. The correct overall behavior is predicted as well as the subtle
change in spectral dependence at the half width points which has already been
seen in the experimental data of Yu et al. (14).
NonExponential Relaxation of Nuclear Magnetization
Although it is agreed that (i) the low temperature NMR lineshapes indiÂ¬
cate a random distribution of molecular orientations (for both the local axes
and the alignment o =(3J â€” 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 temÂ¬
peratures (T < 0.1K). Is it a very rapid, but smooth variation (25,27,70)
with temperature, corresponding to a collective freezing of the orientational
fluctuations or a slow, smooth dependence (9,14,28) with no evidence of any
strong collective behavior? In order to resolve this problem it is important
39
to understand two unusual propertiesthe spectral inhomogeneity of the reÂ¬
laxation rate 2^ 1 across the NMR absorption spectrum (14,29,30,72) and the
nonexponential decay of the magnetization of a given isochromat (14). The
spectral inhomogeneity has been discussed previously and the purpose of this
part is to show that the strong departure from exponential decay for the magÂ¬
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)
Â«*)!>* [l (%=)>'
1SR OJqT
The frequency of a particular component of the NMR absorption line is
given by
A u = Â±DP2{cc)o (3.25)
2
and this can be satisfied by very different values of o and a; e.g. Av = \D
occurs for o â€” â€” l,P2 = Â° = ~ l;o = â€”Â§> P2 = Â§â€¢â€¢â€¢â€¢! the only
constraints being that a and P2 lie within their limits; â€” 2 < o < 0, and
i < P2 < 1. Obviously, different pairs of a and P2 (for fixed Av) can result
in very different values of and Tig ft. 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
V, which give the variation of T\ for Au = \D, \D and 0, respectively. The
relative contribution of these rates can be determined from the probabilities
40
n(a),n(P2) of finding o and P2. At low temperatures, the analysis of the line
shape indicates that a good approximation for IT(cr) is a triangular distribution
I1(
very glassy) which requires that II(P2) oc â€”, 1 The calculated rates
T^{o, P2) have a relative weight P = II(
component = ^^D\a\P2.
Magnetization ratio
M(t) Â£Pe Tl
M(0) _ Â£p
(3.26)
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
/\u are indicated by the symbols.
Table III Variation of Relaxation Rates within A Given Isochromat (Au = \D)
parame.
P2Ã*)
larame.
kl
prob.
n(f2)
prob.
1%)
rates
j.l
SR
rates
yl
1\DD
rates(a)
rr1
13
16
1,5
1.022
1.067
0.058
0.234
0.267
7
8
Ã©
7
1.044
1.143
0.107
0.431
0.492
13
16
13
1.069
1.231
0.144
0.583
0.666
3
4
i
3
1.095
1.333
0.167
0.681
0.777
n
Ã6
V
1.124
1.455
0.170
0.706
0.804
5
8
8
5
1.155
1.600
0.150
0.634
0.720
9
1fi
Ã6
9
1.188
1.778
0.097
0.422
0.478
1
2
2
1.225
2.000
0.0
0.0
0.0
Rates given in units of
The Determination of The Molecular Correlation Time t and T^ at Au = 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 (At^
parame.
6Â¡(Â«)
Darame.
cr
prob.
n(P2)
prob.
nw
rates
j.l
1\ SR
rates
T~ 1
J 1DD
rates
rr1
15
16
~S~
15
1.022
0.533
0.092
0.272
0.325
7^
8
1.044
0.571
0.179
0.526
0.629
13
T6
13
1.069
0.615
0.260
0.760
0.910
3
4
i
3
1.095
0.667
0.333
0.972
1.165
11
16
8
11
1.119
0.727
0.398
1.159
1.389
5
8
4
1.155
0.800
0.450
1.316
1.576
^I
1G
5
1.188
0.889
0.486
1.435
1.715
1
2
1
1.225
1.000
0.500
1.500
1.789
7^
16â€”
â€”8~
7
1.265
1.143
0.482
1.485
1.763
J
A
3
1.309
1.333
0.417
1.337
1.577
16
8
5
1.359
1.600
0.275
0.938
1.097
nr
4
2
1.414
2.000
0.0
0.0
0.0
Rates given in units of 24^2 T
line at Au = 0 in Fig.9, is obtained t = 1.51 x 10_75. For
T = 1.51 x 10_75
1
TÂ¡
I At/=0â€” 1.49597 X
103
10.0446
T\ At/=0= 6.7144(msec)
(3.27)
T\ at Av = 0 is 6.71 ms which is in excellent agreement with the experimental
value of 6.6 Â± 0.5ms (14,70).
Since D=173.1 khz, in Fig.9 the Ai/ for the calculated are 87,43.5 and
0 khz which are below the values chosen by the Duke group (14). The same
theroy to calculate for Av = 98 and 58 khz and the results are depicted
in Fig. 10. The overall agreement is very satisfactory.
Comparison of the calculated decays M^u{t) with the experimental results
shows that not only is the correct overall deviation from exponential decay
42
predicted, but that thei'e 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 â€” 0)
parame.
M
prob.
11(a)
rates
J.1
1\ SR
rates
T~ 1
11DD
rates(a)
rr1
8
15
0.533
1.467
2.249
3.096
4
1
0.571
1.429
2.245
3.070
0.615
1.385
2.237
3.036
3
0.667
1.333
2.222
2.992
11
0.727
1.273
2.198
2.933
5
0.800
1.200
2.160
2.853
8
9
0.889
1.111
2.099
2.740
1
1.000
1.000
2.000
2.577
~S~
7
1.143
0.857
1.837
2.331
4
3
1.333
0.667
1.556
1.940
8
5
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 cr considered in Table IV in order
to facilitate the comparison.
Rates given in units of npÃ³~
Â° 24
M(t)/M(o)
43
O 15 30 45
t (msec)
Figure 9. Time Dependence of M(t) for Different Ai'
M(t)/M(o)
44
Figure 10. Time Dependence of M(t) for
Au  98 and Au = 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 its 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 3x3 density
matrices pÂ¿ (for each site i). The are completely described by
(1) the molecular dipole mements (Jx)i, {Jy)ii{Jz)i and
(2) the quadrupole moments (JxJy)i, (JyJz)iâ€¢â€¢â€¢â€¢ 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 operatiors with 0 < L < 2J for general J and the
associated multipole moments
hm = Tr{Pnlm) (41)
For simplicity the site index has been dropped. The expansion of the single
particle density operator in terms of the multipole moments is given by
2 J L
0 = (2 > T 1) E E (2i+1)iLWnÂ£M (42)
1 â€™ L=0M=L
There are three conditions imposed on p: Hermiticity and both weak and
strong positivity conditions(34).
45
46
p is a Hermitian operator aiul
lLM = ((43)
Tl00 is a unit matrix operator and t00 = 1 = Trp
The weak positivity conditions are given by
27TT  rr("2)  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 + l)1; then the matrix describes a
completely unpolarized state and Tr(p2) = (2J + l)1.
For density matrices of spinâ€”^ particles, condition (4.4) is the only conÂ¬
dition imposed by the positivity condition. But for J > further conditions
are imposed on the density matrix and on the multipole 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 multipole moments and thus on the allowed values
of the local order parameters for orthoH2 molecules in the solid mixtures.
It is useful to construct orthonormal matrix representations of the irreÂ¬
ducible operators Him in the representation (J2,JZ). For J=1 these are given
47
by the following 3x3, matrix operators with rows (and columns) labelled by
the eigenvalues 1,0, â€” 1 of Jz.
and
1 1 Z1 0 0
Hio = fiJz = 7= 0 0 0
â€¢ v ... â– V? \ o o l
i i (Â° 1
nn = 7=J+ =Â° o 1
V2 V2 \ 0 0 0
1 /_ ,2 .9> f
n20 = ^^) = ^
1 0 0
02 0
0 0 1
1 1 (0 1 o
n2i  ~{'hJ+ + J+Jz) = â€”j= ooi
2 V2 \o 0 0
1 fÂ° 0
,^22 = 7yJ+ ~ 0 0 0
2 \o o o/
Mirt
nL,M = (i) nl
M
ÃœLL'ÃœMM1
(4.6)
(4.7)
(4.8)
(4.9)
The quadrupole operators Il2Af transform analogously to the spherical harÂ¬
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 symÂ¬
metry for each molecule. The natural choice for the zaxis is along the net com
l
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 rnements in the above notation:
P â€” ^ ^ Mnffn + ^ ^ QmJ^2r
m
where nn =m)
â€” /rrt
(4.10)
48
The expression (4.1U) is identical with that was derived by ref.35 and ref.73.
We still can choose x and y axes such that QÂ¿ is real. i.e.
1
(4.11)
and
(JxJy 4* JyJx) â€” 0 (412)
Q2 measures the departure from axial symmetry about the z axis and is someÂ¬
times called the eccentricity (29, 30). It can be shown that with this choice of
local reference frame Qi and Q\ also vanish and the density matrix may be
written as
1
3
1
(1
0
0 \
0
0
0 +
Vo
0
1
where
MO
V6V
0
Q2
{Jz)
0
0
Q2 \
0
7gQoJ
(4.13)
Q0 = r((3Jt2))
(4.14)
Qo is the alignment (29, 30) along the zaxis. Sometimes it is convenient to
define the alignment and the eccentricity as
13
â€™â€¢ =
Qo
(4.15)
V =(Jx  Jy) = 2Q2 (4.16)
Both o' and ?/ have maximum amplitude of unity. In terms of these parameters
p becomes
1
21
/I
0
0 >
0
0
1
+
VO
0
1)
3l
0
0
3Â°
in A
2â€
0
0 &')
(4.17)
49
V
a'
Figure 11. The Allowed Values of/r =(JZ),
o And 77 for Spin1 Particle
50
where p =(JZ).
The three eigenvalues of p are
x 12,
A i â€” â€” T o
1 3 3
= iia'Â±V/*2 + i72
3 3 9 V ** '
(4.18)
The strong positivity conditions, An > 0, are therefore seen to restrict the
allowed values of the local order parameters o',r\ and p to the interior of a
cone (Fig. 11) in the 3D parameter space. The vertex of the cone is located at
o' = 1, p = r/ = 0, corresponding to the pure state \ ip) =\ Jz = 0), and the
base of the cone is defined by o' â€” â€” \ and p2 + rÂ¡2 = 1 which corresponds
to the pure state  rp) = cos7  Jz = 1) + sin'y  Jz â€” â€” 1) with (Jz) = cos2~Â¡
and (J+) = sin2') (The polar angle 7 generates the points on the circle of the
coneâ€™s baseplate).
The positivity domain shown in Fig. 11 is the same as those obtained by
Minnaert (34) using the EbhardCood theorem and similar to those given by
W.Lakin (32) in his analysis of the states of polarization of the deuteron.
Having established the physical considerations which determine the limited
range of allowed values for the order parameters, we now turn to the special
case of solid hydrogen.
Applications to Solid Hydrogen
In the absence of interactions which break time reversal symmetry, the
expectation value (Ja) must vanish for all a in solid hydrogen. This is the
socalled â€œquenchingâ€ 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 must
accordingly vanish. The separation of the rotational energy levels is given by
Ej = BJ(J f 1) with 13=85.37K, and in the solid at low temperatures only the
lowest J values, J = 0 for parai^ and J = 1 for orthoH2, need be considered.
At low temperatures the anisotropic forces between the ortho molecules lift
the rotational degeneracy and the molecules align themselves with respect to
one another to minimize their interaction energy. For high ortho concentrations
one observes a periodic alignment in a Pa% configuration with four interpeneÂ¬
trating simple cubic sublattices, the molecules being aligned parallel to a given
body diagonal in each sublattice and the order parameters, o'a =(l â€”  ja),
are the same at each site o'a = 1.
The long range periodic order for the molecular alignments is lost below
a critical concentration of approximately 55% and the NMR studies indicate
that there is only short range orientational ordering with a broad distribution of
local symmetry axes and local order parameters throughout the sample. This
purely local ordering has been referred to as a quadrupolar glass in analogy
with the spin glasses, but unlike the dipolar spin glasses, there is no welldefined
transition from the disordered state to the glass regime. In order to describe
the ortho molecules in the glass regime, where there is a large number of sites
with various values of o', 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 parainters:
52
(1) From the previous discussion, the allowed values of o' and rÂ¡ lie within
a triangle bounded by the three lines (Fig.12 AABC).
J L ,
 + o' > 0
3 3â€œ
'â€¢r a Â± T] > 0
3 3 2
(4.19)
which represent the strong positivity conditions for the eigenvalues of p when
the angular momentum is quenched.
(2) Since (Ja) = 0 for all a, we are free to choose the zaxis which was
previously fixed by the net component of the angular momentum. The natural
choice for the local reference axes is now the set of principal axes for the
quadrupolar tensor
Qafi â€” JpJa)
(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 â€œallowedâ€ portion of the [o', rj) 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 A CFE by suitable rotations (i.e.
relabel the axes).
The important point is that the pure states
and
'Iâ€™C =1 Jz = o)
VM = ~^(l Jz = 1)+ I Jz = 1))
(4.21)
(4.22)
53
are not inequivalenl (Â¡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 can be obtained from by a
rotation of the axes by about x axis.
The rotation operator
Â«.(f * (Â£)*+<(Â£)+/ (Â«3)
and
= e*2 xÂ¡)A
(4.24)
The rotation Rx{^f) leaves the state F invariant and maps D onto the point
G in parameter space.
We can furthermore show that the rotation Rx(j) maps the following
triangular regions of parameter space onto one another
A CDF Â» A AGF
ACER AAEF
and
ABDF > ABGF
(4.25)
It is also seen by considering the transformation
,3tt,
(4.26)
using the matrix representation
/ I _L
2 ^
v/2 u v5
i t i
V 2 ^ 2 J
(4.27)
54
Table VI Special Points in Orientational Order Parameter Space, (see Fig. 12)
Label
A
parame.
(ini
Wave Function
II i>+1DIA/2
B
(VV
IIÂ«1 D/Vi
C
EÂ°L
DS
D
(Li)L
HD DV210)/2
E
M;
l*'(l 0+ 1 â€”1Â» â€” Vi 1 0)]/2
F
(0.0)
FT + 0U) + (i + 0li)^2o)l/v^6
G
H.O).
K1 + 0 ID+ (i + .')1)1/2
Rx{^Â£): 0c *â– Vâ€™4,V>Ã¼ 4â€™g'ixIjb,%1â€™f,xIje flxed
Rz{\): 'Pa * 'Pu, * fixed
The Rx{y) transforms the points (cr',77) into new points (o",r/) given by
(4.28)
which corresponds to the mapping given by eqs.(4.25).
There is some confusion in the literature concerning the values of o'. In Van
Kranendonkâ€™s(75) book the negative value of o' had been ruled out as it was
unphysical. Hut this is not strictly correct. Van Kranendonkâ€™s remarks refer
to considerations of the pure states  Jz = 0) and  Jz = Â±1) only. He does not
discuss either the density matrix approach or the formulation of the intrinsic
quadrupolar order parameters o' and rÂ¡ needed to describe the orientational
degrees of freedom of the orthoH2 molecules.
The results of the relabeling transformation are particularly easy to underÂ¬
stand if we consider the pair (S = o', N = ^77) which transform orthonor
inally when the axes are rotated. The parameter space (S, N) is shown in
55
Fig.13. The relation Rx(^f) 'in (.S', iV) parameter space through the line BZ
with the transformed points given by
S'
N'
(4.29)
Similarly, the relabeling (x, y, z) â€”> (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 orthoHÂ¿ molecules in the solid state. This is not the only choice that
can be made for a â€œprimitiveâ€ area of inequivalent values of We may
also choose the sum of ACFK and AGF J 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
D(7T20 (i))oZ
(4.30)
As previously discussed, considering the transformation to the local frame
we obtained
Ai/Â¿ = Â±D[o'ip2{cosOi) + ^rÂ¡isin29lcos2l] (4.31)
where (0Â¿, <^>Â¿) are the polar angles defining the orientation of the applied magÂ¬
netic field with respect to the local molecular symmetry axes.
56
("iÂ»'1)
â–º (T
B
Figure 12. The Allowed Values of o And 77
for Spin1 Particle
57
Figure 13. Diagram of Allowed S And N
58
We propose that the assumption of axial symmetry can be tested experÂ¬
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 o' is constant and 77 = 0, but it has not previously been
considered for the glass phase.
In zero applied magnetic field the degeneracy of the nuclear spin levels is
lifted only by the intramolecular spinspin interactions.
Hdd(Ã) = hD[^c',(3I22t  if) + ifcUh + /Â£,)] (4.32)
The tensorial operators for the rotational degrees of freedom have been reÂ¬
placed by the expectation values o' and 77. An applied radiofrequency field
can induce magnetic dipole transitions between the nuclear spin levels in analÂ¬
ogy with the socalled pure quadrupole resonance absorption(51 Chapter VIII).
The eigenstates of Hjjq are  Â¡z â€” 0) and  Â±) = ^( Iz â€” l)Â±  h = â€”1).
For each molecule i, three resonance lines can be expected corresponding to
the transitions  +) â€”> 0),  â€”) â€”> 0) and  +) â€”* â€”) with frequencies
v\ = D(o[ + t?J, uf = D[o[  Jt?Â¿) and vf = D^, respectively. If
axial symmetry is a good approximation, there is only one line at = â€”DoÂ¿
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 o
X. Li, H. Meyer and A. J. Berlinsky (23) have proposed a model for the
distribution function of o'. 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
Pi ~ 2^~ 4 + 4 + 4)
p? = Â¿<4 4 + 4>
Pi =(i  4> = \(Jl + 4  4)
The pure states p= 1 correspond to rpawave functions for a = x,y,or z.
(a)
This leads to a natural parameterization of the p1 given by
where the â€œenergiesâ€ Eai 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 ]T)a Eai â€”
0. i.e.
P(EX, Ey, Ez)
hjEHE*+E?)/A2
2ttA2
fi{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
as a function of the degree of local orientational order parameter measured
I"Â»;/)â„¢'
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 â€œtwo level system,â€
corresponding to JZi = Â±1 and JZi â€” 0 separeted by an energy gap AÂ¿.
60
The probability distribution of A is Gaussian as following
P(A) =
nD2
e 2o'2
(4.33)
The order parameter a =(3 J2 â€” 2) will be
We can prove that
2e kt  2
= _a_
2e kt + 1
M4 _ 15 (tr4)
M2
7 (<Â«2Â»2
where
roo
(o2) = /
Jo
(2e irr  2)2 / 2 _
(2e kt + l)2 V ttD2
A2
dA
(4.34)
(4.35)
(4.36)
let % = X
, fÂ°Â° (2e~KfX  2)2 Ã2 =x?_
(Â° )~  __d_ v wV~e 2 (437)
â€¢/o (2e ktX f 1)2 V 7T
D X
<<>4>
fÂ°Â° (2e kt
Jo (2 e~^r
2)4 12 x2
e~ dX
(4.38)
(2e ktx + l)4 V X
By using numerical integral method the curve in Fig.14 shows the ratio
A versus y/ (ct2) (order).
(a2)
Considering the intermolecular and the intramolecular dipolar broadening,
the fourth moment M4 and the second mement M2 should be (51)
where
,, rrp;,/*]2}
3Y{[*', \U[,IX ]]2}
Mi = T7m
_ ^intra __ ^inter
x; 1  3cÂ°s2<>Â«(3/*/â€˜  /â€˜ . 71)
4 > > r LI
(4.39)
(4.40)
fc.t
61
KnUr=irhhY.(1 3T2s',]nsi
ij
r3
i]
Following a quite complicated but straightforward calculation we can prove
that
M2 = M\ntra + Mrjnier
(4.41)
M4 = M\ntra + M\nter + 4 MlntraMl2nter
(4.42)
For a Gaussian shape one has Mnter = 3(M2 er)Â¿ and
= M'tn,ra + 3(Aij",tr)2 + 4M'2n,râ€˜M'2nt"
(4.43)
Taking the same value of Mnier = 20[khz)2 as in ref.23 we obtain the curve
IHU U Â¡ â€”
in Fig. 15. The upper one is ^MintrÂ«y2 versus y (cr2) and the lower one is p^2
\/Wz)
versus
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
experimental lineshape is 3.15 (18) and the value of maximum of calculated
lineshape is 5.60.
62
MODEL (INTRA)
ORDER
Figure 14. Diagram of
P?
versus
vV>
63
MODEL
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)
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 occuring after application of resonant rf 90Â° â€” r â€” xp^ pulse
sequences and rf 90Â° â€” r â€” xp^ â€” tw â€” pulse sequences.
The purpose of this chapter is to develop the theory of spin echoes of solid
#2 in orientationally ordered phase and discuss the lowfrequency dynamics of
orientational glasses.
Solid Echoes
There are several publications which have presented calculations for the
amplitude of solid echo responses to 90Â° â€” r â€” xp^ pulse sequences (39, 78, 79).
We would like to examine the time dependence of the nuclear operators
and the conditions for the focussing of solid echoes.
The Time Dependence of The Nuclear Operators
A general twopulse sequence denoted by (j â€” r â€” xp^) is sketched in Fig. 16.
In NMR experiments one observes only the orthomolecules (7 = 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)(Ix + tiy)],
where p is the density operator at time t and 1 the nuclear spin operator.
64
65
Ro^)
RW.tp)
/V
Figure 16. Solid Echo (twopulse sequence)
66
Ro(?) R()
tuy Â«>T tyv â™¦ 2T
Figure 17. Stimulated Echo (threepulse sequence)
67
In equilibrium the expression of density matrix is given by
eWo
Hieq Tr(eWo)
Let Z be the direction of applied magnetic field.
\ = ~uohIzl
(5.1)
where ui0 is the Larmor frequency.
Since
Pie
(! + Jet hi)
a + ^U,)
(5.2)
IV(1+ &/â€žâ€¢) 3' ' KT
We know that the radio frequency field is responsible for nonequilibrium
behavior of the system.
In the rotating frame after the first 90Â° pulse, the density matrix p(0+)
would be
p(0+) = Hldtpi{0)ehlHldt (5.3)
If the radio frequency field is in the y direction of the rotation frame, we
obtained
is./ ,l, hojQ
/(<>+) = e*t4i(i + ^)eH4
Â«(0+) = I(1 + ^TIl]
(5.4)
This represents that the first pulse R0 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.
nâ€˜dd = E ^4,+ e
2
Â¿O
(5.5)
68
where
and
Â°Â¡ =(34  2)
t2o ~ v^3^1 ~~
, , jr [T HV?(t')dt'!/, . ^0 T x i rT
p(r_) = e Â«w v ' (1 + Â¡r=;Ix)eh Jo
Tjsec
Hdd
(t')dt'
KT
(5.6)
It is assumed that the second pulse is a pulse and the phase of that is
(the angle between radio frequency field and y axis in rotating frame). After
the second pulse the density matrix is given by
p(T+) _ eii/>(Ixsin)p^T_jeiip(Ixsin+IyCos)
(5.7)
The nuclear spin density matrix at time t after second pulse is therefore
p{t + t) = tt(Ã)i2(^)u(0^(l + ^/x)Â«t(r)iÃ¼t(^)ttt(Ã) (5.8)
where
_ e\iAwIzi^DP2(e)<7(3I^2)]t
R((f>) = ^(Ixsint+Iycoscf))
u(r) = e[iAuhi\DP2(e)o{3Pz2))T
and the signal
S(t + r) a Tr{p(t + r)/+}
Tr{p(t + r)/+}
(5.9)
(5.10)
(5.11)
= lj^Tr{u(t)R((f>)u{T)Ixu\T)rt{)u\t)I+}
= \tfrT (m\u{t) \m2){m2\ R{4>) \m3)(m3\u{T) \m4)
mim8
(ra4  Ix  m5)(m5  u^r)  m6)(m6  ^(0)  m7)
69
(m7  ut(i)  m8)(m8  1+  mi) (5.12)
Formation of Solid Echoes
After a lengthy calculation the results of eq.(5.12) are still quite compliÂ¬
cated (for general
the same as that in ref.39. We would like to give the results in some special
conditions.
(1)xp = 0 (apply one pulse)
Tr{p(t)I+} = e~lAujtcos[^DP2{0)(Jt} (5.13)
This is called the Free Induction Decay.
(2) cp = 0,xp â€”  This is a 90Â° â€” r â€” 90Â° pulse sequence
Tr{p(t +t)I+}
= â€”c'^cos^DP^e^t  r)](l  e*A"2') (5.14)
Obviously, Tr{p{t + r)/+} has a maximum value at t = t. This is what
is the meant by a solid echo. Another question is that according to eq.(5.14)
the inphase echo (Aw = 0) does not exist. This is in disagreement with the
experimental results. The reason lies in neglecting intermolecular dipoledipole
interaction and the difficulty of getting exact Aw = 0 in experiments.
(3)<^> =1,^=1 This is a 90^ â€” t â€” 90Â£ pulse sequence
Tr{p(t + r)I+}
= ~^iAutcos\\DP2(e)a(t  t)](1 + eA"2") (5.15)
From eq.(5.15), same as the second case, Tr{p(t + obtaines maximum
value at t = t, forming the solid echo.
70
Stimulated Echoes
It is already shown that in certain cases a sequence of three 90Â° 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 â€œfingerprintâ€ of the local molecular
orientations at a given time with those at some later time (less than T\) and
thereby used to detect ultraslow molecular reorientations.
Applications to the study of the molecular dynamics on cooling into the
quadrupolar glass phase of solid hydrogen will be discussed.
Formation of Stimulated Echoes
A threepulse sequence is sketched in Fig. 17. As the analysis of solid echoes,
the nuclear spin density matrix at different time are given by
/>(<>) = Peq Â» j(l +
/>(<>+) = (1 + "ix)
â€ž(r_) = e~iS: Â«ZtlWp{0+)eU; "{?(*')<Â«'
p[t+) = eâ€™i'l>(I*ain+IyC08{Ixsin+IyC03)
p[(tw + r)_] = e hndd twp(r+)ehndd
/#Â«,+Â»â– )+] = ei*'lI**in*,+eoaW p[(tw + r)_
p{tw + t + t) = e~hHddt p\[tw + T)+]ehHdd l
The signal
S{tw + r f t) a Tr{p(tw + t + t)I+} (5.16)
The first preparatory pulse creates a transverse magnetization in the rotatÂ¬
ing frame. Under the influence of the Hamiltonian given by equation(5.15), the
71
evolution during the short time r(
states described by p(r_) which contains both transverse magnetization and
transverse alignment. The transverse components are transferred by a second
pulse R(4>) 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 â€œfingerprintâ€ of the local alignments o. After the waiting
period tw
pulse of rotation angle ip'. The spin polarization and spin alignment â€˜storedâ€™
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 evoÂ¬
lutionary period r and the transverse signal focuses to stimulated echo after a
delay time r following the third pulse.
Tr{p(tw + t + f)/+}
1 hu 0
3 ~KT
y] (mi I e hHdd * I m2)(m2 \ I m3)(m3 I e hHdd tw  7714)
miâ€”mi2
(m4 I R() I m5)(m5  e~hHddT  m6)(m6  Ix \ m7)
(m7 I ehndd I m8)(mg  R(4>) \ mg)(mg \ e* dd w  m 10)
(mw I B+(0') I mu)(mn   m12){m12  1+  m,)
(5.17)
Since the calculation is extremely tedious and the results for general (p,'
and general ip,xp' are very complicated, we only investigate the results of calÂ¬
culations by taking two special cases.
(1)
= o, 4f = f
= f, = f
72
This is a 90Â° â€” r â€” 90Â° â€” tw â€” 90Â° pulse sequence
Tr{p(tw + t + Â£)/+}
= ^e~^tw)cos[DP2(0)o(t â€” r)]
2 KT 2
H â€”^e~l^'u}ticos(Au>T)sin2(Aijjtw)cos[DP2(6)o(t + r)]
2 KT 2
â€” ^^e_ÃAwÃsÂ¿n(Au;f)sm(Au;Ãly)cos[^T)P2(^)0'(Ã â€” tw + r)] (5.18)
3 KT 2
(2)
f,^ = f
0 = f,^ = f
This is a 90Â° â€” r â€” 90^ â€” tw â€” 90Â° pulse sequence
Tr{p(ÃK; + r + 07+}
= ^e_Ã^wÃÂ¿sm2(AcuÃU!)sÂ¿n(Ac<;r)cos[DP2(^)cr(7 ~ 01
3 ii T 2
+ ^^e~l^
3 ii T 2
+ â€”^le~^lJjtcos(Aut)cos(Aujtw)cos[DP2(6)cr(t â€” tw + 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 â€” t. The results
are sketched in Fig. 18. The appearance of multipole echoes has been seen in
experiments (Fig.19) for orthopara hydrogen mixtures at low temperatures.
Engelsberg et al. (81) have presented results for nuclear spin stimulated
echoes in glasses. The curve of 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 + 2t (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 â€” r â€” \ â€” tw â€” ^)
stimulated in the quadrupolar glass phase of solid hydrogen (r = 25ps, 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 ^â€¢ For
experimental curve only a very slight modificaion (indicated by the arrow) was
observed. We will discuss this phenomenon later.
The experimental curve in Fig.22 gives the relation of stimulated echo
versus waiting time tw of solid hydrogen (orthoconcentration x = 0.54, T =
220mK). The time scale is logarithmic. The logarithmic decay behavior can
be understood in terms of the motional damping of stimulated echoes.
LowFrequency Dynamics of Orientational Glasses
The orientational glasses (20, solid orthopara H2 mixtures (6), N2/A mixÂ¬
tures (83) and the KBri_xK(CN)x mixed crystals (82, 84, 85)) form a subÂ¬
group of the general family of spinglasses which continue to generate intense
interest because of the apparently universal low temperature properties obÂ¬
served for a very diverse range of examples (dilute magnetic alloys, mixed crysÂ¬
tals, dilute mixtures of rotors, partially doped semiconductors (86), Josephson
junction arrays (87) and others). The most apparent striking universal feaÂ¬
tures (20) are an apparent freezing of the local degrees of freedom on long
time scales without any average periodic long range order, characteristic slow
relaxations and historydependence following external field (magnetic, electric,
elasticstrain....) perturbations, and a very large number of stable low energy
states.
74
Figure 18. Sketch of The Results of Calculations
75
Figure 19. Experimental Curve (x = 23%,T â€” 38mK)
76
Nuclear spin stimulated echoes in glasses
3635
Figure 20. Spin Echoes in Borosilicate Glass at 4.2K
77
78
Waiting tmt,tm (at)
Figure 22. The Observed Decay of Stimulated Echo
(x = 0.54, T = 220mK)
(squaresir = 12.5/us; circles and triangles:
t = 25 fxs)
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 mixtures
(for X < 55%) leads automatically to the existence of local electric field graÂ¬
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 mixÂ¬
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 Huseâ€™s recent picture (89), we provide a explanation of the low
frequency relaxation and the lowtemperature specific heat of solid ortho para
hydrogen mixtures.
In the scaling model of Fisher and Huse the lowenergy excitations which
dominate the longdistance and longtime correlations are given by clusters of
coherently reoriented spins. Their basic assumptions are:
(1) Density of states at zero energy for droplets (d dimension) length scale
L as L~6, where 0 < 6 <
(2) Free energy barriers Eg for cluster formation scale as Eg ~ with
0 <4>
With these assumptions, Fisher and Huse show that the autocorrelation
function
CM =((s,(o)sM)t(s>)?)c
_e_
decays as (logi) for f â€”> oo.
(5.20)
80
For our system, assuming axial symmetry, the quasistatic local orientaÂ¬
tional order parameters are the alignments ox =(3J â€” 2)t and the correspondÂ¬
ing autocorrelation functions Ct(f) â€” aÂ¿(0)aÂ¿(Ã) can be studied directly by
NMR.
For a 90Â°y â€” t â€” 90Â°y â€” tw â€” 90Â° pulse sequence we assume at t â€” t, order
parameter a = cr(r) and at t = t ( tw,a â€” cr(r + tw) for each ortho molecule
Tr{p(tw + t + t)I+}
â€” ^^Â¿e~'lAulticos(Au)T)cos2(Autw)cos[]DP2(O)a(T + tw)t  \dP2{0)o(t)t\
SKI 2 2
+ ^Â¿rÂ£e~lAlJjticos(AujT)sin2(AÃ¼jtw)cos[^DP2(0)a(T + tw)t + ^DP2(^)cr(r)r]
SKI 2 2
â€” â€”^e~lAu,tsin(A
3 KT
cos[^DP2{0)o(t + tw)t  ^DP2{9)o(t + tw)tw + ^DP2{0)(j(t)t} (5.21)
Considering t â€” r, the stimulated echo amplitude
A a((cos[ZVat(r)]cos[Â£>ir
where the double brackets refer to an average of configuration and Dx =
DP2(cos0{). The important point is that if the local order parameters
remain fixed during tw, there is no damping of the stimulated echo, while
(Tj changes due to local reorientations, then the contribution to the echo is
severely attenuated. The product Dt can in practice be made very large and
this method can therefore be used to study ultraslow motions in solids. We
believe that in Fig.21 the departure portion from ^ (indicated by an arrow) is
due to slow motion.
A barrier will have a characteristic life time given by an Arrhenius Law
81
or tunneling rate T
E
T(Eb) = r0e"^
(5.23)
where Tq is the characteristic attempt frequency for clusters of this size. In
the long time limit Tq is resonably welldefined because it is associated with a
characteristic cluster size. In a time t the only barriers crossed will be those satÂ¬
isfying 0 < EB < Emax[t) where Emax{t) = KBTlog Any barriers crossed
lead to significant changes in the local order parameters and the amplitude of
the stimulated echo is then simply
fc
A(t) = /
Je,
P(EB)dEB
max(t)
(5.24)
At low temperatures, assuming a constant density of barrier heights P(EB),
we find
A{t)=lKBTP0\og(U (5.25)
The prefactor Pq can be determined from the low temperature behavior of the
heat capacity.
For the orthomolecules with angular momentum J â€” 1, we can assoÂ¬
ciate a simple two level system with the energy states for a given molecules;
the states JÂ¿ = Â±1 being separated from the state J^ = 0 by a gap 3AÂ¿ (The
states J^ = Â±1 are degenerate if there are no interactions which break time
reversal symmetry). At low energies we can, following the above arguments,
identify the low energy excitations (which determine Cv at low T) with a broad
quasiconstant distribution P(A) for 0 < A < Aq for the spins in a cluster.
Identifying P(A = 0) with Pq, the density of low energy barriers, we have
NxR
18 A2
ts2t2 3A 3A
^ 1 4e kbT + eKsT
dA
+ 4
(5.26)
where x is the orthoH2 concentration.
let u = 3^r
cv
NxR
2 KBT
82
o in
f3t<Â£
Jo
u
4e u + eu + 4
du
(5.27)
set t =
let
^=^1*
 = it
V 3 J0
f t v?du
Jo 4eu + eu
+ 4
u2du
(5.28)
(5.29)
4e_u + eu + 4
The resulting C(, (Figure 23) has a linear temperature dependence at low T
and a peak at TpÂ¿. = 0.70^ in close resemblance to the temperature behavior
observed by Haase et al. (90). From the peak position in the experimental
data, = 1.27, PqK â€” 0.86 and for the stimulated echo decay this value
gives
(5.30)
Â¿calc!*) = 1 O.43log10(^)
to
and the observed decay
Aobs.it) = 1 O.55log10(â€”)
to
(5.31)
at T = 0.22K for x â€” 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 paÂ¬
rameters centered on o' = 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 Ãq are much shorter
than the spectraldiffusion time scale (~ sec) seen by the recovery of holes
burnt in the NMR lineshape. We therefore find it difficult to attribute the
logarithmic decay seen in H2 to spectral diffusion across the NMR linshape.
84
I
Cv' â€” t
t
Figure 23. Calculated Curve of C'v â€” t
CHAPTER 6
SUMMARY AND CONCLUSIONS
The orthopara mixtures of solid #2 are studied theoretically for fee lattices
{X > 0.55) of finite site by Akira Mishima and Hiroshi Miyagi (92). The sysÂ¬
tematic theoretical studies of nuclear magnetism for quadrupolar glass regime
are carried out in this dissertation.
We have developed a theory of the nuclear spinlattice relaxation of ori
entationally ordered ortho hydrogen molecules for the case of local ordering
in the quadrupolar glass phase of solid hydrogen. We have investigated the
temperature dependence of T\. It shows that Gaussian Free Induction Decay
is a quite good approximation. The calculations indicate a strong spectral inÂ¬
homogeneity of the relaxation rate T^(Au) throughout the NMR absorption
line. The detailed dependence is much stronger than the simple dependence
T~l{Av) oc (2 + c) given by earlier estimates by A. B. Harris et al. (l). There
is also a strong variation with the orientation of the local symmetry axis with
respect to the external field. The variation is in agreement with that found
by Hardy and Berlinsky (93) for the longrange ordered Pa$ phase. If the
crosssection relaxation between different isochromats is taken into account,
the results of calculations of spectral inhomogeneity is in good agreement with
the experimental results.
In addition to the spectral inhomogeneity, the relaxation is also found
to be strongly nonexponential. This can be understood easily. As a result
of the glassy nature of the system, there is a broad distribution of both local
orientational order parameters (a) and the orientation (ct) of the local symmetry
85
86
axes for the molecular alignments. The existence of these distributions at
low temperatures means that a given frequency Au in the spectrum comes from
all the different allowed combinations of o and a that satisfy Ai/ â€” DP2[ol)o.
The dependence of the relaxation rates on a and P2 then leads to a distribution
of local relaxation rates for a fixed Au. It is this distribution in rates that leads
to the observed nonexponential behavior of M(t). Calculations based on the
expected probability distributions for o and P2 yield results in good agreement
with the results reported in the publications (14).
The most important conclusion of the study of T\ is that the spectral inhoÂ¬
mogeneity and nonexponential recovery both results in an order of magnitude
variation of the nuclear spin relaxation and must therefore be correctly underÂ¬
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 o' =( 1 â€” Â§JÃ), and
(3) the eccentricity r? = (jJ â€” Jy).
The positivity conditions for the density matrix show that the only allowed
values of [o',r]) are those enclosed in a triangle in (cr,,r/) space whose vertices
are the pure states  Jz = 0) and  Jz â€” Â±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 2o' > rÂ¡ > 0. Orientational states with
negative o' are not excluded on theoretical grounds.
Studies of molecular dynamics have shown the existence of paralibrons,
i.e. collective excitations that exist in large clusters with welldeveloped short
range order.
Nuclear spin stimulated echoes have proved to be very effective for the
study of ultraslow molecular motions both in the molecular orientational
glasses and in ordinary glasses (81). The existence of multiple echoes which is
given by theoretical calculations is also in good agreement with experimental
results of solid hydrogen and ordinary glass at low temperature.
We have offered a unified explanation of the slow relaxational behavior and
low temperature heat capacity of the quadrupolar glass phase of solid hydrogen
in terms of the density of lowenergy excitations in the system.
Since the stimulated echoes are damped by any change in the orientational
states during tw, they can be used to detect very slow molecular motions. The
phenomeno 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 Huseâ€™s recent picture of the short range spinglasses and the domination
of the longterm relaxation by low energy largescalecluster excitations.
In analogy with spin glasses (e.g. alloys like Cu â€” Mn where a random
configuration of spins condenses at low temperature) the linear specific heat
with temperature at low T can be understood in terms of the suggestion (94)
that the dominant contribution to the specific heat will be from clusters of
particles for which the energy barrier is sufficiently great so that resonant
tunneling between the two local minima does not occur, but sufficiently small so
that tunneling between the two levels can take place and thermal equilibration
88
can occur during the time span of the specific heat experiment. We obtained
a very good linear dependence curve at low T by using a numerical integral
method.
Having combined with the results of logarithmic decay of stimulated echoes
and linear lowtemperature specific heat, a very good numerical expression of
the amplitude of the stimulated echo as a function of waiting time tw for
x â€” 54%, T = 0.22K is obtained.
Most people agree that molecular orientations of random orthopara hydroÂ¬
gen mixtures become frozen at low temperature and there is no evidence for
any welldefined phase transition on cooling from the completely disordered
high temperature phase. However, the gradual transition to the glass state
in these systems involves strong cooperativity as evidenced by Monte Carlo
calculations. Studies of the nuclear relaxation and molecular dynamics have
shown quantitatively how important the cooperativity is. Further theoretical
study of the slow motions needs to be carried out to improve the understanding
of the nature of the quadrupole ordering in random mixtures at low temperaÂ¬
tures and their relation to the family of glasses. Further experiments at very
low temperatures will also provide a deeper understanding of the properties of
the quadrupolar glass state.
APPENDIX
FLUCTUATIONDISSIPATION THEOREM
The fluctuationdissipation theorem is extremely important because it reÂ¬
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:
= (A.i)
= (A.2)
/Â«(<) = 4 (A.3)
in
Using an identity due to Kubo the response function may also be written
as
fP
fM = / ds{Mi(ihs)Mk(t)) (A.4)
Jo
fkM =  i* is(M,(,hs)Mk(t)) (A.5)
Jo
Here ihs plays the role of time. Such formulas appear in quantum statistical
mechanics because of the formal similarity between the way time and temperÂ¬
ature must be treated. In the calculation of a canonical partition function one
must take into account the factor e~P^. in calculating the time evolution of
operators one must consider operators of the form . By writing t = ihs
one can see that s will play a role identical to (3 in all formal developments.
89
90
From eq.(2.35) and eq.(A.5) it follows that
rP rP
Fki(t) = ds(Mi(ihs)Mk(t))  lim / ds(Mi(ihs)Mk(T))
Jo Tâ€”*oo Jq
(A.6)
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 (M/)(M^). Hence the relaxation
function may be written in its more usual form
F\y(0 = [ ds(AMi(ihs)AMk{t))
Jo
(A.7)
In this form there exists a mathematical identity between the relaxation func
tion and the correlation function gkÂ¿{t), given by
r+oo
FM = 1 B{t T)gkl{T)dT
J â€” oo
(A.8)
where
0kl(i) = J<[AA4(<).AM(O)!+)
(A.9)
the + subscript indicates the anticommutator; or in terms of Fourier transforms
T (r A FÂ¡d[w)
Sk,H = Ã‰M
(A.10)
where
B(t) = Ajr log\coth{ )]
(A.ll)
and
1 hu> (/3hu>\
B{u) 2 V 2 J
(A.12)
The derivation of these identities is called the quantummechanical fluctuation
dissipation theorem. This theorem relates the response of step function disturÂ¬
bance to the correlation function of fluctuations in the equilibrium ensemble.
In the classical limit (h â€”> 0) B(t) â€”> Â¡36(i), and one obtains the classical
fluctuationdissipation theorem
Fkl (0 =
(A.13)
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BIOGRAPHICAL SKETCH
Ying Lin was born in China, on June 9, 1946. She was educated in several
cities in China. She entered the department of physics at The University
of Science and Technology of China as a undergraduate student in 1962 and
was awarded a B.S in 1968(because the â€œcultural revolution,â€ graduation was
delayed).
During 19681978 she worked in a factory (First Tractor Manufactory) and
at a institute (Chinese Electronic Engineering Design Institute) as an engineer.
After the â€œcultural revolutionâ€ she made up her mind to study physics again.
She passed the entrance exam and became a graduate student in The General
Research Institute for Nonferrous Metals in Peking in 1979. She obtained an
M.S. of Engineering in 1981. After that she decided to go to the United States
for further studies.
In September of 1981 she came to the United States. Since September of
1981 she has been a graduate student in the Department of Physics at the
University of Florida. She began research under Professor Neil S. Sullivan
at the end of 1983 and worked on the NMR theory of solid hydrogen at low
temperatures.
96
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope
and quality, as a dissertation for the degree of Doctor of Philosophy.
Neil Samuel Charles Sullivan, Chairman
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope
and quality, as a dissertation for the degree of Doctor of Philosophy.
E. Raymond Andew
Graduate Research Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope
and quality, as a dissertation for the degree of Doctor of Philosophy.
Pradeep Kumar
Associate Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope
and quality, as a dissertation for the degree of Doctor of Philosophy.
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope
and quality, as a dissertation for the degree of Doctor of Philosophy.
David B. Tanner
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope
and quality, as a dissertation for the degree of Doctor of Philosophy.
This dissertation was submitted to the Graduate Faculty of the Department
of Physics in the College of Liberal Arts and Sciences and to the Graduate
School and was accepted as partial fulfillment of the requirements for the degree
of Doctor of Philosophy.
April 1988
Dean, Graduate School
98
UNIVERSITY OF FLORIDA
3 1262 08556 7831
UNIVERSITY OF FLORIDA
3 1262 08556 7831
PAGE 1
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