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1. STATISTICAL THEORY OF THE EQUILIBRIUM
ELASTIC AND THERMAL PROPERTIES OF CRYSTALS
V. B. Nemstov
A study is made of the equilibrium elastic and thermal properties of a crystal that consists
of particles with rotational degrees of freedom. To describe its strain and stress states,
asymmetric strain and stress tensors are used. On the basis of microscopic expressions
for the strain and stress tensors by means of a local-equilibrium ensemble, statistical ex-
pressions are obtained for the specific heat of the deformed medium for fixed strain tensors,
for four isothermal tensors of elastic moduli, temperature stress coefficients, and coeffi-
cients of thermal expansion. These properties are described by static correlation func-
tions of dynamical quantities.
Introduction
The dynamic theory of a crystal lattice is in constant use in the microscopic theory of solids (see,
for example, [1, 2]). Recently, the Born model of a crystal has been successfully used to construetatheory
of inhomogeneous elastic media with spatial dispersion [3-5]. However, because of the purely dynamic
approach, the theory does not contain thermodynamic relations. These are introduced when the system is
treated statistically.
In recent years the methods of statistical thermodynamics have been greatly developed and it has
been possible to construct a consistent theory of both the equilibrium and nonequilibrium behavior of macro-
scopic systems [6].
A deformed state of a solid is a state of incomplete thermodynamic equilibrium [7], for whose descrip-
tion the local-equilibrium distribution can be used [6].
In the present paper we develop a theory of the equilibrium elastic and thermal properties of a crystal
on the basis of the local-equilibrium ensemble. We consider systems with rotational degrees of freedom
msymmetric media). They consist of nonspherical molecules whose interaction is described by potential
that depends not only on the distance between their centers of inertia but also on the orientation of the part-
icles (noncentral forces).
The deformed state of an asymmetric medium is determined by two strain tensors. The fundamental
feature of this approach is that the dynamical variables of the strain tensors are included in the local-equi-
librium distribution.
The method of the local-equilibrium operator is of great generality and has great advantages over the
method of averaging with respect to a state of complete equilibrium. In the present paper, these advantages
make it possible to allow consistently for spatial dispersion of the equilibrium elastic and thermal properties
of an inhomogeneous asymmetric medium.
The study of systems with rotational degrees of freedom is of independent interest [8]. Such systems
include, in particular, liquid crystals, which have provoked particular interest in recent years (see, for
example, [9]).
Irreversible processes in these systems have been considered by the nonequilibrium statistical oper-
ator method [10], and studies have been made of the viscous and high-frequency elastic properties [11, 12]
and also the equilibrium thermoelastic characteristics of spatially homogeneous asymmetric media [13].
S. M. Kirov Belorussian Technology Institute. Translated from Teoreticheskaya i Matematicheskaya
Fizika, Vol. 14, No. 2, pp. 262-271, February, 1973. Original article submitted January 12, 1972.
9 1974 Consultants Bureau, a division of Plenum Publishing Corporation, 227 g'est 17th Street, New York, :V. Y. I00tt.
No part of this publication may be reproduced, stored in a retrieval system, or transmitLed, in any form or by any means,
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196

2. It has been shown that macroscopic manifestation of rotational degrees of freedom is the cause of non-
trivial features in the propagation of sound [14], in Rayleigh scattering of light [15], in dielectric properties
[16], and in piezo electric and optical effects [17].
Local-Equilibrium Distribution for a Deformed Solid
Suppose the molecuIes have fixed mean equilibrium positions of their centers of mass and orientation.
SmalI displacements and angles of rotation of the particles are measured from these mean positions and
orientations.
To construct the dynamical quantities of the strain tensors we extend the method of continualization
of the dispIacement field of a discrete point lattice [3, 41 to systems with rotational degrees of freedom.
We introduce the dynamical quantities of the displacement field and the field of the angles of rotation
of the particles of the medium by the relations
N 2r
u(r)= v ~ u~8(r_ q0,), q~(r) = v ~, ~6(r_ q0,), (1)
where u u and q)v are the vectors of a small displacement and the rotation angle of the particle with number
v; v is the volume per particle; N is the number of particles; 5(r) is the delta function; q0u is the mean
equilibrium radius vector of particle v; and r is the radius vector of a point of space.
On account of the translational invariance of the lattice and the periodic boundary condition, the ad-
missible values of the wave vector can be restricted to the unit ceil of the reciprocal lattice (or the Brillouin
zone). Then the Fourier transforms of u(r) andq~(r) can be written as
N
= vB(k)~ u" exp{ik, q0~},ll(k)
(2)
(k) = vB (k) ~. q~"exp{~k 9qo~}.r
If k belongs to the unit cell of the reciprocal lattice, then B(k) = 1 and t~(k) = 0 otherwise.
Allowing for the transition of the discrete distribution of the wave vectors into a continuous distribu-
tion in the limit of an infinite lattice, we can represent the Fourier transforms of the functions (2) in the
form
/g
u (r) = v ~, u"~L(r - q0"),
v=l
N
~(r) = v}-~ +'6~(r - q0").
(3)
The function 6B/r ) is the Fourier transform of B(k) [3, 4]. It vanishes at the sites of the lattice and is equal
to v -1 for r = 0. The expressions (3) give the desired continual representation of the fields of the displace-
ments and rotation angles of the particles at the lattice sites. At the tattice sites, the fields u(r) and ~o(r)
take the values Uu and ~ou [3, 4].
On the basis of (3) we determine the dynamical quantities of the strain tensors:
e~ = Oa~ / Ox~ -- e,~p.,, ~ = OrO~/ Ox~. (4)
These dynamical quantities will be used to give a microscopic description of the deformed state of the
medium in the approximation of small strains. The expressions for the strain tensors have been derived
statistically already [11].
If the mean positions of the centers of mass of the molecuIes are not fixed (liquid or nematie liquid
crystal), the medium does not exert an elastic resistance to the quasistatic shear strain. Then instead of
the tensor elk one must use the dynamicaI quantity of the mass density (m is the particle mass):
N
(r) = m ~ 6 (r -- q~).P
v~t
197

3. The theory allows a generalization to the case of finite strains, for whose tensors one can also intro-
duce microscopic expressions.
In addition to the strain tensors, the dynamical quantity e of the energy density [6] will also determine
the state of the system. It is convenient to regard the state variables as the components of a 19-dimensional
vector: Pi{e, eli, E22, E33, El2, E2I, El3, E31, a23, E32, Tll, ")/22, T33, Y12, T21, 713, T31, T23, 'Y32}"
On the basis of the dynamical variables Pi(i = 1, 2,..., 19), we construct a local-equilibrium distribu-
tion, using the principle of maximality of the information entropy [6]. The local-equilibrium distribution
has the form
O,=Q~-~exp {- S[~(r)e(r)+~ F~.(r)P=(r)]dr}, (6)
m=2
where Ql is a statistical integral; fl(r) is the reciprocal local temperature. The thermodynamic parameters
and F m are determined from the condition that the loeal~equilibrium mean values of the dynamical vari-
ables P be equal to their given mean values. These thermodynamic parameters are the auxiliary fields
m
that give rise to the given mean values of the dynamical variables [18].
In the Fourier representation, the local-equilibrium distribution is written in the form
pL=Qz-,exp{_Z~F,~(k)p.~(_k~}. (7)
k m=l
Using (7) to determine the mean value of a certain quantity M, we can show [6] that
0<M(k)>~_=-(M(k),P~(-k~)), (8)
OF.(k,)
where
(M(k), P=(-k,)) = ([M(k) -- (M(k))~] [P.(-k,) - <P.(--k,)),])L. (9)
On the basis of (8) for M = Pm' we obtain
,0<P~ (k))L=- (P~(k),P~(-k,)), (10')
OF~(k,)
aF~(kt) ,= _ (pp) _~,k, (10")
where (PP)--kk is the matrix that is the inverse of the static correlation matrix (Pn(kl, Pm(-k~)). Thenthe
nm
derivative
,0<M(k))~ 0<M(k))z OF~(k~)
n,k 2
with allowance for (8) and (10") is determined by
0<M(k)h 2 -~a <P,~(k=)>~ (M(k), P~ (-- k2) ) (PP)-k~..,k'" (11)
We establish an expression for the derivative of <M(k))l with respect to thereciprocal temperature at
fixed values of (Pm(k))/(m = 2 ..... 19):
- - OF~ (ko) ~ /<v.~h"
If we fix (Pro(k))/, then the Fn(k2) become functions of/3(kl~, which enables us to determine the derivatives
Orn/aP.
Then using (8), we obtain
a(M (k))~) <er~>L= _ (34 (k), e (-- kl)) +
(a~ (kl)
1.9
(M (k), p~(- k~)) (p,~ (k,), ~(-- k~))(pp):~:~. (1_2)
198

4. The expressions for the derivatives (ii) and (125 serve as the basis for the statistical determination
of the characteristics of the equilibrium elastic and thermal properties of the crystal. In what follows, to
simplify the formulas we shall not specify the dependence of quantities on k, northe subscript l in the mean
values with respect to Pl"
Elastic and Thermal Properties of a Crystal
Setting M in (12) equal to the Fourier transform of e(k) , we establish an expression for the specific
heat of the deformed medium at fixed strain tensors:
ae t p -,
where k is Boltzmann's constant. If we set Pj = -p~kS, we obtain the well-known expression for the spe-
cific heat of a liquid at constant volume [19, 20].
In t13), Ca, T(k, kl) is the Fourier transform of the kernel of the integral relation between the energy
density e(r) and the local temperature T(r') at another point of the region of space occupied by the medium.
The expression (13) refers to the general case of a spatially inhomogeneous medium.
Note that the energy density includes the kinetic energy of the rotational motion in addition to the
well-known expression.
Using (12), we can also determine the temperature stress coefficients:
For this, we'must take M to be components of the microscopic tensors of the ordinary and the moment
stresses. The Fourier transforms of the latter are defined by
I ~" /e ~k'~
m 2 ~. ik 9R~ '
~, v • Mi.~Xjv. (e - l) ] eikq
rL m 2 ik. R~~
(14)
where p[ and s.~ are the components of the momentum and the angular momentum of a particle; R~P " = q~
1
-qV is the radius vector joining the centers of mass of particles t~ and v; and XiV/z are its components;
F vg' and M v~ are the force and the couple applied to the particle v by the particle tz. To obtain (14) one
must apply a Fourier transformation to the Poisson brackets in Eqs. (5) of [11].
The flux density tensor of the intrinsic angular momentum, ~rij, can be expressed in terms of the
lattice variables and the force constants in the same way as is done for the momentum flux tensor in [21].
To simplify the notation in what follows, we shall use a single-index notation for the components of
the stress tensors: ~'i (i = 1 ..... 9), ~i (i = 10 ..... 185, and also for their temperature coefficients: k.
1
and t~i-
The statistical expressions for the temperature stress coefficients have the form
'[ L ]~n = -- (~., e) -- (Pj, e) (~., P~)(PP)~~kT i
i,5~2
t9
As in the case of the specific heat, the expreesions (15) are the Fourier transforms of the kernels
of nonlocal integral relations between the stress tensors and the temperature. It is important that the tem-
perature stress coefficients include the new coefficients ~n' which are not present in theories of media
with a central interaction of the molecules.
199

5. As is done for liquids in [20], it is convenient to replace the energy density by a quantity ~ such that
its static correlation functions with the strain tensors vanish. This requirement is satisfied by
19
0 = e- Z Pj(e,P,)(PP) if'.
Then the set of variables can be ordered as follows: Pi{ett, e22, e~3, a12, ~21, e13, %1, e23, e32, Tll, 722,
733, 712, ~21, ~/13, 731, 72~, ~32, 0}.
The static correlation matrix (Pi(k), Pj(-kl) ) has a diagonal block structure. The upper block is
formed by the correlation functions of the strain tensors, and the remaining part of the matrix has the
correlation function (e(k)O(-kl)) on the principal diagonal. For a nongyrotropic medium the upper block
splits up into two diagonal blocks, since (eij(k)Tmn(-k~)} = 0. On the basis of Frobenius's formula [22] it
is easy to show that the reciprocal matrix (PP)-~.tk also has a diagonal block structure, and that its blocks
~3
are matrices that are the inverses of the corresponding blocks of the original matrix.
Taking for M the components of the microscopic stress tensors (14) and using (11), we establish ex-
pressions for the Fourier transforms of the kernels of integral nonlocal relations between the stress tensors
and the strain tensors at fixed temperature T(k). We shall call these transforms the isothermal equilibrium
elasticity moduli.
In the single-subscript notation for the stress and strain tensors, the four tensors of the elasticity
moduli are determined by
A~J~ ~ ! ~'.v ~ ('~' P~) (PP) "j-' (i, ] -- t .... 9),
m~t
i8
Bo o- O<p,> = (z.P.,)(PP)~.7' (i=t .... 9, j ~ t0,..., t8),
c"~= ~ / = ~, (~''p") (PP)"c' (~ = io ..... i8, ] = f ..... 9),
= ( "
D@ ~a--'Tc~-fflr,=E(n,,P,.)(PP),.:-'-.-,. -- (i,]---- 10 ..... 18). (16)
For a nongyrotropic medium it fellows from symmetry considerations that the elastic properties of a
medium are characterized by two elastic moduli tensors, their expressions in the usual notation having the
form
0 --1
A,~.~= (~,j~) (~) ,~,,,,,, (17')
Q --1
If we set epq = -p(k), in (17'), we obtain the well-known expression for the isothermal dilational
modulus of elasticity of a liquid (see, for example, [19, 20]).
The tensors A ~ and C~ determine the elasticity of a medium with respect to a deformation due to in-
homogeneity of the displacement field. The tensors B ~ and Do characterize the elasticity of the medium
with respect to the orientational deformation. The statistical expressions for them can be used to describe
the orientational elasticity of liquid crystals. If the medium has an inversion center, the orientational
elasticity is determined by the single tensor D ~
It is important that not only the specific heat and the temperature stress coefficients but also the
moduli of elasticity can be represented as expressions containing the static correlation functions of the
fluctuations of the dynamical quantities, for which molecular expressions are known. Another important
feature of the theory is the fact that the thermoelastic characteristics of the medium are determined with
allowance for spatial dispersion for a spatially inhomogeneous medium.
Let us now consider the possibility of concrete calculation of the elastic and thermal properties of a
medium on the basis of Eqs. (13), (15), and (17). One of the possibilities is associated with an expansion
of the dynamical quantities of the stress tensors and the energy density in a series in the strain tensors.
200

6. Then the determination of the correlation functions of these quantities with the strain tensors reduces to
calculating the correlation functions of the strain tensors. And these - the correlation functions of the
strain tensors - can be expressed in terms of the correlation functions of the displacements and the rota-
tion angles of the particles. To calculate the latter, Green's functions can be used directly.
The calculation of mean values with respect to the local-equilibrium ensemble presents considerable
difficulties. Considering states that differ only slightly from a state of complete equilibrium, we average
with respect to the equilibrium ensemble. Then for the equilibrium moduli of elasticity we obtain simple
expressions if we restrict ourselves to the linear terms in the expansion of the stress tensors (14) in a
series in the strain tensors. The product of the correlation matrix of the strain tensors and its reciprocal
gives the identity matrix, and the moduli of elasticity are equal to the coefficients of the strain tensors in
the given expansions. In [11], the author determined these coefficients, ignoring spatial dispersion, in a
calculation of the high-frequency moduli of elasticity. (In the notation of [11], these are the coefficients of
the strain tensors in the expressions for ATik/V and AIIik/V.)
Using the expressions for these coefficients, we find in the linear approximation without allowance
for spatial dispersion
1 N 'vp, v~
Ai,j,~ = OX]~ X.,
i ~ OM~"v
B~ ox7x?x:i~in
v~ ";B
C o __ 1 0 M~__Xk v,,
ikj,~ 2 V ~,~ OX~~ X~,
D o f s OM~~ ~ ~.
-- X~ X,~ , (18)
where V is the volume of the system. The formulas contain the mean equilibrium values of the spatial
coordinates and the angular variables of the particles. Since the mean values of the momenta and the
angular momenta vanish, kinetic parts are not present in the expressions for the moduli of elasticity.
The expression for the tensor A~ in the case of a central interaction of the particles is identical with the
well-known expression of the dynamic theory of crystal lattices [1].
It is interesting that the equilibrium moduli of elasticity in this approximation are identical to within
the kinetic parts to the high-frequency moduli of elasticity of a medium without the initial stresses [11, 12].
From this we conclude that the difference between the equilibrium and the high-frequency moduli of elasti-
city of a solid is due not only to their kinetic parts but, principally, to the nonlinear terms in the expansion
of the stress tensors in a series in the strain tensors.
In the same way one can establish a formula for the moduli of elasticity in the linear approximation
with allowance for spatial dispersion by expanding the expressions (14) in a series in the strain tensors.
The second possibility is associated with the fact that the static correlation functions of the stress
tensors with the strain tensors can be readily calculated in general form by means of the stationarity con-
dition
</)(k)A (--k,) > =--<B(k) A(--ki) > (t)=dB/dt).
Choosing B to be successively the density of the momentum and the intrinsic angular momenta and A to be
the strain tensors, and using the conservation laws for these densities, we reduce the calculation of these
static correlation functions to the calculation of the correlation hmctions of the momentum and angular
momentum densities. Therefore, the determination of the moduli of elasticity reduces to ealculating the
static correlations of the strain tensors.
Note that the expressions obtained for the speeific heat, the temperature stress coefficients, and the
isothermal moduli of elasticity enables us to calculate the adiabatic moduli of elastieity and the coefficients
of thermal expansion. For this we must use the thermodynamic relations [13], which are written down be-
low for the Fourier transforms of the corresponding quantities:
201

7. (19)
A ~ Co
o D o
(20)
where A, B, C, and D are the adiabatic moduli of elasticity; cq4m and fikm are the coefficients of thermal
expansion. For an asymmetric medium,
a~j= OT !,: ~i~=
Note that in (19) and (20) there is also summation over the wave vector.
The set of coefficients of thermal expansions includes the new coefficients fii'" Since flij = (0s/~ij)T,~-
(s is the entropy density) [13], the coefficients of thermal expansion determine a n~w piezocaloric effect
due to the presence of moment stresses.
Equations (19) and (20) generalize the well-known relations [23] for an elastic body with symmetric
stress tensor. The generalization is associated with not only the appearance of new characteristics of the
media but also with the allowance for nonlocality. For a nongyrotropic medium the relations (20) simplify,
since the tensors B ~ and Co vanish.
I am very grateful to D. N. Zubarevand L. A. Rott and also to the participants of the seminars they
conduct for helpful discussions.
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202