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THE THIRD LAW OF THERMODYNAMICS AND THE LOW TEMPERATURE PROPERTIES OF
DISORDERED SOLIDS
R. Dandoloff, R. Zeyher
To cite this version:
R. Dandoloff, R. Zeyher. THE THIRD LAW OF THERMODYNAMICS AND THE LOW TEMPER-
ATURE PROPERTIES OF DISORDERED SOLIDS. Journal de Physique Colloques, 1981, 42 (C4),
pp.C4-205-C4-208. �10.1051/jphyscol:1981442�. �jpa-00220899�
JOURNAL DE PHYSIQUE
CoZZoque C4, suppZ6ment au nO1O, Tome 42, octobre 1981 page C4-205
THE THIRD LAW OF THERMODYNAMICS AND THE LOW TEMPERATURE PROPERTIES OF DISORDERED SOLIDS
R. Dandoloff and R. Zeyher
Max-PZanck-Institut fiir Festkorperforsc~ung, 7 Stuttgart 80, F. B. G.
Abstract.- It is shown that if the entropy can be derived from a restricted or unrestricted partition function both the second and the third law of ther- modynamics hold. The consequences of this result for the low temperature be- havior of solids are discussed. Moreover it is proposed that
379
leading term of the entropy of disordered solids at low temperatures is 'LT due to a se- cond diffusing mode caused by the conservation of mass.Introduction.- The third law of thermodynamics (TLT) states that the entropy of a system vanishes if the temperature tends to zero. Recently it has been shown by P.V. Giaquinta et al. (1) that a violation of the TLT causes the specific heat to be linear at low temperatures. The argument uses the following thermodynamic equations:
K and K represent the adiabatic and isothermal compressibility, respectively,
S T
and C and C the specific heat at constant pressure and volume, respectively. We see fyom ~ q . ~ ( l ) that if &m Efo then C -C ". T;
5
remains finite in this limit.P
v
From Eqs. (1)-(3) we deduce that C and Cv are separatly proportional to T at low temperatures. Eq. ( 2 ) shows that
&g
Eifo is equivalent with the violation of the TLT.
It has also been pointed out that the two-level-model of Anderson, Halperin and Varma (2) and Phillips (3) yields a linear specific heat only if the TLT is violated. The thermal expansion 6 is given in the model of two-level-systems by the following expression ( 4 ) :
with A2 =
: A +
A2;A
is the asymmetry of the double well potential, A. characteri- zes the height of the potential barrier. Taking an average over all possible. A
and
A
one gets for B the following expression:B = - -
noDkgln2. ( 5 )v
n is the density of states and D is a deformation potential coupling the phonons a8d the two-level-systems. According to Eq. (5)
B
is independent of temperature.Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981442
C4-206 JOURNAL DE PHYSIQUE
Consequently does not tend to zero for T-to if the deformation potential D is nonzero. There is no experimental evidence that D vanishes at low energies ( 5 ) . Thus the TLT is violated in this model.
In order to clarify the applicability of the TLT to solids we have investigated the statistical foundation of the TLT. The "usual" statistical argument in favor of the validity of the TLT (6) is based on the properties of a finite solid consisting of N particles localized in a volume V. The spectrum of the eigenenergies is dis- crete. If the degeneracy of the ground state increases less than exponentially for large N's we have
(N)
lim lim (VrT) = o.
N (6)
N-Mo W o
The above argumentation was seriously questioned by H.B.G. Casimir (7), M.J. Klein (8) and R.B. Griffiths (9). They pointed that the temperature must be lowered to an unattainably small (of the order of 10 K for a macroscopic s+stem) value in order to be sure that the system is effectively in its ground state. The order of the limits in (6)ljyp1ies that the relative fluctuations in the energy density are -1 instead of W-
.
For an infinitely large system the contribution of the ground state to the partition function is neglegible at any nonzero temperature. Thus So, the limiting entropy density at T=o may depend crucially also on the distribution of low-energy excitations. As a result the statement (6) is thermodynamically irre- levant and certainly does not describe the experimental observations related to the TLT.We have investiqated the correct mathematical formulation of the TLT, namely (N)
lim lim = constant.
N 17)
P o N-
The simple argument leading to Eq.(6) cannot be applied to Eq.(7) because there will be in general a continuum of states with energies smaller than kBT for every tem- perature T>o
.
The Hamiltonian H of our system of M atoms forming a solid with a volume V is:
+ +
u(i) are the displacements from the rest positions ;(O) (i), p(i) the corresponding momenta, and m(i) is the mass of the i-th atom. We are dealing with a solid and therefore we assume that there is at least one set of rest positions for the atoms for which the free en r y has a minimum. This can clearly be assumed in the crystal- line case where the ~'~'(i) form a periodic network. Glasses on the other hand are not in their thermodynamically most stable state. One method to describe the glassy state in terms of thermodynamic variables in the usual way is obtained if one assu- mes that all atomic configurations can be divided into accessible and non-accessible canfigurations (for instance periodic arrangements of the atoms). The corresponding restrict d free energy may have an absolute minimum for a random network of posi- tions 2'0) (i)
.
In the u(i) representation the reduction of phase space can be accomplished by adding a suitable potential W' to the original potential W. Finally one has to perform an average over an ensemble because there are many possibilities of reductions of phase space which correspond to equivalent macroscopic descriptions of the glassy state.The free energy per particle, f(v,T), of a solid can be written as (9):
v is the specific volume v=v/N,@ is the sum of all skeleton graphs, G is the dis- placement-displacement Green's function and 1 is the self-energy defined by 6@/6~.
Tr is a trace over the discrete frequencies w =2TinT (n=o,fl,
...
) and over all dis- placements. c > denotes an average over an ensemble of equivalent random networks. nIn the case of an ordered solid < > is to be omitted.
We assume that the thermodynamic limit of the spectral function of G ( ~ ) exists.
The t h ~ ~ ~ o d y n a m i c limit of the right hand side of Eq.(9) is then obtained by repla- cing G by its thermodynamic limit G. We extend also all internal sums over the infinite solid. Using then the fact that can be represented as a trace the right hand side of Eq.(9) becomes independent of N and f is an intensive variable.
We determine only the leading term of f at low temperatures. There is an expli- cit temperature dependence dae to the sum over the discrete frequencies and an im- plicit dependence due to the temperature dependence of the spectral function. Carry-
ing out a low temperature expansion of the right hand side of Eq.(9) one finds that the leading term comes only from the explicit temperature dependence. The leading term in the entropy per particle, s= -(Jf/a~) is given by the following formula:
m
where
- 1
h (&),v) = - lim < Im G (w+iq) Re G (w+in)
+
Im ln (-G-' (&)+in) ) > (1 1)n
SoThe dummy variable x in the integral in Eq.(Io) originates from w/T (h = kg =I). It is important to note that the temperature appears in Eq.(lo) only in the product xT. This is due to the fact that the leading term of s is independent of the tem- perature dependence of the spectral function.
For a finite N the spectral function of the infinite solid can be diagonalized and each term in Eq.(11) becomes diagonal. In the diagonal representation we have for each eigenvalue:
Using Im G $ o it is easy to show that each eigenvalue is nonnegative and therefore h altogether is nonnegative. This remains valid also in the limit NI.m
.
As a con-sequence of
A
)o, the specific heat CV =$s/~T)~
is also nonnegative which means that the system is thermodynamically stable. Using Eq. (12) one can show that h has the upper bound 3(1+
T). A possible divergence of the integral in Eq. (10).can only be due to the behavior of the integrand at small x's. Furthermore only the small &)-dependence of i\ is relevant for the limit TSO. If I\(w) exhibits a jump at GO or if A(w) approaches zero for w$o slower than any power of w the in- tegrand in Eq. (lo) diverges and s is identical +m. However this case cannot occur in real nature because there exist rigorous upper bounds for s (10). In the other case we can interchange the T+o limit and the integral sign in Eq. (lo) and obtain
-
lim s(T,v) = dx xeX
lim A(,,~)] = 0.
T+o
j
r n [ T L 0This means that the TLT holds and that the constant on the right hand side of Eq. ( 7 ) has to be zero.
The basic input in our derivation of the TLT is the quantum mechanical partition function taken in the thermodynamic limit and reformulated in terms of a spectral function. We showed that the limit TSo is given by the limit &)So of the quantity A(&)). Thus we find that the properties of excited states are important and the de- generacy of the ground state is irrelevant in agreement with Ref. (8). We also did not have to impose conditions for the density of excited states as in Ref. (6). As a conclusion of our investigation of the statistical foundation of the TLT we can say that if thermodynamics of a solid can be derived from a quantum mechanical par- tition function, both the second and the third law of thermodynamics are fulfilled.
We would like now to propose a new theory for the low temperature specific heat of disordered solids.which does not contradict the TLT. Our theory is based on the existence of an additional hydrodynamic mode in strongly disordered solids which
JOURNAL DE PHYSIQUE
usually is neglected and does not occur in harmonic or weakly anharmonic solids.
This additional mode is associated with the conservation of mass (11).
Assuming for simplicity only one kind of atoms there are 8 hydrodynamic variab- les associated with the Hamiltonian in Eq. ( 8 ) : The momentum density, the energy density, and the mass density. In addition we have three broken symmetries aJong the three space directions which are restored by the displacement variables u. If each atom is bound to a definite potentiaj. minimum and if there are no unoccupied minima of comparable energy the fluctuations in the mass density are due to the displacements and the mass density and the displacements are not linear independent.
As a result there are only 7 hydrodynamic modes (three propagating sound waves ( which have to be counted twice) and one entropy diffusion mode). However in the general case there is one additional diffusion mode. In the case of an ordered solid this mode describes vacancy diffusion (11) and in the case of disordered solids it describes configurational rearrangements of the atoms (12).
In calculating G at low temperatures the entropy fluctzations can be omitted if the TLT holds. After a Fourier transform G is diagonal in k and the longitudinal part has the following form:
al and a2 are static susceptibilities which can be written as derivatives of the free energy (121, D is the transport coefficient associated with the extra
3 B d e - Inserting G from Eq. (14) into Eq. (11) the leading term in s becomes % T
.
Because of CV ?/T~s/~T) we find that the leading term in CV should also be propor- tional to T
.
In conclusion we have shown that for general reasons a contri- bution % T3l2 is expected in disordered solids if the atoms can make "large" dis- placements.References.
(1) GIAQUINTA P.V., MARCH N.H., PARINELLO M, and TOSI M.P., Phys. Rev. Lett.
39
(1977) 41
(2) ANDERSON P.W., HALPERIN B.I. and VARMA C.M., Philos. Mag.
25
(19721 1(3) PHILLIPS W.A., J. Low Temp. Phys.
1
(1972) 351 (4) PHILLIPS W.A., J. Low Temp. Phys.21
(19731 757(5) LASJAUNIAS J.C., RAVEX A., VANDORPE M., and HUNKLINGER S., Sol. State COUUU.
17 (1975) 1045
(6)
-
MUNSTER A., " Statistical ~hermodynamics", Vol. 11, Springer-Verlag, Berlin, 1969(7) CASIMIR H.B.G., Zeitschr. f. Physik
171
(1963) 246(8) KLEIN M.J., in Rendiconti S.I.F. Enrico Fermi, Corso X, Bologna, 1960, p.1 (9) GRIFFITHS R.B., in " A Critical Review of Thermodynamics", ed. by E.B. Stuart,
B. ~ a l / ~ r , A.J. Brainard, Mono Book Corp., Baltimore, 1970, p. 101
(lo) THIRRING W., "Quantenmechanik QroOer Systeme", Springer-Verlag Wien-New York, 1980
(11) MARTIN P.C., PARODI O., and PERSHAN P.S., Phys. Rev. (1972) 2401 (12) COHEN C., FLEMING P., and GIBBS J.H., Phys. Rev. (1976) 866
