Problems and Solutions on Thermodynamics and Statistical Mechanics

184 Problems d Solutions on Thermodynamics d Statisticd Mechanics

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2024
Consider a glass in which some fraction of its constituent atoms may
occupy either of two slightly different positions giving rise to two energy
levels A; > 0 and -A, for the ith atom.
(a) If each participating atom has the same levels A and -A, calculate
the contribution of these atoms to the heat capacity. (Ignore the usual
Debye specific heat which will also be present in a real solid.)
(b) If the glass has a random composition of such atoms so that all
values of A, are equally likely up to some limiting value A0 > 0, find the
behavior of the low temperature heat capacity, i.e., kT << Ao. (Definite
integrals need not be evaluated provided they do not depend on any of the
parameters.)
(Princeton)
Solution:
(a) The mean energy per atom is Z = A tanh (&). Its contribution
to the specific heat is

d?^2 1

c, = - dT = 4k (&) (eA/kT + e-A/kT)2

Summing up the terms for all such atoms, we have

(^2 1)
cv = 4Nk (&). (eA/kT + e-A/kT)2 '
(b) The contribution to the specific heat of the ith atom is
1
2
ci = 4k (2) (eA,/kT + e-A,/kT)2 *
When kT << A,, we have