Thermodynamics, Statistical Physics, and Quantum Mechanics

THERMODYNAMICS AND STATISTICAL PHYSICS 231

b) If we assume that the N atoms of the solid each have three degrees of

freedom and thesamefrequency then the totalenergy

The specific heat

c)In the high-temperaturelimit of(S.4.101.4) wehave

In regular units

which corresponds to the law of Dulong and Petit, does not depend on

the composition of the material but only on the total number of atoms

and should be a good approximation at high temperatures, especially for

one-componentelements. Prom the numbers in the problem,

Therefore,

Note that, at high enough temperatures, anharmonic effects calculated in
Problem 4.97 may become noticeable. Anharmoniccorrections are usually
negative and linearly proportional to temperature.

At low temperatures (S.4.101.4) becomes

The heat capacity goes to zero as exp whereas the
experimental results give (see Problem4.42). The faster falloff of
the heat capacity is due to the “freezing out” of the oscillations at
given the single natural frequency

d)