Specific heat

For a crystal containing N atoms there will be 3N independent normal modes of oscillation each of frequency ? _i. Einstein considered the case of 3N independent harmonic oscillators of the same frequency, the average energy of which is

Using the definition of the specific heat C _V = ( ?E/ ?T) _V, then

is obtained, where x = h ? /kT. For large values of T, C _V = 3Nk = 3R, in agreement with experiment. This latter result holds approximately for solid compounds also, with the proviso that the dimensions are energy/atom or energy/g atom. However, the variation of C _V with T, especially at low temperatures, does not agree with experiment.

Debye suggested that the solid be treated as a continuous isotropic medium, which leads to the existence of a range of frequencies from zero to some maximum ? _m, to a density of normal modes that varies as the square of the frequency up to some maximum frequency such that the total number of normal modes equals 3N. He then found that

from which the specific heat is found to be

This relation reduces to 3Nk at high temperature and at low temperature to

both of which are in agreement with experiment. Since the Debye temperature ? = h ? _m/k, it is possible to use the experimental data for the specific heat at low temperature to evaluate a value for