A system of vibrations in a crystalline solid lattice can be modeled as an Einstein solid, i.e. by considering N quantum harmonic oscillator potentials along each degree of freedom. Then, the free energy of the system can be written as
where the index &alpha; sums over all the degrees of freedom. In the 1907 Einstein model (as opposed to the later Debye model) we consider only the high-energy limit:
Then
and we have
Define geometric mean frequency by
where g measures the total number of spatial degrees of freedom of the system.
Thus we have
Using energy
we have
This gives heat capacity at constant volume
which is independent of the temperature.
For another more precise derivation, see Debye model.