text: 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.