Ideal harmonic crystal

We may usefully think of a crystalline solid as a single huge molecule in which the motions of the atoms are decomposable into nearly independent normal modes of vibration. Weak interactions between the modes allow energy to flow between them, leading to thermal equilibrium among them, just as the weak interactions and rare collisions between molecules in an otherwise ideal gas are necessary in order to establish the equilibrium properties of the gas including the Maxwell velocity distribution. Just as there is no further reference to those interactions in the equilibrium properties of the ideal gas, so will there be no further reference to the weak, anharmonic interactions between the otherwise independent harmonic oscillators that constitute this ideal crystal.

If the crystal consists of N atoms there will be 3N – 6 such normal modes of vibration, but since N is very large we may ignore the 6, which is associated with the translation and rotation of the crystal as a whole, and say that there will be 3N vibrational modes. Precisely because N is large, it is pointless to try to enumerate the modes and find the frequency νi (i = 1, …, 3N) of each one. Instead, we recognize that there is virtually a continuum of frequencies ν, and we characterize the crystal by a frequency-distribution function g(ν) such that g(ν)dν is the fraction of all the 3N modes that have frequencies in the infinitesimal frequency interval ν to ν + dν.