The Debye–Waller factor for polyatomic solids. Relationships between X-ray and specific-heat Debye temperatures. The Debye–Einstein model

The Debye-Waller factor for a polyatomic crystal is derived in the Debye approximation. If the crystal has a basis of p atoms per lattice point, it is shown that the specific-heat Debye temperature, ΘD, and the X-ray Debye temperature, ΘM, are related by ΘD ≃ ΘMP1/2 in the classical limit. The Debye and Einstein theories are then combined to yield an expression for the Debye-Waller factor of a polyatomic solid. The acoustic phonon modes are described with a Debye approximation, and the optic modes with an Einstein model. For temperatures above the Debye temperature, the Debye and Einstein parts of the Debye-Waller factor have the same dependence on temperature and diffraction vector. Thus, the two parts cannot be distinguished.