Question

Assuming the propagation velocities of longitudinal and transverse vibrations to be the same and equal to v, find the Debye temperature

(a) for a unidimensional crystal, i.e. a chain of identical atoms, incorporating no atoms per unit length;

(b) for a two-dimensional crystal, i.e. a plane square grid consisting of identical atoms, containing no atoms per unit area;

(c) for a simple cubic lattice consisting of identical atoms, containing no atoms per unit volume.

Answer 1

To determine the Debye temperature we cut the high-frequency modes in such a way as to get the total number of modes correctly.*

(a) In a linear crystal with n0l atoms, the number of modes of transverse vibrations in any given plane cannot exceed n0l. Then

(b) In a square lattice, the number of modes of transverse oscillations cannot exceed n0S. Thus

(c) In a cubic crystal, the maximum number of transverse waves must be 2n0V (two for each atom). Thus