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 n o atoms per unit length;
(b) For a two-dimensional crystal, i.e. a plane square grid consisting of identical atoms, containing n o atoms per unit area;
(c) For a simple cubic lattice consisting of identical atoms, containing no atoms per unit volume.