General Form

jueves, 20 de mayo de 2010

General Form

General Form of the Density of States

The diagram below shows a portion of two adjacent surfaces of constant frequency corresponding to frequencies w and w+dw. Owing to dispersion these surfaces are not normal to the direction of the wavevector k. Consider a small pill box bounded by the surfaces w and w+dw centred around the point k. The unit vector normal to the frequency surfaces is n. The pill box has area dS n.
The number of allowed values of k for which the phonon frequency is between w and w+dw is

= Volume of k space/volume occupied by one state (mode)

The integral is extended over the volume of the shell in k space bounded by the two

Surfaces of equal w in k space.

surfaces on which the phonon frequency is constant, one surface on which the frequency is w and the other on which the frequency is w+dw. This is straight forward where k is small, i.e. no dispersion, since the constant frequency surfaces will be spherical. However, for the general situation where k may be large, one has to deal with a much more complicated shape. The problem for us is to evaluate the volume between these surfaces. Owing to dispersion these surfaces are not normal to the direction of the wave vector k.