Response Function for the Anharmonic Crystal

Up to now we have considered the response function for a system of damped harmonic oscillators. If we allow for anharmonic interactions between phonons, the response function will be modified. The theory of crystal dynamics in which anharmonic effects are weak gives the following expression for the response function [5.47] 

In this expression, w.(q) is the harmonic frequency of the mode ( ) and 2 j(q,(ja) is known as the (complex) self-enevgy of this mode and is given 

Since we assume that Aj(q,o)) and rj(q,o)) are small compared with o)j(q), the expression (5.139) can also be written in the form 

From this expression it is obvious that due to anharmonic interactions, each harmonic frequency o)j(q) is replaced by a complex frequency 

It therefore follows that the normal coordinate of the harmonic mode, 

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