Frequency Widths and Shifts from Perturbation Theory

3 Frequency Widths and Shifts from Perturbation Theory 

It remains to calculate rj(q,u) and Aj(q,w) which appear in the response function (5.141). We start from the Hamiltonian H = Hq + H, where Hq is the harmonic part given by (3.71) and H is the anharmonic part taken to be 

H3 and describe the cubic and quartic anharmonic interactions higher-order terras are neglected in the Taylor expansion of the potential energy. H is given by 

In this list, term 1, for example, is an operator which acts on the complete phonon state. to give a reduction of unity in the occupation number for the mode (q,j) and an increase of unity in each of the phonons (-q J ) and (-q2J2). In other words, one phonon splits into two. The complete classification is given in Table 5.1. 

From Table 5.1, it follows that there are four basic processes (i) absorption of a phonon with the creation of two other phonons, (ii) absorption of two phonons with the creation of another, (iii) simultaneous disappearance 

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