Multi-Phonon Absorption and Anharmonicity

The absorption coefficient for a multi-phonon combination can be expressed as the product of three terms. The first one is the matrix element of the coupling term between the phonons involved in the process. It is non-zero only for specific phonon combinations determined by selection rules derived from symmetry considerations. The second one describes the temperature 

The temperature-dependent term represents the difference in the occupation numbers of the phonon states involved in the process. As phonons are bosons, the occupation number for a phonon of pulsation u at temperature T is given by the Bose-Einstein statistics as 

For two-phonon processes involving branches t and t and phonons with wave vectors q and q, the temperature-dependent term is  

Similar relationships can be obtained for three-phonon processes. At low temperature, n (iv, T) is much smaller than unity and the above expression tends to unity for summation processes, and to zero for the difference processes, which are, therefore, not observed at low temperature. At higher temperature, the absorption intensity increases for both processes [45], at a difference with the one-phonon process, which is temperature-independent. 

---