Finite linewidths and "forbidden" three-phonon interactions

We have used thermodynamic perturbation theory to study the effect of the finite energy spread of phonon excitations upon the possibility of three-phonon interactions which could not otherwise conserve energy. It is shown that a proper treatment of this effect is equivalent to a consideration of higher order processes in which the interacting phonons occur in virtual intermediate states, but that such processes cannot in general be adequately taken into account simply by assigning a lifetime to one or more of the interacting phonons. It is also shown that the higher order terms of conventional perturbation theory should be modified, whether or not the three-phonon process is energetically allowed; if it is, these modified terms should be substituted for the three-phonon contribution rather than added to it. We give a qualitative discussion of applications to liquid helium II and to dielectric solids.