Self-energy of phonons in an anharmonic crystal to O(λ<SUP>4</SUP>)

General expressions for all the contributions of O(λ4) to the self-energy of phonons in an anharmonic crystal are obtained. Results obtained are valid for all temperatures. Expressions are deduced for the self-energy in the high-temperature limit which gives the width of the phonons. Relevance of these results in the study of temperature dependence of the width of the fundamental lattice vibration absorption peak in the ionic crystals is pointed out