The frequency shift and lifetime of a localized vibration mode

Abstract

A localized vibration mode due to a light impurity atom in a crystal is an exact eigenstate of the crystal when the latter is studied in the harmonic approximation. When the anharmonic nature of the interatomic forces is taken into account, it is found that the frequency of the localized mode suffers a complex shift from its value in the harmonic approximation. The real part of this shift describes an actual change in the frequency of this mode, while the imaginary part is conventionally related to the reciprocal of the lifetime of this mode. In this paper expressions are obtained for the real and imaginary parts of the complex shift of the frequency of a localized vibration mode associated with an isotopic impurity in an arbitrary cubic Bravais host crystal. Cubic and quartic anharmonic terms in the expansion of the crystal potential energy in powers of the atomic displacements are retained in the present calculation. Our results are exact to the lowest nonvanishing order in the anharmonic force constants. To display some of the qualitative features of the formal results, the expressions obtained are evaluated in an approximate fashion in the high temperature limit for the special case that the host crystal is a monatomic linear chain.