Localized Mode in an Anharmonic Crystal

Abstract

We consider a harmonic crystal with a single isotopic impurity whose coupling to its neighbors includes a cubic anharmonic term. The impurity is chosen to be light enough that a localized mode exists. The Hamiltonian for this system is truncated, in a physically reasonable way, so as to yield an exactly solvable Schrödinger equation. The solutions give a new insight into the nature of the localized mode, which in this approximation is exactly analogous to an unstable particle in the Lee model of field theory. The wave functions are exhibited and the lifetime is calculated for a light impurity in a simple cubic lattice at $T=0\mathrm{}$.