Bounds on Localized Modes in the Crystal Impurity Problem

TL;DR

This paper establishes simple bounds on localized vibrational modes in cubic crystals with point defects, providing conditions based on mass and force-constant changes for the emergence of localized modes.

Contribution

It introduces a general trace condition constraint applicable to various cubic crystal structures, offering new criteria for localized mode formation due to defects.

Findings

Localized modes occur if defect mass is less than half of the host atom mass.

Localized modes can arise if the defect force-constant exceeds twice the host atom's self force-constant.

The derived bounds are valid for any combination of force-constants and neighbor interactions.

Abstract

Using general properties of the crystal site representation normal mode matrix, we provide some very simple bounds on localized modes in simple, body-centered and face-centered cubic crystals with substitutional point defects. We derive a trace condition constraint on the net change in crystal eigenfrequencies caused by the introduction of a defect, with the condition being a completely general one which holds for any combination of central and non-central crystal force-constants and for all-neighbor interactions. Using this condition we show that the sufficient condition for producing localized modes in an arbitrary cubic crystal by a mass change at the defect site is that the defect mass be less than one half of that of the host atom mass which it replaces, and that the sufficient condition for producing localized modes in an arbitrary cubic crystal by force-constant changes alone is that the defect site self force-constant be greater than twice that of the pure crystal self force-constant of the host atom which it replaces.