Bounds on Localized Modes in the Crystal Impurity Problem

Using general properties of the crystal site representation normal mode matrix, we provide some simple bounds on localized modes in crystals with point defects. I. DERIVATION OF THE TRACE CONDITION The substitutional insertion of a point defect impurity into an otherwise perfect crystal will typically (see e.g. [1]) modify the spectrum of the 3N normal modes of the crystal, leading in certain circumstances (such as the insertion of impurities which are lighter in mass than the host atoms which they replace or which are more strongly coupled to the host crystal atoms than the ones they replace) to the generation of modes with frequencies which lie beyond the band maximum ωmax of the crystal. Such modes will not be plane waves which propagate throughout the crystal, but will instead fall off exponentially fast away from the defect site and thus be localized to it. Moreover, with the rest of the crystal atoms not participating appreciably in such localized modes, the intensity of the defect in such modes will be N times larger than the intensity it would otherwise have had in a crystal plane wave mode, to thus give the localized mode enough intensity to render it observable. While such modes could be of relevance for phenomena such as the Mössbauer effect associated with the insertion of Mössbauer active defects into host crystals, historical recoil-free fraction Mössbauer studies only involved an averaging over all the lattice modes of the system, to thereby only allow one to infer the possible presence of localized modes indirectly. However, with the advent of dedicated synchrotron rings it became possible to monitor Mössbauer active systems mode by mode directly; and via nuclear resonant inelastic x-ray scattering studies, modes lying beyond the host crystal phonon band maximum have now explicitly been seen in the Fe/Cu [2] and Fe/NiAl [3] systems. Consequently, knowing whether the Email address: [email protected] Email address: [email protected] 1 insertion of defects into crystal hosts might generate localized modes can be of great value for such studies. In this paper we use a very straightforward trace technique to enable us to identify some very general conditions under which such localized modes can be produced, and since the very introduction of a defect breaks the translation invariance of the lattice, we will work entirely in the coordinate space crystal site representation. To derive the trace condition we recall that in the harmonic approximation the equations of motion for the displacements from equilibrium euα(l) of the atoms of a pure 3Ndimensional crystal lattice are given by