Study of the possible effects of defects in a soft lattice

To understand a phonon kink in the (110) LA phonon dispersion near $q\sim \mathrm{}(0.09,0.09,0)\mathrm{}{q}_{\mathrm{ZB}}$ in ${\mathrm{Nb}}_3$Sn discovered recently by Axe and Shirane, a computer calculation has been done on a model of vacancies in a soft lattice. The model consists of a two-dimensional square lattice with a vacancy concentration of 1% arranged periodically. The interatomic forces are arranged so that part of the phonon modes of the perfect lattice are unstable. The atoms around the vacancies are allowed to relax and the phonon frequencies of the resulting system are calculated. We found that (i) the distortion around the vacancy can break the symmetry of the crystal, and (ii) the small-$q$ portion of the phonon spectrum consists of two flat portions with different slopes joined smoothly to each other.