Electronic polarizabilities of ions and birefringence in rutile (TiO2)

Abstract

The ele$\stackrel{\mathrm{`}}{\mathrm{c}}$tronic polarizabilities of ions in the Ti${\mathrm{O}}_2$ crystal have been calculated by taking into account the effect of the Madelung potential as originally pointed out by Ruffa. The calculation is based on the free-ion polarizabilities reported by Pauling. The estimated electronic polarizabilities are shown to be consistent with the polarizability-radius-cube relation according to which the electronic polarizability of an ion should vary as the cube of its radius. The ionic radius of the ${\mathrm{Ti}}^{4+}$ ion thus obtained agrees rather closely with that deduced from x-ray maps of the electron-density distribution. Our calculations suggest an explanation of the observed birefringence in rutile by assuming the anisotropy of the oxygen-ion polarizability to arise from a spheroidal shape of the ion with semiaxes which lead to the observed $\frac{c}{a}$ ratio of the rutile crystal.