Polarisabilities of ions in rutile (TiO₂) calculated with Madelung potentials determined by Ewald's method

Abstract

We have been interested in why only materials with a high dielectric constant such as Ti02, W03, and H-Nb 0 are able t o make u s e of crystallographic s h e a r (CS) report /4/, t o clarify this question, we calculated, by using Madelung's method (i. e., direct summation), the Madelung potential at each site in rutile (Ti02) which was treated as an ionic crystal. Using the Madelung potential at 4+ the titanium site, we then determined the electronic polarisability of the T i ion following Ruffa' s theory /5/. Moreover, we calculated the ionic polarisa- 2-bility of the Tih ion and the electronic and ionic polarisabilities of the 0 ion using the electronic polarisability of the Ti4+ ion and the static and optical dielectric constants of stoichiometric rutile determined experimentally. The electronic polarisability of the 02-ion was found to be large in value compared with that of other 3d transition metal oxides in which CS planes are not observed when they are reduced. We suggested in our previous paper /4/ that the high dielectric constant is due to the high electronic polarisability of the 02-ion and the formation of CS planes in non-stoichiometric rutile is possibly connected with the high electronic polarisability of the oxygen. Ewald' s method. However, Ewald' s method is generally employed in calculating Madelung potentials of ions in ionic crystals for the following two reasons. As Redlack and Grindlay suggested /6/, Ewald' s method is applicable to any Bravais lattice and the Madelung potentials obtained by this approach are independent 1) Tokiwadai, Hodogaya-ku, Yokohama 240, Japan.2 ) Bradford BD 7 1 DP, West Yorkshire, Great Britain.

Energy calculations for planar faults in reduced rutile (TiO 2 )

“…The large relaxation energy due to polarization in the formation of c.s. planes m ust be attributed to the large polarizabilities, especially the electronic polariz ability of the oxygen ion, as we have suggested in the past (Iguchi & Tilley 1977 a;Iguchi et al 1980Iguchi et al , 1981Iguchi et al , 1983. This leads to the contention th a t only mater with not only a high static dielectric constant, b u t also large refractive indices, such as Ti0 2, W 0 3 and H -N b 20 5 are able to make use of c.s.…”

“…planes and polarizabilities in these materials was to be expected (Iguchi & Tilley 1977 a;Tilley 1977). F urther studies have suggested th a t an im portant feature in this respect is th a t crystals must show a large electronic polarizability for the oxygen ions (Iguchi et al 1980(Iguchi et al , 1981(Iguchi et al , 1983(Iguchi et al , 1984. Relaxation due to this polarizati is likely to have an im portant role to play in the formation and stabilization of c.s.…”

“…The displacement of the ion results in a dipole moment on th a t ion. In such a case the energy of the crystal is given by the summation of Eu , E iep, E w and the relax the last quantity we have treated rutile as an ionic crystal in the same way as in the past (Iguchi et al 1980(Iguchi et al , 1981(Iguchi et al ,1983Sawatari et al 1982). The relaxation energy is then calculated by using the polarizable point ion shell model (Hatcher & Dienes 1961, 1964Wilson et al 1970;Dienes et al 1975;Harding 1981).…”