A classical electrostatic model for tetragonal single crystals of barium titanate

Experimental observations of the ionic coordinates in single crystals of tetragonal barium titanate, together with theoretical estimates of the ionic charges, are taken as the starting point of a classical analysis of an electrostatic model of the crystal in which each ion is represented by a point charge carrying a point dipole; this dipole represents that arising from the electronic polarizability of the ion of the crystal. The positions occupied by the ions are such that their charges cause ionic polarization of the unit cells of the crystal. The charges also cause an electric field to exist at each ion; its calculation is based upon the Lorentz formula for internal field, but with a crucial difference in the manner of its application from the manner in which it previously has been applied. The ion exhibits electronic polarization caused not only by the field acting on it due to ionic charges, but also by that due to the electronic dipoles created at all other ions; the electronic polarization process is consequently highly interactive, These considerations lead to the derivation of an equation which must be satisfied if the spontaneous polarization is to be predicted; a similar procedure leads to another independent equation for the prediction of the refractive index. The electronic polarizabilities of the ions are constituents of each of these equations, and the insertion into them of literature values for the electronic polarizabilities of the barium and titanium ions permits the evaluation of the electronic polarizabilities of the oxygen ions in their two different crystallographic positions. The fields acting at, and the electrostatic forces acting on, each ion are then calculated.