Electric Field Gradients in Point-Ion and Uniform-Background Lattices

The lattice contribution to the field gradient in ionic crystals and metals is a quantity which has a well-defined value. However, for an actual evaluation, the field gradient is usually broken up into a number of conditionally convergent series with poor convergence. Rapidly convergent expressions for these series, and consequently, for the field gradient can be obtained by applying the method of plane-wise summation. This method is applied to the field gradient in ionic crystals with tetragonal and hexagonal symmetry and to the field gradient in tetragonal and hexagonal close-packed metal structures. As an example, an expression for the field gradient at the position of the anion is derived for ionic crystals with the Cd${\mathrm{I}}_2$ structure. This expression is numerically evaluated for Co${\mathrm{Br}}_2$, Fe${\mathrm{Br}}_2$, Mg${\mathrm{Br}}_2$, Mn${\mathrm{Br}}_2$, Ca${\mathrm{I}}_2$, Cd${\mathrm{I}}_2$, Co${\mathrm{I}}_2$, Fe${\mathrm{I}}_2$, Ge${\mathrm{I}}_2$, Mg${\mathrm{I}}_2$, and Mn${\mathrm{I}}_2$. Rather extensive numerical results are also presented for both close-packed metal structures, including values for the field gradient in Li, Be, Zn, In, and Rh.