A new method for the evaluation of lattice dynamical Coulomb sums

Abstract

In modelling ionic crystals with the rigid-ion model under the adiabatic and harmonic approximations, the series involving the long-range Coulomb forces appear to be only conditionally convergent. The approach most commonly used to sum these series is the Ewald method, which separates them into two rapidly convergent parts in real and reciprocal space. Instead, the technique described here makes use of infinite sheets of charge to cancel effectively the long-range moments of these forces. The original series can then be rewritten in a form which rapidly converges, but still only in real space. The new method was tested by successfully duplicating Kellermann's model (1940) results, which made use of the Ewald method. The new technique is restricted to lattice dynamical problems as opposed to static problems but may be applied to any crystal lattice and incorporated into any lattice dynamical model which also makes use of the harmonic approximation.