Internal Field in General Dipole Lattices

The calculation of the internal field in dipole lattices necessitates summing the contributions due to all dipoles in the crystal. These dipole sums are conditionally convergent, which means that they depend on the order of summation (i.e., the shape of the crystal). Rapidly converging expressions for these sums are obtained with the method of planewise summation, for lattices of arbitrary symmetry and arbitrary dipole orientation. With these expressions one can evaluate the internal field at an arbitrary point of complex dipole lattices. As an example, the internal electric field at the positions of the water molecules is evaluated for potassium ferrocyanide [${\mathrm{K}}_4$Fe${\mathrm{}\left(\mathrm{CN}\right)\mathrm{}}_6$·3${\mathrm{H}}_2$O].