On the Decomposition of Lattice-Periodic Functions

In a study of the classical problem of the calculation of the potential due to an ionic crystal lattice, and in particular, generalizations of the Ewald method (Ref. 1) along the lines of Nijboer and Dewette (Refs. 3, 4), W. J. C. Grant (Ref. 2) proposed the following problem: Suppose we say that an ionic crystal lattice is primitive if for a suitable choice of origin there exist three vectors xi , .f 2 , .f3 such that the charge at the point nxxx + n2.f'2 + n3x3 is just qo(-- 1 for some fixed q0 and for all triples of integers (ni , n 2 , n3) and that the charge at all other points is zero. (For example, the ordinary NaCl lattice is primitive with the Xi taken to be the unit coordinate vectors and q0 = 1.) The question is then: Which crystal lattices can be decomposed into finite sums of primitive lattices? Different primitive lattices in the decomposition may have different origins and by the sum of two lattices we mean, of course, the component-wise sum. The object of this paper is threefold: (i) The problem is extended to its natural n-dimensional analogue. (ii) Rather simple necessary and sufficient conditions are given for the existence of the desired decomposition. 1191