Explicit Fourier representations of non-ideal hypercentric P.D.F.'s of E

Explicit non-ideal hypercentric distributions of the magnitude of the normalized structure factor have been derived and investigated for the space group P. One of the distribution types investigated arises when the asymmetric unit of P consists of several identical centrosymmetric motifs, interrelated by additional non-crystallographic centres of symmetry. The ideal version of such distributions was studied by Rogers & Wilson [Acta Cryst. (1953), 6, 439-449]. The other distribution type studied originates from atomic arrangements in which the asymmetric unit is composed of several unrelated centrosymmetric fragments that may differ in their sizes and chemical compositions. Explicit non-ideal probability density functions (p.d.f.'s) of the magnitude of the normalized structure factor were formulated for both above types of distribution as Fourier series, were evaluated numerically and were compared with appropriately simulated distributions of |E|. The computations were carried out for a range of atomic compositions, and numbers of unrelated centrosymmetric fragments that comprise the asymmetric unit; both of these factors have a significant influence on the deviation of the hypercentric p.d.f.'s from the values predicted on the basis of the central-limit theorem approximation.