Exact random-walk models in crystallographic statistics. VII. An all-space-group study of the effects of atomic heterogeneity on the P.d.f.'s of E

Abstract

Exact expressions have been found for the probability density functions (p.d.f.'s) of the magnitude of the normalized structure factor for all the two-dimensional and most three-dimensional space groups [Part VI: Rabinovich, Shmueli, Stein, Shashua & Weiss (1991). Acta Cryst. A47, 328-335]. The results of that investigation are used in the present article to examine some effects of atomic heterogeneity, in the various space-group symmetries, on the p.d.f.'s. Some typical comparisons are made between p.d.f.'s based on the central limit theorem and p.d.f.'s computed from exact formulae. In addition, the exact results are compared to histograms of simulated values of magnitude of [E]. It is found that the p.d.f.'s for some space groups are influenced rather strongly by the presence of outstandingly heavy scatters, but they are quite insensitive to the presence of such scatterers in other space groups. The often made general statement 'The presence of outstandingly heavy scatterers may invalidate the indications of Wilson's statistics' is made more precise here, insofar as it depends on the particular space group.