Exact random-walk models in crystallographic statistics. VI. P.D.F.'s of magnitude of [E] for all plane groups and...

An exact calculation of the probability density function (p.d.f.) of magnitude of [E], the magnitude of the normalized structure factor, can be developed in terms of Fourier and Fourier-Bessel series whose coefficients can be expressed in terms of the characteristic function. This article provides the formulae for atomic contributions to such characteristic functions. The results presented in this study are applicable to all the plane groups and to 206 three dimensional space groups. Only the space groups isomorphous to the cubic point groups 432, 43m and m3m were omitted due to the complexity of the resulting expressions and the small deviations of the corresponding densities from the central-limit-theorem approximation, which have been observed in simulations for extreme atomic heterogeneities. Representative derivations illustrating the problems and techniques of their solution are provided. All the theoretical results have been computed numerically and compared with simulated distributions. Some results of these computations are illustrated in the accompanying paper, Part VII of this series [Rabinovich, Shmueli, Stein, Shashua & Weiss (1991). Acta Cryst. A47, 336-340].