Exact random-walk models in crystallographic statistics. V. Non-symmetrically bounded distributions of...

Derivations of exactly formulated Fourier-series representations of probability density functions (p.d.f.'s) of the magnitude of the structure factor are based on the fact that the structure factor and its real and imaginary parts are bounded. In some situations the bounds of the real and imaginary parts of the structure factor are different, and this gives rise to modified forms of the p.d.f.'s. Three physical examples that call for such modifications are investigated: (i) effect of dispersive scatterers on the p.d.f. of a centrosymmetric structure factor, (ii) effect of the presence of a centrosymmetric fragment in the asymmetric unit of a non-centrosymmetric space group, and (iii) effect of heavy scatterers in special positions of a non-centrosymmetric space group, where the imaginary part of the trigonometric structure factor for these special positions vanishes by symmetry. The general form of an exact Fourier p.d.f., taking account of such modifications, is derived, and expressions for Fourier coefficients are obtained for selected examples of the above three situations. It is seen that the effects of pseudocentrosymmetry may be most pronounced and those of dispersion are significant mainly in the range of small values of the structure factor.