Exact random-walk models in crystallographic statistics. I. Space groups P and P1

Probability density functions that are exact solutions to classical random-walk problems have been adapted to represent distributions of the magnitude of the normalized structure factor, for the space groups P and P1. The functions are given by readily summable Fourier and Fourier-Bessel series, and account explicitly for the atomic composition of the asymmetric unit. These new probability density functions have been extensively tested by comparison with simulated histograms of |E|, for a wide range of atomic compositions. The most heterogeneous compositions examined are C₁₄U and C₂₉U, for P and P1, respectively. Very good agreement between the simulated and theoretical distributions has been obtained in all these tests, over the entire (useful) range 0 < |E| < 3. A distribution of |E| values, recalculated from published data on a triclinic platinum complex with chloroorganic ligands, has also been compared with the new probability functions and excellent agreement with the (expected) P theoretical distribution has been obtained. The discrepancy between the recalculated distribution and the P1 theoretical rules out the latter space group both by visual comparison and quantitative discrepancy criteria. It is concluded that probability density functions are definitely preferable to moments in attempting to resolve a space-group ambiguity. Measures of discrepancy to be used in such statistical tests are proposed and discussed in some detail.