Probabilistic methods in crystal structure analysis

One of the main goals of modern crystallography is the determination of the detailed internal structure of crystalline matter, at the atomic level. Statistical analyses and, in particular, random-walk models play a central role in inferring structural information from crystallographic data. Such methods are routinely employed by crystallographers in the determination of crystal symmetry from the experimental data, and in the solution of the outstandingly important problem for this discipline, the phase problem. Three classes of approaches are discussed: (a) methods based entirely on the central limit theorem; (b) approximate expansions in terms of orthogonal polynomials that have the central-limit-theorem pdf as their weight function—that is, Gram–Charlier and Edgeworth expansions; and (c) pdf’s that are exactly formulated and reduced to computable forms, represented as Fourier and Fourier–Bessel series. Both univariate and multivariate pdf’s of crystallographic interest are derived and discussed. Some other approximate probabilistic approaches that have been applied to crystallographic problems are also briefly reviewed.