Exact joint distribution of E h , E k and E h+k , and the probability for the positive sign of the triple product in the space group P\overline{1}

Abstract

A recently formulated method of deriving exact Fourier-series representations of joint probability density functions (p.d.f.'s) of several normalized structure factors is applied to the derivation of an exact expression for the conditional probability that the sign of the triple product E h E k E h + k is positive. The relevant joint and conditional probabilities are derived for the space group P\bar 1. The Fourier coefficients of the p.d.f. are given by rapidly convergent series of Bessel functions, and the convergence properties of the Fourier summations are also found to be favourable. The exact conditional probability is compared with the currently employed approximate one, well known as the hyperbolic tangent formula, for several hypothetical structures. The examples illustrate the effects of the number of atoms in the unit cell, the magnitude of the E values and the atomic composition on the exact and approximate probabilities. It is found, in agreement with previous studies, that the hyperbolic tangent formula may indeed significantly underestimate the probability when the number of equal atoms is small and the E values are only moderately large, and when the structure contains outstandingly heavy atoms. The opposite behaviour, i.e. the approximate probability overestimating the exact one, was not observed in the present calculations. For large values of the triple product in equal-atom and heterogeneous models, the agreement between the approximate and exact probabilities is usually good.