Exact joint probability distribution for centrosymmetric structure factors. Derivation and application to the Φ1relationship in the space groupPgif">\overline{1}

Abstract

Recent applications of exact random-walk techniques to crystallographic structure-factor statistics have now been extended to multivariate joint probability density functions of several structure factors. The technique of deriving such multivariate exact density functions is introduced, and is applied to the study of the simplest sign relationship: Σ1, in the space group P\bar 1. An exact expression is obtained for the probability that the sign of the normalized structure factor E(2h 2k 2l) is positive, given the magnitudes of E(2h 2k 2l) and E(h k l), and an extensive numerical examination of this new expression - as compared with the conventional asymptotic formula for this probability - is presented. It is shown that the asymptotic formula usually underestimates the probability that E(2h 2k 2l) is positive, the discrepancies between the exact and asymptotic results being rather serious when the atomic composition of the asymmetric unit is heterogeneous (even moderately so), and when the number of atoms is small; a paucity of atoms leads to significant discrepancies even in the equal-atom case. On the other hand, for large asymmetric units of low heterogeneity and for high E values, the exact and asymptotic expressions agree very well in their predictions. The qualitative behaviour of the new exact expression is consistent with the known features of the Σ1 relationship and its statistical interpretations.