Exact random-walk models in crystallographic statistics. II. The bicentric distribution for the space groupP\overline{1}

Abstract

An exact probability density function for the magnitude of the normalized structure factor |E| has been derived for the space group P\bar 1, taking account of the presence of one non-crystallographic center of symmetry. The function is based on the exact solution of the corresponding random-walk model and its expansion into a Fourier series. The above result is compared with simulated semi-cumulative distributions based on hypothetical structures and very good agreement is obtained for the equal-atom case, as well as for a heterogeneous asymmetric subunit containing fourteen carbon atoms and one uranium atom. The new exact bicentric probability density functions of |E|, for the space group P\bar 1, reduce to the well known asymptotic expressions that are valid for equal-atom structures and a large number of atoms in the asymmetric unit of the space group.