A new joint probability distribution for the triplet phase sum. Inclusion of higher-order terms

A new joint probability distribution of three normalized structure factors is derived for the case of N equal atoms in the space group P1. The distribution appears in a series expansion. Convergence of the series is obtained by inclusion of higher-order terms up to order N^-5, although with the exception of terms up to order N^-1/2 the distribution does not contain all possible terms. Test results show improved estimates of the absolute value of the triplet phase sum when compared with the Cochran distribution. In particular, the systematic errors are reduced appreciably.