Moments of the trigonometric structure factor

The eighth moment of the magnitude of the trigonometric structure factor has been computed for all the space groups and all the reflection subsets giving rise to different functional forms of this quantity. This extension of previously published computations of lower moments permits the construction of four-term generalized distributions of normalized intensity, which are necessary in treating problems arising from highly heterogeneous atomic compositions in various space-group symmetries. The related problem of odd-even mixed partial moments of the trigonometric structure factor has also been investigated, and these mixed moments were found to vanish for all the three-dimensional space groups, confirming the correctness of the hitherto published theoretical statistics. Similar computations for the plane groups showed that non-zero values of the mixed partial odd-even moments are obtained for p3, p31m, p3ml, p6 and p6m. This result calls for some modifications of the statistical formalism to be applied to two-dimensional sets of intensity data. The modifications required for the centrosymmetric case are indicated in some detail.