Addition of higher cumulants to the crystallographic structure-factor equation: a generalized treatment for thermal-motion effects

The usual crystallographic structure-factor equation, with three positional and six anisotropic-temperature-factor coefficients, assumes that the thermal-motion probability density function is centrosymmetric. However, phenomena such as libration and anharmonic vibration can cause skewness. In this study, ten more coefficients per atom representing the third cumulant of the probability density function for thermal-motion are added to the structure-factor equation to permit a determination of the nature of the skewness. The Edgeworth series expansion based on the normal probability density function is used to analyze the results. The equations are generalized to include also the fourth cumulant, which describes kurtosis. The `cumulant-expansion model' for thermal motion is a statistical model without kinematic constraints and provides an unbiased estimate for the skewness of the density function for thermal motion. Application of the model to neutron diffraction data from crystals containing methyl groups (which are undergoing torsional oscillation) confirms the assumption that the density functions for the hydrogen atoms of a methyl group are skewed as an arc about the axis of torsional oscillation. The model has not been applied with X-ray diffraction data; if it were, the resulting parameters would describe the skewness of the combined electron and thermal-motion probability density functions.