The temperature factor of an atom in a rigid vibrating molecule. I. Isotropic thermal motion

The temperature factor of an atom in a molecular crystal has been derived in terms of the rigid-body parameters (u²) and (ω²), where (u²) is the mean-square amplitude of translational displacement of the molecule in any direction and (ω²) is the mean-square amplitude of angular (librational) displacement about any axis through the centre of inertia of the molecule. The analysis, which is appropriate to a cubic crystal containing rigid molecules undergoing isotropic thermal motion, is correct to the second power of (ω²); the second-order treatment is necessary for interpreting accurate Bragg intensity data such as those discussed in a subsequent paper. Substitution of the temperature factor into the structure-factor equation yields an expression containing terms which can be identified with the first, second and third cumulants of the `cumulant expansion' model of thermal motion.