Significance of symmetry in representing real‐ and reciprocal‐space properties of a crystal and of its constituents is discussed in terms of symmetry eigenfunctions. The formation of the site‐symmetrized multipole expansion for the 32 crystal point symmetries is shown, and the mutual relations between the real‐ and reciprocal‐space expansions are discussed. The origin of the symmetry‐based phase relations, equivalences and extinction rules of structure amplitudes is pointed out. Distinction is made between the accurate and approximate ones arising from the true space‐group symmetry and from the approximate free‐particle symmetry of the constituents, respectively. Finally, the symmetry imposed by reality of charge density and its violation through anomalous dispersion are discussed in terms of symmetric and antisymmetric contributions of atomic charge densities.

A rigorous treatment of the constraints imposed by site and body symmetry on the orientational distribution of librating molecules is presented in terms of multipole expansions. Radial densities of the site-symmetric dynamic multipole expansion are linear combinations of the radial densities of the body-symmetric static multipole expansion of the same multipole order. The transformation from static to dynamic radial densities is transmitted by libration matrices, which can be included as parameters in a structure factor model. The matrix elements are connected to the rotational dynamics of the molecule. They are expansion coefficients of the orientational distribution in a basis obtained by site and body symmetrization of the real Wigner functions. Both symmetrizations are reduced to selecting terms from a general basis according to simple index rules. Independence of the kind of density function and Fourier invariance make the formalism useful in combined neutron and X-ray diffraction studies.

No abstract

Accurate X-ray structure factors of CaO, SrO and BaO were measured at 293 K from single crystals and analysed together with the data of Sanger [Acta Cryst. (1969), A25, 694-702] on MgO. The oxygen ion is seen to suffer an average compression and a fourth-order cubic harmonic type deformation, as compared to the atomic superposition model, yielding a lower average charge density in the surrounding region and a charge transfer from [ 111 ] to the [ 100] direction. These features strengthen with increasing atomic number of the cation. Mg and Ca show a weak broadening. About nine electrons are localized around the oxygen position. The tenth electron of 02-is more widely distributed. The electron counts of the cation correspond most closely to Mg +, Ca 2+, Sr 2+ and Ba 2+. The results confirm the dominance of ionic character and increase of ionicity with increasing atomic number of the cation.

Site symmetrized expressions of the atomic temperature factor for all crystallographic point groups have been derived within the framework of the effective one-particle potential formalism. The formulation makes use of the harmonic oscillator eigenfunctions leading to Fourier invariant expressions for the probability density function. The present formalism lends itself for a straightforward application to any crystal structure. The treatment is applied to the analysis of recent neutron diffraction data on zinc. The results are found to be in full accord with those obtained by the use of earlier formalisms.