Section 1.11.6.3. Hidden internal symmetry of the dipole–quadrupole tensors in resonant atomic factors

It is fairly obvious from expressions (1.11.6.3) and (1.11.6.16) that in the non-magnetic case the symmetric and antisymmetric third-rank tensors, and , which describe the dipole–quadrupole contribution to the X-ray scattering factor, are not independent: the antisymmetric part, which is also responsible for optical-activity effects, can be expressed via the symmetric part (but not vice versa). Indeed, both of them can be described by a symmetric third-rank tensor resulting from the second-order Born approximation (1.11.6.3),whereFrom equation (1.11.6.17), one can infer that the symmetry restrictions for and are the same. Then it can be seen that can be expressed via .

For any symmetry, and have the same number of independent elements (with a maximum 18 for site symmetry 1). Thus, one can reverse equation (1.11.6.17) and express directly in terms of :

Using equations (1.11.6.18) and (1.11.6.20), one can express all nine elements of through :according to which the antisymmetric part of the dipole–quadrupole term is a linear function of the symmetric one [however, not vice versa: equations (1.11.6.21) cannot be reversed].

Note that the equations (1.11.6.21) impose an additional restriction on , which applies to all atomic site symmetries:This is, in fact, a well known result: the pseudo-scalar part of vanishes in the dipole–quadrupole approximation used in equation (1.11.6.3). Thus, for point symmetry 1, has only eight independent elements rather than nine. This additional restriction works in all cases of higher symmetries provided the pseudo-scalar part is allowed by the symmetry (i.e. point groups 2, 3, 4, 6, 222, 32, 422, 622, 23 and 432). All other symmetry restrictions on arise automatically from equation (1.11.6.21) taking into account the symmetry of [symmetry limitations on and for all crystallographic point groups can be found in Sirotin & Shaskolskaya (1982) and Nye (1985)].

Let us consider two examples, ZnO and anatase, TiO₂, where the dipole–dipole contributions to forbidden reflections vanish, whereas both the symmetric and antisymmetric dipole-quadrupole terms are in principal allowed. In these crystals, the dipole–quadrupole terms have been measured by Goulon et al. (2007) and Kokubun et al. (2010).

In ZnO, crystallizing in the wurtzite structure, the 3m symmetry of the atomic positions imposes the following restrictions on :where , , , are energy-dependent complex tensor elements [keeping the notations by Sirotin & Shaskolskaya (1982), the x axis is normal to the mirror plane, the y axis is normal to the glide plane and the z axis corresponds to the c axis of ZnO]. If we suppose these restrictions for Zn at , then for the other Zn at , which is related to the first site by the glide plane, there is the following set of elements: . Therefore, the structure factors of the glide-plane forbidden reflections are proportional to .

For the symmetric and antisymmetric parts one obtains from equations (1.11.6.17) and (1.11.6.18) the non-zero componentsand

Physically, we can expect that because survives even for tetrahedral symmetry , whereas is non-zero owing to a deviation from tetrahedral symmetry; in ZnO, the local coordinations of the Zn positions are only approximately tetrahedral.

In the anatase structure of TiO₂, the symmetry of the atomic positions imposes restrictions on the tensors [keeping the notations of Sirotin & Shaskolskaia (1982): the x and y axes are normal to the mirror planes, and the z axis is parallel to the c axis]:where and are energy-dependent complex parameters. If we apply these restrictions to the Ti atoms at and , then for the other two inversion-related Ti atoms at and (centre ), the parameters are and .

For the symmetric and antisymmetric parts one obtains as non-vanishing componentsand

It is important to note that the symmetric part of the atomic factor can be affected by a contribution from thermal-motion-induced dipole–dipole terms. The latter terms are tensors of rank 3 proportional to the spatial derivatives , which take the same tensor form as but are not related to by equations (1.11.6.21). In ZnO, which was studied in detail by Collins et al. (2003), the thermal-motion-induced contribution is rather significant, while for anatase the situation is less clear.