Tables for
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.11, pp. 277-278

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

V. E. Dmitrienko,a* A. Kirfelb and E. N. Ovchinnikovac

aA. V. Shubnikov Institute of Crystallography, Leninsky pr. 59, Moscow 119333, Russia,bSteinmann Institut der Universität Bonn, Poppelsdorfer Schloss, Bonn, D-53115, Germany, and cFaculty of Physics, M. V. Lomonosov Moscow State University, Leninskie Gory, Moscow 119991, Russia
Correspondence e-mail: Hidden internal symmetry of the dipole–quadrupole tensors in resonant atomic factors

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It is fairly obvious from expressions ([link] and ([link] that in the non-magnetic case the symmetric and antisymmetric third-rank tensors, [I_{jkl}^{+}] and [I_{jlk}^{-}], 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 [t_{ijk}=t_{ikj}] resulting from the second-order Born approximation ([link],[\eqalignno{I^{+}_{ijk}&=(t_{ijk}+t_{jik})/2, &(\cr I^{-}_{ijk}&=(t_{ijk}-t_{jik})/2,&(}]where[t_{ijk}= -{\textstyle{{1}\over{2}}}I_{ijk}.\eqno(]From equation ([link], one can infer that the symmetry restrictions for [I^{+}_{ijk}] and [t_{ijk}] are the same. Then it can be seen that [I^{-}_{ijk}] can be expressed via [I^{+}_{ijk}].

For any symmetry, [I^{+}_{ijk}] and [t_{ijk}] have the same number of independent elements (with a maximum 18 for site symmetry 1). Thus, one can reverse equation ([link] and express [t_{ijk}] directly in terms of [I^{+}_{ijk}]:[\eqalignno{t_{111}&=I^{+}_{111},\quad t_{211}=2I^{+}_{121}-I^{+}_{112},\quad t_{311}=2I^{+}_{311}-I^{+}_{113}, &\cr t_{122}&=2I^{+}_{122}-I^{+}_{221},\quad t_{222}=I^{+}_{222},\quad t_{322}=2I^{+}_{232}-I^{+}_{223},&\cr t_{133}&=2I^{+}_{313}-I^{+}_{331},\quad t_{233}=2I^{+}_{233}-I^{+}_{332},\quad t_{333}=I^{+}_{333},&\cr t_{123}&=I^{+}_{123}+I^{+}_{312}-I^{+}_{231},\quad t_{223}=I^{+}_{223},\quad t_{332}=I^{+}_{332},&\cr t_{113}&=I^{+}_{113},\quad t_{231}=I^{+}_{231}+I^{+}_{123}-I^{+}_{312},\quad t_{331}=I^{+}_{331},&\cr t_{112}&=I^{+}_{112},\quad t_{221}=I^{+}_{221},\quad t_{312}=I^{+}_{312}+I^{+}_{231}-I^{+}_{123}. &\cr&&(}]

Using equations ([link] and ([link], one can express all nine elements of [I^{-}_{ijk}] through [I^{+}_{ijk}]:[\eqalignno{I^{-}_{231}&=I^{+}_{123}-I^{+}_{312},\quad I^{-}_{232}=I^{+}_{223}-I^{+}_{232},\quad I^{-}_{233}=I^{+}_{233}-I^{+}_{332}, &\cr I^{-}_{311}&=I^{+}_{311}-I^{+}_{113},\quad I^{-}_{312}=I^{+}_{231}-I^{+}_{123},\quad I^{-}_{313}=I^{+}_{331}-I^{+}_{313}, &\cr I^{-}_{121}&=I^{+}_{112}-I^{+}_{121},\quad I^{-}_{122}=I^{+}_{122}-I^{+}_{221},\quad I^{-}_{123}=I^{+}_{312}-I^{+}_{231}, &\cr&&(}]according to which the antisymmetric part of the dipole–quadrupole term is a linear function of the symmetric one [however, not vice versa: equations ([link] cannot be reversed].

Note that the equations ([link] impose an additional restriction on [I^{-}_{ijk}], which applies to all atomic site symmetries:[I^{-}_{123}+I^{-}_{231}+I^{-}_{312}=0.\eqno(]This is, in fact, a well known result: the pseudo-scalar part of [I^{-}_{ijk}] vanishes in the dipole–quadrupole approximation used in equation ([link]. Thus, for point symmetry 1, [I^{-}_{ijk}] 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 [I^{-}_{ijk}] arise automatically from equation ([link] taking into account the symmetry of [I^{+}_{ijk}] [symmetry limitations on [I^{+}_{ijk}] and [I^{-}_{ijk}] for all crystallographic point groups can be found in Sirotin & Shaskolskaya (1982[link]) and Nye (1985[link])].

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[link]) and Kokubun et al. (2010[link]).

In ZnO, crystallizing in the wurtzite structure, the 3m symmetry of the atomic positions imposes the following restrictions on [t_{ijk}]:[\eqalignno{t_{131}&=t_{223}=e_{15}, &(\cr t_{222}&=-t_{112}=-t_{211}=e_{22}, &(\cr t_{311}&=t_{322}=e_{31}, &(\cr t_{333}&=e_{33},&(}]where [e_{15}], [e_{31}], [e_{22}], [e_{33}] are energy-dependent complex tensor elements [keeping the notations by Sirotin & Shaskolskaya (1982[link]), 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 [\textstyle{1\over 3},\textstyle{2\over 3},z], then for the other Zn at [\textstyle{2\over 3},\textstyle{1\over 3},z+\textstyle{1\over 2}], which is related to the first site by the glide plane, there is the following set of elements: [e_{15},e_{31},-e_{22},e_{33}]. Therefore, the structure factors of the glide-plane forbidden reflections are proportional to [e_{22}].

For the symmetric and antisymmetric parts one obtains from equations ([link] and ([link] the non-zero components[\eqalignno{I^{+}_{131}&=I^{+}_{232}=(e_{15}+e_{31})/2, &(\cr I^{+}_{222}&=-I^{+}_{121}=-I^{+}_{112}=e_{22}, &(\cr I^{+}_{113}&=I^{+}_{223}=e_{15}, &(\cr I^{+}_{333}&=e_{33}&(}]and[I^{-}_{232}=-I^{-}_{311}=I^{+}_{113}-I^{+}_{131}=(e_{15}-e_{31})/2.\eqno(]

Physically, we can expect that [|e_{15}+e_{31}|\gg |e_{15}-e_{31}|] because [e_{15}+e_{31}] survives even for tetrahedral symmetry [\bar{4}3m], whereas [e_{15}-e_{31}] 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 [\bar{4}m2] symmetry of the atomic positions imposes restrictions on the tensors [t_{ijk}] [keeping the notations of Sirotin & Shaskolskaia (1982[link]): the x and y axes are normal to the mirror planes, and the z axis is parallel to the c axis]:[\eqalignno{t_{131}&=-t_{223}=e_{15}, &(\cr t_{311}&=-t_{322}=e_{31},&(}]where [e_{15}] and [e_{31}] are energy-dependent complex parameters. If we apply these restrictions to the Ti atoms at [0,0,0] and [\textstyle{1\over 2},\textstyle{1\over 2},\textstyle{1\over 2}], then for the other two inversion-related Ti atoms at [0,\textstyle{1\over 2},\textstyle{1\over 4}] and [\textstyle{1\over 2},0,\textstyle{3\over 4}] (centre [2/m]), the parameters are [-e_{15}] and [-e_{31}].

For the symmetric and antisymmetric parts one obtains as non-vanishing components[\eqalignno{I^{+}_{131}&=-I^{+}_{232}=(e_{15}+e_{31})/2, &(\cr I^{+}_{113}&=-I^{+}_{223}=e_{15}&(}]and[I^{-}_{232}=I^{-}_{311}=I^{+}_{131}-I^{+}_{113}=(e_{31}-e_{15})/2.\eqno(]

It is important to note that the symmetric part [I^{+}_{ijk}] 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 [{{\partial f^{dd}_{ij}}/{\partial x_{k}}}], which take the same tensor form as [I^{+}_{ijk}] but are not related to [I^{-}_{ijk}] by equations ([link]. In ZnO, which was studied in detail by Collins et al. (2003[link]), the thermal-motion-induced contribution is rather significant, while for anatase the situation is less clear.


Collins, S. P., Laundy, D., Dmitrienko, V., Mannix, D. & Thompson, P. (2003). Temperature-dependent forbidden resonant X-ray scattering in zinc oxide. Phys. Rev. B, 68, 064110.
Goulon, J., Jaouen, N., Rogalev, A., Wilhelm, F., Goulon-Ginet, C., Brouder, C., Joly, Y., Ovchinnikova, E. N. & Dmitrienko, V. E. (2007). Vector part of optical activity probed with X-rays in hexagonal ZnO. J. Phys. Condens. Matter, 19, 156201.
Kokubun, J., Sawai, H., Uehara, M., Momozawa, N., Ishida, K., Kirfel, A., Vedrinskii, R. V., Novikovskii, N., Novakovich, A. A. & Dmitrienko, V. E. (2010). Pure dipole–quadrupole resonant scattering induced by the p–d hybridization of atomic orbitals in anatase TiO2. Phys. Rev. B, 82, 205206.
Nye, J. F. (1985). Physical Properties of Crystals: Their Representation by Tensors and Matrices. Oxford University Press.
Sirotin, Y. & Shaskolskaya, M. P. (1982). Fundamentals of Crystal Physics. Moscow: Mir.

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