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

International Tables for Crystallography (2013). Vol. D, ch. 1.11, p. 278

Section 1.11.6.4. Tensor structure 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:  dmitrien@crys.ras.ru

1.11.6.4. Tensor structure factors

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Once the tensor atomic factors have been determined [either from phenomenological expressions like (1.11.6.16)[link], according to the site-symmetry restrictions, or from given microscopic expressions, e.g. (1.11.4.3)[link]], tensor structure factors are obtained by summation over the contributions of all atoms in the unit cell, as in conventional diffraction theory:[\eqalignno{F_{jm}({\bf H})&=\textstyle\sum\limits_{t,u}o_t D_{jm}^{tu} \exp(-2\pi i{\bf H}\cdot{\bf r}^{tu}) \exp[-W^{tu}({\bf H})], &\cr &&(1.11.6.37)\cr F^{+}_{jmn}({\bf H})&=\textstyle\sum\limits_{t,u}o_t I_{jmn}^{tu+} \exp(-2\pi i{\bf H}\cdot{\bf r}^{tu}) \exp[-W^{tu}({\bf H})], &\cr &&(1.11.6.38)\cr F^{-}_{jmn}({\bf H})&=\textstyle\sum\limits_{t,u}o_t I_{jmn}^{tu-} \exp(-2\pi i{\bf H}\cdot {\bf r}^{tu}) \exp[-W^{tu}({\bf H})], &\cr &&(1.11.6.39)\cr F_{jmnp}({\bf H})&=\textstyle\sum\limits_{t,u}o_t Q_{jmnp}^{tu} \exp(-2\pi i{\bf H}\cdot {\bf r}^{tu}) \exp[-W^{tu}({\bf H})], &\cr &&(1.11.6.40)}]where the index t enumerates the crystallographically different types of scatterers (atoms belonging to the same or different chemical elements), the index u denotes the crystallographically equivalent positions; [o_t \le 1] is a site-occupancy factor, and [W^{tu}({\bf H})] is the Debye–Waller temperature factor. The tensors of the atomic factors, [D_{jm}^{tu}], [I_{jmn}^{tu+}], [I_{jmn}^{tu-}], [Q_{jmnp}^{tu}], are, in general, different for crystallographically equivalent positions, that is for different u, and it is exactly this difference that enables the excitation of the resonant forbidden reflections.

Extinction rules and polarization properties for forbidden reflections are different for tensor structure factors of different ranks, a circumstance that may be used for experimental separation of different tensor contributions (for tensors of rank 2, information is given in Tables 1.11.2.1[link] and 1.11.2.2[link]). In the harmonic approximation, anisotropies of the atomic thermal displacements (Debye–Waller factor) are also described by tensors of rank 2 or higher, but, owing to these, excitations of glide-plane and screw-axis forbidden reflections are not possible.








































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