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

International Tables for Crystallography (2013). Vol. D, ch. 1.11, pp. 270-271

Section 1.11.2.2. Tensorial structure factors and forbidden reflections

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

1.11.2.2. Tensorial structure factors and forbidden reflections

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In spite of its simplicity, equation (1.11.2.1)[link] provides non-trivial restrictions on the tensorial structure factors of Bragg reflections. The sets of allowed reflections, listed in International Tables for Crystallography Volume A (Hahn, 2005[link]) for all space groups and for all types of atom sites, are based on scalar X-ray susceptibility. In this case, reflections can be forbidden (i.e. they have zero intensity) owing to glide-plane and/or screw-axis symmetry operations. This is because the scalar atomic factors remain unchanged upon mirror reflection or rotation, so that the contributions from symmetry-related atoms to the structure factors can cancel each other. In contrast, atomic tensors are sensitive to both mirror reflections and rotations, and, in general, the tensor atomic factors of symmetry-related atoms have different orientations in space. As a result, forbidden reflections can in fact be excited just due to the anisotropy of susceptibility, so that the selection rules for possible reflections change.

It is easy to see how the most general tensor form of the structure factors can be deduced from equation (1.11.2.1)[link]. The structure factor of a reflection with reciprocal-lattice vector [{\bf H}] is proportional to the Fourier harmonics of the susceptibility. The corresponding relations (Authier, 2005[link], 2008[link]) simply have to be rewritten in tensorial form:[F_{jk}({\bf H})=-{{\pi V}\over{r_0\lambda^2}} \chi_{jk}({\bf H})\equiv -{{\pi V}\over{r_0\lambda^2}}\int\chi_{jk}({\bf r}) \exp(-2\pi i{\bf H}\cdot{\bf r})\,{\rm d}{\bf r},\eqno(1.11.2.3)]where [r_0=e^2/mc^2] is the classical electron radius, [\lambda] is the X-ray wavelength and [V] is the volume of the unit cell.

1.11.2.2.1. Glide-plane forbidden reflections

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Considering first the glide-plane forbidden reflections, there may, for instance, exist a glide plane [c] perpendicular to the [x] axis, i.e. any point [x,y,z] is transformed by this plane into [\bar{x},y,z+\textstyle{{1}\over{2}}]. The corresponding matrix of this symmetry operation changes the sign of [x],[R_{jk}^c=R_{jk}^{cT}=\pmatrix{-1&0&0\cr 0&1&0\cr0&0&1}, \eqno(1.11.2.4)]and the translation vector into [{\bf a}^c=(0,0,\textstyle{1\over 2})]. Substituting (1.11.2.4)[link] into (1.11.2.1)[link] and exchanging the integration variables in (1.11.2.3)[link], one obtains for the structure factors of reflections [0k\ell][F_{jk}(0k\ell)=\exp(-i\pi\ell)R^c_{jm}R^{cT}_{nk}F_{mn}(0k\ell).\eqno(1.11.2.5)]If [F_{jk}(0k\ell)] is scalar, i.e. [F_{jk}(0k\ell)=F(0k\ell)\delta_{jk}], then [F(0k\ell)=] [-F(0k\ell)] for odd [\ell], hence [F(0k\ell)] vanishes. This is the well known conventional extinction rule for a [c] glide plane, see International Tables for Crystallography Volume A (Hahn, 2005[link]). If, however, [F_{jk}(0k\ell)] is a tensor, the mirror reflection [x\to -x] changes the signs of the [xy] and [xz] tensor components [as is also obvious from equation (1.11.2.5)[link]]. As a result, the [xy] and [xz] components should not vanish for [\ell=2n+1] and the tensor structure factor becomes[F_{jk}(0k\ell\semi\ell=2n+1)=\pmatrix{ 0&F_1&F_2\cr F_1&0&0\cr F_2&0&0}.\eqno(1.11.2.6)]In general, the elements [F_1] and [F_2] are complex, and it should be emphasized from the symmetry point of view that they are different and arbitrary for different [k] and [\ell]. However, from the physical point of view, they can be readily expressed in terms of tensor atomic factors, where only those chemical elements are relevant whose absorption-edge energies are close to the incident radiation energy (see below).

It is also easy to see that for the non-forbidden (= allowed) reflections [0k\ell\semi\ell=2n], the non-zero tensor elements are just those which vanish for the forbidden reflections:[F_{jk}(0k\ell;\ell=2n)=\pmatrix{F_1&0&0\cr 0&F_2&F_4\cr 0&F_4&F_3}.\eqno(1.11.2.7)]Here the result is mainly provided by the diagonal elements [F_1\approx F_2\approx F_3], but there is still an anisotropic part that contributes to the structure factor, as expressed by the off-diagonal element. In principle, the effect on the total intensity as well as the element itself can be assessed by careful measurements using polarized radiation.

1.11.2.2.2. Screw-axis forbidden reflections

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For the screw-axis forbidden reflections, the most general form of the tensor structure factor can be found as before (Dmitrienko, 1983[link]; see Table 1.11.2.1[link]). Again, as in the case of the glide plane, for each forbidden reflection all components of the tensor structure factor are determined by at most two independent complex elements [F_1] and [F_2]. There may, however, exist further restrictions on these tensor elements if other symmetry operations of the crystal space group are taken into account. For example, although there are [2_1] screw axes in space group [I2_13], [F_1=F_2=0] and reflections [00\ell\semi\ell=2n+1] remain forbidden because the lattice is body centred, and this applies not only to the dipole–dipole approximation considered here, but also within any other multipole approximation.

Table 1.11.2.1| top | pdf |
The indices [\ell] of the screw-axis/glide-plane forbidden reflections ([n = 0, \pm 1, \pm 2,\ldots]) and independent components of their tensorial structure factors [F^{{\bf H}}_{jk}]

Other components: [F^{{\bf H}}_{yy}=-F^{{\bf H}}_{xx}], [F^{{\bf H}}_{zz}=0], [F^{{\bf H}}_{jk}=F^{{\bf H}}_{kj}]. The direction of the z axis is selected along the corresponding screw axes. The last column lists different types of polarization properties defined in Section 1.11.3[link].

Screw axis or glide plane[\ell][F^{{\bf H}}_{xx}][F^{{\bf H}}_{xy}][F^{{\bf H}}_{xz}][F^{{\bf H}}_{yz}]Type
[2_1] [2n+1] 0 0 [F_1] [F_2] I
[3_1] [3n\pm 1] [F_1] [\mp iF_1] [F_2] [\pm iF_2] II
[3_2] [3n\pm 1] [F_1] [\pm iF_1] [F_2] [\mp iF_2] II
[4_1] [4n\pm 1] 0 0 [F_1] [\pm iF_1] I
[4_1] [4n+2] [F_1] [F_2] 0 0 II
[4_2] [2n+1] [F_1] [F_2] 0 0 II
[4_3] [4n\pm 1] 0 0 [F_1] [\mp iF_1] I
[4_3] [4n+2] [F_1] [F_2] 0 0 II
[6_1] [6n\pm 1] 0 0 [F_1] [\pm iF_1] I
[6_1] [6n\pm 2] [F_1] [\pm iF_1] 0 0 II
[6_1] [6n+3] 0 0 0 0  
[6_2] [3n\pm 1] [F_1] [\pm iF_1] 0 0 II
[6_3] [2n+1] 0 0 0 0  
[6_4] [3n\pm 1] [F_1] [\mp iF_1] 0 0 II
[6_5] [6n\pm 1] 0 0 [F_1] [\mp iF_1] I
[6_5] [6n\pm 2] [F_1] [\mp iF_1] 0 0 II
[6_5] [6n+3] 0 0 0 0  
[c] [2n+1] 0 [F_1] [F_2] 0 II

In Table 1.11.2.1[link], resulting from the dipole–dipole approximation, some reflections still remain forbidden. For instance, in the case of a [6_3] screw axis, there is no anisotropy of susceptibility in the [xy] plane due to the inevitable presence of the threefold rotation axis. For [6_1] and [6_5] axes, the reflections with [\ell = 6n + 3] also remain forbidden because only dipole–dipole interaction (of X-rays) is taken into account, whereas it can be shown that, for example, quadrupole interaction permits the excitation of these reflections.

References

Authier, A. (2005). Dynamical Theory of X-ray Diffraction. Oxford University Press.
Authier, A. (2008). In International Tables for Crystallography, Volume B, Reciprocal Space, edited by U. Shmueli, 3rd ed., pp. 626–646. Heidelberg: Springer.
Dmitrienko, V. E. (1983). Forbidden reflections due to anisotropic X-ray susceptibility of crystals. Acta Cryst. A39, 29–35.
Hahn, Th. (2005). Editor. International Tables for Crystallography, Volume A, Space-Group Symmetry, 5th ed. Heidelberg: Springer.








































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