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. 271

Section 1.11.2.2.2. Screw-axis 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

1.11.2.2.2. 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

Dmitrienko, V. E. (1983). Forbidden reflections due to anisotropic X-ray susceptibility of crystals. Acta Cryst. A39, 29–35.








































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