Screw-axis forbidden reflections

Dmitrienko, V. E.; Kirfel, A.; Ovchinnikova, E. N.

doi:10.1107/97809553602060000910

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

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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. Kirfel^b and E. N. Ovchinnikova^c

^a A. 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; see Table 1.11.2.1). 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.

$[\ell]$	$[F^{{\bf H}}_{xx}]$	$[F^{{\bf H}}_{xy}]$	$[F^{{\bf H}}_{xz}]$	$[F^{{\bf H}}_{yz}]$	Type
	0	0			I
$[3n\pm 1]$		$[\mp iF_1]$		$[\pm iF_2]$	II
$[3n\pm 1]$		$[\pm iF_1]$		$[\mp iF_2]$	II
$[4n\pm 1]$	0	0		$[\pm iF_1]$	I
			0	0	II
			0	0	II
$[4n\pm 1]$	0	0		$[\mp iF_1]$	I
			0	0	II
$[6n\pm 1]$	0	0		$[\pm iF_1]$	I
$[6n\pm 2]$		$[\pm iF_1]$	0	0	II
	0	0	0	0
$[3n\pm 1]$		$[\pm iF_1]$	0	0	II
	0	0	0	0
$[3n\pm 1]$		$[\mp iF_1]$	0	0	II
$[6n\pm 1]$	0	0		$[\mp iF_1]$	I
$[6n\pm 2]$		$[\mp iF_1]$	0	0	II
	0	0	0	0
	0			0	II

In Table 1.11.2.1, 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.Google Scholar

International Tables for Crystallography (2013). Vol. D. ch. 1.11, p. 271