International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 |
International Tables for Crystallography (2013). Vol. D. ch. 1.11, pp. 270-271
Section 1.11.2.2.1. Glide-plane forbidden reflectionsaA. 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 |
Considering first the glide-plane forbidden reflections, there may, for instance, exist a glide plane perpendicular to the
axis, i.e. any point
is transformed by this plane into
. The corresponding matrix of this symmetry operation changes the sign of
,
and the translation vector into
. Substituting (1.11.2.4)
into (1.11.2.1)
and exchanging the integration variables in (1.11.2.3)
, one obtains for the structure factors of reflections
If
is scalar, i.e.
, then
for odd
, hence
vanishes. This is the well known conventional extinction rule for a
glide plane, see International Tables for Crystallography Volume A
(Hahn, 2005
). If, however,
is a tensor, the mirror reflection
changes the signs of the
and
tensor components [as is also obvious from equation (1.11.2.5)
]. As a result, the
and
components should not vanish for
and the tensor structure factor becomes
In general, the elements
and
are complex, and it should be emphasized from the symmetry point of view that they are different and arbitrary for different
and
. 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 , the non-zero tensor elements are just those which vanish for the forbidden reflections:
Here the result is mainly provided by the diagonal elements
, 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.
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