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. 277-278
Section 1.11.6.3. Hidden internal symmetry of the dipole–quadrupole tensors in resonant atomic factors
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 |
It is fairly obvious from expressions (1.11.6.3) and (1.11.6.16)
that in the non-magnetic case the symmetric and antisymmetric third-rank tensors,
and
, which describe the dipole–quadrupole contribution to the X-ray scattering factor, are not independent: the antisymmetric part, which is also responsible for optical-activity effects, can be expressed via the symmetric part (but not vice versa). Indeed, both of them can be described by a symmetric third-rank tensor
resulting from the second-order Born approximation (1.11.6.3)
,
where
From equation (1.11.6.17)
, one can infer that the symmetry restrictions for
and
are the same. Then it can be seen that
can be expressed via
.
For any symmetry, and
have the same number of independent elements (with a maximum 18 for site symmetry 1). Thus, one can reverse equation (1.11.6.17)
and express
directly in terms of
:
Using equations (1.11.6.18) and (1.11.6.20)
, one can express all nine elements of
through
:
according to which the antisymmetric part of the dipole–quadrupole term is a linear function of the symmetric one [however, not vice versa: equations (1.11.6.21)
cannot be reversed].
Note that the equations (1.11.6.21) impose an additional restriction on
, which applies to all atomic site symmetries:
This is, in fact, a well known result: the pseudo-scalar part of
vanishes in the dipole–quadrupole approximation used in equation (1.11.6.3)
. Thus, for point symmetry 1,
has only eight independent elements rather than nine. This additional restriction works in all cases of higher symmetries provided the pseudo-scalar part is allowed by the symmetry (i.e. point groups 2, 3, 4, 6, 222, 32, 422, 622, 23 and 432). All other symmetry restrictions on
arise automatically from equation (1.11.6.21)
taking into account the symmetry of
[symmetry limitations on
and
for all crystallographic point groups can be found in Sirotin & Shaskolskaya (1982
) and Nye (1985
)].
Let us consider two examples, ZnO and anatase, TiO2, where the dipole–dipole contributions to forbidden reflections vanish, whereas both the symmetric and antisymmetric dipole-quadrupole terms are in principal allowed. In these crystals, the dipole–quadrupole terms have been measured by Goulon et al. (2007) and Kokubun et al. (2010
).
In ZnO, crystallizing in the wurtzite structure, the 3m symmetry of the atomic positions imposes the following restrictions on :
where
,
,
,
are energy-dependent complex tensor elements [keeping the notations by Sirotin & Shaskolskaya (1982
), the x axis is normal to the mirror plane, the y axis is normal to the glide plane and the z axis corresponds to the c axis of ZnO]. If we suppose these restrictions for Zn at
, then for the other Zn at
, which is related to the first site by the glide plane, there is the following set of elements:
. Therefore, the structure factors of the glide-plane forbidden reflections are proportional to
.
For the symmetric and antisymmetric parts one obtains from equations (1.11.6.17) and (1.11.6.18)
the non-zero components
and
Physically, we can expect that because
survives even for tetrahedral symmetry
, whereas
is non-zero owing to a deviation from tetrahedral symmetry; in ZnO, the local coordinations of the Zn positions are only approximately tetrahedral.
In the anatase structure of TiO2, the symmetry of the atomic positions imposes restrictions on the tensors
[keeping the notations of Sirotin & Shaskolskaia (1982
): the x and y axes are normal to the mirror planes, and the z axis is parallel to the c axis]:
where
and
are energy-dependent complex parameters. If we apply these restrictions to the Ti atoms at
and
, then for the other two inversion-related Ti atoms at
and
(centre
), the parameters are
and
.
For the symmetric and antisymmetric parts one obtains as non-vanishing componentsand
It is important to note that the symmetric part of the atomic factor can be affected by a contribution from thermal-motion-induced dipole–dipole terms. The latter terms are tensors of rank 3 proportional to the spatial derivatives
, which take the same tensor form as
but are not related to
by equations (1.11.6.21)
. In ZnO, which was studied in detail by Collins et al. (2003
), the thermal-motion-induced contribution is rather significant, while for anatase the situation is less clear.
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