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

International Tables for Crystallography (2013). Vol. D. ch. 1.11, pp. 272-274

Section 1.11.3. Polarization properties and azimuthal dependence

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:

1.11.3. Polarization properties and azimuthal dependence

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There are two important properties that distinguish forbidden reflections from conventional (`allowed') ones: non-trivial polarization effects and strong azimuthal dependence of intensity (and sometimes also of polarization) corresponding to the symmetry of the direction of the scattering vector. The azimuthal dependence means that the intensity and polarization properties of the reflection can change when the crystal is rotated around the direction of the reciprocal-lattice vector, i.e. they change with the azimuthal angle of the incident wavevector k defined relative to the scattering vector. The polarization and azimuthal properties, both mainly determined by symmetry, are two of the most informative characteristics of forbidden reflections. A third one, energy dependence, is determined by physical interactions, electronic and/or magnetic, where the role of symmetry is indirect but nevertheless also important (e.g. in splitting of atomic levels etc., see Section 1.11.4[link]).

In the kinematical theory, usually used for weak reflections, one obtains for unpolarized incident radiation the intensity of a conventional reflection as given by[I_{{\bf H}}=A_{{\bf H}}|F({\bf H})|^2\left(1+ \cos^2 2\theta \right)/2,\eqno(]where [\theta] is the Bragg angle, [F({\bf H})] is the scalar structure factor of reflection [{\bf H}], and [A_{{\bf H}}] is a scale factor, which depends on the incident beam intensity, the sample volume, the geometry of diffraction etc. (see International Tables for Crystallography Volume B ), and can be set to [A_{{\bf H}}=1] hereafter.

If the structure factor is a tensor of rank 2, then the reflection intensity obtained with incident and reflected radiation with polarization vectors, respectively, [{\bf e}] and [{\bf e}^{\prime}] (prepared and analysed by a corresponding polarizer and analyser) is given by[ I_{{\bf H}}({\bf e}^{\prime},{\bf e}) =|F_{jk}({\bf H})e_{j}^{\prime *}e_{k}|^2,\eqno(]where the star denotes the complex conjugate. The maximum of this expression is reached when [{\bf e}^{\prime}] is equal to the polarization of the diffracted beam. In general, the polarization of the diffracted secondary radiation, [{\bf e}^{\prime}_{{\bf H}}], depends on the incident beam polarization [{\bf e}]:[{\bf e}^{\prime}_{{\bf H}}={\bf C}_{{\bf H}}/\sqrt{|{\bf C}_{{\bf H}}|^2},\eqno(]where[({\bf C}_{{\bf H}})_j= \left[{\bf k}^{2}F_{jk}({\bf H})-k^{\prime}_jk^{\prime}_nF_{nk}({{\bf H}})\right]e_{k}\eqno(](the second term in this expression provides orthogonality between the polarization vector and the corresponding wavevector). If the polarization of the diffracted beam is not analysed, the total intensity of the diffracted beam [I^{\rm tot}_{{\bf H}}({\bf e})] is equal to [I_{{\bf H}}({\bf e}^{\prime}_{{\bf H}},{\bf e})]. If the tensor structure factor is a direct product of two vectors, then the polarization of the diffracted beam does not depend on the incident polarization.

The polarization analysis of forbidden reflections frequently uses the linear polarization vectors [{\boldsigma}] and [{\boldpi}]. Vector [{\boldsigma}] is perpendicular to the scattering plane, whereas the vectors [{\boldpi}] and [{\boldpi}^\prime] are in the scattering plane so that [{\boldsigma},{\boldpi},{\bf k}] and [{\boldsigma},{\boldpi}^\prime,{\bf k}^\prime] form right-hand triads. Note that the components of the polarization vectors, [{\boldsigma}=(\sigma_x,\sigma_y,\sigma_z)] etc., change with the azimuthal angle if the crystal is rotated about the scattering vector.

In special cases, circular polarizations are very useful and sometimes even indispensable, because they enable us to distinguish right- and left-hand crystals or to unravel interferences between magnetic and electric scattering (see below).

If the incident radiation is [{\boldsigma}]- or [{\boldpi}]-polarized or non-polarized, then the total reflection intensities for these three cases are given by the following expressions:[I_{\boldsigma}=I_{{\bf H}}({\boldsigma},{\boldsigma}) +I_{{\bf H}}({\boldpi}^\prime,{\boldsigma}), \eqno(][I_{\boldpi}=I_{{\bf H}}({\boldsigma},{\boldpi}) +I_{{\bf H}}({\boldpi}^\prime,{\boldpi}),\eqno(][I_{{\bf H}}=(I_{\boldsigma}+I_{\boldpi})/2.\eqno(]A more general approach uses the Stokes parameters for the description of partially polarized X-rays and the Müller matrices for the scattering process (see a survey by Detlefs et al., 2012[link]). This issue will, however, not be discussed further since there is no principal difference to conventional optics.

Let us consider the polarization and azimuthal characteristics of screw-axis forbidden reflections listed in Table[link]. These characteristics are rather different for two types of reflections: type I reflections are those for which [F_{xx}=F_{yy} =F_{xy}=0], while all other reflections constitute the rest, type II.

The type-I forbidden reflections have the simplest polarization properties. From equations ([link]–([link] and Table[link], one obtains [I_{{\bf H}}({\boldsigma},{\boldsigma})= I_{{\bf H}}({\boldpi^\prime},{\boldpi}) = 0] and [I_{{\bf H}}=I_{\boldsigma}=] [I_{\boldpi}=I_{{\bf H}}({\boldsigma},{\boldpi})= I_{{\bf H}}({ {\boldpi}^\prime},{\boldsigma})], where [I_{{\bf H}}({{\boldpi}^\prime},{\boldsigma})] is given by[\eqalignno{I_{{\bf H}}({\boldpi}^\prime,{\boldsigma})&=[|F_1|^2\sin^2\varphi+ |F_2|^2\cos^2\varphi, &(\cr &\quad -Re(F_1F_2^*)\sin 2\varphi] \cos^2 \theta &(}]for a [2_1] screw axis and[I_{{\bf H}}({\boldpi}^\prime,{\boldsigma})=|F_1|^2 \cos^2 \theta \eqno(]for [4_1], [4_3], [6_1] and [6_5] screw axes, where [\varphi] is the azimuthal angle of crystal rotation about the scattering vector [{\bf H}]. Thus, [{\boldsigma}]-polarized incident radiation results in reflected radiation with [{\boldpi}] polarization and vice versa; and unpolarized incident radiation gives unpolarized reflected radiation.

Note that there is no azimuthal dependence of intensity in ([link]. Nevertheless, the phase of the diffracted beams changes with azimuthal rotation, as might be observed via interference with another scattering process, for example, with multiple (Renninger) diffraction. Such measurements could also be useful for determining the phases of the complex [F_1] and [F_2] above.

The polarization properties of type-II reflections are quite distinct from those of type-I reflections. The intensities belonging to various polarization channels, i.e. combinations of primary and secondary beam polarizations ([{\boldsigma} \to {\boldsigma}], [{\boldsigma} \to {\boldpi}^\prime] etc.), exhibit different azimuthal symmetries for different screw axes.

For [3_1] and [3_2] screw axes, the azimuthal symmetry is threefold:[\eqalignno{I_{\boldsigma}&=|F_1|^2(1+\sin^2\theta)+|F_2|^2\cos^2\theta+D(\varphi),&\cr I_{\boldpi}&=|F_1|^2\sin^2\theta (1+\sin^2\theta)+|F_2|^2\cos^2\theta+D(\varphi), &\cr I_{{\bf H}}&=|F_1|^2(1+\sin^2\theta)^2/2+|F_2|^2\cos^2\theta+D(\varphi) ,&\cr&&(}]where [D(\varphi)=\sin 2\theta\,[Re(F_1F_2^*)\cos 3\varphi\mp Im(F_1F_2^*)\sin 3\varphi]]. The [\mp] sign corresponds to [F_{xy}=\pm iF_{xx}] in Table[link].

For [4_1], [4_3], and [4_2] screw axes, the symmetry is fourfold:[\eqalignno{I_{\boldsigma}&=|F_1|^2B(\varphi)+|F_2|^2C(\varphi) &\cr&\quad +Re(F_1F_2^*)\cos^2\theta\sin 4\varphi, &\cr I_{\boldpi}&=\sin^2\theta\big[|F_1|^2C(\varphi)+|F_2|^2B(\varphi) &\cr&\quad +Re(F_1F_2^*)\cos^2\theta\sin 4\varphi\big], &\cr I_{{\bf H}}&=(I_{\boldsigma}+I_{\boldpi})/2,&(}]where [B(\varphi)=1-\cos^2\theta\sin^2 2\varphi] and [C(\varphi)=1-\cos^2\theta\cos^2 2\varphi].

No azimuthal dependence exists for the screw axes [6_1], [6_2], [6_4] and [6_5]:[\eqalignno{I_{\boldsigma}&=|F_1|^2(1+\sin^2\theta), &\cr I_{\boldpi}&=|F_1|^2(1+\sin^2\theta)\sin^2\theta, &\cr I_{{\bf H}}&=|F_1|^2(1+\sin^2\theta)^2/2.&(}]

Unlike the type-I reflections, the intensities of the type-II reflections are different for [{\boldsigma}]- and [{\boldpi}]-polarized incident beams. What is more interesting is that type-II reflections are `chiral', i.e. their intensities differ for right-hand and left-hand circularly polarized incident radiation. As an example, we take the type-II back-reflections ([\theta=\pi/2]) for three- and sixfold screw axes. We find from Table[link] and equations ([link] and ([link] that only the beams with definite circular polarization (right-hand if [F_{xy}= iF_{xy}] and left-hand if [F_{xy}=-iF_{xy}]) are reflected and that the back-reflected radiation has the same circular polarization in both cases. For opposite polarization, the reflection is absent. Thus, under these circumstances, the crystal may be regarded as a circular polarizer or analyser. If [\theta \,\lt\, \pi/2], the eigen-polarizations are elliptic and the axial ratio of the polarization ellipse is equal to [\sin \theta ] for the sixfold screw axes (whereas for the three- and fourfold screw axes, this ratio depends on the parameters [F_1] and [F_2]).

The chirality of type-II reflections can be used to distinguish enantiomorphous crystals. Although this was suggested many years ago, its potential was only recently proved by experiments, first on α-quartz, SiO2, and berlinite, AlPO4 (Tanaka et al., 2008[link]; Tanaka, Kojima et al., 2010[link]), later for tellurium (Tanaka, Collins et al., 2010[link]). All three candidates crystallize in the space groups [P3_121] or [P3_221]. The case of tellurium is particularly interesting because standard X-ray diffraction methods for absolute structure determination fail in elemental crystals.

The non-trivial polarization and azimuthal properties discussed above are, in most cases, determined by symmetry, and they are used as evidence confirming the origin of the forbidden reflections. They are also used for obtaining detailed information about anisotropy of local susceptibility and, hence, about structural and electronic properties. For instance, careful analysis of polarization and azimuthal dependences allows one to distinguish between different scenarios of the Verwey phase transition in magnetite, Fe3O4 – a longstanding and confusing problem (see Hagiwara et al., 1999[link]; García et al., 2000[link]; Renevier et al., 2001[link]; García & Subías, 2004[link]; Nazarenko et al., 2006[link]; Subías et al., 2012[link]).


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