Magnetic X-ray structure factor as a function of photon polarization

Coppens, P.; Su, Z.; Becker, P. J.

doi:10.1107/97809553602060000615

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 8.7, pp. 733-734

Section 8.7.4.10.2. Magnetic X-ray structure factor as a function of photon polarization

P. Coppens,^a Z. Su^b and P. J. Becker^c

^a 732 NSM Building, Department of Chemistry, State University of New York at Buffalo, Buffalo, NY 14260-3000, USA,^bDigital Equipment Co., 129 Parker Street, PKO1/C22, Maynard, MA 01754-2122, USA, and ^cEcole Centrale Paris, Centre de Recherche, Grand Voie des Vignes, F-92295 Châtenay Malabry CEDEX, France

8.7.4.10.2. Magnetic X-ray structure factor as a function of photon polarization

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Some geometrical definitions are summarized in Fig. 8.7.4.1, where parallel $[(\|)]$ and perpendicular $[(\perp)]$ polarizations will be chosen in order to describe the electric field of the incident and diffracted beams. In this two-dimensional basis, vectors A and B of (8.7.4.98) can be written as (2 × 2) matrices: $[\matrix{ {\bf A} = \sin^2\theta \left(\matrix{ 0 &-(\hat{\bf k} + \hat{\bf k}') \cr (\hat{\bf k}+ \hat{\bf k}') & 2(\hat{\bf k} \times \hat{\bf k}')}\right) &{\perp \atop \|} \cr \vphantom{} \cr{ \hfil\, i\rightarrow\, \, \, \, \, \, \, \,\perp\hfil \qquad \, \, \, \, \, \, \, \,\,\,\,\,\,\,\,\parallel\hfil} &\uparrow f\cr} ]$ (i and f refer to the incident and diffracted beams, respectively); $[\let\normalbaselines\relax\openup3pt{\bf B} = \left(\matrix{ \hat{\bf k} \times\hat{\bf k}' \cr 2\hat{\bf k}\sin^2\theta} \matrix{-2\hat{\bf k}'\sin^2\theta \cr \hat {\bf k} \times\hat{\bf k}'}\right). \eqno (8.7.4.100)]$ By comparison, for the Thomson scattering, $[\hat{\boldvarepsilon}\cdot\hat{\boldvarepsilon}' = \left(\matrix{1&0 \cr0&\cos2\theta}\right). \eqno (8.7.4.101)]$ The major difference with Thomson scattering is the occurrence of off-diagonal terms, which correspond to scattering processes with a change of polarization. We obtain for the structure factors $[A_{if}]$ : $[\eqalign{ A_{\perp\perp} &= F-i\,{\hbar\omega \over mc^2}(\hat{\bf k}\times\hat{\bf k}')\cdot {\bf M}_S \cr A_{\perp\|} &= -2i\, {\hbar\omega \over mc^2}\, \sin^2\theta\{(\hat{\bf k}+\hat{\bf k}')\cdot{\bf M}_L+\hat{\bf k}'\cdot {\bf M}_S\} \cr A_{\|\perp} &= 2i\, {\hbar \omega \over mc^2}\, \sin2\theta\{(\hat{\bf k} + \hat{\bf k}')\cdot {\bf M}_L +\hat{\bf k}'\cdot {\bf M}_S\} \cr A_{\|\|} &= F\cos^2\theta - i{\hbar\omega \over mc^2}\, (\hat{\bf k}\times\hat{\bf k}')\cdot\{4\sin^2 \theta\,{\bf M}_L{\bf M}_S\}.} \eqno (8.7.4.102)]$ For a linear polarization, the measured intensity in the absence of diffracted-beam polarization analysis is $[I_\alpha = |A_{\perp\perp}\cos\alpha+A_{\|\perp}\sin\alpha|^2 + |A_{\perp\|}\cos \alpha+A_{\|\|}\sin\alpha|^2, \eqno (8.7.4.103)]$ where α is the angle between E and $[\hat{\boldvarepsilon}]$ . In the centrosymmetric system, without anomalous scattering, no interference term occurs in (8.7.4.103). However, if anomalous scattering is present, F = F′ + iF′′, and terms involving [F''M_S] or [F''M_L] appear in the intensity expression.

Figure 8.7.4.1| top | pdf |

Some geometrical definitions.

The radiation emitted in the plane of the electron or positron orbit is linearly polarized. The experimental geometry is generally such that $[(\hat{\bf k},\hat{\bf k}')]$ is a vertical plane. Therefore, the polarization of the incident beam is along $[\hat{\boldvarepsilon}]$ (α = 0). If a diffracted-beam analyser passes only $[\|]$ components of the diffracted beam, one can measure $[|A_{\perp\|}|^2]$ , and thus eliminate the charge scattering.

For non-polarized radiation (with a rotating anode, for example), the intensity is $[I=\textstyle{1\over2}[|A_{\|\|}|^2+|A_{\|\perp}|^2+ |A_{\perp\|}|^2 + |A_{\perp\perp}|^2]. \eqno (8.7.4.104)]$ The radiation emitted out of the plane of the orbit contains an increasing amount of circularly polarized radiation. There also exist experimental devices that can produce circularly polarized radiation. For such incident radiation, $[E_\| = \pm iE_\perp \eqno (8.7.4.105)]$ for left- or right-polarized photons. If E′ is the field for the diffracted photons, $[\eqalign{ E'_\perp &=A_{\perp\perp} + iA_{\|\perp} \cr E'_\| &= A_{\|\perp} + iA_{\|\|}.} \eqno (8.7.4.106)]$ In this case, `mixed-polarization' contributions are in phase with F, leading to a strong interference between charge and magnetic scattering.

The case of radiation with a general type of polarization is more difficult to analyse. The most elegant formulation involves Stokes vectors to represent the state of polarization of the incident and scattered radiation (see Blume & Gibbs, 1988).

References

Blume, M. & Gibbs, D. (1988). Polarization dependence of magnetic X-ray scattering. Phys. Rev. B, 37, 1779–1789.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 8.7, pp. 733-734