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

International Tables for Crystallography (2006). Vol. C, ch. 8.7, p. 733

Section 8.7.4.10.1. Introduction

P. Coppens,a Z. Sub and P. J. Beckerc

a732 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

8.7.4.10.1. Introduction

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In addition to the usual Thomson scattering (charge scattering), there is a magnetic contribution to the X-ray amplitude (de Bergevin & Brunel, 1981[link]; Blume, 1985[link]; Brunel & de Bergevin, 1981[link]; Blume & Gibbs, 1988[link]). In units of the chemical radius [r_e] of the electron, the total scattering amplitude is [A_x=F_C+F_M, \eqno (8.7.4.95)]where FC is the charge contribution, and FM the magnetic part.

Let [\hat{\boldvarepsilon}] and [\hat{\boldvarepsilon}'] be the unit vectors along the electric field in the incident and diffracted direction, respectively. k and k′ denote the wavevectors for the incident and diffracted beams. With these notations, [F_C=F({\bf h})\, \hat{\boldvarepsilon}\cdot \hat{\boldvarepsilon}', \eqno (8.7.4.96)]where F(h) is the usual structure factor, which was discussed in Section 8.7.3[link] [see also Coppens (2001[link])]. [F_M = -i\, {h\omega \over mc^2}\, \{{\bf M}_L({\bf h})\cdot {\bf A}+{\bf M}_S({\bf h})\cdot {\bf B}\}. \eqno (8.7.4.97)][{\bf M}_L] and [{\bf M}_S] are the orbital and spin-magnetization vectors in reciprocal space, and A and B are vectors that depend in a rather complicated way on the polarization and the scattering geometry: [\eqalignno{ {\bf A} &= 4\sin^2\theta \,\hat{\boldvarepsilon}\times\hat{\boldvarepsilon}' - (\hat{\bf k}+\hat{\boldvarepsilon})(\hat{\bf k}\cdot\hat{\boldvarepsilon}') + (\hat{\bf k}' \times \hat{\boldvarepsilon}')(\hat{\bf k}'\cdot\hat{\boldvarepsilon}) \cr {\bf B} &= \hat{\boldvarepsilon}'\times \hat{\boldvarepsilon} + ({\bf k}'+{\boldvarepsilon}')({\bf k}\cdot{\boldvarepsilon}) - (\hat{\bf k}\times\hat{\boldvarepsilon})(\hat{\bf k}\cdot\hat{\boldvarepsilon}') \cr &\quad - (\hat{\bf k}'\times\hat{\boldvarepsilon}') \times (\hat{\bf k}\times \hat{\boldvarepsilon}). & (8.7.4.98)}]For comparison, the magnetic neutron scattering amplitude can be written in the form [F^{\rm neutron}_M = [{\bf M}_L({\bf h}) + M_S({\bf h})]\cdot {\bf C}, \eqno (8.7.4.99)]with [{\bf C}=\hat{\bf h}\times{\boldsigma}\times \hat{\bf h}].

From (8.7.4.99)[link], it is clear that spin and orbital contributions cannot be separated by neutron scattering. In contrast, the polarization dependencies of [{\bf M}_L] and [{\bf M}_S] are different in the X-ray case. Therefore, owing to the well defined polarization of synchrotron radiation, it is in principle possible to separate experimentally spin and orbital magnetization.

However, the prefactor [(h\omega/mc^2)\sim10^{-2}] makes the magnetic contributions weak relative to charge scattering. Moreover, FC is roughly proportional to the total number of electrons, and FM to the number of unpaired electrons. As a result, one expects [|F_M/F_C|] to be about 10−3.

It should also be pointed out that FM is in quadrature with FC. In many situations, the total X-ray intensity is therefore [I_x = |F_C|^2 + |F_M|^2.]Thus, under these conditions, the magnetic effect is typically 10−6 times the X-ray intensity.

Magnetic contributions can be detected if magnetic and charge scattering occur at different positions (antiferromagnetic type of ordering). Furthermore, Blume (1985[link]) has pointed out that the photon counting rate for [|F_M|^2] at synchrotron sources is of the same order as the neutron rate at high-flux reactors.

Finally, situations where the `interference' [F_CF_M] term is present in the intensity are very interesting, since the magnetic contribution becomes 10−3 times the charge scattering.

The polarization dependence will now be discussed in more detail.

References

Bergevin, F. de & Brunel, M. (1981). Diffraction of X-rays by magnetic materials. I. General formulae and measurements on ferro- and ferrimagnetic compounds. Acta Cryst. A37, 314–324.
Blume, M. (1985). Magnetic scattering of X-rays. J. Appl. Phys. 57, 3615–3618.
Blume, M. & Gibbs, D. (1988). Polarization dependence of magnetic X-ray scattering. Phys. Rev. B, 37, 1779–1789.
Brunel, M. & de Bergevin, F. (1981). Diffraction of X-rays by magnetic materials. II. Measurements on antiferromagnetic Fe2O3. Acta Cryst. A37, 324–331.
Coppens, P. (2001). The structure factor. International tables for crystallography, Vol. B, edited by U. Shmueli, Chap. 1.2. Dordrecht: Kluwer Academic Publishers.








































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