(International Tables for Crystallography) Tensorial properties of local crystal susceptibilities

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2013). Vol. D. ch. 1.11, p. 275

Section 1.11.5. Non-resonant magnetic scattering

V. E. Dmitrienko,^a ^* A. Kirfel^b and E. N. Ovchinnikova^c

^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: dmitrien@crys.ras.ru

1.11.5. Non-resonant magnetic scattering

| top | pdf |

Far from resonance ( $[\hbar \omega\gg E_c-E_a]$ ), the non-resonant parts of the scattering factor, [f_0] and $[f_{ij}^{\rm mag}]$ , described by the first two terms in (1.11.4.3) are the most important. In the classical approximation (Brunel & de Bergevin, 1981), there are four physical mechanisms (electric or magnetic, dipolar or quadrupolar) describing the interaction of an electron and its magnetic moment with an electromagnetic wave, causing the re-emission of radiation. The non-resonant magnetic term $[f^{\rm magn}]$ is small compared to the charge (Thomson) scattering owing (a) to small numbers of unpaired (magnetic) electrons and (b) to the factor $[\hbar\omega/mc^2]$ of about 0.02 for a typical X-ray energy $[\hbar\omega=10\ {\rm keV}]$ . This is the reason why it is so difficult to observe non-resonant magnetic scattering with conventional X-ray sources (de Bergevin & Brunel, 1972, 1981; Brunel & de Bergevin, 1981), in contrast to the nowadays normal use of synchrotron radiation.

Non-resonant magnetic scattering yields polarization properties quite different from those obtained from charge scattering. Moreover, it can be divided into two parts, which are associated with the spin and orbital moments. In contrast to the case of neutron magnetic scattering, the polarization properties of these two parts are different, as described by the tensors (Blume, 1994) $[A_{ijk}=-2(1-{\bf k}\cdot{\bf k}^{\prime}/k^2)\epsilon_{ijk},\eqno(1.11.5.1)]$ $[\eqalignno{B_{ijk}&=\epsilon_{ijk}-\big[\epsilon_{ilk}k^{\prime}_l k^{\prime}_j -\epsilon_{jlk}k_l k_i +\textstyle{{1}\over{2}}\epsilon_{ijl}(k^{\prime}_l k_k+k_l k^{\prime}_k)&\cr &\quad- \textstyle{{1}\over{2}}[{\bf k}\times{\bf k}^{\prime}]_i\delta_{jk} - \textstyle{{1}\over{2}}[{\bf k}\times {\bf k}^{\prime}]_j\delta_{ik}\big]/k^2,&(1.11.5.2)}]$ where $[\epsilon_{ijk}]$ is a completely antisymmetric unit tensor (the Levi-Civita symbol).

Being convoluted with polarization vectors (Blume, 1985; Lovesey & Collins, 1996; Paolasini, 2012), the non-resonant magnetic term can be rewritten as $[\eqalignno{&f^{\rm magn}_{\rm nonres}({\bf G})&\cr&\quad=-i{{\hbar\omega}\over{mc^{2}}}\big\langle a\big|\textstyle\sum\limits_{p} ({\bf A}\cdot [{\bf G}\times {\bf P}_{p}]/\hbar k^{2}+{\bf B} \cdot{\bf s}_p) \exp({i{\bf G}\cdot{\bf r}_{p}})\big|a\big\rangle,&\cr&&(1.11.5.3)}]$ with vectors $[\bf A]$ and $[\bf B]$ given by $[{\bf A}=[{\bf e}^{\prime *}\times{\bf e}],\eqno(1.11.5.4)]$ $[\eqalignno{{ \bf B}&=[{\bf e}^{\prime *}\times{\bf e}]-\{[{\bf k}\times{\bf e}]({\bf k}\cdot{\bf e}^{\prime *})-[{\bf k}^{\prime}\times{\bf e}^{\prime *}]({\bf k}^{\prime}\cdot{\bf e})&\cr &\quad+ [{\bf k}^{\prime}\times{\bf e}^{\prime *}]\times[{\bf k}\times{\bf e}]\}/k^2.&(1.11.5.5)}]$ According to (1.11.5.4) and (1.11.5.5), the polarization dependences of the spin and orbit contributions to the atomic scattering factor are significantly different. Consequently, the two contributions can be separated by analysing the polarization of the scattered radiation with the help of an analyser crystal (Gibbs et al., 1988). Usually the incident (synchrotron) radiation is σ-polarized, i.e. the polarization vector is perpendicular to the scattering plane. If due to the orientation of the analysing crystal only the σ-polarized part of the scattered radiation is recorded, we can see from (1.11.5.4) that the orbital contribution to the scattering atomic factor vanishes, whereas it differs from zero considering the $[\sigma\to\pi]$ scattering channel.

References

Bergevin, F. de & Brunel, M. (1972). Observation of magnetic superlattice peaks by X-ray diffraction on an antiferromagnetic NiO crystal. Phys. Lett. A, 39, 141–142.Google Scholar

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.Google Scholar

Blume, M. (1985). Magnetic scattering of X-rays. J. Appl. Phys. 57, 3615–3618.Google Scholar

Blume, M. (1994). Resonant X-ray diffraction and polarization analysis at the iron K-edge. In Resonant Anomalous X-ray Scattering. Theory and Applications, edited by G. Materlik, C. J. Sparks & K. Fischer, pp. 91–97. Amsterdam: North-Holland.Google Scholar

Brunel, M. & de Bergevin, F. (1981). Diffraction of X-rays by magnetic materials. II. Measurements on antiferromagnetic Fe₂O₃. Acta Cryst. A37, 324–331.Google Scholar

Gibbs, D., Harshmann, D. R., Isaacs, E. D., McWhan, D. B., Mills, D. & Vettier, C. (1988). Polarization and resonance properties of magnetic X-ray scattering in holmium. Phys. Rev. Lett. 61, 1241–1244.Google Scholar

Lovesey, S. W. & Collins, S. P. (1996). X-ray Scattering and Absorption by Magnetic Materials. Oxford: Clarendon Press.Google Scholar

Paolasini, L. (2012). Magnetism in condensed matter: resonant and magnetic scattering by synchrotron radiation. In Italian School on Magnetism, Pavia 2012, pp. 1–83. Universita di Pavia.Google Scholar

International Tables for Crystallography (2013). Vol. D. ch. 1.11, p. 275