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

International Tables for Crystallography (2013). Vol. D, ch. 1.11, p. 274

Section 1.11.4. Physical mechanisms for the anisotropy of atomic X-ray susceptibility

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:

1.11.4. Physical mechanisms for the anisotropy of atomic X-ray susceptibility

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Conventional non-resonant Thomson scattering in condensed matter is the result of the interaction of the electric field of the electromagnetic wave with the charged electron subsystem. However, there are also other mechanisms of interaction, e.g. interaction of electromagnetic waves with spin and orbital moments, which was first considered by Platzman & Tzoar (1970[link]) for molecules and solids. They predicted the sensitivity of X-ray diffraction to a magnetic structure of a crystal, as later observed in the pioneering works of de Bergevin & Brunel (de Bergevin & Brunel, 1972[link], 1981[link]; Brunel & de Bergevin, 1981[link]). It is reasonable to describe all X-ray–electron interactions by the Pauli equation (Berestetskii et al., 1982[link]), which is a low-energy approximation to the Dirac equation (typical X-ray energies are [\hbar\omega\ll mc^2] [\approx 0.5\ {\rm MeV}] where m is the electron mass). The equation accounts for charge and spin interaction with the electromagnetic field of the wave, and spin–orbit interaction (Blume, 1985[link], 1994[link]) using the following Hamiltonian:[\eqalignno{H^{\prime}&={{e^2}\over{2mc^2}}\sum_{p}{\bf A}^2({\bf r}_{p})- {{e}\over{mc}}\sum_p{\bf P}_p\cdot{\bf A}({\bf r}_{p})&\cr& \quad-{{e\hbar}\over{mc}}\sum_{p}{\bf s}_{p}\cdot[\nabla\times{\bf A}({\bf r}_{p})] &\cr&\quad-{{e^2\hbar}\over{2(mc^2)^{2}}}\sum_{p}{\bf s}_{p}\cdot [\dot {\bf A}({\bf r}_{p})\times{\bf A}({\bf r}_{p})], &(}]where [{\bf P}_{p}] is the momentum of the pth electron, and [{\bf A}({\bf r}_{p})] is the vector potential of the electromagnetic wave with wavevector [{\bf k}] and polarization [{\bf e}].

Here and below [{\bf A}\!=\!\textstyle\sum_{{\bf k},\alpha}({{2\pi\hbar c^2}/{V\omega_k}})^{1/2} [{\bf e}({\bf k}\alpha)c({\bf k}\alpha)\exp({i{\bf k}\!\cdot\!{\bf r}})] + [{\bf e}^*({\bf k}\alpha)c^+({\bf k}\alpha)\exp({-i{\bf k}\cdot{\bf r}})]], where [V] is a quantization volume, index [\alpha] labels two polarizations of each wave, [{\bf e}({\bf k}\alpha)] are the polarizations vectors, and [c({\bf k}\alpha)] and [c^+({\bf k}\alpha)] are the photon annihilation and creation operators.

Considering X-ray scattering by different atoms in solids as independent processes [in Section 1.2.4[link] of International Tables for Crystallography Volume B, this is called `the isolated-atom approximation in X-ray diffraction'; the validity of this approximation has been discussed by Kolpakov et al. (1978[link])], the atomic scattering amplitude [f], which describes the scattering of a wave with wavevector [{\bf k}] and polarization [{\bf e}] into a wave with wavevector [{\bf k}^{\prime}] and polarization [{\bf e}^{\prime}], can be written as[ f({\bf k},{\bf e},{\bf k}^{\prime},{\bf e}^{\prime})= -{{e^2}\over{mc^2}}f_{jk}({\bf k}^{\prime},{\bf k})e_j^{\prime *}e_k, \eqno(]where the tensor atomic factor [f_{jk}({\bf k}^{\prime},{\bf k})] depends not only on the wavevectors but also on the atomic environment, magnetic and orbital moments etc. From equation ([link] and with the help of perturbation theory (Berestetskii et al., 1982[link]), the atomic factor [f_{jk}({\bf k}^{\prime},{\bf k})] can be expressed as[\eqalignno{&f_{jk}({\bf k}',{\bf k})&\cr&\quad=\sum_{a}p_{a}\Bigg\{ \big\langle a\big|\sum_{p}\exp({i\bf{G}\cdot{\bf r}_{p}})\big|a\big\rangle\delta_{jk} &\cr &\quad\quad-i{{\hbar\omega}\over{mc^{2}}}\big\langle a\big|\sum_{p}\exp({i{\bf G}\cdot{\bf r}_{p}})\left(-i{{[{\bf G}\times{\bf P}_{p}]_{l}}\over{\hbar H^{2}}}A_{jkl}+s_{l}^{p}B_{jkl}\right)\big|a\big\rangle &\cr &\quad\quad-{{1}\over{m}}\sum_{c}\left({{E_{a}-E_{c}}\over{\hbar\omega}}\right){{\langle a|O_{j}^{+}({\bf k}')|c\rangle\langle c|O_{k}({\bf k})|a\rangle}\over{E_{a}-E_{c}+\hbar\omega-i{{\Gamma}\over{2}}}} &\cr&\quad\quad+{{1}\over{m}}\sum_{c}\left({{E_{a}-E_{c}}\over{\hbar\omega}}\right){{\langle a|O_{k}({\bf k})|c\rangle\langle c|O_{j}^{+}({\bf k}')|a\rangle}\over{E_{a}-E_{c}-\hbar\omega}}\Bigg\}, &\cr &&(}]where the first line describes the non-resonant Thomson scattering and [\Gamma] is the energy width of the excited state [|c\rangle]. The second line gives non-resonant magnetic scattering with the spin and orbital terms given by the rank-3 tensors [B_{jkl}] ([link] and [A_{jkl}] ([link], respectively. Compared to the second-to-last line, where the energy denominator can be close to zero, the last line is usually neglected, but sometimes it has to be added to the non-resonant terms, in particular at photon energies far from resonance. The third term gives the dispersion corrections also addressed as resonant scattering, magnetic and non-magnetic. In equation ([link], [E_{a}] and [E_{c}] are the ground and excited states energies, respectively; [p_{a}] is the probability that the incident state of the scatterer [|a\rangle] is occupied; and [{\bf G}={\bf k}-{\bf k}'] is the scattering vector (in the case of diffraction [|{\bf G}|=4\pi\sin\theta/\lambda], where [\theta] is the Bragg angle). The vector operator [{\bf O}({\bf k})] has the form[{\bf O}({\bf k})=\textstyle\sum\limits_{p}\exp({i{\bf k}\cdot {\bf r}_{p}})({\bf P}_{p}- i\hbar[{\bf k}\times {\bf s}_{p}]). \eqno(]The second term in this equation is small and is frequently omitted.

In general, the total atomic scattering factor looks like[\eqalignno{f_{jk}({\bf k}^{\prime},{\bf k},E)&= [f_{0}(|{\bf k}^{\prime}-{\bf k}|)+f_{0}^{\prime}(E)+if_{0}^{\prime\prime}(E)]\delta_{ij}&\cr &\quad+f_{jk}^{\prime}({\bf k}^{\prime},{\bf k},E)+if_{jk}^{\prime\prime}({\bf k}^{\prime},{\bf k},E)+f_{jk}^{\rm mag},&\cr&&(}]where [f_{0}] is the ordinary Thomson (non-resonant) factor, [f_{0}^{\prime}(E)] and [f_{0}^{\prime\prime}(E)] are the isotropic corrections to the dispersion and absorption, which become stronger near absorption edges ([\sim10^{-1}f_{0}]), and [f_{ij}^{\prime}({\bf k}^{\prime},{\bf k},E)] and [f_{ij}^{\prime\prime}({\bf k}^{\prime},{\bf k},E)] are the real and imaginary contributions accounting for resonant anisotropic scattering and are sensitive to the local symmetry of the resonant atom and its magnetism. In the latter case, one should add the tensor [f_{ij}^{\rm mag}] ([\sim10^{-2}][10^{-3}f_{0}]) describing magnetic non-resonant scattering, which is also anisotropic (see the next section).


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