Atom-centred expansion

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. 729-730

Section 8.7.4.5.1. Atom-centred expansion

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

8.7.4.5.1. Atom-centred expansion

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We first consider the case where spins are localized on atoms or ions, as it is to a first approximation for compounds involving transition-metal atoms. The magnetization density is expanded as $[{\bf m}_S({\bf r}) = \textstyle\sum\limits_j\langle {\bf S}_j\rangle\,\langle s_j({\bf r}-{\bf R}_j)\rangle, \eqno (8.7.4.54)]$ where $[\langle{\bf S}_j\rangle]$ is the spin at site j, and $[\langle s_j\rangle]$ the thermally averaged normalized spin density $[f_j({\bf h})]$ , the Fourier transform of $[s_j({\bf r})]$ , is known as the `magnetic form factor'. Thus, $[{\bf M}({\bf h})=\textstyle\sum\limits_j\langle{\bf S}_j\rangle\;f_j({\bf h}) T_j \exp \,(2\pi{\bf h}\cdot {\bf R}_j), \eqno (8.7.4.55)]$ where [T_j] and $[{\bf R}_j]$ are the Debye–Waller factor and the equilibrium position of the jth site, respectively.

Most measurements are performed at temperatures low enough to ensure a fair description of [T_j] at the harmonic level (Coppens, 2001). [T_j] represents the vibrational relaxation of the open-shell electrons and may, in some situations, be different from the Debye–Waller factor of the total charge density, though at present no experimental evidence to this effect is available.

8.7.4.5.1.1. Spherical-atom model

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In the crudest model, $[s_j({\bf r})]$ is approximated by its spherical average. If the magnetic electrons have a wavefunction radial dependence represented by the radial function U(r), the magnetic form factor is given by $[f(h)=\textstyle\int\limits^\infty_0\,U^2(r)4 \pi r^2\,{\rm d} r \,j_0(2\pi hr) = \langle\, j_0\rangle, \eqno (8.7.4.56)]$ where [j_0] is the zero-order spherical Bessel function. For free atoms and ions, these form factors can be found in IT IV (1974).

One of the important features of magnetic neutron scattering is the fact that, to a first approximation, closed shells do not contribute to the form factor. Thus, it is a unique probe of the electronic structure of heavy elements, for which theoretical calculations even at the atomic level are questionable. Relativistic effects are important. Theoretical relativistic form factors can be used (Freeman & Desclaux, 1972; Desclaux & Freeman, 1978). It is also possible to parametrize the radial behaviour of U. A single contraction-expansion model [κ refinement, expression (8.7.3.6)] is easy to incorporate.

8.7.4.5.1.2. Crystal-field approximation

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Crystal-field effects are generally of major importance in spin magnetism and are responsible for the spin state of the ions, and thus for the ground-state configuration of the system. Thus, they have to be incorporated in the model.

Taking the case of a transition-metal compound, and neglecting small contributions that may arise from spin polarization in the closed shells (see Subsections 8.7.4.9 and 8.7.4.10), the normalized spin density can be written by analogy with (8.7.3.76) as $[s({\bf r}) = \textstyle\sum\limits^5_{i=1}\, \textstyle\sum\limits^5_{j\ge i}\,D_{ij}\, d_i ({\bf r}) d_j ({\bf r}), \eqno (8.7.4.57)]$ where $[D_{ij}]$ is the normalized spin population matrix. If $[\rho_{d\uparrow}]$ and $[\rho_{d\downarrow}]$ are the densities of a given spin, $[s({\bf r}) = {\rho_{d\uparrow}-\rho_{d\downarrow} \over n_\uparrow - n_\downarrow}, \eqno (8.7.4.58)]$ the d-type charge density is $[\rho_d({\bf r})=\rho_{d\uparrow}+\rho_{d\downarrow} \eqno (8.7.4.59)]$ and is expanded in a similar way to s(r) [see (8.7.3.76)], $[\rho_d({\bf r})=\textstyle\sum\limits_i\, \textstyle\sum\limits_{j\ge i}\,P_{ij} d_i d_j\semi \eqno (8.7.4.60)]$ writing $[\rho_{d \sigma} = \textstyle\sum\, P^\sigma_{ij}\, d_i d_j \eqno (8.7.4.61)]$ with σ = $[ \uparrow]$ and $[\downarrow]$ , one obtains $[\matrix{ P_{ij} = P^\uparrow_{ij} +P^\downarrow_{ij} \cr \vphantom{}\cr D_{ij} = (P^\uparrow_{ij} - P^\downarrow_{ij})/(n_\uparrow - n_\downarrow).} \eqno (8.7.4.62)]$ Similarly to $[\rho_d({\bf r})]$ , [s(r)] can be expanded as $[s({\bf r})= U^2 (r)\, \textstyle\sum\limits^4_{l=0}\; \textstyle\sum\limits^l_{m=0}\; \textstyle\sum\limits_p\, D_{lmp}\, y_{lmp}(\theta, \varphi), \eqno (8.7.4.63)]$ where U(r) describes the radial dependence. Spin polarization leads to a further modification of this expression. Since the numbers of electrons of a given spin are different, the exchange interaction is different for the two spin states, and a spin-dependent effective screening occurs. This leads to $[\rho_{d\sigma}({\bf r}) = \kappa^3_\sigma U^2(\kappa_\sigma r)\textstyle \sum\limits_{lmp}\, P^\sigma_{lmp}\, y_{lmp}, \eqno (8.7.4.64)]$ with σ = $[\ \uparrow]$ or $[\downarrow]$ , where two κ parameters are needed.

The complementarity between charge and spin density in the crystal field approximation is obvious. At this particular level of approximation, expansions are exact and it is possible to estimate d-orbital populations for each spin state.

8.7.4.5.1.3. Scaling of the spin density

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The magnetic structure factor is scaled to $[F_N({\bf h})]$ . Whether the nuclear structure factors are calculated from refined structural parameters or obtained directly from a measurement, their scale factor is not rigorously fixed.

As a result, it is not possible to obtain absolute values of the effective spins 〈S_n〉 from a magnetic neutron scattering experiment. It is necessary to scale them through the sum rule $[\textstyle\sum\limits _n\, \langle{\bf S}_n\rangle = {\boldmu}, \eqno (8.7.4.65)]$ where $[\boldmu]$ is the macroscopic magnetization of the sample.

The practical consequences of this constraint for a refinement are similar to the electroneutrality constraint in charge-density analysis (§8.7.3.3.1).

References

Coppens, P. (2001). The structure factor. International tables for crystallography, Vol. B, edited by U. Shmueli, Chap. 1.2. Dordrecht: Kluwer Academic Publishers.Google Scholar

Desclaux, J. B. & Freeman, A. J. (1978). J. Magn. Magn. Mater. 8, 119–129.Google Scholar

Freeman, A. J. & Desclaux, J. P. (1972). Neutron magnetic form factor of gadolinium. Int. J. Magn. 3, 311–317.Google Scholar

International Tables for X-ray Crystallography (1974). Vol. IV. Birmingham: Kynoch Press.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 8.7, pp. 729-730