Spherical-atom model

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, p. 729

Section 8.7.4.5.1.1. Spherical-atom model

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

8.7.4.5.1.1. 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.

References

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, p. 729