Spin density for an assembly of localized systems

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. 727

Section 8.7.4.3.3. Spin density for an assembly of localized systems

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

8.7.4.3.3. Spin density for an assembly of localized systems

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A complex magnetic system can generally be described as an ensemble of well defined interacting open-shell subsystems (ions or radicals), where each subsystem has a spin $[\hat{\bf S}_n]$ , and [S^2_n] is assumed to be a good quantum number. The magnetic interaction occurs essentially through exchange mechanisms that can be described by the Heisenberg Hamiltonian: $[{\scr H}=-\textstyle\sum\limits_{n\lt m}\,J_{nm}\,\hat{\bf S}_n\cdot \hat{\bf S}_m - \textstyle\sum\limits_n\, {\bf B}_0\cdot{\bf S}_n, \eqno (8.7.4.30)]$ where $[J_{nm}]$ is the exchange coupling between two subsystems, and B₀ an applied external field (magneto-crystalline anisotropic effects may have to be added). Expression (8.7.4.30) is the basis for the understanding of magnetic ordering and phase diagrams. The interactions lead to a local field B_n, which is the effective orienting field for the spin S_n.

The expression for the spin-magnetization density is $[{\bf m}_s({\bf r})=\textstyle\sum\limits_n\langle{\bf S}_n\rangle s_n({\bf r}). \eqno (8.7.4.31)]$ The relative arrangement of 〈S_n〉 describes the magnetic structure; $[s_n({\bf r})]$ is the normalized spin density of the nth subsystem.

In some metallic systems, at least part of the unpaired electron system cannot be described within a localized model: a band-structure description has to be used (Lovesey, 1984). This is the case for transition metals like Ni, where the spin-magnetization density is written as the sum of a localized part [described by (8.7.4.31)] and a delocalized part [described by (8.7.4.29)].

References

Lovesey, S. W. (1984). Theory of neutron scattering from condensed matter. Oxford: Clarendon Press.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 8.7, p. 727