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International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C. ch. 8.7, p. 732
Section 8.7.4.7.1. Vector fieldsa 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 |
The vector potential field is defined as ![[{\bf A}({\bf r}) = \int\, \displaystyle{\bf m_s({\bf r}')\times ({\bf r}-{\bf r}') \over |{\bf r}-{\bf r}'|^3} \,{\rm d}{\bf r}'. \eqno (8.7.4.87)]](/teximages/cbch8o7/cbch8o7fd229.svg)
In the case of a crystal, it can be expanded in Fourier series: ![[{\bf A}({\bf r})={2i\over V}\, \sum_{\bf h}\, \left\{ \displaystyle{{\bf M}_S({\bf h})\times {\bf h} \over h^2}\right\} \exp\,(-2\pi i{\bf h}\cdot{\bf r})\semi \eqno (8.7.4.88)]](/teximages/cbch8o7/cbch8o7fd230.svg)
the magnetic field is simply ![[\eqalignno{ {\bf B}({\bf r}) &={\boldnabla}\times{\bf A}({\bf r}) \cr &=-{4\pi \over V}\, \sum_{\bf h}\, \left[\displaystyle{{\bf h}\times{\bf M}_S\,({\bf h})\times{\bf h} \over h^2}\right] \exp\,(-2\pi i{\bf h}\cdot{\bf r}). &(8.7.4.89)}]](/teximages/cbch8o7/cbch8o7fd231.svg)
One notices that there is no convergence problem for the h = 0 term in the B(r) expansion.
The magnetostatic energy, i.e. the amount of energy that is required to obtain the magnetization ms, is ![[\eqalignno{ E_{ms} &= -\textstyle{1\over2}\, \int\limits_{\rm cell} {\bf m}_s({\bf r})\cdot {\bf B}({\bf r}) \,{\rm d} r \cr &={2\pi\over V}\, \sum_{\bf h}\, \displaystyle{ [{\bf M}_s(-{\bf h})\times{\bf h}]\cdot [{\bf M}_S({\bf h})\times{\bf h}] \over h^2}. & (8.7.4.90)}]](/teximages/cbch8o7/cbch8o7fd232.svg)
It is often interesting to look at the magnetostatics of a given subunit: for instance, in the case of paramagnetic species.
For example, the vector potential outside the magnetized system can be obtained in a similar way to the electrostatic potential (8.7.3.30)![[link]](/graphics/greenarr.gif)
: ![[{\bf A}({\bf r}') = \int\ \displaystyle{ [{\boldnabla}\times {\bf m}_S({\bf r})] \over |{\bf r} - {\bf r}'|}\, {\rm d} {\bf r}. \eqno (8.7.4.91)]](/teximages/cbch8o7/cbch8o7fd233.svg)
If ![[r'\gg r, 1/|{\bf r}-{\bf r}'|]](/teximages/cbch8o7/cbch8o7fi310.svg)
can be easily expanded in powers of 1/r′, and A(r′) can thus be obtained in powers of 1/r′. If ms(r) = 〈S〉s(r), ![[{\bf A}({\bf r}') = \langle{\bf S}\rangle \times \int \displaystyle{ {\boldnabla}s ({\bf r}) \over |{\bf r}-{\bf r}'|}\, {\rm d}{\bf r}. \eqno (8.7.4.92)]](/teximages/cbch8o7/cbch8o7fd234.svg)
