Polarized neutron scattering of centrosymmetric crystals

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. 728

Section 8.7.4.4.4. Polarized neutron scattering of centrosymmetric crystals

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

8.7.4.4.4. Polarized neutron scattering of centrosymmetric crystals

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If $[\boldlambda]$ is assumed to be in the vertical Oz direction, M(h) will in most situations be aligned along Oz by an external orienting field. If α is the angle between M and h, and $[x = {r_0M({\bf h}) \over F_N({\bf h})}, \eqno (8.7.4.43)]$ with [F_N] expressed in the same units as [r_0] , one obtains, for centrosymmetric crystals, $[R = {1+2x\sin^2\alpha+x^2 \sin^2 \alpha\, \over 1-2x \sin^2 \alpha+x^2 \sin^2\alpha\,}. \eqno (8.7.4.44)]$ If $[x\ll 1]$ , $[R\sim 1+ 4x \sin^2 \alpha. \eqno (8.7.4.45)]$ For $[x\sim0.05]$ and α = π/2, R now departs from 1 by as much as 20%, which proves the enormous advantage of polarized neutron scattering in the case of low magnetism.

Equation (8.7.4.44) can be inverted, and x and its sign can be obtained directly from the observation. However, in order to obtain M(h), the nuclear structure factor $[F_N({\bf h})]$ must be known, either from nuclear scattering or from a calculation. All systematic errors that affect $[F_N({\bf h})]$ are transferred to M(h).

For two reasons, it is not in general feasible to access all reciprocal-lattice vectors. First, in order to have reasonable statistical accuracy, only reflections for which both $[I_\uparrow]$ and $[I_\downarrow]$ are large enough are measured; i.e. reflections having a strong nuclear structure factor. Secondly, $[\sin\alpha]$ should be as close to 1 as possible, which may prevent one from accessing all directions in reciprocal space. If M is oriented along the vertical axis, the simplest experiment consists of recording reflections with h in the horizontal plane, which leads to a projection of m(r) in real space. When possible, the sample is rotated so that other planes in the reciprocal space can be recorded.

Finally, if α = π/2, $[I_{\uparrow\downarrow}]$ vanishes, and neutron spin is conserved in the experiment.

References

International Tables for Crystallography (2006). Vol. C. ch. 8.7, p. 728