International
Tables for
Crystallography
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 2.3, pp. 69-70

Section 2.3.2.5. Magnetic form factors and magnetic scattering lengths

C. J. Howarda* and E. H. Kisia

aSchool of Engineering, University of Newcastle, Callaghan, NSW 2308, Australia
Correspondence e-mail:  chris.howard@newcastle.edu.au

2.3.2.5. Magnetic form factors and magnetic scattering lengths

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For a complete treatment of the magnetic interaction between the neutron and an atom carrying a magnetic moment, and the resulting scattering, the reader is referred elsewhere [Marshall & Lovesey, 1971[link]; Squires, 1978[link]; Section 6.1.2[link] of Volume C (Brown, 2006[link] a[link])]. The magnetic moment of an atom is associated with unpaired electrons, but may comprise both spin and orbital contributions. The magnetic interaction between the neutron and the atom depends on the directions of the scattering vector and the magnetic moment vector according to a triple vector product. The direction of polarization of the neutron must also be taken into account. For an unpolarized incident beam, the usual case in neutron powder diffraction, it is a useful consequence of the triple vector product that the magnetic scattering depends on the sine of the angle that the scattering vector makes with the magnetic moment on the scattering atom (see Section 2.3.4 and Chapter 7 in Kisi & Howard, 2008[link]). The extent of the unpaired electron distribution (usually outer electrons) implies that the scattering diminishes as a function of Q, an effect that can be described by a magnetic form factor. For a well defined direction for the magnetic moment M, and with a distribution of moment that can be described by a normalized scalar m(r), the form factor as a function of the scattering vector h [defined in equation (1.1.17)[link] in Chapter 1.1]6 is the Fourier transform of m(r),[f\left({\bf h}\right)=\textstyle\int m({\bf r})\exp\left(2\pi i{\bf h}\cdot{\bf r}\right)\,{\rm d}{\bf r},]where m(r) can comprise both spin and orbital contributions [Section 6.1.2[link] of Volume C (Brown, 2006a[link])]. The tabulated form factors are based on the assumption that the electron distributions are spherically symmetric, so that [m({\bf r})=m(r)={U}^{2}(r)], where U(r) is the radial part of the wave function for the unpaired electron. In the expansion of the plane-wave function [\exp(2\pi i{\bf h}\cdot{\bf r})] in terms of spherical Bessel functions, we find that the leading term is just the zeroth-order spherical Bessel function [{j}_{0}(2\pi hr)] with a Fourier transform[\langle {j}_{0}(h)\rangle =4\pi \textstyle\int\limits_{0}^{\infty }{U}^{2}\left(r\right){j}_{0}\left(2\pi hr\right){r}^{2}\, {\rm d}r.]

This quantity is inherently normalized to unity at h = 0, and may suffice to describe the form factor for spherical spin-only cases. In other cases it may be necessary to include additional terms in the expansion, and these have Fourier transforms of the form[\langle {j}_{l}(h)\rangle =4\pi \textstyle\int\limits_{0}^{\infty }{U}^{2}(r){j}_{l}(2\pi hr){r}^{2}\, {\rm d}r]with l even; these terms are zero at h = 0 (Brown, 2006[link] a[link]). In practice these quantities are evaluated using theoretical calculations of the radial distribution functions for the unpaired electrons [Section 4.4.5[link] of Volume C (Brown, 2006b[link])].

Form factors can be obtained from data tabulated in Section 4.4.5[link] of Volume C (Brown, 2006b[link]). Data are available for elements and ions in the 3d- and 4d-block transition series, for rare-earth ions and for actinide ions. These data are provided by way of the coefficients of analytical approximations to [\langle {j}_{l}(h)\rangle ], the analytical approximations being[\langle {j}_{0}(s)\rangle =A\exp\left(-a{s}^{2}\right)+B\exp\left(-b{s}^{2}\right)+C\exp\left(-c{s}^{2}\right)+D]and for l ≠ 0[\langle {j}_{l}(s)\rangle ={s}^{2}\left[A\exp\left(-a{s}^{2}\right)+B\exp\left(-b{s}^{2}\right)+C\exp\left(-c{s}^{2}\right)+D\right],]where s = h/2 in Å−1. These approximations, with the appropriate coefficients, are expected to be coded in to any computer program purporting to analyse magnetic structures. Although the tabulated form factors are based on theoretical wave functions, it is worth noting that the incoherent scattering from an ideally disordered (i.e., paramagnetic) magnetic system will display the magnetic form factor directly.

It is often convenient to define a (Q-dependent) magnetic scattering length[p=\left({{{e}^{2}\gamma }\over{2{m}_{e}{c}^{2}}}\right)gJf,]where me and e are the mass and charge of the electron, γ (= μn) is the magnetic moment of the neutron, c is the speed of light, J is the total angular momentum quantum number, and g is the Landé splitting factor given in terms of the spin S, orbital angular momentum L, and total angular momentum quantum numbers by[g=1+{{J\left(J+1\right)+S\left(S+1\right)-L(L+1)}\over{2J(J+1)}}.]

For the spin-only case, L = 0, J = S, so g = 2. The differential magnetic scattering cross section per atom is then given by [{q}^{2}{p}^{2}] where [|q|=\sin\alpha ], α being the angle between the scattering vector and the direction of the magnetic moment. This geometrical factor is very important, since it can help in the determination of the orientation of the moment of interest; there is no signal, for example, when the moment is parallel to the scattering vector. Further discussion appears in Chapters 2 (Section 2.3.4) and 7 in Kisi & Howard (2008[link]).

References

Brown, P. J. (2006a). Magnetic scattering of neutrons. International Tables for Crystallography, Volume C, Mathematical, Physical and Chemical Tables, 1st online ed., edited by E. Prince, pp. 590–593. Chester: International Union of Crystallography.Google Scholar
Brown, P. J. (2006b). Magnetic form factors. International Tables for Crystallography, Volume C, Mathematical, Physical and Chemical Tables, 1st online ed., edited by E. Prince, pp. 454–461. Chester: International Union of Crystallography.Google Scholar
Kisi, E. H. & Howard, C. J. (2008). Applications of Neutron Powder Diffraction. Oxford University Press.Google Scholar
Marshall, W. & Lovesey, S. W. (1971). Theory of Thermal Neutron Scattering: The Use of Neutrons for the Investigation of Condensed Matter. Oxford: Clarendon Press.Google Scholar
Squires, G. L. (1978). Introduction to the Theory of Thermal Neutron Scattering. Cambridge University Press.Google Scholar








































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