Orbital contribution to the magnetic scattering

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, pp. 730-731

Section 8.7.4.6. Orbital contribution to the magnetic scattering

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

8.7.4.6. Orbital contribution to the magnetic scattering

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Q_L (h) is given by (8.7.4.10) and (8.7.4.12). Since $[{\boldnabla}\psi]$ in (8.7.4.11) does not play any role in the scattering cross section, we can use the restriction $[{\bf j}({\bf r})={\boldnabla}\times {\bf m}_L({\bf r}), \eqno (8.7.4.71)]$ where $[{\bf m}_L({\bf r})]$ is defined to an arbitrary gradient. It is possible to constrain $[{\bf m}_L({\bf r})]$ to have the form $[{\bf m}_L({\bf r})=\hat {\bf r}\times {\bf v}({\bf r}),]$ since any radial component could be considered as the radial component of a gradient. With spherical coordinates, (8.7.4.71) becomes $[{\bf j}=-{1\over r}\, {\partial \over \partial r}[r {\bf v}], ]$ which can be integrated as $[{\bf r}\times{\bf v}=-\textstyle\int\limits^\infty_r\,y \hat{\bf r} \times {\bf j}(y \hat{\bf r})\,{\rm d} y\semi]$ one finally obtains $[{\bf m}_L({\bf r}) ={1\over r}\; \int\limits^{y=\infty}_{y=r}\; \hat{\bf r}y\times{\bf j}(\hat{\bf r} y)\, {\rm d} y. \eqno (8.7.4.72)]$ With f(x) defined by $[f(x) =-{1\over x^2}\, \int\limits^{ix} _0 \, tl^t\, {\rm d} t, \eqno (8.7.4.73)]$ the expression for M_L(h) is obtained by Fourier transformation of (8.7.4.72): $[M_L({\bf h})=\textstyle{1\over2}\int{\bf r}\times{\bf j}({\bf r})\,f({\bf h}\cdot{\bf r}) \,{\rm d} {\bf r},]$ which leads to $[{\bf M}_L({\bf h}) = \textstyle{1\over 4} \left\langle\sum\limits_j\, \left\{{\bf l}_j\; f(2\pi{\bf h}\cdot {\bf r}_j)+f (2\pi{\bf h}\cdot{\bf r}_j){\bf l}_j\right\} \right\rangle \eqno (8.7.4.74)]$ by using the definition (8.7.4.9) of j.

This expression clearly shows the connection between orbital magnetism and the orbital angular momentum of the electrons. It is of general validity, whatever the origin of orbital magnetism.

Since f(0) = 1, $[{\bf M}_L(0)=\textstyle{1\over2}\langle{\bf L}\rangle, \eqno (8.7.4.75)]$ as expected.

8.7.4.6.1. The dipolar approximation

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The simplest approximation involves decomposing j(r) into atomic contributions: $[{\bf j}({\bf r})=\textstyle\sum\limits_n\; {\bf j}_n({\bf r} - {\bf R}_n). \eqno (8.7.4.76)]$ One obtains $[{\bf M}_L=\textstyle\sum\limits_n{\bf M}_{L,n}({\bf h})\exp\, (2\pi i {\bf h} \cdot{\bf R}_n). \eqno (8.7.4.77)]$ $[{\bf M}_{L,n}({\bf h})]$ is the atomic magnetic orbital structure factor .

We notice that $[f(2\pi{\bf h}\cdot{\bf r})]$ as defined in (8.7.4.73) can be expanded as $[f(2\pi{\bf h}\cdot {\bf r}) = 4\pi \textstyle\sum\limits_l \textstyle\sum\limits_m\, (i)^l\gamma_l(2\pi hr)\,Y_{lm}(\hat r)\, Y^*_{lm}(\hat h), \eqno (8.7.4.78)]$ where $[\gamma_l(x)={2\over x^2}\,\int\limits^x_0\, tj_l(t)\,{\rm d} t. \eqno (8.7.4.79)]$ [j_l] is a spherical Bessel function of order l, and $[Y_{lm}]$ are the complex spherical harmonic functions.

If one considers only the spherically symmetric term in (8.7.4.78), one obtains the `dipolar approximation', which gives $[{\bf M}^D_{L,n}=\textstyle{1\over2}\langle {\bf L}_n\rangle\langle\gamma_{0n}\rangle, \eqno (8.7.4.80)]$ with $[\langle \gamma_{0n}\rangle = \textstyle\int\limits^\infty_0\, 4\pi r^2U^2_n(r)\gamma_0(2\pi h r) \,{\rm d} r,]$ and $[\gamma_0(x) ={2\over x^2}(1-\cos x). \eqno (8.7.4.81)]$ [U_n(r)] is the radial function of the atomic electrons whose orbital momentum is unquenched. Thus, in the dipolar approximation, the atomic orbital scattering is proportional to the effective orbital angular momentum and therefore to the orbital part of the magnetic dipole moment of the atom.

Within the same level of approximation, the spin structure factor is $[{\bf M}_S=\textstyle\sum\limits_n\,{\bf M}_{S,n}({\bf h})\exp\,(2\pi i{\bf h}\cdot{\bf R}_n), \eqno (8.7.4.82)]$ with $[{\bf M}_{S,n}({\bf h}) = \langle {\bf S}_{\bf n}\rangle\langle\,j_{0n}\rangle,]$ and $[\langle\,j_{0n}\rangle = \textstyle\int\limits^\infty_0\, 4\pi r^2U^2_n(r)\,j_0(2\pi h r) \,{\rm d} r.]$ Finally, the atomic contribution to the total magnetic structure factor is $[{\bf M}^D_n = \langle{\bf S}_n\rangle\langle\,j_{0n}\rangle +\textstyle{1\over 2}\langle {\bf L}_n\rangle \langle\gamma_{0n}\rangle. \eqno (8.7.4.83)]$ If 〈J_n〉 is the total angular momentum of atom n and [g_n] its gyromagnetic ratio, (8.7.4.35)–(8.7.4.37) lead to: $[{\bf M}^D_n = \langle{\bf J}_n\rangle \left\{[g_n-1]\langle\,j_{0n}\rangle + {2-g_n \over 2}\, \langle\gamma_{0n}\rangle \right\}. \eqno (8.7.4.84a)]$

Another approach, which is applicable only to the atomic case, is often used, which is based on Racah's algebra (Marshall & Lovesey, 1971). At the dipolar approximation level, it leads to a slightly different result, according to which 〈γ_0n〉 is replaced by $[\langle \gamma_{0n}\rangle \sim \langle\, j_{0n}\rangle + \langle\, j_{2n}\rangle. \eqno (8.7.4.85)]$ The two results are very close for small h where the dipolar approximation is correct. With (8.7.4.35)–(8.7.4.37), (8.7.4.84a) can also be written as $[{\bf M}^D_n = {\bf M}^D_{S,n} + \langle{\bf S}_n\rangle\, {2-g_n \over 2(g_n-1)}\, \langle\gamma_{0n}\rangle, \eqno (8.7.4.84b)]$ where the second term is the `orbital correction'. Its magnitude clearly depends on the difference between [g_n] and 2, which is small in 3d elements but can become important for rare earths.

8.7.4.6.2. Beyond the dipolar approximation

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Expressions (8.7.4.74) and (8.7.4.78) are valid in any situation where orbital scattering occurs. They can in principle be used to estimate from the diffraction experiment the contribution of a few configurations that interact due to the $[{\bf L}\cdot{\bf S}]$ operator. In delocalized situations, (8.7.4.74) is the most suitable approach, while Racah's algebra can only be applied to one-centre cases.

8.7.4.6.3. Electronic structure of rare-earth elements

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When covalency is small, the major aims are the determination of the ground state of the rare-earth ion, and the amount of delocalized magnetization density via the conduction electrons.

The ground state |ψ〉 of the ion is written as $[|\psi\rangle = \textstyle\sum\limits_M\, a_M|JM\rangle, \eqno (8.7.4.86)]$ which is well suited for the Johnston (1966) and Marshall & Lovesey (1971) formulation in terms of general angular-momentum algebra. A multipolar expansion of spin and orbital components of the structure factor enables a determination of the expansion coefficient [a_M] (Schweizer, 1980).

References

Johnston, D. F. (1966). Theory of the electron contribution to the scattering of neutrons by magnetic ions in crystals. Proc. Phys. Soc. London, 88, 37–52.Google Scholar

Marshall, W. & Lovesey, S. W. (1971). Theory of thermal neutron scattering. Oxford University Press.Google Scholar

Schweizer, J. (1980). In Electron and magnetization densities in molecules and crystals, edited by P. Becker. New York: Plenum.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 8.7, pp. 730-731