Magnetization densities from neutron magnetic elastic 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. 725-726

Section 8.7.4.2. Magnetization densities from neutron magnetic elastic 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

8.7.4.2. Magnetization densities from neutron magnetic elastic scattering

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The scattering process is discussed in Section 6.1.3 and only the features that are essential to the present chapter will be summarized here.

For neutrons, the nuclear structure factor $[F_N({\bf h})]$ is given by $[F_N({\bf h})=\textstyle\sum\limits_j b_j T_j \exp (2\pi i {\bf h}\cdot {\bf R}_j). \eqno (8.7.4.3)]$ [b_j] , [T_j] , $[{\bf R}_j]$ are the coherent scattering length, the temperature factor, and the equilibrium position of the jth atom in the unit cell.

Let σ be the spin of the neutron (in units of $[\hbar/2]$ ). There is a dipolar interaction of the neutron spin with the electron spins and the currents associated with their motion. The magnetic structure factor can be written as the scalar product of the neutron spin and an `interaction vector' Q(h): $[F_M({\bf h})={\boldsigma}\cdot{\bf Q}({\bf h}). \eqno (8.7.4.4)]$ Q(h) is the sum of a spin and an orbital term: $[{\bf Q}_s]$ and $[{\bf Q}_L]$ , respectively. If r₀ = γr_e ~ 0.54 × 10⁻¹² cm, where γ is the gyromagnetic factor (= 1.913) of the neutron and [r_e] the classical Thomson radius of the electron, the spin term is given by $[{\bf Q}_s({\bf h})=r_0\hat{\bf h}\times{\bf M}_s({\bf h})\times\hat{\bf h}, \eqno (8.7.4.5)]$ $[\hat{\bf h}]$ being the unit vector along h, while M_s(h) is defined as $[{\bf M}_s({\bf h}) = \left\langle \textstyle\sum\limits_j {\boldsigma}_j \exp\, (2\pi i {\bf h}\cdot {\bf r}_j) \right\rangle, \eqno (8.7.4.6)]$ where $[{\boldsigma}_j]$ is the spin of the electron at position $[{\bf r}_j]$ , and angle brackets denote the ensemble average over the scattering sample. $[{\bf M}_s({\bf h})]$ is the Fourier transform of the spin-magnetization density $[{\bf m}_s({\bf r})]$ , given by $[{\bf m}_s({\bf r})=\left\langle \textstyle\sum\limits_j\, {\boldsigma}_j\delta({\bf r}-{\bf r}_j) \right\rangle. \eqno (8.7.4.7)]$ This is the spin-density vector field in units of 2μ_B.

The orbital part of Q(h) is given by $[{\bf Q}_L({\bf h})=-{ir_0 \over 2\pi h}\, \hat{\bf h}\times \left\langle \sum_j\, {\bf p}_j \exp\,(2\pi i{\bf h}\cdot{\bf r}_j)\right\rangle, \eqno (8.7.4.8)]$ where $[{\bf p}_j]$ is the momentum of the electrons. If the current density vector field is defined by $[{\bf j}({\bf r}) = -{e\over 2m} \left\langle \sum_j\, \left\{{\bf p}_j\delta({\bf r}-{\bf r}_j)+\delta({\bf r}-{\bf r}_j){\bf p}_j \right\} \right\rangle, \eqno (8.7.4.9)]$ Q_L(h) can be expressed as $[{\bf Q}_L({\bf h})=-{ir_0\over 2\pi h}\, \hat{\bf h}\times {\bf J}({\bf h}), \eqno (8.7.4.10)]$ where J(h) is the Fourier transform of the current density j(r).

The electrodynamic properties of j(r) allow it to be written as the sum of a rotational and a nonrotational part: $[{\bf j}({\bf r}) = {\boldnabla} \psi + {\boldnabla} \times [{\bf m}_L({\bf r})], \eqno (8.7.4.11)]$ where $[{\boldnabla}\psi]$ is a `conduction' component and $[{\bf m}_L({\bf r})]$ is an `orbital-magnetization' density vector field.

Substitution of the Fourier transform of (8.7.4.11) into (8.7.4.10) leads in analogy to (8.7.4.5) to $[{\bf Q}_L({\bf h})=r_0 \hat{\bf h}\times {\bf M}_L({\bf h})\times \hat{\bf h}, \eqno (8.7.4.12)]$ where M_L(h) is the Fourier transform of m_L(r). The rotational component $[{\boldnabla}\psi]$ of j(r) does not contribute to the neutron scattering process. It is therefore possible to write Q(h) as $[{\bf Q}({\bf h})=r_0 \hat{\bf h}\times {\bf M}({\bf h})\times \hat{\bf h}, \eqno (8.7.4.13)]$ with $[{\bf M}({\bf h})={\bf M}_s({\bf h})+{\bf M}_L({\bf h}) \eqno (8.7.4.14)]$ being the Fourier transform of the `total' magnetization density vector field, and $[{\bf m}({\bf r})={\bf m}_s({\bf r})+{\bf m}_L({\bf r}). \eqno (8.7.4.15)]$ As Q(h) is the projection of M(h) onto the plane perpendicular to h, there is no magnetic scattering when M is parallel to h. It is clear from (8.7.4.13) that M(h) can be defined to any vector field V(h) parallel to h, i.e. such that h × V(h) = 0.

This means that in real space m(r) is defined to any vector field v(r) such that $[{\boldnabla} ]$ × v(r) = 0. Therefore, m(r) is defined to an arbitrary gradient.

As a result, magnetic neutron scattering cannot lead to a uniquely defined orbital magnetization density. However, the definition (8.7.4.7) for the spin component is unambiguous.

However, the integrated magnetic moment $[\boldmu]$ is determined unambiguously and must thus be identical to the magnetic moment defined from the principles of quantum mechanics, as discussed in §8.7.4.5.1.3.

Before discussing the analysis of magnetic neutron scattering in terms of spin-density distributions, it is necessary to give a brief description of the quantum-mechanical aspects of magnetization densities.

References

International Tables for Crystallography (2006). Vol. C. ch. 8.7, pp. 725-726