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The dipolar approximation
International Tables for Crystallography (2006). Vol. C, Section 8.7.4.6.1, p. 731 [ doi:10.1107/97809553602060000615 ]
The dipolar approximation 8.7.4.6.1. The dipolar approximation The simplest approximation involves decomposing j(r) into atomic contributions: One obtains is the atomic magnetic orbital structure factor. We notice that as defined in (8.7.4.73) can be expanded as where is a spherical Bessel function of order l, and are the complex spherical ...
Orbital contribution to the magnetic scattering
International Tables for Crystallography (2006). Vol. C, Section 8.7.4.6, pp. 730-731 [ doi:10.1107/97809553602060000615 ]
Orbital contribution to the magnetic scattering 8.7.4.6. Orbital contribution to the magnetic scattering QL(h) is given by (8.7.4.10) and (8.7.4.12). Since in (8.7.4.11) does not play any role in the scattering cross section, we can use the restriction where is defined to an arbitrary gradient. It is possible to ...
Other types of model
International Tables for Crystallography (2006). Vol. C, Section 8.7.4.5.3, p. 730 [ doi:10.1107/97809553602060000615 ]
Other types of model 8.7.4.5.3. Other types of model One may wish to take advantage of the fact that, to a good approximation, only a few molecular orbitals are involved in s(r). In an independent particle model, one expands the relevant orbitals in terms of atomic basis functions (LCAO ...
General multipolar expansion
International Tables for Crystallography (2006). Vol. C, Section 8.7.4.5.2, p. 730 [ doi:10.1107/97809553602060000615 ]
General multipolar expansion 8.7.4.5.2. General multipolar expansion In this subsection, the localized magnetism picture is assumed to be valid. However, each subunit can now be a complex ion or a radical. Covalent interactions must be incorporated. If are atomic basis functions, the spin density s(r) can always be written as ...
Scaling of the spin density
International Tables for Crystallography (2006). Vol. C, Section 8.7.4.5.1.3, p. 730 [ doi:10.1107/97809553602060000615 ]
Scaling of the spin density 8.7.4.5.1.3. Scaling of the spin density The magnetic structure factor is scaled to . Whether the nuclear structure factors are calculated from refined structural parameters or obtained directly from a measurement, their scale factor is not rigorously fixed. As a result, it is not possible to ...
Crystal-field approximation
International Tables for Crystallography (2006). Vol. C, Section 8.7.4.5.1.2, pp. 729-730 [ doi:10.1107/97809553602060000615 ]
Crystal-field approximation 8.7.4.5.1.2. Crystal-field approximation Crystal-field effects are generally of major importance in spin magnetism and are responsible for the spin state of the ions, and thus for the ground-state configuration of the system. Thus, they have to be incorporated in the model. Taking the case of ...
Spherical-atom model
International Tables for Crystallography (2006). Vol. C, Section 8.7.4.5.1.1, p. 729 [ doi:10.1107/97809553602060000615 ]
Spherical-atom model 8.7.4.5.1.1. Spherical-atom model In the crudest model, is approximated by its spherical average. If the magnetic electrons have a wavefunction radial dependence represented by the radial function U(r), the magnetic form factor is given by where is the zero-order spherical Bessel function. For free atoms ...
Atom-centred expansion
International Tables for Crystallography (2006). Vol. C, Section 8.7.4.5.1, pp. 729-730 [ doi:10.1107/97809553602060000615 ]
Atom-centred expansion 8.7.4.5.1. Atom-centred expansion We first consider the case where spins are localized on atoms or ions, as it is to a first approximation for compounds involving transition-metal atoms. The magnetization density is expanded as where is the spin at site j, and the thermally averaged normalized ...
Modelling the spin density
International Tables for Crystallography (2006). Vol. C, Section 8.7.4.5, pp. 729-730 [ doi:10.1107/97809553602060000615 ]
Modelling the spin density 8.7.4.5. Modelling the spin density In this subsection, the case of spin-only magnetization is considered. The modelling of is very similar to that of the charge density. 8.7.4.5.1. Atom-centred expansion | | We first consider the case where spins are localized on atoms or ions, as it ...
Error analysis
International Tables for Crystallography (2006). Vol. C, Section 8.7.4.4.7, p. 729 [ doi:10.1107/97809553602060000615 ]
Error analysis 8.7.4.4.7. Error analysis In the most general case, it is not possible to obtain x, and thus M(h) directly from R. Moreover, it is unlikely that all Bragg spots within the reflection sphere could be measured. Modelling of M(h) is thus of crucial importance. The analysis of ...
International Tables for Crystallography (2006). Vol. C, Section 8.7.4.6.1, p. 731 [ doi:10.1107/97809553602060000615 ]
The dipolar approximation 8.7.4.6.1. The dipolar approximation The simplest approximation involves decomposing j(r) into atomic contributions: One obtains is the atomic magnetic orbital structure factor. We notice that as defined in (8.7.4.73) can be expanded as where is a spherical Bessel function of order l, and are the complex spherical ...
Orbital contribution to the magnetic scattering
International Tables for Crystallography (2006). Vol. C, Section 8.7.4.6, pp. 730-731 [ doi:10.1107/97809553602060000615 ]
Orbital contribution to the magnetic scattering 8.7.4.6. Orbital contribution to the magnetic scattering QL(h) is given by (8.7.4.10) and (8.7.4.12). Since in (8.7.4.11) does not play any role in the scattering cross section, we can use the restriction where is defined to an arbitrary gradient. It is possible to ...
Other types of model
International Tables for Crystallography (2006). Vol. C, Section 8.7.4.5.3, p. 730 [ doi:10.1107/97809553602060000615 ]
Other types of model 8.7.4.5.3. Other types of model One may wish to take advantage of the fact that, to a good approximation, only a few molecular orbitals are involved in s(r). In an independent particle model, one expands the relevant orbitals in terms of atomic basis functions (LCAO ...
General multipolar expansion
International Tables for Crystallography (2006). Vol. C, Section 8.7.4.5.2, p. 730 [ doi:10.1107/97809553602060000615 ]
General multipolar expansion 8.7.4.5.2. General multipolar expansion In this subsection, the localized magnetism picture is assumed to be valid. However, each subunit can now be a complex ion or a radical. Covalent interactions must be incorporated. If are atomic basis functions, the spin density s(r) can always be written as ...
Scaling of the spin density
International Tables for Crystallography (2006). Vol. C, Section 8.7.4.5.1.3, p. 730 [ doi:10.1107/97809553602060000615 ]
Scaling of the spin density 8.7.4.5.1.3. Scaling of the spin density The magnetic structure factor is scaled to . Whether the nuclear structure factors are calculated from refined structural parameters or obtained directly from a measurement, their scale factor is not rigorously fixed. As a result, it is not possible to ...
Crystal-field approximation
International Tables for Crystallography (2006). Vol. C, Section 8.7.4.5.1.2, pp. 729-730 [ doi:10.1107/97809553602060000615 ]
Crystal-field approximation 8.7.4.5.1.2. Crystal-field approximation Crystal-field effects are generally of major importance in spin magnetism and are responsible for the spin state of the ions, and thus for the ground-state configuration of the system. Thus, they have to be incorporated in the model. Taking the case of ...
Spherical-atom model
International Tables for Crystallography (2006). Vol. C, Section 8.7.4.5.1.1, p. 729 [ doi:10.1107/97809553602060000615 ]
Spherical-atom model 8.7.4.5.1.1. Spherical-atom model In the crudest model, is approximated by its spherical average. If the magnetic electrons have a wavefunction radial dependence represented by the radial function U(r), the magnetic form factor is given by where is the zero-order spherical Bessel function. For free atoms ...
Atom-centred expansion
International Tables for Crystallography (2006). Vol. C, Section 8.7.4.5.1, pp. 729-730 [ doi:10.1107/97809553602060000615 ]
Atom-centred expansion 8.7.4.5.1. Atom-centred expansion We first consider the case where spins are localized on atoms or ions, as it is to a first approximation for compounds involving transition-metal atoms. The magnetization density is expanded as where is the spin at site j, and the thermally averaged normalized ...
Modelling the spin density
International Tables for Crystallography (2006). Vol. C, Section 8.7.4.5, pp. 729-730 [ doi:10.1107/97809553602060000615 ]
Modelling the spin density 8.7.4.5. Modelling the spin density In this subsection, the case of spin-only magnetization is considered. The modelling of is very similar to that of the charge density. 8.7.4.5.1. Atom-centred expansion | | We first consider the case where spins are localized on atoms or ions, as it ...
Error analysis
International Tables for Crystallography (2006). Vol. C, Section 8.7.4.4.7, p. 729 [ doi:10.1107/97809553602060000615 ]
Error analysis 8.7.4.4.7. Error analysis In the most general case, it is not possible to obtain x, and thus M(h) directly from R. Moreover, it is unlikely that all Bragg spots within the reflection sphere could be measured. Modelling of M(h) is thus of crucial importance. The analysis of ...
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