International Tables for Crystallography (2006). Vol. C, ch. 8.7, pp. 713-734
doi: 10.1107/97809553602060000615

Chapter 8.7. Analysis of charge and spin densities

Contents

  • 8.7. Analysis of charge and spin densities  (pp. 713-734) | html | pdf | chapter contents |
    • 8.7.1. Outline of this chapter  (p. 713) | html | pdf |
    • 8.7.2. Electron densities and the n -particle wavefunction  (p. 713) | html | pdf |
    • 8.7.3. Charge densities  (pp. 714-725) | html | pdf |
      • 8.7.3.1. Introduction  (p. 714) | html | pdf |
      • 8.7.3.2. Modelling of the charge density  (pp. 714-715) | html | pdf |
      • 8.7.3.3. Physical constraints  (p. 715) | html | pdf |
        • 8.7.3.3.1. Electroneutrality constraint  (p. 715) | html | pdf |
        • 8.7.3.3.2. Cusp constraint  (p. 715) | html | pdf |
        • 8.7.3.3.3. Radial constraint  (p. 715) | html | pdf |
        • 8.7.3.3.4. Hellmann–Feynman constraint  (p. 715) | html | pdf |
      • 8.7.3.4. Electrostatic moments and the potential due to a charge distribution  (pp. 716-721) | html | pdf |
        • 8.7.3.4.1. Moments of a charge distribution  (pp. 716-718) | html | pdf |
          • 8.7.3.4.1.1. Moments as a function of the atomic multipole expansion  (pp. 716-717) | html | pdf |
          • 8.7.3.4.1.2. Molecular moments based on the deformation density  (p. 717) | html | pdf |
          • 8.7.3.4.1.3. The effect of an origin shift on the outer moments  (pp. 717-718) | html | pdf |
          • 8.7.3.4.1.4. Total moments as a sum over the pseudoatom moments  (p. 718) | html | pdf |
          • 8.7.3.4.1.5. Electrostatic moments of a subvolume of space by Fourier summation  (p. 718) | html | pdf |
        • 8.7.3.4.2. The electrostatic potential  (pp. 718-720) | html | pdf |
          • 8.7.3.4.2.1. The electrostatic potential and its derivatives  (pp. 718-719) | html | pdf |
          • 8.7.3.4.2.2. Electrostatic potential outside a charge distribution  (p. 720) | html | pdf |
          • 8.7.3.4.2.3. Evaluation of the electrostatic functions in direct space  (p. 720) | html | pdf |
        • 8.7.3.4.3. Electrostatic functions of crystals by modified Fourier summation  (pp. 720-721) | html | pdf |
        • 8.7.3.4.4. The total energy of a crystal as a function of the electron density  (p. 721) | html | pdf |
      • 8.7.3.5. Quantitative comparison with theory  (pp. 721-722) | html | pdf |
      • 8.7.3.6. Occupancies of transition-metal valence orbitals from multipole coefficients  (pp. 722-723) | html | pdf |
      • 8.7.3.7. Thermal smearing of theoretical densities  (pp. 723-724) | html | pdf |
        • 8.7.3.7.1. General considerations  (p. 723) | html | pdf |
        • 8.7.3.7.2. Reciprocal-space averaging over external vibrations  (pp. 723-724) | html | pdf |
      • 8.7.3.8. Uncertainties in experimental electron densities  (pp. 724-725) | html | pdf |
      • 8.7.3.9. Uncertainties in derived functions  (p. 725) | html | pdf |
    • 8.7.4. Spin densities  (pp. 725-734) | html | pdf |
      • 8.7.4.1. Introduction  (p. 725) | html | pdf |
      • 8.7.4.2. Magnetization densities from neutron magnetic elastic scattering  (pp. 725-726) | html | pdf |
      • 8.7.4.3. Magnetization densities and spin densities  (pp. 726-727) | html | pdf |
        • 8.7.4.3.1. Spin-only density at zero temperature  (p. 726) | html | pdf |
        • 8.7.4.3.2. Thermally averaged spin-only magnetization density  (pp. 726-727) | html | pdf |
        • 8.7.4.3.3. Spin density for an assembly of localized systems  (p. 727) | html | pdf |
        • 8.7.4.3.4. Orbital magnetization density  (p. 727) | html | pdf |
      • 8.7.4.4. Probing spin densities by neutron elastic scattering  (pp. 727-729) | html | pdf |
        • 8.7.4.4.1. Introduction  (pp. 727-728) | html | pdf |
        • 8.7.4.4.2. Unpolarized neutron scattering  (p. 728) | html | pdf |
        • 8.7.4.4.3. Polarized neutron scattering  (p. 728) | html | pdf |
        • 8.7.4.4.4. Polarized neutron scattering of centrosymmetric crystals  (p. 728) | html | pdf |
        • 8.7.4.4.5. Polarized neutron scattering in the noncentro­symmetric case  (p. 728) | html | pdf |
        • 8.7.4.4.6. Effect of extinction  (pp. 728-729) | html | pdf |
        • 8.7.4.4.7. Error analysis  (p. 729) | html | pdf |
      • 8.7.4.5. Modelling the spin density  (pp. 729-730) | html | pdf |
        • 8.7.4.5.1. Atom-centred expansion  (pp. 729-730) | html | pdf |
          • 8.7.4.5.1.1. Spherical-atom model  (p. 729) | html | pdf |
          • 8.7.4.5.1.2. Crystal-field approximation  (pp. 729-730) | html | pdf |
          • 8.7.4.5.1.3. Scaling of the spin density  (p. 730) | html | pdf |
        • 8.7.4.5.2. General multipolar expansion  (p. 730) | html | pdf |
        • 8.7.4.5.3. Other types of model  (p. 730) | html | pdf |
      • 8.7.4.6. Orbital contribution to the magnetic scattering  (pp. 730-731) | html | pdf |
        • 8.7.4.6.1. The dipolar approximation  (p. 731) | html | pdf |
        • 8.7.4.6.2. Beyond the dipolar approximation  (p. 731) | html | pdf |
        • 8.7.4.6.3. Electronic structure of rare-earth elements  (p. 731) | html | pdf |
      • 8.7.4.7. Properties derivable from spin densities  (pp. 731-732) | html | pdf |
        • 8.7.4.7.1. Vector fields  (p. 732) | html | pdf |
        • 8.7.4.7.2. Moments of the magnetization density  (p. 732) | html | pdf |
      • 8.7.4.8. Comparison between theory and experiment  (p. 732) | html | pdf |
      • 8.7.4.9. Combined charge- and spin-density analysis  (p. 732) | html | pdf |
      • 8.7.4.10. Magnetic X-ray scattering separation between spin and orbital magnetism  (pp. 733-734) | html | pdf |
        • 8.7.4.10.1. Introduction  (p. 733) | html | pdf |
        • 8.7.4.10.2. Magnetic X-ray structure factor as a function of photon polarization  (pp. 733-734) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 8.7.4.1. Some geometrical definitions  (p. 733) | html | pdf |
    • Tables
      • Table 8.7.3.1. Definition of difference density functions  (p. 714) | html | pdf |
      • Table 8.7.3.2. Expressions for the shape factors S for a parallelepiped with edges δ x , δ y , and δ z   (p. 719) | html | pdf |
      • Table 8.7.3.3. The matrix M −1 relating d -orbital occupancies P ij to multipole populations P lm   (p. 722) | html | pdf |
      • Table 8.7.3.4. Orbital–multipole relations for square-planar complexes (point group D 4 h )  (p. 723) | html | pdf |
      • Table 8.7.3.5. Orbital–multipole relations for trigonal complexes  (p. 723) | html | pdf |