Electrons

Schwarz, K.

doi:10.1107/97809553602060000912

International Tables for Crystallography (2013). Vol. D. ch. 2.2, pp. 314-333
https://doi.org/10.1107/97809553602060000912

Chapter 2.2. Electrons

2.2. Electrons (pp. 314-333) | html | pdf | chapter contents |
K. Schwarz
- 2.2.1. Introduction (p. 314) | html | pdf |
- 2.2.2. The lattice (p. 314) | html | pdf |
  - 2.2.2.1. The direct lattice and the Wigner–Seitz cell (p. 314) | html | pdf |
  - 2.2.2.2. The reciprocal lattice and the Brillouin zone (p. 314) | html | pdf |
- 2.2.3. Symmetry operators (pp. 314-315) | html | pdf |
  - 2.2.3.1. Transformation of functions (pp. 314-315) | html | pdf |
  - 2.2.3.2. Transformation of operators (p. 315) | html | pdf |
  - 2.2.3.3. The Seitz operators (p. 315) | html | pdf |
  - 2.2.3.4. The important groups and their first classification (p. 315) | html | pdf |
- 2.2.4. The Bloch theorem (pp. 315-316) | html | pdf |
  - 2.2.4.1. A simple quantum-mechanical derivation (pp. 315-316) | html | pdf |
  - 2.2.4.2. Periodic boundary conditions (p. 316) | html | pdf |
  - 2.2.4.3. A simple group-theoretical approach (p. 316) | html | pdf |
- 2.2.5. The free-electron (Sommerfeld) model (p. 317) | html | pdf |
- 2.2.6. Space-group symmetry (pp. 317-318) | html | pdf |
  - 2.2.6.1. Representations and bases of the space group (pp. 317-318) | html | pdf |
  - 2.2.6.2. Energy bands (p. 318) | html | pdf |
- 2.2.7. The vector and the Brillouin zone (p. 318) | html | pdf |
  - 2.2.7.1. Various aspects of the $[{\bf k}]$ vector (p. 318) | html | pdf |
  - 2.2.7.2. The Brillouin zone (BZ) (p. 318) | html | pdf |
  - 2.2.7.3. The symmetry of the Brillouin zone (p. 318) | html | pdf |
- 2.2.8. Bloch functions (p. 319) | html | pdf |
- 2.2.9. Quantum-mechanical treatment (pp. 319-320) | html | pdf |
  - 2.2.9.1. Exchange and correlation treatment (p. 319) | html | pdf |
  - 2.2.9.2. The choice of basis sets and wavefunctions (p. 319) | html | pdf |
  - 2.2.9.3. The form of the potential (p. 319) | html | pdf |
  - 2.2.9.4. Relativistic effects (p. 320) | html | pdf |
- 2.2.10. Density functional theory (pp. 320-321) | html | pdf |
- 2.2.11. Band-theory methods (pp. 321-323) | html | pdf |
  - 2.2.11.1. LCAO (linear combination of atomic orbitals) (p. 321) | html | pdf |
  - 2.2.11.2. TB (tight binding) (pp. 321-322) | html | pdf |
  - 2.2.11.3. The pseudo-potential schemes (p. 322) | html | pdf |
  - 2.2.11.4. APW (augmented plane wave) and LAPW methods (p. 322) | html | pdf |
  - 2.2.11.5. KKR (Korringa–Kohn–Rostocker) method (p. 322) | html | pdf |
  - 2.2.11.6. LMTO (linear combination of muffin-tin orbitals) method (p. 322) | html | pdf |
  - 2.2.11.7. CP (Car–Parrinello) method (p. 322) | html | pdf |
  - 2.2.11.8. Order N schemes (pp. 322-323) | html | pdf |
- 2.2.12. The linearized augmented plane wave method (pp. 323-324) | html | pdf |
- 2.2.13. The local coordinate system (pp. 324-325) | html | pdf |
  - 2.2.13.1. Crystal harmonics (pp. 324-325) | html | pdf |
  - 2.2.13.2. Interpretation for bonding (p. 325) | html | pdf |
- 2.2.14. Characterization of Bloch states (pp. 325-327) | html | pdf |
  - 2.2.14.1. Characterization by group theory (p. 325) | html | pdf |
  - 2.2.14.2. Energy regions (p. 325) | html | pdf |
  - 2.2.14.3. Decomposition according to wavefunctions (pp. 325-326) | html | pdf |
  - 2.2.14.4. Localized versus itinerant electrons (p. 326) | html | pdf |
  - 2.2.14.5. Spin polarization (p. 326) | html | pdf |
  - 2.2.14.6. The density of states (DOS) (pp. 326-327) | html | pdf |
- 2.2.15. Electric field gradient tensor (pp. 327-330) | html | pdf |
  - 2.2.15.1. Introduction (p. 327) | html | pdf |
  - 2.2.15.2. EFG conversion formulas (pp. 327-328) | html | pdf |
  - 2.2.15.3. Theoretical approach (pp. 328-330) | html | pdf |
- 2.2.16. Examples (pp. 330-332) | html | pdf |
  - 2.2.16.1. F.c.c. copper (pp. 330-331) | html | pdf |
  - 2.2.16.2. The rutile TiO₂ (p. 331) | html | pdf |
  - 2.2.16.3. Core electron spectra (p. 332) | html | pdf |
- 2.2.17. Conclusion (p. 332) | html | pdf |
- References | html | pdf |
- Figures
  - Fig. 2.2.5.1. Plane waves (p. 317) | html | pdf |
  - Fig. 2.2.7.1. The Brillouin zone (BZ) and the irreducible wedge of the BZ for the f.c.c. direct lattice (p. 318) | html | pdf |
  - Fig. 2.2.12.1. Schematic partitioning of the unit cell into atomic spheres (I) and an interstitial region (II) (p. 323) | html | pdf |
  - Fig. 2.2.14.1. Relativistic radial wavefunctions (large component) of the uranium atom (p. 326) | html | pdf |
  - Fig. 2.2.15.1. Unit cell of the high-temperature superconductor YBa₂Cu₃O₇ with four non-equivalent oxygen sites (p. 329) | html | pdf |
  - Fig. 2.2.16.1. Character of energy bands of f.c.c. copper in the $[\Delta]$ direction (p. 330) | html | pdf |
  - Fig. 2.2.16.2. Decomposition of the Cu d bands into the $[e_{g}]$ and $[t_{2g}]$ manifold (p. 331) | html | pdf |
  - Fig. 2.2.16.3. The local coordinate system in rutile for titanium (small spheres) and oxygen (large spheres) (p. 331) | html | pdf |
  - Fig. 2.2.16.4. Schematic transitions in X-ray emission and absorption spectra (p. 331) | html | pdf |
- Tables
  - Table 2.2.13.1. Picking rules for the local coordinate axes and the corresponding combinations ( $[\ell mp]$ ) of non-cubic groups taken from Kurki-Suonio (1977) (p. 324) | html | pdf |
  - Table 2.2.13.2. LM combinations of cubic groups as linear cominations of $[y_{\ell mp}]$ 's (given in parentheses) (p. 324) | html | pdf |
  - Table 2.2.15.1. Partial O 2p charges (in electrons) and electric field gradient tensor O EFG (in 10²¹ V m⁻²) for YBa₂Cu₃O₇ (p. 329) | html | pdf |
  - Table 2.2.16.1. $[W_{\ell\ell^{\prime}}]$ factors for X-ray emission spectra showing the $[\Delta\ell=\pm 1]$ selection rule (p. 332) | html | pdf |

Chapter 2.2. Electrons

Contents