Phonons

Eckold, G.

doi:10.1107/97809553602060000638

International Tables for Crystallography (2006). Vol. D. ch. 2.1, pp. 266-293
https://doi.org/10.1107/97809553602060000638

Chapter 2.1. Phonons

2.1. Phonons (pp. 266-293) | html | pdf | chapter contents |
G. Eckold
- 2.1.1. Introduction (p. 266) | html | pdf |
- 2.1.2. Fundamentals of lattice dynamics in the harmonic approximation (pp. 266-274) | html | pdf |
  - 2.1.2.1. Hamiltonian and equations of motion (pp. 266-267) | html | pdf |
  - 2.1.2.2. Stability conditions (p. 267) | html | pdf |
  - 2.1.2.3. The dynamical matrix (pp. 267-268) | html | pdf |
  - 2.1.2.4. Eigenvalues and phonon dispersion, acoustic modes (pp. 268-270) | html | pdf |
  - 2.1.2.5. Eigenvectors and normal coordinates (p. 270) | html | pdf |
  - 2.1.2.6. Amplitudes of lattice vibrations (pp. 270-271) | html | pdf |
  - 2.1.2.7. Density of states and the lattice heat capacity (pp. 271-272) | html | pdf |
  - 2.1.2.8. Thermal expansion, compressibility and Grüneisen parameters (pp. 272-274) | html | pdf |
- 2.1.3. Symmetry of lattice vibrations (pp. 274-291) | html | pdf |
  - 2.1.3.1. Symmetry constraints for the dynamical matrix (pp. 274-281) | html | pdf |
    - 2.1.3.1.1. Example (pp. 277-281) | html | pdf |
  - 2.1.3.2. Symmetry of dispersion planes (p. 281) | html | pdf |
  - 2.1.3.3. Symmetry properties of eigenvectors (pp. 281-283) | html | pdf |
    - 2.1.3.3.1. Example (p. 283) | html | pdf |
  - 2.1.3.4. Symmetry coordinates (pp. 283-286) | html | pdf |
    - 2.1.3.4.1. Example (pp. 284-286) | html | pdf |
  - 2.1.3.5. Degeneracy of lattice vibrations (pp. 286-289) | html | pdf |
    - 2.1.3.5.1. Accidental degeneracy (p. 286) | html | pdf |
    - 2.1.3.5.2. Time-reversal degeneracy (pp. 286-288) | html | pdf |
    - 2.1.3.5.3. Example (pp. 288-289) | html | pdf |
  - 2.1.3.6. Compatibility relations (pp. 289-290) | html | pdf |
    - 2.1.3.6.1. Example (pp. 289-290) | html | pdf |
  - 2.1.3.7. Optical selection rules (pp. 290-291) | html | pdf |
    - 2.1.3.7.1. Example (p. 291) | html | pdf |
- 2.1.4. Conclusion (p. 291) | html | pdf |
- 2.1.5. Glossary (pp. 291-292) | html | pdf |
- References | html | pdf |
- Figures
  - Fig. 2.1.2.1. Definition of position vectors (p. 266) | html | pdf |
  - Fig. 2.1.2.2. Definition of the force acting on atom ( $[\kappa l]$ ) when atom ( $[\kappa' l']$ ) is displaced by $[{\bf u}_{\kappa'l'}]$ (p. 267) | html | pdf |
  - Fig. 2.1.2.3. Phonon dispersion of b.c.c. hafnium for wavevectors along the main symmetry directions of the cubic structure (p. 268) | html | pdf |
  - Fig. 2.1.2.4. Phonon dispersion of Nd₂CuO₄ along the main symmetry directions of the tetragonal structure (p. 269) | html | pdf |
  - Fig. 2.1.2.5. Low-frequency part of the phonon dispersion of deuterated naphthalene at 6 K (p. 269) | html | pdf |
  - Fig. 2.1.2.6. Energy levels of a quantum-mechanical harmonic oscillator (p. 271) | html | pdf |
  - Fig. 2.1.2.7. Phonon dispersion and density of states for GaAs (p. 271) | html | pdf |
  - Fig. 2.1.2.8. Schematic representation of the true phonon density of states (solid line) along with the Debye approximation (dotted line) (p. 272) | html | pdf |
  - Fig. 2.1.2.9. Temperature dependence of the heat capacity at constant volume according to the Debye model (p. 272) | html | pdf |
  - Fig. 2.1.3.1. Transformation of atomic displacements by a symmetry operation (p. 275) | html | pdf |
  - Fig. 2.1.3.2. Relation between interaction of symmetry-related atoms (p. 275) | html | pdf |
  - Fig. 2.1.3.3. Symmetry-related atoms in different primitive cells (p. 276) | html | pdf |
  - Fig. 2.1.3.4. Projection along the tetragonal z axis of the example structure given in Table 2.1.3.1 (p. 278) | html | pdf |
  - Fig. 2.1.3.5. Symmetry of the dispersion surface (p. 281) | html | pdf |
  - Fig. 2.1.3.6. The dispersion relation is an even function of q (p. 281) | html | pdf |
  - Fig. 2.1.3.7. Matrix of symmetry coordinates at $[{\bf q}={\bf 0}]$ for the example structure given in Fig. 2.1.3.4 and Table 2.1.3.1 (p. 287) | html | pdf |
  - Fig. 2.1.3.8. (a) Accidental degeneracy of phonons with different symmetry (p. 287) | html | pdf |
  - Fig. 2.1.3.9. Low-frequency part of the phonon dispersion of KLiSO₄ at room temperature (space group P6₃) (p. 289) | html | pdf |
  - Fig. 2.1.3.10. Illustration of the compatibility relations for phonons in a tetragonal crystal with space group for wavevectors along [ $[\xi 00 ]$ ] (p. 290) | html | pdf |
  - Fig. 2.1.3.11. Principle of infrared absorption (p. 290) | html | pdf |
  - Fig. 2.1.3.12. Principle of Raman spectroscopy (p. 290) | html | pdf |
- Tables
  - Table 2.1.3.1. Example structure in space group (p. 278) | html | pdf |
  - Table 2.1.3.2. Atom transformation table (p. 278) | html | pdf |
  - Table 2.1.3.3. Character table of the point group (p. 283) | html | pdf |
  - Table 2.1.3.4. Irreducible representations of the point group (p. 288) | html | pdf |
  - Table 2.1.3.5. Irreducible representations of the space group for $[{\bf q}={\bf 0}]$ (the $[\Gamma]$ point) (p. 289) | html | pdf |
  - Table 2.1.3.6. Irreducible representations of the space group for $[{\bf q}={\bf c}^*/2]$ (the A point) (p. 289) | html | pdf |
  - Table 2.1.3.7. Irreducible representations of the space group for $[{\bf q}={\bf 0}]$ (the $[\Gamma]$ point) (p. 290) | html | pdf |
  - Table 2.1.3.8. Character table of the space group for $[{\bf q}={\bf 0}]$ (the $[\Gamma]$ point) (p. 291) | html | pdf |

Chapter 2.1. Phonons

Contents