International Tables for Crystallography (2013). Vol. D, ch. 2.1, pp. 286-313
doi: 10.1107/97809553602060000911

Chapter 2.1. Phonons

Contents

  • 2.1. Phonons  (pp. 286-313) | html | pdf | chapter contents |
    • 2.1.1. Introduction  (p. 286) | html | pdf |
    • 2.1.2. Fundamentals of lattice dynamics in the harmonic approximation  (pp. 286-294) | html | pdf |
      • 2.1.2.1. Hamiltonian and equations of motion  (pp. 286-287) | html | pdf |
      • 2.1.2.2. Stability conditions  (p. 287) | html | pdf |
      • 2.1.2.3. The dynamical matrix  (pp. 287-288) | html | pdf |
      • 2.1.2.4. Eigenvalues and phonon dispersion, acoustic modes  (pp. 288-290) | html | pdf |
      • 2.1.2.5. Eigenvectors and normal coordinates  (p. 290) | html | pdf |
      • 2.1.2.6. Amplitudes of lattice vibrations  (pp. 290-291) | html | pdf |
      • 2.1.2.7. Density of states and the lattice heat capacity  (pp. 291-292) | html | pdf |
      • 2.1.2.8. Thermal expansion, compressibility and Grüneisen parameters  (pp. 292-294) | html | pdf |
    • 2.1.3. Symmetry of lattice vibrations  (pp. 294-311) | html | pdf |
      • 2.1.3.1. Symmetry constraints for the dynamical matrix  (pp. 294-301) | html | pdf |
        • 2.1.3.1.1. Example  (pp. 297-301) | html | pdf |
      • 2.1.3.2. Symmetry of dispersion planes  (p. 301) | html | pdf |
      • 2.1.3.3. Symmetry properties of eigenvectors  (pp. 301-303) | html | pdf |
        • 2.1.3.3.1. Example  (p. 303) | html | pdf |
      • 2.1.3.4. Symmetry coordinates  (pp. 303-306) | html | pdf |
        • 2.1.3.4.1. Example  (pp. 304-306) | html | pdf |
      • 2.1.3.5. Degeneracy of lattice vibrations  (pp. 306-309) | html | pdf |
        • 2.1.3.5.1. Accidental degeneracy  (p. 306) | html | pdf |
        • 2.1.3.5.2. Time-reversal degeneracy  (pp. 306-308) | html | pdf |
        • 2.1.3.5.3. Example  (pp. 308-309) | html | pdf |
      • 2.1.3.6. Compatibility relations  (pp. 309-310) | html | pdf |
        • 2.1.3.6.1. Example  (pp. 309-310) | html | pdf |
      • 2.1.3.7. Optical selection rules  (pp. 310-311) | html | pdf |
        • 2.1.3.7.1. Example  (p. 311) | html | pdf |
    • 2.1.4. Conclusion  (p. 311) | html | pdf |
    • 2.1.5. Glossary  (pp. 311-312) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 2.1.2.1. Definition of position vectors  (p. 286) | 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. 287) | 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. 288) | html | pdf |
      • Fig. 2.1.2.4. Phonon dispersion of Nd 2 CuO 4 along the main symmetry directions of the tetragonal structure  (p. 289) | html | pdf |
      • Fig. 2.1.2.5. Low-frequency part of the phonon dispersion of deuterated naphthalene at 6 K  (p. 289) | html | pdf |
      • Fig. 2.1.2.6. Energy levels of a quantum-mechanical harmonic oscillator  (p. 291) | html | pdf |
      • Fig. 2.1.2.7. Phonon dispersion and density of states for GaAs  (p. 291) | 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. 292) | html | pdf |
      • Fig. 2.1.2.9. Temperature dependence of the heat capacity at constant volume according to the Debye model  (p. 292) | html | pdf |
      • Fig. 2.1.3.1. Transformation of atomic displacements by a symmetry operation  (p. 295) | html | pdf |
      • Fig. 2.1.3.2. Relation between interaction of symmetry-related atoms  (p. 295) | html | pdf |
      • Fig. 2.1.3.3. Symmetry-related atoms in different primitive cells  (p. 296) | 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. 298) | html | pdf |
      • Fig. 2.1.3.5. Symmetry of the dispersion surface  (p. 301) | html | pdf |
      • Fig. 2.1.3.6. The dispersion relation is an even function of q   (p. 301) | 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. 307) | html | pdf |
      • Fig. 2.1.3.8. ( a ) Accidental degeneracy of phonons with different symmetry  (p. 307) | html | pdf |
      • Fig. 2.1.3.9. Low-frequency part of the phonon dispersion of KLiSO 4 at room temperature (space group P 6 3 )  (p. 309) | html | pdf |
      • Fig. 2.1.3.10. Illustration of the compatibility relations for phonons in a tetragonal crystal with space group [P4mm] for wavevectors along [[\xi 00 ]]  (p. 310) | html | pdf |
      • Fig. 2.1.3.11. Principle of infrared absorption  (p. 310) | html | pdf |
      • Fig. 2.1.3.12. Principle of Raman spectroscopy  (p. 310) | html | pdf |
    • Tables
      • Table 2.1.3.1. Example structure in space group [P4mm]   (p. 298) | html | pdf |
      • Table 2.1.3.2. Atom transformation table  (p. 298) | html | pdf |
      • Table 2.1.3.3. Character table of the point group [4mm]   (p. 303) | html | pdf |
      • Table 2.1.3.4. Irreducible representations of the point group [3]   (p. 308) | html | pdf |
      • Table 2.1.3.5. Irreducible representations of the space group [P6_3] for [{\bf q}={\bf 0}] (the [\Gamma] point)  (p. 309) | html | pdf |
      • Table 2.1.3.6. Irreducible representations of the space group [P6 _3] for [{\bf q}={\bf c}^*/2] (the A point)  (p. 309) | html | pdf |
      • Table 2.1.3.7. Irreducible representations of the space group [P4mm] for [{\bf q}={\bf 0}] (the [\Gamma] point)  (p. 310) | html | pdf |
      • Table 2.1.3.8. Character table of the space group [P4mm] for [{\bf q}={\bf 0}] (the [\Gamma] point)  (p. 311) | html | pdf |