Solid Hydrogen pp 131-172 | Cite as

Lattice Vibrations and Elastic Properties

  • Jan Van Kranendonk


The solid hydrogens, together with solid He4 and He3, form a class of solids known as quantum crystals. These solids are characterized by the property that the amplitude of the zero-point lattice vibrations is an appreciable fraction of the lattice constant, as a result of the small mass of the molecules and the relatively weak intermolecular forces. In a normal crystal the lattice vibrations can be treated by expanding the potential energy in powers of the displacements of the molecules from their equilibrium positions, but in quantum crystals this procedure leads to imaginary phonon frequencies in at least part of the BZ, and must be replaced by a self-consistent phonon formalism not based on an expansion in powers of the displacements. Only the main concepts of the theory of quantum crystals are discussed here, and for further details we refer to the available reviews.1, 2 The elastic properties of a solid are closely related to the long-wavelength phonon modes, and a generalized Debye model, parameterized in terms of the elastic constants, is introduced to describe the anisotropy in the propagation and polarization properties of the phonon modes in hcp solid H2 and D2. This model is useful for calculating certain properties of these solids depending on the anisotropy of the phonon field.


Coherent State Lattice Vibration Harmonic Approximation Solid Hydrogen Quantum Crystal 
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  1. 1.
    T. R. Koehler, Lattice dynamics of quantum crystals, in Dynamical Properties of Solids, Vol. 2, pp. 1–104, edited by G. K. Horton and A. A. Maradudin. North-Holland Publ. Co. Amsterdam (1975).Google Scholar
  2. 2.
    H. R. Glyde, Solid Helium, in Rare Gas Solids, edited by M. L. Klein and J. A. Venables, Chapter 7. Academic Press Inc. Ltd., London (1976).Google Scholar
  3. 3.
    A. A. Maradudin, E. W. Montroll, G. Weiss, and I. P. Ipatova, Theory of lattice dynamics in the harmonic approximation, Academic Press Inc., New York (1971).Google Scholar
  4. 4.
    T. H. K. Barron and M. L. Klein, Perturbation theory of anharmoniccrystals, in Dynamical Properties of Solids, I, 391–449. North-Holland Publ. Co. Amsterdam (1975).Google Scholar
  5. 5.
    N. R. Werthamer, Self-consistent phonon theory of rare gas solids, in Rare Gas Solids, ed. by M. L. Klein and J. A. Venables, Ch. 5. Academic Press Inc. (London) Ltd. (1976).Google Scholar
  6. 6.
    L. H. Nosanow, Theory of quantum crystals, Phys. Rev. 146, 120–133 (1966).ADSCrossRefGoogle Scholar
  7. 7.
    T. R. Koehler, New theory of lattice dynamics at 0 K, Phys. Rev. 165, 942–950 (1968).ADSCrossRefGoogle Scholar
  8. 8.
    H. Horner, Lattice dynamics of strongly anharmonic crystals with hard core interactions, Z. Phys. 242, 432–443 (1971).ADSCrossRefGoogle Scholar
  9. 9.
    J. Noolandi and J. Van Kranendonk, The use of coherent states in the theory of quantum crystals, Can. J. Phys. 50, 1815–1825 (1972).ADSCrossRefGoogle Scholar
  10. 10.
    R. J. Glauber, Coherent and incoherent states of the radiation field, Phys. Rev. 131, 2766–2788 (1963).MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    M. L. Klein and T. R. Koehler, Self-consistent phonon spectrum of hcp H2 and D2, J. Phys. C3, L102–104 (1970).ADSGoogle Scholar
  12. 12.
    S. Luryi and J. Van Kranendonk, Elastic constants and anisotropic pair correlations in solid hydrogen and deuterium, Can. J. Phys. 57, 136–146 (1979).ADSCrossRefGoogle Scholar
  13. 13.
    J. F. Nye, Physical Properties of Crystals. Oxford University Press, London (1957).MATHGoogle Scholar
  14. 14.
    V. V. Goldman, Elastic constants of solid hexagonal close-packed hydrogen and deuterium, J. Low Temp. Phys. 24, 297–313 (1976).ADSCrossRefGoogle Scholar
  15. 15.
    M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Oxford University Press, London (1954).MATHGoogle Scholar
  16. 16.
    E. L. Pollock, T. A. Bruce, G. V. Chester, and J. A. Krumhansl, High-pressure behavior of solid H2, D2, He3, He4, and Ne20, Phys. Rev. B5, 4180–4190 (1972).ADSGoogle Scholar
  17. 17.
    A. B. Harris, A. J. Berlinsky, and W. N. Hardy, Interactions between ortho-H2 molecules in nearly pure para-H2, Can. J. Phys. 55, 1180–1210 (1977).ADSCrossRefGoogle Scholar
  18. 18.
    S. Luryi and J. Van Kranendonk, Renormalized interactions in solid hydrogen and analysis of the ortho-pair level structure, Can. J. Phys. 57, 307–326 (1979).ADSCrossRefGoogle Scholar
  19. 19.
    M. Nielsen, Phonons in solid hydrogen and deuterium studied by inelastic coherent neutron scattering, Phys. Rev. B7, 1626–1635, (1973).ADSGoogle Scholar
  20. 20.
    P. J. Thomas, S. C. Rand, and B. P. Stoicheff, Elastic constants of parahydrogen determined by Brillouin scattering, Can. J. Phys. 56, 1494–1501 (1978).ADSCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1983

Authors and Affiliations

  • Jan Van Kranendonk
    • 1
  1. 1.University of TorontoTorontoCanada

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