Quantum theory of Lattice Vibrations
The quantum mechanical theory of lattice vibrations in solids is reviewed and summarized. The formalism is developed first in the classical manner, and the various symmetries of the normal mode eigenvalues are discussed. The transition to the quantum mechanical formalism is done by introducing operator forms for the appropriate physical (observable) quantities. As an extension of the quantum mechanical formalism, the thermodynamic properties of the lattice vibrations (entropy, free energy, etc.) are investigated. An extensive treatment of critical points in the phonon density of states is given; this includes a discussion of the so-called Van Hove singularities, and a cataloguing of the types of singularities involved. Finally, the lattice vibrational equation of state is formulated, and the effects of the boundary conditions on the frequency distribution are discussed.
KeywordsSaddle Point Quantum Theory Brillouin Zone Lattice Vibration Reciprocal Lattice
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