A Theory of the Lattice Vibration of Anharmonic Solids

The lattice vibration of anharmonic or “non-ideal” solids is studied using a method of double-time Green's function, without assuming the smallness of the anharmonicity of atomic vibrations from the outset. To encompass disordered and amorphous solids and molecular systems as well, a system under consideration is taken to be arbitrary in structure and in composition. In treating equations satisfied by Green's functions a decoupling approximation is employed, which leads to a self-consistent phonon theory. A set of hierarchy equations are thereby obtained which give renormalized eigenfrequencies of phonons in terms of “effective interatomic or intermolecular potentials” or “effective force constants” in an implicit manner. In contrast to almost all current self-consistent phonon theories in which attention is focused to the case of quantum crystals, the theory is designed to reformulate theory of lattice dynamics and therefore to study the anharmonic vibrational properties of solids in general. In solving the hierarcy equations the first and the second order approximations are employed which give renormalized harmonic phonons and renormalized anharmonic phonons with three-phonon scattering processes taken into account, respectively. A detailed discussion is given of the properties of effective potentials. It is shown that the effect of the anharmonicity of atomic vibrations is to modulate a bare potential in a manner similar to the random thermal modulation of X rays or neutrons in solids, thus giving rise to a decrease in the minimum value and an increase in the minimum position of the effective potential as the anharmonicity increases; its asymptotic expression reduces to a mean intermolecular potential energy in liquids or in imperfect gases. As applications and implications of results obtained here, a brief discussion is given of some general properties of effective potential, the dynamical instability or the melting of solids, phonons in liquids, anharmonic vibrational properties of phonon impurity modes in crystals, impurity- or defect-induced anharmonic vibrations in solids, etc.