Phonons in time-dependent hartree approximation with application to order-disorder ferroelectrics

The time-dependent Hartree approximation introduced into lattice dynamics by Fredkin and Werthamer is used to study the properties of long-wavelength acoustic phonons. An equation for the acoustic polarization and velocity including internal strain effect is derived, which has a form similar to the familiar result of harmonic lattice dynamics, but with temperature-dependent coefficients arising from anharmonic effects. Certain symmetry properties relating internal strain to elastic and piezoelectric constants derived previously for a harmonic lattice are also valid in the time-dependent Hartree approximation. These include the result that only Raman-active modes contribute to internal strain, and only modes simultaneously Raman- and infrared-active produce piezoelectricity. The time-dependent Hartree approximation provides a convenient formalism for the discussion of lattice vibrations in strongly anharmonic crystals such as order-disorder ferroelectrics. In particular, the piezoelectric and elastic anomalies which may accompany incipient instabilities in optical-phonon modes of order-disorder ferroelectrics are discussed. Also, a two-level approximation for the soft-optical-phonon branch in the time-dependent Hartree formalism is shown to be equivalent to results obtained previously by means of a dynamical pseudospin formulation. © 1968 The American Physical Society.