Phonons in TimeDependent Hartree Approximation with Application to OrderDisorder Ferroelectrics
Abstract
The timedependent Hartree approximation introduced into lattice dynamics by Fredkin and Werthamer is used to study the properties of longwavelength 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 temperaturedependent 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 timedependent Hartree approximation. These include the result that only Ramanactive modes contribute to internal strain, and only modes simultaneously Raman and infraredactive produce piezoelectricity. The timedependent Hartree approximation provides a convenient formalism for the discussion of lattice vibrations in strongly anharmonic crystals such as orderdisorder ferroelectrics. In particular, the piezoelectric and elastic anomalies which may accompany incipient instabilities in opticalphonon modes of orderdisorder ferroelectrics are discussed. Also, a twolevel approximation for the softopticalphonon branch in the timedependent Hartree formalism is shown to be equivalent to results obtained previously by means of a dynamical pseudospin formulation.
 Publication:

Physical Review
 Pub Date:
 November 1968
 DOI:
 10.1103/PhysRev.175.1062
 Bibcode:
 1968PhRv..175.1062M