Towards a Quantum Many-Body Theory of Lattice Dynamics. II. Collective Fluctuation Approximation

A novel viewpoint towards the lattice dynamics of an anharmonic crystal, put forward in a previous paper of the same title, is enlarged and extended. This viewpoint first focuses attention on the motion of a single atom in a static environment, and develops the collective modes of the crystal as a whole from a superposition of the motions of the individual atoms. It is shown that of the many collective modes for a given wave vector, three are identifiable as one-phonon modes in that only these contribute to the displacement-displacement response, and a simple expression for the eigenfrequencies of these modes is exhibited. The other modes are shown to be associated with single-particle transitions, and their contribution to the neutron structure function $S(\mathrm{k}, \omega )$ is derived in the special case of a purely harmonic lattice. The many-body approximation is extended to include collective fluctuations in the equilibrium state. Serious difficulties in principle are encountered, associated with maintaining translational invariance, but are partially overcome by an ad hoc procedure. Collective-mode frequencies renormalized in this way are compared with those obtained from an alternative theory which deals exclusively and from the outset with collective coordinates only.