Nonlinear Vibrational Excitations in a Diatomic Lattice.

In this thesis we examine the dynamics and the statistical mechanics of nonlinear vibrational excitations in a diatomic model for a solid which may undergo a displacive structural phase transition. The model consists of a diatomic chain of harmonically coupled nearest neighbour atoms including a nonlinear on-site potential on one species. The Hamiltonian describing the diatomic model leads to two coupled nonlinear partial differential field equations. We identify a number of travelling wave solutions of these equations including linearized phonons, kink and pulse type solitary waves, and nonlinear periodic waves. The dispersion relation for the linearized phonons exhibits two branches, a quasi-acoustic branch and a quasi-optical branch. The pulse solitary wave is shown to be unstable to small perturbations whereas the kink solitary wave has a finite creation energy and is long lived in the presence of small perturbations. Molecular-dynamics (MD) experiments we have performed show that the kinks exhibit soliton like properties. In particular, phonon wavepackets (both acoustic and optical) pass through the kinks, and kinks pass through one another, with little distortion. The thermodynamical properties of the nonlinear diatomic chain are computed using (i) a two component transfer integral operator equation and (ii) an ideal gas phenomenology incorporating the linearized phonons and the nonlinear kinks as elementary excitations. The agreement between the two expressions for the free energy density thus obtained formally establishes that the low temperature excitation spectrum for the nonlinear diatomic chain is dominated by both the familiar linear phonon excitations and by nonlinear vibrational (kink) excitations. The nonlinear vibrational excitations are shown to be responsible for a central peak in the dynamical structure factor for the nonlinear diatomic lattice. Molecular-dynamics simulations of our model reveal that the height of the peak increases and the width of the peak decreases as the temperature goes to the critical temperature. At small wavevectors the central peak splits producing a new excitation branch in addition to the usual soft-mode phonon sideband. The present results may have relevance for the central peak observed in the scattering cross section from inelastic neutron scattering experiments performed on SrTiO(,3).