A model of coupled harmonic oscillator in Zwanzig-type chain. Phonon approach (Q2923127)

This paper develops the wanzig theory to surface ferromagnetic excitations in thin films. The author proposes a new approach to the theoretical study of interactions between atoms in 1D chains of coupled harmonic oscillators and processes that occur during an initial impulse imparted to any one of the chain atoms. First, a simple model of 1D chain of atoms in the crystal lattice is formulated and some preliminaries of Zwanzig's procedure to the chain of coupled harmonic oscillations are provided. With this aim, the Hamiltonian is written for the system consisting of five monoatomic 2D layers, parallel to the surface plane. The Hamiltonian is extended by allowing one to introduce boundary conditions and effective external forces. Then, the specific role of a third layer in the considered model is analyzed by writing the corresponding system of differential-difference equations. By introducing a generating function, the author obtains the differential equation for the generating function and, as a result, the explicit formulae for the first relative displacement in the case of a chain, structured from 5 atoms. Finally, numerical results are examined for the example of a 5-atom Zwanzig-type chain.