VIBRATIONAL DYNAMICS OF ATOMS IN A ONE-DIMENSIONAL CRYSTAL LATTICE

A crystal is composed of a periodic arrangement of atoms and molecules that vibrate around their equilibrium positions due to thermal motion at finite temperatures. When atoms are displaced from these positions, they experience a restoring force that follows Hooke's law and is proportional to the displacement. This study analyzes the vibrational dynamics of a one-dimensional lattice, where the atoms are modeled as masses connected by springs with a force constant. The motion of atoms in this system is governed by the equations of motion, and the resulting lattice vibrations are examined in the context of phase velocity, sound velocity, and Brillouin zones. The analysis is further extended to a lattice consisting of two different kinds of atoms, leading to the formation of distinct acoustic and optical modes. The dispersion relations and the role of wave vectors defined by cyclic boundary conditions are explored, with emphasis on the periodic nature of the ω-q curve and the significance of the first Brillouin zone.