Parameterized and systematically assembled operators for lattice defect dynamics

In this paper, we develop a calculation method for determining the phonon or vibration spectra of lattices with defects. The dynamical matrices of lattices containing defects are calculated by introducing defects systematically into the dynamical matrices of pristine, defect-free lattices using linear operators. Each operation effectively modifies or removes an individual bond or mass. Then, complex defect configurations can be constructed through reiterative application of the operators. The proposed method may be applied to systems containing any interaction type or bond order, is amenable to parameterization with continuous variables, and is suited to the study of periodic structures and atomic lattices. For verification, the method is used to calculate the dilute limit in the optical mode of a point vacancy and the periodic-to-aperiodic convergence rate as a function of the increasing unit cell size. Additional demonstrations are then shown using two dimensional lattices containing a vacancy point defect and a tri-vacancy defect cluster.