It is well understood that defects adversely affect the electro-mechanical properties of materials. Ideally, defect compositions of device materials could be measured, but present technology in the field of atomic defect detection is either destructive in nature, or is unable to determine the precise atomic composition of materials. In the adjacent field of damage detection in large-scale truss networks, algorithms based on spectral measurements have successfully been employed to locate damaged members. Already similar principles have been applied to material lattices experimentally by using Raman Spectroscopy to qualitatively approximate defect densities within materials. However, the applications have largely been limited to surface defects or two-dimensional materials, and the host lattices and defect types are primarily studied anecdotally.

This thesis details a numerical method for determining the precise phonon or vibration spectra of material 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 modifies or removes an individual bond or interaction. Complex defect configurations can be composed through reiterative application of the operators. The proposed methods may be applied to systems containing any interaction type or bond order, including space trusses and atomic lattices. The method is demonstrated by numerically determining the convergence rate of phonon properties in the dilute limit of a single point vacancy. Then the same methodology is applied to two-dimensional atomic lattices with central forces,

two-dimension truss networks with distributed mass, as well as three-dimensional

atomic lattices with non-linear many body potentials. In each example, the defect

structure and properties are shown to alter the spectral properties of the materials.