Effect of Defects on Lattice Vibrations

The theory of the effect of localized defects such as impurities, holes, and interstitials on the vibrations of crystal lattices is developed. Although most of the analysis is concerned with one-dimensional chains, the general approach to defects in three-dimensional lattices is outlined through the example of a simple cubic lattice with nearest-neighbor interactions. Many types of defects cause localized normal modes whose effect dies out rapidly with distance from the defect. Mathematical techniques, which involve the use of Green's functions, are discussed for the theory of these localized modes. The vibrational frequencies of these modes are displaced out of the band of frequencies of a perfect lattice. The theory of interaction of two defects as a function of their distance of separation is developed for the range of very low temperatures through the calculation of the change of zero-point energy of a lattice as a result of the introduction of a defect pair. Defects attract each other in a monatomic lattice. The attraction between two mass defects in a linear chain is inversely proportional to the cube of their distance of separation. The effect of a localized defect mode in a simple cubic lattice diminishes as with the distance r as r^-1exp (-Ar).