The Theory of the One-Dimensional Lattice Defects

A general method of calculating the frequency shift due to lattice defects is developed for a one dimensional lattice with an arbitrary number of lattice points. The method is based on the Fourier transform of the equation of motion. It is shown that the frequency spectrum is determined by the roots of 5${\times}$5 secular equation, the coefficients of which depend on defects in the mass and the force constant as well as the number of the lattice points. For the limiting case of infinite lattice, the dimension of the secular equation reduces to three and the result agrees with that of Montroll and Potts.