tMatrix and Phase Shifts in Solid-State Scattering Theory

We are concerned with the scattering of an excitation in a solid by a potential of finite range. The appropriate scattering theory is formulated using the $t$ matrix. A general expression for $t$-matrix elements is obtained involving only finite sums. Phase shifts are introduced and are related to the change in density of states produced by the scattering. The phase shifts satisfy a form of Levinson's theorem. The relation between the phase shifts and the $t$ matrix is examined with the aid of the optical theorem, and it is shown that in the case of the scattering of long-wavelength excitations in a spherical band (subject to certain limitations described in the text), the usual relation between the phase shifts and the scattering amplitude can be recovered. As an application, the change in the density of states and the excitation lifetime are obtained for spin waves in a Heisenberg ferromagnet with a small concentration of magnetic defects.