Long-range interactions in free-free transitions

The soft-photon approximation for bremsstrahlung derived some time ago by Low and extended by Feshbach and Yennie to allow for resonances is further generalized, in the framework of nonrelativistic scattering theory, to apply to the case where the scattering potential has a long-range Coulomb tail. The derivation is based on an asymptotic evaluation of the matrix element in configuration space in which electric-dipole, magnetic-dipole, and electric-quadrupole components of the particle-field interaction are accounted for. The $M1\mathrm{}$ and $E2\mathrm{}$ contributions give rise to retardation corrections, of order $\frac{v}{c}$, to the leading $E1\mathrm{}$ term. The Coulomb tail has the effect of introducing certain factors into the approximate bremsstrahlung amplitude which depend logarithmically on the photon frequency. The result obtained here has an immediate application to the problem of Coulomb scattering in a low-frequency laser field and provides a generalization of earlier work based on the electric-dipole approximation.