Second-order singularities in transfer ionization

In processes in which one electron is transferred from a target to a projectile and the other electron is ionized, called transfer ionization, two second-order terms arise in the transition amplitude at high collision velocities. There is a kinematic singularity and a weaker Thomas singularity. These second-order singular terms are formulated to first order in the projectile electron and first order in the electron-electron interactions. Using standard peaking approximations, the T matrix is reduced to an approximate closed-form expression. Our expression is qualitatively similar to the expression found earlier by J. S. Briggs and K. Taulbjerg [J. Phys. B 12, 2565 (1979)], who used a different formulation and evaluation and omitted an exchange contribution which we include. In this paper we evaluate cross sections for both the Thomas peak in the angle of scattering of the projectile and a kinematic ridge in the momentum distributions, ${\mathrm{k}}_2$, of the ejected electron, both of which in principle could be observed experimentally. The ridge in transfer ionization has not been previously discussed in the literature, nor has it been observed experimentally.