Relativistic eikonal theory of electron capture

The eikonal theory of electron capture by relativistic projectiles is developed. It is shown that as long as the electron spin is not measured, a density-matrix formalism can greatly simplify the calculation. The exact eikonal cross section is expressed in terms of two-dimensional integrals. For 1s-1s capture an approximate closed-form expression valid for not too high nuclear charges is also given. This form explicitly displays contributions from magnetic capture and from relativistic modifications of the electron orbits. At each stage of the development the corresponding Oppenheimer-Brinkman-Kramers (OBK) result can be retrieved as a special case. It is concluded that the multiple-scattering contributions described by the eikonal approach are crucial in reducing the calculated cross section significantly below the OBK and second-Born-approximation values. In this way, good agreement with experimental data is obtained.