Approximation Scheme for Electron Capture into Arbitrary Principal Shells of Energetic Bare Projectiles

An expression given by Fock for the density matrix in momentum space of an electron in the nth hydrogenic shell is used to derive a prescription for calculating capture cross sections into arbitrary principal shells of energetic bare projectiles. Application to an eikonal treatment yields a simple formula showing that capture cross sections scale to the Oppenheimer-Brinkman-Kramers values with a Z-independent factor between 0.1 and 0.4, in agreement with experimental data.