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 $n\mathrm{th}$ 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.