Electron Capture by Protons in Hydrogen and Effect of an Electric Field

By treating an electric field in the direction of the incident proton as a perturbation, the electron capture cross section in the Brinkman-Kramers approximation up to the first order in the field is obtained. The cross section for transition between principal quantum numbers, in particular for capture from the ground state into the ground state, is quadratic in the field. The experimental verification of this symmetry is desirable. The cross section in the zeroth-order approximation for transition between two Stark levels is independent of the quantum numbers of these levels and depends only on their principal quantum numbers. The inclusion of the first order, which is linear in the field, splits the cross section into different values. The simplicity of the zeroth-order cross-section formula between Stark levels allows through a transformation the evaluation of the cross section between the optical levels as a polynomial in the incident energy. A general expression for the capture from the state $\mathrm{nl}$ into the state $n^{\prime }{l}^{\prime }$ is given. An expression for the momentum distribution of the hydrogen atom in an electric field, correct up to the first order, is given. It is found that this distribution for the Stark levels of the atom in the zeroth order is the same as the known momentum distribution for their related principal quantum number. The momentum distributions for the principal quantum numbers are found to be quadratic in the field.