A closed formula for low velocity K-shell ionization cross section and the universal function in PWBA

Bethe (1930) obtained, in the framework of the PWBA, a double differential cross secton for K-shell ionization in the energy and momentum transfers from the projectile to the target atom. The difficulties encountered for further simplifications of Bethe's formula, which preserves its usefulness, point to the direction of numerical integrations over the momentum and the energy transfers. Numerical tables of nonrelativistic PWBA calculations for direct Coulomb ionization of inner shells by heavy-charged particles are extensively used for inner shell ionization studies. These tables give the cross sections in terms of two dimensionless parameters, the scaled projectile velocity and the scaled target binding energy. However, since a long time ago, an approximately analytical formula has been looked for, which would give the total cross section in terms of a few powers. It will be extremely useful to dispose of analytical expressions which are valid in the entire range of the adiabatic region. An infinite power series in the scaled parameters would be an ideal form for the cross section. In this paper such an expression is proposed