United-atom approximation in the problem of $\Sigma -\Pi$ transitions during close atomic collisions

We consider a very simple approximation, in which the splitting of the energy of a $P$ state in a united atom into $\Sigma$ and $\Pi$ quasimolecular terms for small internuclear distances $R$ depends quadratically on $R$, and the colliding atoms pass one another along a straight or a hyperbolic trajectory. In this case the transition probability for a given scattering angle or impact parameter depends on only one parameter—the Massey parameter. This probability is computed numerically along with the elastic scattering phase shifts. Approximate formulas are obtained for both the adiabatic limit (in which the parameter is large and the flight is slow) and the sudden-perturbation limit for which the parameter is small, the flight is fast, and the process reduces to a sudden rotation of the internuclear axis. We also obtain, in the adiabatic case, the first term in the expansion of the factor in front of the exponential. In the intermediate range of the parameter, simple analytic approximations, ensuring transition to the limiting cases, are proposed. Analytic expressions for the total transition cross sections are obtained in the limits of large and small collision velocities. The results of the calculations are applicable to $\Sigma -\Pi$ transitions of electrons and holes in both outer and inner shells for close collisions in a broad energy range, where, owing to a scale transformation, all cases reduce to one. Reasonable agreement is found with other more complicated calculations for the collisions of specific atoms.