Three-Parameter Formula for the Electronic Stopping Cross Section at Nonrelativistic Velocities

The Firsov formalism for calculating the electronic stopping cross section has been modified by (i) giving a precise quantum-mechanical definition of the electron flux across the Firsov plane in terms of the bound-state wave functions and (ii) by including the relative motion of the Firsov plane in the flux calculation. This formalism is then extended in a semiphenomenological manner to the velocity region where the relative velocity of the colliding atoms is of the same order of magnitude as the orbital velocities of their electrons. When hydrogenic $1s$ wave functions are used, three adjustable parameters result from this treatment, one from the modification of the Firsov formalism and two from the extension to higher velocities. It is found that adjustment of these parameters to give a best fit to experimental data yields an expression which accurately gives the electronic stopping cross section $S_e$ for any collision partners, and at all nonrelativistic velocities. The three parameters are considered as adjustable for the purposes of this paper, but it is shown here that the low-energy parameter $Z$ is calculable from first principles. In addition, one of the high-energy parameters $a$ is shown to be a linear function of the target atomic "radius."