Transport cross sections based on a screened interaction potential: Comparison of classical and quantum-mechanical results

Standard classical and quantum-mechanical methods are used to characterize the momentum-transfer cross section needed in energy-loss calculations and simulations for heavy, swift charges moving in an electron gas. By applying a well-known, finite-range screened Coulombic potential energy to model the two-body collision, the quantitative applicability range of the classical cross section is investigated as a function of charge $\left(Z\right)$, screening length $\left(R\right)$, and scattering relative velocity $\left(v\right)$. The a posteriori condition $(Z∕R)∕v^2<1$, as an upper bound for heavy charges, is deduced for this applicability range from the comparative study performed.

Figure 1

Transport cross sections as a function of $Z$, at a fixed value of the screening parameter $(R=1)$, and for different relative momenta ($k=1.5$, 3, and 5). Open circles refer to quantum-mechanical calculations and are obtained from Eq. (6). The solid curves are based on Eq. (2), while the dashed ones on the first-order Born approximation given by Eq. (7).

Figure 2

The same as in Fig. 1, at the $R=2$ value for the screening parameter.

Figure 3

The same as in Fig. 1, at the $R=3$ value for the screening parameter. Solid circles refer to quantum-mechanical calculations with a rescaled Yukawa potential and are obtained from Eq. (6). (see the text for further details.)