Notes on Barkas-Andersen effect

Barkas effect in stopping of fast diclusters in solids

“…Therefore, quantitative estimates of the Barkas effect for single protons in Al, which only consider valence electrons, must be amended with inner-shell contributions in order to achieve quantitative agreement with experiments (14,15). That task was successfully accomplished in (14) by using the oscillator model of stopping medium (8,23). Accordingly, our results for the Barkas effect in the stopping of a swift dicluster, which are based on the hydrodynamic model alone, should be considered mostly as qualitative estimates.…”

“…This simple description of aluminium is amended here by using finite damping rate η = ω p /10. However, it must be stressed that, while the hydrodynamic model is suitable for the description of valence electrons in aluminium, there are also contributions from the inner electron shells that become increasingly important with increasing proton speeds, as shown in (8). Therefore, quantitative estimates of the Barkas effect for single protons in Al, which only consider valence electrons, must be amended with inner-shell contributions in order to achieve quantitative agreement with experiments (14,15).…”

“…In order to tackle this issue, one may resort to the so-called Barkas-Anderson effect, which gives a perturbative correction to the linear theory (6)(7)(8). This effect was named after W.H.…”

“…Namely, linear theory of stopping for individual point-like particles with the charge of Z proton charges gives the stopping force of the order ∼ Z 2 , whereas nonlinear effects within the Barkas effect give a correction to the linear theory for point-like charges of the order ∼ Z 3 (11), which accounts for the most of the observed charge-sign dependence (9,10). While these nonlinear effects in stopping of single charges have been studied over the years (6,8,(12)(13)(14)(15), only a few studies appeared on such effects on stopping of clusters of charges, and most of those studies were limited to slow diatomic particles (16)(17)(18). The stopping of fast clusters was mostly treated by linear theories (4,5,(19)(20)(21), with only two notable exceptions (22,23).…”

“…In this work we adopt the hydrodynamic model within the Thomas-Fermi-von Weizsäcker (TFW) approximation for the internal energy of valence electrons in a metal target, which was used by Crawford et al (7) to study Barkas effect on swift protons moving in an Al target. While the linearised hydrodynamic model was shown to reproduce the results of several other linear models for the stopping of single point charges at the medium to large speeds (7), there exist more sophisticated methods that are better suited for studying either the Barkas effect for single charges (8,14,15) or the stopping of clusters in the linear regime (21,23) over broader ranges of the respective parameter spaces. However, the simplicity and computational tractability of the hydrodynamic model enables us to tackle the combined Barkas and vicinage effects for a swift proton dicluster and obtain results that may provide new insight into this difficult problem, at least at a qualitative level.…”