Barkas-Andersen effect

The theoretical literature on the Barkas-Andersen effect is extensive. The present discussion focuses on the role of quantal V5. classical arguments. 

Traditionally the effect is incorporated into stopping theory via expansion in powers of Zj. As pointed out by Lindhard [21], there are two independent dimensionless parameters containing Zj, namely, the Bohr parameter k (equation (17)) and the Barkas parameter 

While the Bloch correction represents a series expansion in powers of k, the leading term in the Barkas-Andersen correction was found [22] to be oc B. 

One must expect the presence of mixed terms of the form k B in the expansion. The term of lowest order a —2, d = l), contributing oczf to the stopping cross section, would indicate a difference between the Barkas-Andersen correction evaluated from the Born series and the Bohr model, respectively. While such a comparison has not been performed in general terms, a numerical evaluation for the specific case of Li in C revealed a negligible difference [24]. 

In practice, series expansions of the stopping cross section in powers of Zi are only useful at high beam velocities where neither the Bloch nor 

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Fig. 3. Barkas-Andersen effect predicted by binary stopping theory. Plotted are stopping numbers for singly charged Ar, Li, and H ions and for their anti-ions in Si. Also shown are the respective ion/anti-ion ratios. From Ref. [23].

Clever schemes have been developed to treat the Barkas-Andersen effect for light ions in an electron gas ([27,34] and others) in what is called the nonlinear quantum regime. While there is little doubt that there must be a lower velocity limit for the validity of Bohr-like stopping theory, a reliable estimate of this limit does not seem available, nor is there a demonstration of where and in what manner quantum mechanics is an indispensable feature. 

Moreover, estimates for an electron gas can hardly be representative for insulating materials. There have been speculations on the relevance of promotion effects [35], but theoretical estimates of the Barkas-Andersen effect in insulating solid materials have been performed only on the basis of classical theory so far... 

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