Correction Barkas

In 1963, Barkas noticed the difference in stopping powers when measured with positively and negatively charges projectiles [16], which implied an odd power Lj contribution to the Born series, and the Zj contribution to the stopping cross-section is now referred to as the Barkas correction. 

In conclusion, it seems that most recent theoretical calculations, independent of whether the target is described as a harmonic oscillator or as an electron gas, find that close and distant collisions are both responsible for the Barkas correction term to the Bethe stopping power. However, some doubt as to the importance of the close collisions still remains (Sung and Ritchie [8.14]). 

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]. 

While more than a handful theoretical schemes are available to nonpermrba-tively evaluate the Barkas-Andersen correction quantum mechanically, binary stopping theory developed recently [32] fulfills the task on the basis of the Bohr stopping model the only quantum feature added is the inverse-Bloch correction (18) which does not differentiate between particle and antiparticle. Figure 4 demonstrates that with regard to comparison with experimental antiproton stopping data, classical theory is fully competitive with various quantum theories. 

A large Barkas-Andersen correction had to be expected for Zj 1. Available estimates referred to the first term in a perturbation expansion in Zj. Such an estimate could not be expected to apply to higher-Zj ions. 

This reasoning was followed by Flory and Rehner in their analysis of swelling networks. We may also calculate the equilibrium from a free energy which does not contain 0, provided the result is corrected afterwards by means of equation (78). Clearly, this method is equivalent, since /c>/7z == ( 0/dV)Vo = —upo, where a represents the hydrostatic pressure due to network interlinking. Barkas and Cassie have used this method to correct the sorption isotherm of wood and of wool. In Fig. 21 the drawn curve represents the sorption of water by wool measured directly. The swelling involved in this sorption, however, is opposed by forces whose Sorption magnitude may be estimated from stress-strain data in the stretch of wool. If the vapour pressure is corrected according to equation (78), one finds the much lower vapour pressures of the dotted curve. This curve extends no further than to 50% relative vapour pressure. 

Stopping power vs. relative momentum, py = p/Mc, for muons in copper. The solid curve indicates the total stopping power, the dash-dotted and dashed lines the Bethe-Bloch equation with and without density effect correction. The vertical bands separate the validity regions of various approximations indicated in the figure. The dotted line denoted with p. indicates the Barkas effect. In the Bethe-Bloch region the stopping power scales with the particle mass and Z/A of the medium... 

In the following, we will regard only small values of q and assume that Lf decreases so fast with i that the series can be truncated at Lj. The functions Lj and Lj contain the contributions from higher order Bom terms and hence reflect the importance of intermediate states in the excitation processes. The first correction term qL gives rise to a different slowing-down of positive and negative particles and is usually called the Barkas term. It will be discussed in the following sections. 

The Barkas term. The leading correction term to the first Bom result gives rise to the Barkas effect and has been the center of much attention during the last decades. The theoretical efforts... 

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