Stopping number

Bethe s formula requires that the velocity of the incident particle be much larger than that of the atomic electrons, a condition not easily fulfilled by the K-electrons except in the lightest elements. The required correction, called the shell correction, is denoted by subtracting a quantity C from the stopping number. In the penetration of high-Z material, even L-shell correction may be required. In that case, C denotes the sum total of all shell corrections. The subject of shell correction has been extensively treated by several authors, and various graphs and formulas are available for its evaluation (see, e.g., Bethe andAshkin, 1953). 

Fermi (1940) pointed out that as /)—-1 the stopping power would power would approach °° were it not for the fact that polarization screening of one medium electron by another reduced the interaction slightly. This effect is important for the condensed phase and is therefore called the density correction it is denoted by adding -Z<5/2 to the stopping number. Fano s (1963) expression for 8 reduces at high velocities to... 

At moderate energies, the electron can acquire relativistic speeds. Including this effect as well as corrections due to shell and density effects, the electron stopping number may be written as... 

Figure 1 also makes it clear that there must be a fairly wide variety of systems and bombardment conditions where neither equation (7) nor (11) can provide accurate predictions of stopping forces. A considerable number of improvements of the theory can be made and has been made, and the main purpose of this paper is to identify to what extent such improvements can be based safely on classical, Bohr-type arguments. For simplicity of notation, reference will be made in the following to the stopping number L defined by... 

In Bethe theory the shell correction ALsheii is conveniently defined as the difference between the stopping number LBom in the Born approximation and the Bethe logarithm LBethe —in (2mv /I). Fano [12] wrote the leading correction in the form... 

Conceptionally the situation is much clearer in classical theory because the cross-over toward negative stopping numbers of the Bohr logarithm L ohi — In(Cmu /ZiUof) can easily be avoided [11]. Orbital motion can be incorporated into the initial conditions [17], although the actual evaluation in Ref. [17] was carried through only to the leading term in u . ... 

A fairly general way to evaluate shell corrections is based on kinetic theory [1]. Here it is assumed that shell corrections account for orbital motion and nothing else. The theoretical basis is a relation between the stopping number Lq for a target at rest and the stopping number L for a moving target [1],... 

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].

The basic stopping power formula of Bethe has a structure similar to that of Bohr s classical theory [cf. Eq. (2)]. The kinematic factor remains the same while the stopping number is given hy B = Zln(2mv /7) for incident heavy, nonrelativistic particles. The Bethe... 

The stopping power is generally normalized by the target scatterer density n, to produce the stopping cross-section, SCv). If one removes the primary velocity dependence and constants from the cross-section, one obtains the stopping number, L(y), where interesting physics is concentrated. These quantities are related as... 

In the pure classical description, the stopping number L is determined by the integration of equation (41)... 

Formally, expression (58) reproduces that obtained in the Lindhard-Sharff model [18] the stopping number for uniform electron gas is averaged over the density distribution in atomic shell. Moreover, if the linear response model is considered (C = G, the factor G is defined by equation (51)) the basic expression of LS model is reproduced exactly ... 

The stopping cross section is related to the stopping number L(v) by... 

The function f e) is usually not completely known. It can either be obtained semi-empirically or can be calculated using the Bragg rule which states that the stopping number of a molecule is the sum of the stopping... 

Fig. 3. Range—energy plot for low-energy electrons in water according to (i) Lea, Action of Radiation on Living Cells, Macmillan, New York, 1947, p. 24 (ii) calculation based on stopping numbers given in J. O. Hirschfelder and J. L. Magee, Phys. Rev., 73 (1948) 207 (Table VI with correction) and (iii) F. Seitz, Phys. Fluids,...

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