The Bethe Theory

Chapter 2 Interaction of Radiation with Matter Energy Transfer 

For knock-on collisions, one uses the Rutherford cross section for free electrons, and the number of free electrons is taken equal to the integral of the oscillator strength up to the energy loss e (dispersion approximation). Thus, 

The contribution of knock-on collisions to the stopping power is now given by 

In this equation, the small lower limit is of no consequence. Also, 2mv2 is sufficiently large to include essentially all oscillator strengths. Therefore, this equation simplifies to 

In Bohr s theory, only estimates of maximum and minimum impact parameters are necessary. Better computations are required for determining the transverse distribution of lost energy or the effect of secondary electrons. The minimum impact parameter according to classical mechanics is ze2/mv2 from angular momentum consideration in quantum mechanics, it is h /mv. In practice, the larger of these two is taken. Also, the impulse approximation used by Bohr for the maximum impact parameter is not an absolute rule energy transfer beyond bmax falls off exponentially (Orear et al., 1956 Mozumder, 1974). 

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Oddershede s earlier results [3-5] calculate the directional values of the dipole oscillator strength distribution for use in the Bethe theory [9], which is valid for high-energy projectiles. Our approach, since we have not implemented the possibility of treating unbound electrons, is restricted to calculating stopping cross... 

It was a fairly common belief that such a theory should be a modification of the Bethe theory. Although the fact that the screening regime (uclassical theory must be more powerful than the Bom approximation in this regime was evidently not taken at face value. 

In the Bethe theory of stopping, the GOS characterize the excitation spectrum of a target for a given momentum transfer q and they are given by... 

Because the semiclassical theories can be used to calculate differential cross sections with relative ease for close collisions between the incident charged particle and the bound electron, and the Bethe theory provides a straightforward method to describe low-energy electrons ejected in distant collisions, it is only natural to combine the best characteristics of the two approaches to derive a comprehensive description of electron impact ionization. 

M. Inokuti, Inelastic collisions of fast charged particles with atoms and molecules - the Bethe theory revisited, Rev. Mod. Phys. 43 (1971) 297. 

Various review papers have described and updated many of the developments and applications emerging from the Bethe theory [16-20],... 

The dielectric formulation by Lindhard is basically consistent with the Bethe theory, but still requires some particular comments. The energy loss in the electron gas model is decomposed in individual and collective... 

In the projectile velocity range v < v0, the Bethe theory of stopping expressed by (5.15) breaks down, and a different approach to electronic stopping theory is needed. Three major models of electronic stopping in this velocity regime have been developed over the years, all of which give the result that the stopping cross-section is proportional to the projectile velocity. 

In the projectile velocity range v < Vq, the Bethe theory of stopping expressed by... 

As was discussed in section 2, Barkas et al. [1.4] found that the ranges of equivelocity -n and ti were not the same. Later, Andersen et al. [8.3, 8.4] observed that the stopping powers of fast, equivelocity protons and a particles did not scale exactly as q It is therefore clear that the q scaling of the Bethe theory does not give a satisfactory description of the slowing-down for non-asymptotic projectile velocities. 

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