Bohr stopping power

FIGURE 2.2 Bohr s semiclassical description of stopping power in terms of the impact parameter. The coulomb interaction is taken at its peak value over the track segment AOB and zero outside. 

STOPPING POWER, LET, AND FLUCTUATIONS 2.1. Bohr s Theory and the Bragg Rule... 

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... 

Despite the apparent similarity of the Bohr and the Bethe stopping power formulae, the conditions of their validity are rather complimentary than the same. Bloch [23] pointed out that Born approximation requires the incident particle velocity v ze jh, the speed of a Is electron around the incident electron while the requirement of Bohr s classical theory is exactly the opposite. For heavy, slow particles, for example, fission fragments penetrating light media, Bohr s formula has an inherent advantage, although the typical transition energy has to be taken as an adjustable parameter. 

Despite the fact that Bohr s stopping power theory is useful for heavy charged particles such as fission fragments, Rutherford s collision cross section on which it is based is not accurate unless both the incident particle velocity and that of the ejected electron are much greater than that of the atomic electrons. The quantum mechanical theory of Bethe, with energy and momentum transfers as kinematic variables, is based on the first Born approximation and certain other approximations [1,2]. This theory also requires high incident velocity. At relatively moderate velocities certain modifications, shell corrections, can be made to extend the validity of the approximation. Other corrections for relativistic effects and polarization screening (density effects) are easily made. Nevertheless, the Bethe-Born approximation... 

The modern form of the stopping power includes two corrections. The first correction applies at high energies at which polarization of electrons by the electric held of the moving ion tends to shield distant electrons this correction depends on the electron density it is subtractive and given the symbol 8. The second correction applies at low energies when the collisions are no longer adiabatic, similar to the limit applied by Bohr. This correction is termed the shell correction as it depends... 

Further developments for slow ions included the application of the density functional theory (DFT) by Echenique et al. [31-33], which yields a more sophisticated (and also non-linear) treatment of many-body effects in dense media. This theory explains also the oscillatory behavior of the stopping powers in the range of low velocities (v < Vq, Vq being the Bohr velocity). But the question of extending the DFT calculations to intermediate or large velocities is still a complicated numerical problem. 

Bloch [8.1] derived a formula for the stopping power which is valid for all values of X, and which is therefore a synthesis of the quantal result of Bethe [2.8] and the classical stopping power deduced by Bohr [6.21]. Bloch [8.1] found that the transition between the classical and the quantal results can be accounted for by setting... 

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