Bethe stopping formula

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). 

In the region 104-109 eV, where the energy loss is via electronic excitation and ionization, Bethe s formula with corrections (Eq. 2.11) describes the stopping power quite accurately. In the interval 104-106 eV, the decrease of stopping power with energy is attributable to the v-2 term. It reaches a minimum of —0.02 eV/A at -1.5 MeV then it shows a relativistic rise before the restricted part rides to the Fermi plateau at -40 MeV. 

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. 

Example Problem Evaluate the stopping power of beryllium metal for 1808+ ions with a kinetic energy of 540 MeV (E/A = 30MeV) using the Bethe-Bloch formula. 

The stopping powers of polystyrene for ions were calculated from those of C and H [32a] based on the additivity rule. For 20 KeV electrons, the stopping power was calculated from that for lOkeV electrons [32b] based on the Bethe-Bloch formula. 

Besides these distinct microscopic features, particle radiation also differs considerably from photon radiation with respect to the macroscopic distribution, i.e., the depth dose distribution. The typical shape is caused by the velocity-dependent stopping power, as described by the Bethe-Bloch-formula [63, 64] ... 

In the shell correction [30], Fano introduced a correction term C/Zm to Bethe s stopping formula, to account for the fact that at low kinetic energies, the velocity of the heavy charged particle is comparable to the electrons of the absorbing material. The C/Zm term corrects for the overestimation of the mean excitation / ionisation potential Im), which depends on both the velocity of the charged particle and the electron of the absorbing medium. 

Figure 1 shows a comparison between experimental stopping data and equations (7) and (11). It is seen that over a broad range of beam energies, Bohr s classical formula is superior to Bethe s when the same /-value is inserted. 

Fig. 1. Stopping force on oxygen ions in aluminium Comparison of Bohr and Bethe formulae with measurements from numerous laboratories compiled in Ref. [6]. From Ref. [7].

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... 

Abstract The effects of interactions of the various kinds of nuclear radiation with matter are summarized with special emphasis on relations to nuclear chemistry and possible applications. The Bethe-Bloch theory describes the slowing down process of heavy charged particles via ionization, and it is modified for electrons and photons to include radiation effects like bremsstrahlung and pair production. Special emphasis is given to processes involved in particle detection, the Cherenkov effect and transition radiation. Useful formulae, numerical constants, and graphs are provided to help calculations of the stopping power of particles in simple and composite materials. 

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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