Bethe stopping power

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. 

A series of inelastic scattering processes with statistical energy losses results in a slowing down of electrons, which can be described by a mean energy loss per unit path length (Bethe stopping power S)... 

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]). 

Figure 5 Stopping power for protons on He calculated with the standard basis (basis A), with two consistent bases (B and C), and in the Bethe approximation using the kinetic theory [17, 18],...

Thus, the average stopping power is proportional to the initial energy except for corrections due to atomic collisions (electronic excitation) near 108 eV. For a medium of nuclear charge Ze and mass number A, the radiation length is given by (Bethe and Ashkin, 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. 

Jens Oddershede s ideas on stopping power theory and their impact and consequences have been briefly reviewed. We have centered our analysis on the relevance of the orbital implementation of the kinetic theory (KT) of stopping and the Bethe and Thomas-Reiche-Khun sum rules, since they have influenced profoundly the development of our research along these lines. 

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... 

L(0) = Z In 7, where I is the mean excitation potential appearing in Bethe s stopping power equation [Eq. (4)]. 7,(2) is proportional to the logarithm of average excitation energy, which is also involved in the Lamb shift [26]. 7,(—1) has been shown to be an optical... 

One aspect of the eondensed phase regarding the delocalization of the deposited energy has been alluded to in See. 1.3. Here we will eonsider the modifications on the oseillator strength and the mean exeitation potential, due to eondensation, whieh would enter in Bethe s stopping power theory [see Eqs. (4-7)]. 

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 details of the stopping power equations have been very well summarized elsewhere [22,34-38]. Most applications involve ions of low charge and high velocity in which the Bethe formalism is valid. The nonrelativistic stopping power equation of Bethe for heavy ions is given approximately by. 

In this condition, Bethe formulated the stopping power for electron according to the Born approximation. Stopping power is a property of irradiated materials and gives the amount of energy deposited per unit path length, —dU/dx. 

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 principal parameter characterizing the molecular stopping power in Bethe s theory is the average ionization potential Im, which depends only on the properties of the molecule. There are different ways of... 

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