Question 6.4.Q2: The Bethe equation for collision stopping power Scol of an a......

(a) Parameters of the general Bethe equation incorporating the Fano corrections are as follows:

N_e   electron density \left(N_e = ZN_A/A \right) expressed in number of electrons per gram of absorber medium with Z the atomic number and A the atomic mass of the absorber. In the first approximation, Z/A ≈ 0.5 for all elements with the notable exception of hydrogen for which Z/A ≈ 1. A closer look at Z/A shows that for Z ≥ 2 it slowly decreases from 0.5 for low Z elements to 0.38 for high Z elements. For example, Z/A for helium-4 is 0.5, for cobalt-60 it is 0.45, and for uranium-235 it is 0.39.

z_e   charge of the heavy charged particle CP (for proton z = 1; for α particle z = 2).

m_ec^2   rest energy of the electron \left(m_ec^2 = 0.511\ MeV \right).

β   velocity of the heavy CP normalized to speed of light c = 3\times 10^8 m/s in vacuum.

I   mean ionization/excitation potential of the absorber.

C   shell correction constant.

δ   density correction.

C_1   is a collision stopping power constant independent of absorbing medium as well as of the physical characteristics of the CP. It is defined with the following expression

where r_e is the classical radius of electron defined as

r_{ e }  =\left(\frac{e^2}{4 \pi \varepsilon_0}\right) \frac{1}{\left(m_{ e } c^2\right)}  =  2.818  fm .

\bar{B}_{\mathrm{col}}  is the so-called atomic stopping number that depends directly on velocity β of the charged particle and indirectly on the atomic number Z of the absorber through the mean ionization/excitation potential I . It is given as

\bar{B}_{\mathrm{col}}=\left\{\ln \frac{2 m_{\mathrm{e}} c^2}{I}+\ln \frac{\beta^2}{1-\beta^2}-\beta^2-\frac{C}{Z}-\delta\right\} .           (6.19)

(b) Shell correction. Bethe’s derivation of S_{col} for heavy CPs traversing an absorber assumes that the velocity υ of the CP is much larger than the velocity υorb of orbital electrons of the absorber atoms. At high kinetic energy E_K of the CP this assumption \left(v \gg v_{\text {orb }}\right) is correct; however, at low E_{\mathrm{K}} \text { where } v \leq v_{\text {orb }} it does not hold, since orbital electrons do not participate in energy transfer from the CP when υ ≤ υ_{orb}. This effect causes an overestimate in the mean ionization/excitation potential I at low E_K and, consequently, results in an underestimate in S_{col} calculated from an uncorrected Bethe equation.
Since K shell electrons are the fastest of all orbital electrons in an absorber atom, they are the first to be affected by low CP velocity with decreasing CP velocity, as the CP penetrates deeper into the absorber. Often thus, the shell correction is addressed as the K shell correction and all possible higher shell corrections are ignored.
The shell correction term C/Z that Fano introduced to correct for the overestimate in I is a function of the absorbing medium as well as of the incident particle velocity υ; however, for the same absorbing medium and the same particle velocity, it is the same for all particles including electrons and positrons.

(c) Density effect correction. Fano introduced a second correction term δ to the Bethe collision stopping power equation to account for the polarization or density effect in condensed absorbing media. The effect influences the soft (distant) collision interactions by polarizing the condensed absorbing medium thereby decreasing the collision stopping power of the condensed medium in comparison with the same absorbing medium in the gaseous state. For heavy CPs the density correction is important at relativistic energies and negligible at intermediate and low energies; however, for electrons and positrons it plays a role in stopping power formulas at all energies.

(d) Dependence of S_{col} on the absorbing medium. S_{col} depends on atomic number Z of the absorber in two ways: (1) directly through the electron density N_e = ZN_A/A of the absorber and (2) indirectly through the mean ionization/excitation potential I of the absorber.
S_{col} is directly proportional to Z/A and this implies that S_{col} decreases with increasing Z as a result of the slight Z/A dependence on Z. Note: for hydrogen Z/A = 1, but for all other elements, it is close to 0.5 ranging from 0.5 for low Z elements and, with increasing Z, slowly decreasing to ∼0.4 for high Z elements.
The indirect dependence of S_{col} on absorber Z is brought about through the −ln I term in the stopping number \bar{B}_{\mathrm{col}}, since I depends on Z, ranging from 19 eV for hydrogen (Z = 1) to ∼900 eV for uranium (Z = 92). Thus, both the direct and indirect dependence of S_{col} on atomic number Z of the absorber causes S_{col} to diminish with increasing Z, however, the decrease in S_{col} is only slight despite the two orders of magnitude range in atomic number Z of the absorber.

(e) Dependence of S_{col} on physical characteristic of the charged particle.

(1) As shown in (6.17), S_{col} depends on CP velocity υ and charge ze but does not depend either directly or indirectly on the rest mass m_0c^2 of the CP. A given absorbing material will have the same S_{col} for all heavy CPs of a given kinetic energy E_K and charge ze.

(2) Discussion of S_{col} dependence on velocity υ must address three ranges in CP velocity: classical velocity υ at low kinetic energy, intermediate velocity at intermediate kinetic energy, and relativistic velocity at high energy. Each one of these three velocity ranges is characterized with its own effect on S_{col}. As evident from (6.17), S_{col} depends on CP velocity β = υ/c through the 1/β^2 term as well as through the \left\{\ln \left[\beta^2 /\left(1-\beta^2\right)\right]-\beta^2\right\} term contained in the atomic stopping number [مatex]\bar{B}_{\mathrm{col}} \text {. }[/latex]
At low kinetic energies E_K the Fano shell correction must be incorporated in the Bethe equation to account for the low velocity υ of the CP and for non-participation of inner shell electrons in the stopping power process. In the intermediate energy region S_{col} is governed by the 1/β^2 term that is proportional to 1/E_K and decreases rapidly with increasing E_K. In the high-energy relativistic region, where β ≈ 1, collision stopping power S_{col} rises slowly with E_K as a result of the slow rise in the \left\{\ln \left[\beta^2 /\left(1-\beta^2\right)\right]-\beta^2\right\} term which slowly increases with E_K.

(3) As far as the charge dependence of S_{col} is concerned, we see from (6.17) that S_{col} is linearly proportional to z^2 where ze stands for the charge of the CP. For example, z = 1 for proton and deuteron; z = 2 for α particle but z = 1 for singly ionized helium atom; and z = 1 for singly ionized carbon atom, z = 2 for doubly ionized carbon atom, and z = 6 for carbon nucleus. This implies, for example, that S_{col} of an absorber will differ by a factor of 4 in the case of proton and α particle of the same velocity β, i.e., S_{col}(α) = 4S_{col}(p) for the same β.