Question 6.4.Q6: Specific ionization j is defined as the number of primary an......

(a) Mean specific ionization j of air for 10 MeV proton is determined using the following steps:

(1) Use the Bethe equation (6.17) for heavy CP to determine mass collision stopping power S of air for 10 MeV proton as follows

S_{\mathrm{col}}=C_1 \frac{N_{\mathrm{e}} z^2}{\beta^2} B_{\mathrm{col}},           (6.59)

with parameters defined in Prob. 131.
(2) Determine the electron density N_e of air for use in (6.59).
(3) Determine β² for 10 MeV proton for use in (6.59).
(4) Calculate the atomic stopping number B_{col} defined in (6.19) for use in (6.59).

\bar{B}_{\text{col}} =\left\{\ln\frac{2m_ec^2}{I}+ \ln\frac{β^2}{1  −  β^2}− β^2 − \frac{C}{Z}−δ\right\}. (6.19)

(5) Multiply S_{col} of (6.59) by density of standard air \left(ρ_{\text{air}} = 1.293\times 10^{−3}\ g/cm^3 \right) to get the linear collision stopping power s_{col}.
(6) Multiply s_{col}\ by\ \bar{W}_{air}, the mean energy required to produce an ion pair in air by a proton, to get the mean specific ionization j of air for a 10 MeV proton.

(1) Before we embark on calculation of the mass collision stopping power S_{col} we must determine the parameters of (6.59).

(2) Electron density N_e of air is calculated using the following composition of air per weight and molecular mass (see Table T8.2): nitrogen N: 75.8 %, M_N = 14.0067; oxygen O: 22.6 %, M_O = 15.9994; argon Ar: 0.93 %, M_{Ar} = 39.948; and carbon dioxide CO_2: 0.03 %, M_{CO2} = 44.
In one gram of air we thus have the following four constituents of importance:
(0.758 g) of N + (0.226 g) of O + (0.0093 g) of Ar + (0.0003 g) of CO_2 and each one of the four components contributes the following number of electrons:
Nitrogen:

Electron density N_e of air is given by the sum of the components listed in (6.60) through (6.63) to yield N_e = 2.99\times 10^{23} electron/g.

(3) Velocity β of a 10 MeV proton is determined using the standard relationship (see T2.7)

\beta^2=\frac{v^2}{c^2}=1-\frac{1}{\left(1+\frac{E_{\mathrm{K}}}{m_{\mathrm{p}} c^2}\right)^2}=1-\frac{1}{\left(1+\frac{10}{938.3}\right)^2}=0.021           (6.64)

(4) Atomic stopping number B_{col} is calculated as follows

Finally, the mass collision stopping power S_{col} of air for a 10 MeV proton is

in excellent agreement with the value of 40.1 MeV · cm²/g that the NIST provides for dry air near sea level and 10 MeV proton at http://physics.nist.gov/cgi-bin/Star/ ap_table.pl.

(5) Linear collision stopping power s_{col} is calculated by multiplying the mass collision stopping power S_{col} by density ρ

s_{\mathrm{col}}=\rho S_{\mathrm{col}}=\left(1.293 \times 10^{-3} \mathrm{~g} / \mathrm{cm}^3\right) \times\left(40.1  \mathrm{MeV} \cdot \mathrm{cm}^2 / \mathrm{g}\right)=0.0518  \mathrm{MeV} / \mathrm{cm}             (6.67)

Since \bar{W}_{air} is 35 eV for protons in air, we get the following result for the mean specific ionization j of air for 10 MeV proton

j=\frac{s_{\text {col }}}{\bar{W}_{\text {air }}}=\frac{0.0518 \times 10^6 \mathrm{eV} / \mathrm{cm}}{35 \mathrm{eV} / \mathrm{ip}}=1480  \mathrm{ip} / \mathrm{cm}=148  \mathrm{ip} / \mathrm{mm} .             (6.68)

(b) As is evident from Fig. 6.7 (see point B), the maximum possible specific ionization j_{max} in air is produced when S_{col} against proton kinetic energy E_K is at its maximum and, for proton in standard air, this maximum occurs at E_K = 0.084 MeV and amounts to S_{\mathrm{col}}^{\max }=736  \mathrm{MeV} \cdot \mathrm{cm}^2 / \mathrm{g}.
Based on the discussion in (a), we now calculate j_{max} as follows

in excellent agreement with the value of j_{max} = 2750 ip/mm stated by Evans (p. 656).

(c) To plot the specific ionization j against the residual range R_{residual} we first complete Table 6.9A which provides the NIST data on S_{col} and RCSDA for protons (0.001 MeV to 10 MeV) in air.

Results of our calculation of j and R_{residual} are displayed in Table 6.9B. In Fig. 6.8 we plot the specific ionization j of Table 6.9B against residual range R_{residual} for the last 5 mm of the 10 MeV proton in air where the proton energy is already significantly diminished, rapidly approaches 0 and the ionization reaches its maximum value at the Bragg peak. Note that j_{max} occurs around the residual range of 1.2 mm where the proton energy falls to E_K = 0.084 MeV.