Question 6.5.Q1: Mass collision stopping power Scol of a stopping medium (abs......
Mass collision stopping power S_{col} of a stopping medium (absorber) for light charged particles (CP) of kinetic energy E_K is given as follows
S_{\mathrm{col}}=C_{\mathrm{e}} \frac{N_{\mathrm{e}}}{\beta^2}\left\{\ln \frac{E_{\mathrm{K}}^2}{I^2}+\ln \left(1+\frac{\tau}{2}\right)+F^{ \pm}(\tau)-\delta\right\}, (6.81)
where
C_e is the stopping power constant for light CP (electron and positron).
N_e is the electron density of the absorber: N_e = ZN_A/A.
r_e is the classical electron radius constant: r_{\mathrm{e}}=e^2 /\left(4 \pi \varepsilon_0 m_{\mathrm{e}} c^2\right)=2.818 \mathrm{fm}.
τ is the electron kinetic energy E_K normalized to electron rest mass m_{\mathrm{e}} c^2=0.511 \mathrm{MeV} \text {, i.e., } \tau=E_{\mathrm{K}} /\left(m_{\mathrm{e}} c^2\right).
F^{ \pm}(\tau) is the stopping power function of electron \left(F^{-}\right) \text {and positron }\left(F^{+}\right) given as follows
F^{-}(\tau)=\frac{1}{(\tau+1)^2}\left[1+\frac{\tau^2}{8}-(2 \tau+1) \ln 2\right] (6.82)
and
F^{+}(\tau)=2 \ln 2-\frac{\tau(\tau+2)}{12(\tau+1)^2}\left[23+\frac{14}{\tau+2}+\frac{10}{(\tau+2)^2}+\frac{4}{(\tau+2)^3}\right]. (6.83)
(a) Use (6.81) to calculate the mass collision stopping power S_{col} of lead absorber (Z = 82; A = 207.2 g/mol; I = 823 eV) for electrons with kinetic energies E_K of 100 keV, 1 MeV, and 10 MeV.
(b) Compare S_{col} of lead calculated in (a) for electron kinetic energies E_K of 100 keV, 1 MeV, and 10 MeV with data obtained from the NIST or from Appendix E of Attix.
(c) Plot the density effect parameter δ of lead against electron kinetic energy E_K in the range from E_K = 10 keV to E_K = 100 MeV. You can obtain the density effect parameter δ on-line from the NIST or from Appendix E of Attix.
(a) The problem will be solved in six steps, with the first five steps used to prepare suitable data for use in the last step that involves calculation of S_{col} with (6.81). The six steps are as follows:
(1) Determine electron density N_e of the lead absorber
N_{\mathrm{e}}=\frac{Z N_{\mathrm{A}}}{A}=\frac{82 \times\left(6.022 \times 10^{23} \mathrm{~mol}^{-1}\right)}{207.2 \mathrm{~g} / \mathrm{mol}}=2.388 \times 10^{23} \text { electron } / \mathrm{g} (6.84)
(2) Determine the normalized kinetic energy τ for 100 keV, 1 MeV, and 10 MeV electrons using the following relationship
\tau=\frac{E_{\mathrm{K}}}{m_{\mathrm{e}} c^2} (6.85)
The calculated values of τ for electrons with kinetic energy E_K of 100 keV, 1 MeV, and 10 MeV are summarized in row 2 of Table 6.11.
(3) Determine normalized electron velocity β for 100 keV, 1 MeV, and 10 MeV electrons using the following expressions
\begin{aligned} & \beta^2=1-\frac{1}{\left(1+\frac{E_{\mathrm{K}}}{m_{\mathrm{e}} c^2}\right)^2}=1-\frac{1}{(1+\tau)^2} \quad \text { and } \\ & 1-\beta^2=\frac{1}{\left(1+\frac{E_{\mathrm{K}}}{m_{\mathrm{e}} c^2}\right)^2}=\frac{1}{(1+\tau)^2} .\quad (6.86) \end{aligned}The calculated values of β, β², and 1 − β² are summarized in rows 3, 4, and 5 of Table 6.11, respectively.
(4) Determine stopping power function for electron F^{−}(τ)] using (6.82) for 100 keV, 1 MeV, and 10 MeV electron. The calculated values of F^{−}(τ) are summarized in row 6 of Table 6.11.
(5) From the NIST (http://physics.nist.gov/PhysRefData/Star/Text/ESTAR.html) obtain the density effect parameter δ of lead absorber for 100 keV, 1 MeV, and 10 MeV electrons. The respective values of δ are listed in row 7 of Table 6.11.
(6) Finally, employ (6.81) in conjunction with Table 6.11 that summarizes the supporting data required for use with (6.81) to determine S_{col} of lead for three electron kinetic energies E_K of 100 keV, 1 MeV, and 10 MeV.
Electron in lead: E_K = 100 keV; τ = 0.1957; β = 0.548
Electron in lead: E_K = 1 MeV; τ = 1.957; β = 0.941
Electron in lead: E_K = 10 MeV; τ = 19.57; β = 0.9988
(b) Table 6.12 displays mass collision stopping power S_{col} of lead for electrons with kinetic energies of 100 keV, 1 MeV, and 10 MeV: row 2 displays values calculated in (a) and row 3 values obtained from the NIST. The agreement between the calculated values of (a) and tabulated values from the NIST is excellent suggesting that (6.81) may be used in conjunction with (6.82) for estimation of mass collision stopping powers for electrons.
Figure 6.11 shows the mass collision stopping power S_{col} of water, aluminum, and lead for electrons against electron kinetic energy E_{K} with heavy solid lines and, for comparison, mass radiation stopping powers for same absorbers are shown with light solid lines. Similarly to stopping power behavior for heavy CPs, the data for electrons show that higher atomic number absorbers have lower S_{col} than lower atomic number absorbers at same electron kinetic energies as a result of the Z/A term in electron density as well as the (−ln I) term, where I is the mean ionization/excitation potential of the absorbing medium.
(c) The density effect parameter δ is a function of the composition and density of the absorber as well as of the velocity υ of the light CP traversing the absorber. Figure 6.12 shows δ plotted against electron or positron kinetic energy E_K of lead obtained from the NIST and also provided in Appendix E of Attix. Data used in (a) for the calculation of the mass collision stopping power S_{col} of lead for 100 keV, 1 MeV, and 10 MeV electrons are shown as points (1), (2), and (3), respectively. The dependence of S_{col} on stopping medium results from two factors in the collision stopping power expression given in (6.81). The two factors are the electron density N_e = ZN_A/A and the mean ionization/excitation potential I , both lowering S_{col} with an increasing atomic number Z of the stopping medium.
Table 6.11 Summary of parameters used in calculation of mass collision stopping power S_{col} of lead for electrons with kinetic energy E_K of 100 keV, 1 MeV, and 10 MeV
\begin{array}{lllll} \hline 1 & \text { Electron kinetic energy } E_{\mathrm{K}} & 100 \mathrm{keV} & 1 \mathrm{MeV} & 10 \mathrm{MeV} \\ \hline 2 & \text { Normalized kinetic energy } \tau & 0.1957 & 1.957 & 19.57 \\ \hline 3 & \text { Normalized electron velocity } \beta & 0.548 & 0.941 & 0.9988 \\ \hline 4 & \beta^2 & 0.30 & 0.886 & 0.9976 \\ \hline 5 & 1-\beta^2 & 0.70 & 0.114 & 0.0024 \\ \hline 6 & \text { Stopping power function } F^{-}(\tau) & 0.028 & -0.220 & 0.0497 \\ \hline 7 & \text { Density effect parameter } \delta & 0.0074 & 0.181 & 1.52 \\ \hline 8 & \text { Calculated } S_{\mathrm{col}}\left(\mathrm{MeV} \cdot \mathrm{cm}^2 / \mathrm{g}\right) & 1.969 & 0.994 & 1.198 \\ \hline \end{array}
Table 6.12 Comparison of mass collision stopping powers S_{col} of lead for electrons of kinetic energy of 100 keV, 1 MeV, and 10 MeV. Data calculated with (6.81) are displayed in row 2; data obtained from the NIST are shown in row 3; and data obtained from Appendix E of Attix are shown in row 4
\begin{array}{lllll} \hline 1 & \text { Kinetic energy } E_{\mathrm{K}} \text { of electron } & 100 \mathrm{keV} & 1 \mathrm{MeV} & 10 \mathrm{MeV} \\ \hline 2 & S_{\mathrm{col}}\left[\text { calculated with }(6.81) \text { in MeV } \cdot \mathrm{cm}^2 / \mathrm{g}\right] & 1.969 & 0.994 & 1.198 \\ \hline 3 & S_{\mathrm{col}}\left[\text { from the NIST in MeV } \cdot \mathrm{cm}^2 / \mathrm{g}\right] & 1.964 & 0.994 & 1.201 \\ \hline 4 & S_{\mathrm{col}}\left[\text { from Appendix E of Attix in MeV } \cdot \mathrm{cm}^2 / \mathrm{g}\right] & 1.964 & 0.994 & 1.201 \\ \hline \end{array}
