Question 6.10.Q2: Johns and Cunningham provide expressions for unrestricted an......
Johns and Cunningham provide expressions for unrestricted and restricted mass collision stopping powers S_{col}\ and\ L_Δ, respectively, in the following format for electrons
S_{\mathrm{col}}=C_{\mathrm{e}} \frac{N_{\mathrm{e}}}{\beta^2}\left\{\ln \frac{E_{\mathrm{K}}^2\left(E_{\mathrm{K}}+2 E_0\right)}{2 E_0 I^2}+\frac{E_{\mathrm{K}}^2}{8\left(E_{\mathrm{K}}+E_0\right)^2}-\frac{\left(2 E_{\mathrm{K}}+E_0\right) E_0 \ln 2}{\left(E_{\mathrm{K}}+E_0\right)^2}\right.\\ \left.+1-\beta^2-\delta\right\}\quad (6.157)and
where Δ is the delta ray electron threshold energy which is equal to kinetic energy of the delta ray electron whose kinetic energy is just large enough to allow it to escape from the region of interest. The other parameters of (6.157) and (6.158) are defined in Prob. 145.
(a) Use expressions (6.157) and (6.158) to calculate S_{col}\ and\ L_Δ, respectively, of water for electrons with the following kinetic energy E_K in MeV: 0.01, 0.1, 1, 10, and 100 and the following delta ray threshold energies Δ in keV: 1, 10, and 100. The density effect parameter δ, as provided by the NIST [http://physics.nist.gov/cgi-bin/Star/e_table.pl] is as follows:
E_K = 0.01 MeV: δ = 0; 0.1 MeV: 0; 1 MeV: 0.243; 10 MeV: 2.992; 100 MeV: 7.077; electron density: N_{\mathrm{e}}=3.343 \times 10^{23} electron/g (Prob. 133); mean ionization/excitation potential: 75 eV (see Prob. 132).
(b) Insert your results onto Fig. 6.29 that plots unrestricted mass collision stopping power S_{col} of water as well as restricted stopping power L_Δ (for Δ of 1 keV, 10 keV, and 100 keV) of water against electron kinetic energy E_K in the kinetic energy range from 1 keV to 100 MeV.
(c) Calculate and plot the ratio L_Δ/S_{col} for the five kinetic energies E_K and three delta ray threshold energies Δ of (a) and (b). Comment on the meaning of the ratio L_Δ/S_{col}.
(d) Show that (6.158) for restricted stopping power L_Δ transforms into (6.157) for unrestricted stopping power S_{\text {col }} \text { when } \Delta=\Delta\left(E_{\mathrm{K}}\right)_{\max }=\frac{1}{2} E_K is used in (6.158).
(a) Expressions (6.157) for unrestricted mass collision stopping power S_{col} and (6.158) for restricted mass collision stopping power L_Δ were used to calculate S_{col} and L_Δ of water for the following kinetic energies E_K in MeV: 0.01, 0.1, 1, 10, and 100 and three delta ray threshold energies Δ of 1 keV, 10 keV, and 100 keV. Results are tabulated in Table 6.22 that also lists appropriate β^2 as well as the density effect parameter δ from the NIST.
(b) Figure 6.30 shows the unrestricted mass collision stopping power S_{col} (heavy solid curve) as well as restricted stopping power L_Δ (light solid curves) for delta ray threshold energies Δ of 1 keV, 10 keV, and 100 keV of water for electrons in the kinetic energy range from 1 keV to 100 MeV obtained from the NIST. Superimposed onto the curves are S_{col} calculated from (6.157) and L_Δ calculated from (6.158). Agreement between calculated data and data from the NIST is excellent.
(c) Ratio L_Δ/S_{col} of water for data of Table 6.22 is shown in Table 6.23 and plotted in Fig. 6.31. The following notable properties of L_Δ/S_{col} are apparent:
(1) For E_K ≤ 2Δ, where E_K is kinetic energy of the electron and Δ is threshold delta ray energy, L_Δ = S_{col} and thus L_Δ/S_{col} = 1. Conclusion: No energy escapes the volume of interest.
(2) As E_K increases beyond 2Δ, L_Δ becomes increasingly smaller in comparison to S_{col} indicating that an increasingly larger portion of incident kinetic energy escapes the volume of interest.
(3) Ratio L_Δ/S_{col} can be considered the proportion of electron kinetic energy E_K that is absorbed locally. The proportion of E_K that escapes the volume of interest is 1 − L_Δ/S_{col}.
(d) To show that (6.158) for restricted stopping power L_Δ transforms into (6.157) for unrestricted stopping power S_{col} when delta ray threshold Δ attains its maximum possible value of \Delta=\Delta\left(E_{\mathrm{K}}\right)_{\max }=\frac{1}{2} E_{\mathrm{K}} \text { we insert } \Delta=\frac{1}{2} E_{\mathrm{K}} into (6.158) and get
Table 6.22 Unrestricted mass collision stopping power S_{col} (heavy solid curve) calculated from (6.157) as well as restricted stopping power L_Δ (light solid curves) calculated from (6.158) for delta ray threshold energies Δ of 1 keV, 10 keV, and 100 keV of water for electrons in kinetic energy range from 1 keV to 100 MeV
\begin{array}{lcclclll} \hline 1 & \begin{array}{l} \text { Kinetic } \\ \text { energy } \\ E(\mathrm{MeV}) \end{array} & \beta^2=\frac{\nu^2}{c^2} & \begin{array}{l} \delta \\ \text { from } \\ \text { NIST } \end{array} & \begin{array}{l} S_{\mathrm{col}} \\ (\mathrm{MeV} / \mathrm{cm}) \end{array} & \begin{array}{l} L_{\Delta=1 \mathrm{keV}} \\ (\mathrm{MeV} / \mathrm{cm}) \end{array} & \begin{array}{l} L_{\Delta=10 \mathrm{keV}} \\ (\mathrm{MeV} / \mathrm{cm}) \end{array} & \begin{array}{l} L_{\Delta=100 \mathrm{keV}} \\ (\mathrm{MeV} / \mathrm{cm}) \end{array} \\ \hline 2 & 0.01 & 0.03802 & 0 & 22.600 & 19.630 & – & – \\ \hline 3 & 0.10 & 0.30055 & 0 & 4.122 & 3.111 & 3.741 & – \\ \hline 4 & 1.00 & 0.85563 & 0.243 & 1.852 & 1.256 & 1.477 & 1.695 \\ \hline 5 & 10.00 & 0.99764 & 2.992 & 1.971 & 1.212 & 1.409 & 1.606 \\ \hline 6 & 100.00 & 0.99997 & 7.077 & 2.206 & 1.248 & 1.442 & 1.639 \\ \hline \end{array}
Table 6.23 Ratio L_Δ/S_{col} of water for data of Table 6.22 where L_Δ is the restricted mass collision stopping power and S_{col} is the unrestricted mass collision stopping power
\begin{array}{lllllllll} \hline 1 & \begin{array}{l} \text { Kinetic } \\ \text { energy } \\ E(\mathrm{MeV}) \end{array} & S_{\mathrm{col}} & L_{\Delta=1 \mathrm{keV}} & \frac{L_{\Delta=1 \mathrm{keV}}}{S_{\mathrm{col}}} & L_{\Delta=10 \mathrm{keV}} & \frac{L_{\Delta=10 \mathrm{keV}}}{S_{\mathrm{col}}} & L_{\Delta=100 \mathrm{keV}} & \frac{L_{\Delta=100 \mathrm{keV}}}{S_{\mathrm{col}}} \\ \hline 2 & 0.01 & 22.600 & 19.630 & 0.869 & – & – & – & – \\ \hline 3 & 0.10 & 4.122 & 3.111 & 0.755 & 3.741 & 0.908 & – & – \\ \hline 4 & 1.00 & 1.852 & 1.256 & 0.678 & 1.477 & 0.800 & 1.695 & 0.915 \\ \hline 5 & 10.00 & 1.971 & 1.212 & 0.615 & 1.409 & 0.715 & 1.606 & 0.815 \\ \hline 6 & 100.00 & 2.206 & 1.248 & 0.566 & 1.442 & 0.654 & 1.639 & 0.743 \\ \hline \end{array}
