Question 6.6.Q2: The ratio between the two components Scol/Srad of light char......
The ratio between the two components S_{col}/S_{rad} of light charged particle (CP) stopping power at a given kinetic energy E_K of the light CP depends on E_K as well as on the atomic number Z of the stopping material and can be approximated with the following empirical expression
\frac{S_{\mathrm{col}}}{S_{\mathrm{rad}}}=\frac{E_{\mathrm{K}}^{\mathrm{crit}}}{E_{\mathrm{K}}}, (6.103)
where E_{\mathrm{K}}^{\mathrm{crit}}, the so-called critical kinetic energy, is the kinetic energy of the light CP at which the two components of S_{\text {tot }} \text { are identical, i.e., } S_{\text {col }}\left(E_{\mathrm{K}}^{\text {crit }}\right)= S_{\mathrm{rad}}\left(E_{\mathrm{K}}^{\text {crit }}\right)=\frac{1}{2} S_{\mathrm{tot}}\left(E_{\mathrm{K}}^{\mathrm{crit}}\right).
(a) Figure 6.18 shows an unlabeled graph of S_{col} in one group and S_{rad} in another group. Each group of curves covers seven stopping materials: helium, carbon, aluminum, copper, silver, lead, and uranium against E_K of the light CP. Identify the two groups of curves and for each curve provide the stopping material.
(b) Verify the validity of (6.103) for the following three stopping materials: carbon, copper, and lead. Obtain the stopping power and E_{\mathrm{K}}^{\mathrm{crit}} data for the three stopping materials from appropriately labeled Fig. 6.18.
(c) For the three stopping materials of (b) plot your results and compare the NIST data on S_{\mathrm{col}} / S_{\mathrm{rad}} \text { with the ratio } E_{\mathrm{K}}^{\text {crit }} \text { in the } E_{\mathrm{K}} range from 0.01 MeV to 1000 MeV. Make general comments on the validity of (6.103).
(a) Before we can identify the stopping power curves in Fig. 6.18 we should consider the following points:
(1) Both S_{col} as given in (6.70) and S_{rad} as given in (6.3) depend upon atomic number Z of the stopping material as well as on kinetic energy E_K of the light CP.
S_{\text{col}} = 4π\left(\frac{e^2}{4πε_0}\right)^2\frac{z^2N_e}{m_ec^2β^2}\left\{\ln\frac{2m_ec^2}{I}+\ln\frac{β^2}{1 − β^2}−β^2\right\} (6.70)
S_{\text{rad}} = αr^2_eZN_e(E_K +m_ec^2)B_{\text{rad}}, (6.3)
(2) Dependence of S_{col} on Z is manifested directly through electron density N_e that is proportional to Z/A and indirectly on ionization/excitation potential I through the (−ln I) term. Both Z/A and (−ln I) diminish S_{col} with increasing Z.
(3) Dependence of S_{rad} on Z is manifested directly through the ZN_e factor that indicates direct proportionality of S_{rad} with Z, since N_e through Z/A exhibits only a slight dependence on Z. Note that in the first approximation Z/A ≈ 0.5 for all elements, however, in reality Z/A ranges from 0.5 at low Z down to ∼0.4 at high Z with only one notable exception of hydrogen for which Z/A = 1.
(4) Dependence of S_{col}\ on\ E_K is divided into 3 regions: at relatively low kinetic energy, S_{col} is inversely proportional to E_K (i.e., goes as 1/E_K), reaches a broad minimum at a few MeV as the light CP approaches speed of light c, and then slowly rises at relativistic energies above 10 MeV.
(5) Dependence of S_{rad}\ on\ E_K is manifested directly through the \left(E_K + m_ec^2\right) term and indirectly through the B_{\text{rad}} term that is constant for E_K < m_ec^2 and exhibits a slow rise with E_K increasing above 1 MeV.
(b) We now use Fig. 6.19 that is based on data from the NIST to determine S_{col}\ and\ S_{rad} for carbon, copper, and lead at the following kinetic energies in MeV: 0.01, 0.1, 1, 10, 100, and 1000. We also determine E^{crit}_K following the same procedure that was followed in Prob. 141. Our results are listed in Table 6.14 in which we also give the ratios S_{col}/S_{rad}\ and\ E^{crit}_K /E_K. Since the table represents 3 stopping materials, one each for low Z, intermediate Z, and high Z, we can draw some general conclusions about the general validity of (6.103), as proscribed in (c)
(c) To visualize better the results of (6.103) presented in Table 6.14 we plot and compare the two ratios S_{col}/S_{rad} (shown as data points) and E^{crit}_K /E_K (shown with solid line) in Fig. 6.20. It is evident that the two ratios agree well for electron kinetic energy E_K above 1 MeV; however, for E_K < 1 MeV the agreement breaks down and S_{col}/S_{rad} < E^{crit}_K /E_K, the lower the kinetic energy the worse is the agreement, so that at kinetic energy E_K = 0.001 MeV ratio E^{crit}_K /E_K exceeds ratio S_{col}/S_{rad} by at least 50 %. We thus conclude that (6.103) is a reasonable approximation for estimation of the ratio S_{col}/S_{rad} at electron kinetic energy exceeding 1 MeV, however, for E_K < 1 MeV (6.103) is not reliable, especially for kinetic energy that is much smaller than 1 MeV.
Table 6.14 Ratios S_{col}/S_{rad}\ and\ E^{crit}_K /E_K of (6.103) of carbon, copper, and lead at various kinetic energies E_K of electron. Data are from the NIST
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline {\begin{array}{l} E_{\mathrm{K}} \\ (\mathrm{MeV}) \end{array}} & {\begin{array}{l} \text { Carbon } Z=6 \\ E_{\mathrm{K}}^{\mathrm{crit}}=96 \mathrm{MeV} \end{array}} &&&& {\begin{array}{l} \text { Copper } Z=29 \\ E_{\mathrm{K}}^{\text {crit }}=24.3 \mathrm{MeV} \end{array}} &&&& {\begin{array}{l} \text { Lead } Z=82 \\ E_{\mathrm{K}}^{\text {crit }}=10 \mathrm{MeV} \end{array}} \\ \hline & S_{\mathrm{rad}} & S_{\mathrm{col}} & \frac{S_{\mathrm{col}}}{S_{\mathrm{rad}}} & \frac{E_{\mathrm{K}}^{\text {crit }}}{E_{\mathrm{K}}} & S_{\mathrm{rad}} & S_{\mathrm{col}} & \frac{S_{\mathrm{cal}}}{S_{\text {rad }}} & \frac{E_{\mathrm{K}}^{\mathrm{crit}}}{E_{\mathrm{K}}} & S_{\text {rad }} & S_{\mathrm{col}} & \frac{S_{\mathrm{col}}}{S_{\mathrm{rad}}} & \frac{E_{\mathrm{K}}^{\text {crit }}}{E_{\mathrm{K}}} \\ \hline 0.01 & 0.003 & 20 & 6667 & 9600 & 0.012 & 13.2 & 1100 & 2430 & 0.02 & 8.5 & 425 & 1000 \\ \hline 0.1 & 0.0035 & 3.7 & 1057 & 960 & 0.017 & 2.7 & 159 & 243 & 0.05 & 2 & 44.4 & 100 \\ \hline 1 & 0.01 & 1.6 & 160 & 96 & 0.046 & 1.3 & 28.3 & 24.3 & 0.13 & 1 & 7.7 & 10 \\ \hline 10 & 0.15 & 1.8 & 12 & 9.6 & 0.57 & 1.4 & 2.5 & 2.43 & 1.2 & 1.2 & 1 & 1 \\ \hline 100 & 2 & 2 & 1 & 0.96 & 7 & 1.7 & 0.24 & 0.243 & 14 & 1.4 & 0.1 & 0.1 \\ \hline 1000 & 22 & 2.1 & 0.1 & 0.1 & 76.5 & 1.85 & 0.024 & 0.024 & 155 & 1 & 0.01 & 0.01 \\ \hline \end{array}
