Question 6.6.Q1: For a given atomic number Z of the stopping medium, kinetic ......

(a) We identify the stopping power curves of Fig. 6.14 through accounting for the following two facts related to stopping powers of light CPs:

(1) For all stopping materials irrespective of atomic number Z the collision component $S_{col}$ of $S_{tot}$ predominates at kinetic energies $E_K$ < 10 MeV, while the radiation component $S_{rad}$ of $S_{tot}$ predominates at $E_K$ > 100 MeV. In the intermediate region, where $E_K$ is between 10 MeV and 100 MeV, the region of predominance depends on the atomic number Z of the stopping medium. Based on this we conclude that the heavy curves in Fig. 6.14 represent $S_{col}$ of the 7 stopping media and the light curves represent $S_{rad}$ of the 7 stopping media (see Fig. 6.15).

(2) At a given kinetic energy $E_K$ of the light CP, $S_{col}$ is inversely proportional to Z of the stopping medium as a result of two properties of the stopping medium: electron density $N_e = ZN_A/A$ and mean ionization/excitation potential I , as evident from (6.17). Both $N_e$ and ln I decrease $S_{col}$ with increasing Z. On the other hand, as shown in (6.3), $S_{rad}$ is proportional to $ZN_e$ which suggests a linear proportion of $S_{rad}$ with Z, since Z/A in Ne is essentially constant for all stopping media.

$S_{\text{rad}} = αr^2_eZN_e(E_K +m_ec^2)B_{\text{rad}},$ (6.3)

In Fig. 6.14 we thus arrange the $S_{col}$ curves in the order of increasing Z from top (He) to bottom (U) and the $S_{rad}$ curves in reverse order with U on the top and He on the bottom, as shown in Fig. 6.15.

(b) We are now ready to determine critical kinetic energy $E_{\mathrm{K}}^{\mathrm{crit}}$ for the seven materials using data from Fig. 6.15. $E_{\mathrm{K}}^{\mathrm{crit}}$ is defined as the intercept between $S_{col}$ and $S_{rad}$ curves for the given stopping material and we can now read $E^{crit}_K$ directly from the graph. The results for the 7 stopping media are listed in Table 6.13 in row (3). Also listed in the table is atomic number Z of the stopping medium in row (2) as well as in row (6) the critical energy $E^{crit}_K$ determined from (6.102).
The measured and calculated $E^{crit}_K$ data are plotted in Fig. 6.16 against atomic number Z of the various stopping materials with data points and solid curve, respectively. The agreement between the two sets appears to be reasonable, suggesting that (6.102) is a good and simple empirical approximation for determination of $E^{crit}_K$ .

However, a comparison between rows (3) and (5) of Table 6.13 suggests otherwise, especially for stopping media of low Z. We therefore re-plot the data in the form of $E_{\mathrm{K}}^{\text {crit }} \times Z$ and get a better picture on the discrepancy between (6.102) and the NIST data, as shown in Fig. 6.17. The grey area in the figure shows a region of ±10 % agreement with (6.102) and we note that for Z < 30 data calculated with (6.102) exceed the NIST data by more than 10 % and the discrepancy margin increases as Z decreases.
Figure 6.17 shows that (6.102) must be used with caution, especially at low atomic number Z where it overestimates the critical kinetic energy $E_{\mathrm{K}}^{\mathrm{crit}}$ by ∼50 %. At large Z (6.102) provides a more reliable means for estimation of the critical kinetic energy $E_{\mathrm{K}}^{\mathrm{crit}}$ achieving an accuracy of about ±5 %.