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 %.