Question 6.4.Q4: In general, the total stopping power for a given charged par......

(a) To calculate the mass stopping power of water for a 100 MeV proton we use the Bethe mass collision stopping power equation (T6.42) that reads

S_{\mathrm{col}}=4 \pi N_{\mathrm{e}}\left(\frac{e^2}{4 \pi \varepsilon_0}\right)^2 \frac{z^2}{m_{\mathrm{e}} c^2 \beta^2}\left\{\ln \frac{2 m_{\mathrm{e}} c^2}{I}+\ln \frac{\beta^2}{1-\beta^2}-\beta^2\right\}=C_1 \frac{N_{\mathrm{e}} z^2}{\beta^2} B_{\mathrm{col}},         (6.33)

where

N_e   is the electron density \left(N_e = ZN_A/A \right) in number of electrons per gram of absorber medium with Z the atomic number and A the atomic mass of water.

z   is the number of electronic charges on the heavy CP (for proton z = 1; for α particle z = 2).

β   is velocity of the CP normalized to speed of light c in vacuum.

I  is the mean ionization/excitation potential of water (I = 75 eV).

B_{col}   is the so-called atomic stopping number that depends directly on velocity β of the CP and indirectly on the atomic number Z of the absorber (water in our case) through the mean ionization/excitation potential I and is given as

B_{\mathrm{col}}=\left\{\ln \frac{2 m_{\mathrm{e}} c^2}{I}+\ln \frac{\beta^2}{1-\beta^2}-\beta^2\right\}           (6.34)

C_1   is a collision stopping power constant independent of absorbing medium as well as of the characteristics of the charged particle. It is expressed as [see (6.18) in Prob. 131]

Before we use (6.33) to calculate the stopping power of water, we must determine the electron density N_e of water and the velocity β of the 100 MeV proton.

(1) Calculation of electron density N_e is carried out as follows: 1 mole of water \left(H_2O \right) equals to 18.0153 g of water [see (T1.22)] and, by definition, contains 6.022\times 10^{23} molecules of water, each molecule containing 2 hydrogen atoms and one oxygen atom. Thus, 1 g of water contains \frac{1}{18.0153}\times 6.022\times 10^{23} molecules of water and, since each molecule of water contains 10 electrons, we conclude that the electron density of water N_e is

(2) Calculation of 100-MeV-proton velocity β is carried out using the standard expression (T2.7) relating relativistic particle velocity β = υ/c with kinetic energy E_K of the particle. The expression is easy to derive from the basic definition of relativistic kinetic energy E_K given as follows

E_{\mathrm{K}}=(\gamma-1) m_0 c^2=\left(\frac{1}{\sqrt{1-v^2 / c^2}}-1\right) m_0 c^2           (6.37)

from which we get the following expression for β

\beta^2=\frac{v^2}{c^2}=1-\frac{1}{\left(1+\frac{E_{\mathrm{K}}}{m_{\mathrm{p}} c^2}\right)^2}=1-\frac{1}{\left(1+\frac{100}{938.3}\right)^2}=0.183 \text {, }            (6.38)

indicating that the velocity υ of a 100 MeV proton is 0.428c.

(3) Calculation of the atomic stopping number B_{col} proceeds as follows

(4) We now use (6.33) to get the final answer for stopping power of water and 100 MeV protons

(b) We now repeat, for 1 MeV and 10 MeV protons in water, the calculation carried out in (a), for each energy first calculating β² using (6.38) and B_{col} using (6.34) and then using (6.33) to get the final results for the mass stopping power.
For 1 MeV protons we get the following results for β^2,\ B_{col},\ and\ S_{col}.

\beta^2=\frac{v^2}{c^2}=1-\frac{1}{\left(1+\frac{E_{\mathrm{K}}}{m_{\mathrm{p}} c^2}\right)^2}=1-\frac{1}{\left(1+\frac{1}{938.3}\right)^2}=2.128 \times 10^{-3},           (6.41)

B_{\mathrm{col}}=\ln \frac{2 m_{\mathrm{e}} c^2}{I}+\ln \frac{\beta^2}{1-\beta^2}-\beta^2=9.520-6.150-2.128 \times 10^{-3}=3.367           (6.42)

and

For 10 MeV protons the results are as follows

\beta^2=\frac{v^2}{c^2}=1-\frac{1}{\left(1+\frac{E_{\mathrm{K}}}{m_{\mathrm{p}} c^2}\right)^2}=1-\frac{1}{\left(1+\frac{10}{938.3}\right)^2}=0.021 \text {, }            (6.44)

and

(c) To determine the kinetic energy E_K of a deuteron for which the stopping power of water is the same as that for a 100 MeV proton, we take a closer look at the functional dependence of (6.33) on physical properties of the charged particle and note that S_{col} has five components: C_1,N_e, z, β,\ and\ B_{col}. Of these, C_1,\ N_e,\ and\ B_{col} are independent of the charged particle, so that, for the mass stopping power of water to be the same for proton and deuteron, the two charged particles should have the same z (they do, since for both particles z = 1) and the same β. As shown in (6.38), β^2 = 0.183 for a 100 MeV proton, therefore we use (6.37) and determine the deuteron kinetic energy E_K that corresponds to β^2 = 0.183 as follows

Thus, for the same stopping power in water, the kinetic energy E_K of the charged particle is proportional to the rest energy of the charged particle.

Since the rest energy of the deuteron m_dc^2 is roughly twice the rest energy m_pc^2 of the proton, its kinetic energy E_K for the same velocity β is twice as high. Thus, the mass stopping power of water is the same (7.30 MeV · cm²/g) for a 100 MeV proton and a 200 MeV deuteron because both charged particles have the same velocity β as well as the same atomic number z = 1.

(d) To calculate the stopping power of water for α particle having the same velocity as the proton, we again evaluate the functional dependence of (6.33) and note that, in addition to charged particle velocity β, the mass stopping power also depends on the square of the atomic number z. Since z = 2 for α particle compared to z = 1 for proton, we conclude that the mass stopping power of water is 4 times as large (]4\times 7.30 MeV · cm²/g = 29.2 MeV · cm²/g) for α particle as that for proton at the same velocity β of the two particles (see point D in Fig. 6.6).
From the discussion in (c) we also note that, at same velocity β, kinetic energy E_K(α) of an α particle is approximately 4-times as large as that of a proton

(e) For 1 MeV, 10 MeV, and 100 MeV protons we calculated with Bethe equation (6.33) the following stopping powers of water: 269.7 MeV · cm²/g, 45.9 MeV · cm²/g, and 7.30 MeV · cm²/g, respectively, while the NIST, accounting for all known corrections, gives the following respective results: 260.8 MeV · cm²/g, 45.7 MeV · cm²/g, and 7.29 MeV · cm²/g. For an α particle with kinetic energy E_K(α) = 400 MeV the NIST provides a stopping power of water of 29.2 MeV · cm²/g, while our rudimentary calculation yields 29.2 MeV·cm²/g. Thus, the agreement with the basic Bethe equation (6.33) and the one incorporating all currently known corrections is quite good.
Stopping power of water for proton and α particle available from the NIST is shown in Fig. 6.6 in the energy range from 1 keV to 1000 MeV and our calculated results are superimposed on the graph with data points. The good agreement between our calculation and the NIST data is evident.