Question 6.8.Q2: According to the derivation of (T6.74), the CSDA range R^M0 ......

(a) The CSDA range R_{\mathrm{CSDA}}\left[\left(E_{\mathrm{K}}^{M_0}\right)\right] \text { of a heavy CP with rest mass } M_0 and charge ze traversing a stopping medium (absorber) is defined as

R_{\mathrm{CSDA}}\left[\left(E_{\mathrm{K}}^{M_0}\right)\right]=\int_0^{\left(E_{\mathrm{K}}^{M_0}\right)} \frac{\mathrm{d} E_{\mathrm{K}}}{S_{\mathrm{col}}\left(E_{\mathrm{K}}\right)}          (6.115)

where \left(E_{\mathrm{K}}^{M_0}\right) is the incident kinetic energy of the \mathrm{CP} \text { and } S_{\mathrm{col}}\left(E_{\mathrm{K}}^{M_0}\right) is the mass collision stopping power of the absorber for heavy particle of rest mass M_0. For a non-relativistic heavy \mathrm{CP}, S_{\mathrm{col}}\left(E_{\mathrm{K}}^{M_0}\right) is given by the Bethe stopping power equation as

where

C_A is a constant for a given absorber.
υ is the velocity of the heavy CP.
m_e is the rest mass of the electron \left(m_{\mathrm{e}}=0.511 \mathrm{MeV} / c^2\right)
I is the ionization/excitation potential of the stopping medium (absorber).
E_{\mathrm{K}}^{M_0} is the kinetic energy of the heavy \text { CP, i.e., } E_{\mathrm{K}}^{M_0}=\frac{1}{2} M_0 v^2 for non-relativistic heavy CP.

Combining (6.116) with (6.115) we get the following expression for R_{\mathrm{CSDA}}\left[\left(E_{\mathrm{K}}^{M_0}\right)\right].

R_{\mathrm{CSDA}}\left[\left(E_{\mathrm{K}}^{M_0}\right)_0\right]=\frac{2 M_0}{C_{\mathrm{A}} z^2} \int_0^{\left(E_{\mathrm{K}}^{M_0}\right)} \frac{\frac{E_{\mathrm{K}}^{M_0}}{M_0} \mathrm{~d} \frac{E_{\mathrm{K}}^{M_0}}{M_0}}{\ln \left[\frac{4 m_{\mathrm{e}}}{I} \frac{E_{\mathrm{K}}^{M_0}}{M_0}\right]}         (6.117)

which we now expand to make a link between the range R_{\mathrm{CSDA}}\left[\left(E_{\mathrm{K}}^{M_0}\right)\right] of the heavy CP of rest mass M_0 and charge ze and range R_{\mathrm{CSDA}}\left[\left(E_{\mathrm{K}}^{\mathrm{p}}\right)\right] of a proton of rest mass m_p = 938.3 MeV and charge e

where C\left(M_0,z\right) is a correction factor for mass M_0 and charge ze of the heavy CP given in (6.119) and \left(E_{\mathrm{K}}^{\mathrm{p}}\right)_0 is the equivalent incident kinetic energy of a proton that satisfies (6.112) and is related to \left(E_{\mathrm{K}}^{\mathrm{M}_0}\right)_0, as shown in (6.120)

C\left(M_0, z\right)=\frac{M_0}{z m_{\mathrm{p}}}          (6.119)

and

\left(E_{\mathrm{K}}^{\mathrm{p}}\right)_0=\frac{m_{\mathrm{p}}}{M_0}\left(E_{\mathrm{K}}^{M_0}\right)_0 .           (6.120)

(b) The mass/charge correction factor C\left(M_0,z\right) for the five heavy charged particles is determined using (6.113) and listed in column (7) of Table 6.19 which also lists the basic relevant physical properties of the five heavy charged particles in addition to proton.

(c) The equivalent proton incident kinetic energy \left(E_{\mathrm{K}}^{\mathrm{p}}\right)_0 for use in (6.112) in determination of the CSDA range of CPs heavier than the proton is calculated with (6.114) for five CPs (deuteron, triton, α particle, carbon-6 ion, and neon-10 ion), all with incident kinetic energy of 500 MeV and given in column (4) of Table 6.20.

(d) The CSDA range of heavy CPs: proton, deuteron, triton, α particle, carbon ion, and neon ion, all with incident kinetic energy \left(E_{\mathrm{K}}^{M_0}\right)_0 of 500 MeV is determined with (6.112) and results are displayed in Table 6.20 and in Fig. 6.26. We note that 500 MeV proton, deuterons, and tritons have a range that exceeds the penetration in water required for radiotherapy, while carbon ions and neon ions at 500 MeV [i.e., at 42 MeV/u (∼500 : 12) and 25 MeV/u (∼500 : 20), respectively] exhibit CSDA ranges that are too low for use in practical radiotherapy.