Question 6.10.Q1: Many expressions, at first glance unrelated to one another, ......
Many expressions, at first glance unrelated to one another, have been used in the literature to describe the unrestricted mass collision stopping power S_{col} of absorbers for electrons. For example, the ICRU Report 37 uses the following form for S_{col}
while the book by Johns and Cunningham has
where C_e is a constant given as C_{\mathrm{e}}=2 \pi r_{\mathrm{e}}^2 E_0=2.55 \times 10^{-25} \mathrm{MeV} \cdot \mathrm{cm}^2; N_e is the number of electrons per unit mass of absorber \left(N_{\mathrm{e}}=Z N_{\mathrm{A}} / A\right) ; E_{\mathrm{K}} is kinetic energy of the electron; E_0 is rest energy of the electron; τ is kinetic energy E_K of the electron normalized to electron rest energy E_0, i.e., τ = E_K/E_0; δ is the so-called density effect parameter that accounts for density effect in condensed media; and I is the mean ionization/excitation potential of the absorber.
Bichsel recommends the following expression that can be used for both the unrestricted as well as the restricted mass collision stopping power
S_{\mathrm{col}}=C_{\mathrm{e}} \frac{N_{\mathrm{e}}}{\beta^2}\left\{\ln \frac{2(\tau+2) E_0^2}{I^2}+F^{-}(\tau, \zeta)-\delta\right\}, (6.132)
where F^{-} is in general defined as
F^{-}=-1-\beta^2+\ln [(\tau-\zeta) \zeta]+\frac{\tau}{\tau-\zeta}+\frac{1}{(\tau+1)^2}\left[\frac{\zeta^2}{2}+(2 \tau+1) \ln \left(1-\frac{\zeta}{\tau}\right)\right], (6.133)
with ζ a special parameter defined as \zeta=\tau /\left(2 E_0\right) for unrestricted stopping power and as ζ = Δ/E_0 for restricted collision stopping power where Δ is equal to kinetic energy of the delta ray whose kinetic energy is just large enough to allow it to escape from the region of interest.
(a) Show that (6.130) and (6.131) are equivalent.
(b) Show that (6.132) incorporating (6.133) with \zeta=\tau /\left(2 E_0\right) is equivalent to (6.131).
(c) Take (6.132) in conjunction with (6.133), insert ζ = Δ/E_0, and derive the expression for restricted mass collision stopping power given in the book by Johns and Cunningham as
(a) To prove equivalency of (6.130) with (6.131) we compare the terms inside the curly brackets of the two equations and transform (6.130) into (6.131) using the definition of the parameter τ = E_K/E_0. Starting with (6.130) we get
where
A=\ln \frac{E_{\mathrm{K}}^2}{I^2}, (6.136)
B=\ln \left(1+\frac{\tau}{2}\right)=\ln \frac{E_{\mathrm{K}}+2 E_0}{2 E_0}, (6.137)
A+B=\ln \frac{E_{\mathrm{K}}^2}{I^2}+\ln \frac{\left(E_{\mathrm{K}}+2 E_0\right)}{2 E_0}=\ln \frac{E_{\mathrm{K}}^2\left(E_{\mathrm{K}}+2 E_0\right)}{2 E_0 I^2}, (6.138)
C=1-\beta^2=\frac{E_0^2}{\left(E_{\mathrm{K}}+E_0\right)^2}=\frac{1}{(\tau+1)^2}, (6.139)
D=1+\frac{\tau^2}{8}-(2 \tau+1) \ln 2=1+\frac{E_{\mathrm{K}}}{8 E_0}-\frac{2 E_{\mathrm{K}}+E_0}{E_0} \ln 2, (6.140)
C \times D=1-\beta^2+\frac{E_{\mathrm{K}}^2}{8\left(E_{\mathrm{K}}+E_0\right)^2}-\frac{\left(2 E_{\mathrm{K}}+E_0\right) E_0}{\left(E_{\mathrm{K}}+E_0\right)^2} \ln 2 . (6.141)
Inserting parameters A, B, C, and D into (6.135) we now get the following expression for (6.135)
Equation (6.142) is identical to (6.131), substantiating the contention that (6.130) and (6.131) are equivalent to one another. Note: Parameter C of (6.139) is determined from the basic definition of kinetic energy of the incident electron E_K/E_0 = \left(1-\beta^2\right)^{-1 / 2}-1
(b) To prove the equivalency of (6.132) with (6.130) and (6.131) when parameter ζ of (6.132) is equal to the maximum possible energy transfer \Delta E_{max} from the incident electron of kinetic energy E_K to a delta ray electron normalized to electron rest energy E_0.
According to convention on indistinguishable colliding particles, \Delta E_{max} equals to 50 % of the kinetic energy E_K of the incident electron. We thus use ζ = E_K/\left(2E_0\right) and modify (6.132) as follows
\left\{\ln \frac{2(\tau+2) E_0^2}{I^2}+F^{-}\left(\tau=\frac{E_{\mathrm{K}}}{E_0}, \zeta=\frac{E_{\mathrm{K}}}{2 E_0}\right)-\delta\right\}=\left\{G+F^{-}-\delta\right\}, (6.143)
with
G=\ln \frac{2(\tau+2) E_0^2}{I^2}=\ln \frac{\left(E_{\mathrm{K}}+2 E_0\right) E_0}{I^2} (6.144)
and
F^{-}=-1-\beta^2+H+J+K \times L, (6.145)
where
H=\ln [(\tau-\zeta) \zeta]=\ln \left[\left(\frac{E_{\mathrm{K}}}{E_0}-\frac{E_{\mathrm{K}}}{2 E_0}\right) \frac{E_{\mathrm{K}}}{2 E_0}\right]=\ln \frac{E_{\mathrm{K}}^2}{4 E_0^2}, (6.146)
J=\frac{\tau}{\tau-\zeta}=\frac{E_{\mathrm{K}} E_0}{E_0\left(E_{\mathrm{K}}-\frac{1}{2} E_{\mathrm{K}}\right)}=2, (6.147)
K=\frac{1}{(\tau+1)^2}=\frac{E_0^2}{\left(E_K+E_0\right)^2} \quad[\text { see (6.139)], } (6.148)
=\frac{E_{\mathrm{K}}^2}{8\left(E_{\mathrm{K}}+E_0\right)^2}-\frac{\left(2 E_{\mathrm{K}}+E_0\right) \ln 2}{\left(E_{\mathrm{K}}+E_0\right)^2} (6.150)
Function F^{-} of (6.145) can now be expressed as follows
leading to the following expression for (6.143)
Equation (6.152) is identical to terms in curly bracket of (6.131) allowing us to conclude that (6.132) with ζ = E_K/\left(2E_0\right) is equivalent to (6.130) which, as shown in (a), in turn is equivalent to (6.131) for description of the mass unrestricted collision stopping power of various absorbers for electrons.
(c) In this section we use (6.132) in conjunction with (6.133) and insert into (6.133) for parameter ζ the δ ray threshold Δ normalized to electron rest energy E_0, i.e., ζ = Δ/E_0, to derive (6.134) for restricted stopping power L_Δ. First, we write the terms in curly bracket of (6.132) as follows
\left\{\ln \frac{2(\tau+2) E_0^2}{I^2}+F^{-}(\tau, \zeta)-\delta\right\}=\left\{A+F^{-}\left(\tau=\frac{E_{\mathrm{K}}}{E_0}, \zeta=\frac{\Delta}{E_0}\right)-\delta\right\} (6.153)
where
A=\ln \frac{2(\tau+2) E_0^2}{I^2}=\ln \frac{2\left(E_{\mathrm{K}}+2 E_0\right) E_0}{I^2} (6.154)
and
Next, after inserting (6.154) and (6.155) into (6.153) we get the following expression for the restricted stopping power L_Δ, in agreement with the expression (6.134) provided by Johns and Cunningham