ON THE CALCULATION OF ENERGY LOST BY ELECTRONS

K. A. DERGOBUZOV, O. B. EVDOKIMOV

(Presented by the scientific seminar of the Faculty of Physics and Technology)

Infinite plane source in an infinite homogeneous medium

In [1] Spencer proposed a method for calculating the energy lost by electrons in an infinite homogeneous medium. In papers [2, 3] a method for calculating the distribution function was developed, which makes it possible to compute the distribution of the lost energy

\[ D(x)=\int_{4\pi} d\Omega \int_x^1 I(x,s,\Theta)\frac{dT}{ds}(s)\,ds, \tag{1} \]

where \(I(x,s,\Theta)\) is the function of the angular and energy distribution;

\[ \frac{dT}{ds} \]

is the specific ionization loss;

\(x\) is a spatial coordinate, expressed in units of \(R_0\), the total range of electrons obtained in the continuous-slowing-down approximation;

\(s\) is the length of the path traveled by the electron, in the same units;

\(D(x)\) is the distribution of the lost energy.

The advantage of the second method is that it makes it possible easily to program and automate the process of calculating the distribution of the lost energy. Moreover, if it is necessary to compute \(D(x)\), \(I(x,s,\Theta)\) need not be determined. Indeed, let us write the equation for \(I(x,s,\varepsilon)\) [2]:

\[ I(x,s_i,\varepsilon_{i+1}) = \int_{4\pi} I(x-\lambda_i\varepsilon_i,s_{i-1},\varepsilon_i) g(\lambda_i,s_i,\omega_i)\,d\Omega_i, \tag{2} \]

where \(g(\lambda_i,s_i,\omega_i)\) is the angular distribution of electrons on the \((i+1)\)-th segment relative to the direction of motion on the \(i\)-th segment of the trajectory;

\[ s_i=\sum_{i=1}^{l}\lambda_i; \]

\[ \varepsilon=\cos\Theta \]

\(\lambda_i\) is the length of a segment of the trajectory.

Multiply equation (2) by \(-\dfrac{dT}{ds}(s_i)\). Taking into account that

\[ \frac{dT}{ds}(s_i)=\gamma(s_i,\lambda_i)\frac{dT}{ds}(s_{i-1}) \]

and, introducing the notation \(D(x,s_i,\varepsilon_{i+1})=-\dfrac{dT}{ds}(s_i) I(x,s_i,\varepsilon_{i+1})\), we obtain an equation for \(D(x,s,\varepsilon)\):

\[ D(x,s_j,\varepsilon_{j+1})=\int_{4\pi} D(x-\lambda_j \varepsilon_j,s_{j-1},\varepsilon_j)\,q(\lambda_j,s_j,\omega_j)\,d\Omega_j, \tag{3} \]

where

\[ q(\lambda_j,s_j,\omega_j)=\gamma(s_j,\lambda_j)\,g(\lambda_j,s_j,\omega_j). \tag{4} \]

In external form equation (3) coincides with (2); consequently, if \(\alpha_{\kappa L}(s)\) and \(\beta_{\kappa L}(s)\) are the harmonic coefficients of the angular moments \(D_L(x,s)\) of the function \(D(x,s,\varepsilon)\), then the recurrence relations for them will be similar to those for the harmonic coefficients of the angular moments of the function \(I(x,s,\varepsilon)\) [2]:

\[ \alpha_{\kappa L}(s_j)\equiv \alpha_{\kappa L}^{j} =Q_L^j\sum_{L'} \alpha_{\kappa L'}^{j-1} V_{LL'}^j -\beta_{\kappa L'}^{j-1} W_{LL'}^j, \tag{5} \]

\[ \beta_{\kappa L}(s_j)\equiv \beta_{\kappa L}^{j} =Q_L^j\sum_{L'} \beta_{\kappa L'}^{j-1} V_{LL'}^j +\alpha_{\kappa L'}^{j-1} W_{LL'}^j, \]

where

\[ Q_L^j=\gamma(s_i,\lambda_i)G_L^j, \]

and \(V_{LL'}^j\), \(W_{LL'}^j\), and \(G_L^j\) are defined in [2].

Then the distribution of the lost energy is

\[ D(x)=\sum_{j(x)}^{j_m}\left\{ \frac{\alpha_{\kappa 0}^j}{2} +\sum_{\kappa=1}^{\infty}\alpha_{\kappa 0}^j\cos \kappa\pi x +\beta_{\kappa 0}^j\sin \kappa\pi x \right\}\lambda_j, \tag{6} \]

where \(j_m\) and \(j(x)\) are determined from the conditions:

\[ x=\sum_{\kappa=1}^{j(x)}\lambda_\kappa \quad \text{and} \quad 1=\sum_{\kappa=1}^{j_m}\lambda_\kappa . \tag{7} \]

The initial conditions for calculating \(D(x)\) are determined by the fact that, at \(s=0\),

\[ D(x,s,\varepsilon)\big|_{s=0} =-\frac{dT}{ds}(0)\delta(x)\delta(1-\cos\Theta). \tag{8} \]

It follows from this that

\[ \alpha_{\kappa L}(0)=-\frac{dT}{ds}(0), \]

\[ \beta_{\kappa L}(0)=0. \tag{9} \]

Instead of the rectangle summation formula, other summation formulas may also be applied. Formula (6) automatically ensures sufficient accuracy of the numerical integration, since \(\lambda_j\ll 1\) (\(\lambda_j\sim 0.02\)).

Approximate allowance for the presence of a boundary

It is known that the theoretical allowance for the finite dimensions of a medium in calculating particle-transport problems encounters great mathematical difficulties. It is therefore of interest to consider the possibilities for an approximate calculation of the influence of the absorber boundary.

We shall consider the essence of the proposed method using the example of a plane-parallel infinite plate of thickness \(x_0\), perpendicular to the \(x\)-axis, on the front surface of which \((x=0)\) there is a plane source:

\[ Q(x,s,\varepsilon)\big|_{x=0}=I(x,s,\varepsilon)\big|_{s=0}=\delta(x)\delta(1-\varepsilon). \tag{10} \]

The problem consists in calculating \(I_{\parallel}(x,s,\varepsilon)\) for \(0\leq x\leq x_0\).

We shall temporarily assume that the absorber is infinite. On the basis of [2], the electron distribution function in the planes \(x=0\) and \(x=x_0\) can be calculated, i.e. \(I(x,s,\varepsilon)\big|_{x=x_0}\) and \(I(x,s,\varepsilon)\big|_{x=0}\). If it is assumed that each electron passes through the plane \(x=x_0\) in the direction opposite to the \(x\)-axis no more than once, the absence of matter in the regions \(x>x_0\) can be taken into account by introducing a negative electron source

\[ Q(x_0,s_j,\varepsilon)=I(x,\varepsilon,s_j)\big|_{x=x_0}\quad \text{for } \varepsilon<0,\ s_j>x_0. \tag{11} \]

The boundary in the plane \(x=0\) is taken into account analogously, if each electron crosses the plane \(x=0\) from the region \(x<0\) no more than once:

\[ Q(0,s_j,\varepsilon)=I(x,s_j,\varepsilon)\big|_{x=0}\quad \text{for } \varepsilon>0,\ s_j\ne 0. \tag{12} \]

Then the distribution function for the plane-parallel plate \(I_{\parallel}(x,s,\varepsilon)\) will be expressed in terms of the distribution functions for an infinite absorber, calculated respectively for the sources (10), (11), and (12), \(I(x,s,\varepsilon)\), \(I^{-}(x,s,\varepsilon)\), and \(I^{+}(x,s,\varepsilon)\):

\[ I_{\parallel}(x,s,\varepsilon)=I(x,s,\varepsilon)-I^{-}(x,s,\varepsilon)-I^{+}(x,s,\varepsilon). \tag{13} \]

The functions \(I^{-}(x,s,\varepsilon)\) and \(I^{+}(x,s,\varepsilon)\) are calculated similarly to \(I(x,s,\varepsilon)\), but with the initial conditions (11) and (12).

Let \(I_j^{-}(x,s,\varepsilon)\) denote the electron distribution function corresponding to the source (10) of electrons with residual path \(s_j\). The full distribution function \(I^{-}(x,s,\varepsilon)\) will be represented as the superposition of \(I_j^{-}(x,s,\varepsilon)\):

\[ I^{-}(x,s,\varepsilon)=\sum_{j=j(x_0)}^{j_m} I_j^{-}(x,s,\varepsilon), \tag{14} \]

where \(j_m\) and \(j(x_0)\) are determined by (7) for \(x=x_0\).

The initial condition for \(I_j^{-}(x,s,\varepsilon)\) has the form (10):

\[ I_j^{-}(x,s,\varepsilon)\big|_{s=s_j}=\delta(x-x_0)I(x,s_j,\varepsilon)H(\varepsilon), \tag{15} \]

where the piecewise step function is

\[ H(\varepsilon)= \begin{cases} 1, & \varepsilon<0,\\ 0, & \varepsilon>0. \end{cases} \]

Since

\[ I(x,s_j,\varepsilon)= \sum_{L'} \frac{2L'+1}{4\pi}P_{L'}(\varepsilon) \left\{ \frac{a_{0L'}(s_j)}{2} + \sum_{k'=1} a_{k'L'}(s_j)\cos k'\pi x+b_{k'L'}(s_j)\sin k'\pi x \right\}, \]

then

\[ \big[I_j^{-}(x,s_j,\varepsilon)\big]_{L} = 2\pi\int_{-1}^{1} I_j^{-}(x,s_j,\varepsilon)P_L(\varepsilon)\,d\varepsilon = \]

\[ = -\delta(x-x_0)\sum_{L'}\frac{2L'+1}{2} \left\{ \frac{a_{0L'}(s_j)}{2} +\sum_{\kappa'=1} a_{\kappa'L'}(s_j)\cos \kappa'\pi x + b_{\kappa'L'}(s_j)\sin \kappa'\pi x \right\}\gamma_{LL'}, \tag{16} \]

where

\[ \gamma_{LL'}=\int_{-1}^{0} P_L(\varepsilon)P_{L'}(\varepsilon)\,d\varepsilon = \frac{1}{2L+1},\quad L'=L; \]

\[ =0,\ (L-L')\text{ even},\quad L\ne L'; \]

\[ = \frac{(-1)^{\frac12(3L+3L'-1)}L!\,L'!} {2^{L+L'-1}(L-L')(L+L'-1) \left[\left(\frac{L}{2}\right)!\left(\frac{L'-1}{2}\right)!\right]^2}, \]

\(L\) even, \(L'\) odd.

Then

\[ [a_j^{-}(s_j)]_{\kappa L} = \cos \kappa\pi x_0 \sum_{L'}\frac{2L'+1}{2}\gamma_{LL'} \left\{ \frac{a_{0L'}(s_j)}{2} + \sum_{\kappa'=1} a_{\kappa'L'}(s_j)\cos \kappa'\pi x_0 + b_{\kappa'L'}(s_j)\sin \kappa'\pi x_0 \right\}, \tag{17} \]

\[ [b_j^{-}(s_j)]_{\kappa L} = \sin \kappa\pi x_0 \sum_{L'}\frac{2L'+1}{2}\gamma_{LL'} \left\{ \frac{a_{0L'}(s_j)}{2} + \sum_{\kappa'=1} a_{\kappa'L'}(s_j)\cos \kappa'\pi x_0 + b_{\kappa'L'}\sin \kappa'\pi x_0 \right\}. \]

Further, on the basis of (17) and the recurrence relations obtained in [2], one can calculate \([a_j^{-}(s_j)]_{\kappa L}\) and \([b_j^{-}(s_j)]_{\kappa L}\) for all \(i>j\) and, consequently, determine \(I^{-}(x,s,\varepsilon)\) and then compute \(I^{-}(x,s,\varepsilon)\) from (14). The function \(I^{+}(x,s,\varepsilon)\) is calculated analogously.

Thus, under the assumption that each electron passes into the region \(0<x<x_0\) from the regions \(x<0\) and \(x>x_0\) no more than once, it is possible to calculate the distribution function and, consequently, also \(D(x)\) for a plane-parallel plate occupying the space from \(x=0\) to \(x=x_0\).

To account for the boundary by the proposed method, a calculation program for an electronic computer is currently being developed.

REFERENCES

1. L. V. Spencer. Phys. Rev., 98, 1597 (1955).

2. A. A. Vorob'ev, O. B. Evdokimov, B. A. Kononov. A model of the penetration of fast electrons. In the collection Dosimetry of large doses, p. 212. Publishing House of the Academy of Sciences of the Uzbek SSR, Tashkent, 1966.

3. A. A. Vorob'ev, O. B. Evdokimov, B. A. Kononov. On the study of the transport of fast electrons, Proceedings of the Tomsk Polytechnic Institute, vol. 143, p. 70. Tomsk University Press, 1966.