PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC
INSTITUTE named after S. M. KIROV

Vol. 143                  1966

ON THE STUDY OF FAST ELECTRON TRANSPORT

A. A. VOROB’EV, O. B. EVDOKIMOV, B. A. KONONOV

(Submitted by the scientific seminar of Department No. 12).

The elementary processes of the interaction of electrons with matter are at present known fairly well. Meanwhile, for practical purposes, the passage of electrons through thicknesses of matter considerably exceeding interatomic distances is of interest. A number of works [1, 2, 6, 8, etc.] are currently devoted to the theory of fast electron transport. Among them, the most important results were obtained by Spencer [1] in calculating the distribution of energy lost by electrons in an infinite medium, and in the authors’ work [8], devoted to solving the basic problem of electron transport from a plane source in an infinite medium.

The present work is devoted to the exposition of an approximate method (the effective-broken-line model) for calculating the energy distribution functions of electrons at large distances from a plane source in an infinite medium. In the second part of the work, the design of an apparatus and the results of an experimental investigation of the distributions of electrons stopped in aluminum and copper are reported.

I. Model of effective broken lines

Energy distribution function of electrons

Electrons are scattered mainly through small angles. This makes it possible to represent the electron trajectory in the form of a certain broken line, the points of inflection of which characterize the fact of a substantial change in the direction of electron motion due to a large number of small-angle scatterings and rare collisions with deflection through large angles. To simplify the calculation, we shall assume that, despite the scatter in the removal of electrons from a given direction, the points of inflection are located, for all electrons, at quite definite values of the path traversed, $s_i$. The length of the segments of the broken line cannot be too small, since, in order to simplify the problem, we assume that the number of collisions within a section of the trajectory must ensure independence of the direction of motion of the electrons in the $i$-th section from the direction in the $(i-1)$-st section of the trajectory. In view of the indicated assumptions, an exact determination of the magnitudes of the segments appears difficult. Analysis shows, however, that reasonable criteria can be found for determining the lengths of the sections. These include the constancy of the root-mean-square angle, in the general case of the mean magni-

quantities \(\overline{\langle 1-\cos\Theta\rangle_i}\), or constancy of the number of collisions within the limits of the trajectory segment \(\bar{\nu}_i\). Mathematically this means:

\[ \overline{\langle 1-\cos\Theta\rangle_i}\equiv \int_{\bar{s}_{i-1}}^{\bar{s}_i} ds\int RN\sigma(\Theta,s)(1-\cos\Theta)\,d\Omega = \overline{\langle 1-\cos\Theta\rangle_c}. \tag{1} \]

\[ \bar{\nu}_i=\int_{\bar{s}_{i-1}}^{\bar{s}_i}\nu(T(s))\,ds=\nu_c \tag{2} \]

where \(\sigma(s,\Theta)\) is the scattering cross section on an atom;
\(N\) is the number of atoms per unit volume;
\(R\) is the total range of the source electrons;
\(\nu\) is the number of collisions per unit path length;
\(\bar{s}_i\) is the path traversed by the electron in units of the total range after passing through \(i\) segments.

In explicit form, integral (1) can be written with the aid of the approximate scattering function \(S_l\) for \(l=1\), introduced by Spencer [1]. In particular, for \(E\ll mc^2\), (1) gives

\[ \bar{\lambda}_i=(1-\bar{s}_{i-1}) \left[ 1-\exp\left( -\frac{\overline{\langle 1-\cos\Theta\rangle_c}}{d_l} \right) \right], \]

where \(d_l\) is defined in [1].

The angular distribution of particles on each segment of the trajectory can be determined from the theory of multiple scattering of electrons of Lewis [2]. If the angular distribution of electrons is described by the function \(\Phi(s,\Theta)\) [2], then the angular distribution of the trajectory segments is determined by the expression:

\[ \Phi_i(\cos\Theta)=\frac{1}{\lambda_i} \int_{\bar{s}_{i-1}}^{s_i}\Phi(s,\cos\Theta)\cdot ds, \tag{3} \]

where henceforth we shall denote by \(s_i\) the length of the electron trajectory, including part of the \(i\)-th segment; \(\bar{\lambda}_i\) is the total length of the segment; \(\bar{s}_i\) is the length of the electron path, including the \(i\)-th segment, so that

\[ \bar{s}_i=\sum_{k=1}^{i}\bar{\lambda}_k \quad\text{and}\quad s_i=\sum_{k=1}^{i-1}\bar{\lambda}_k+\lambda_i. \]

According to [2], the function \(\bar{\Phi}_i(\cos\Theta)\) can be written, taking (3) into account, as

\[ \Phi_i(\cos\Theta)=\sum_{l=0}\frac{2l+1}{4\pi}A_l^iP_l(\cos\Theta), \]

where, in calculating the functions \(A_l^i\), approximations for the scattering function \(S_l\) given in [1] and [3], and equation (3), are used.

For \(E\ll mc^2\) these coefficients have the form:

\[ A_l^i= \frac{\bar{t}_i^{\,d_l+1}-(\bar{t}_{i-1}-\lambda_i)^{d_l+1}} {\lambda_i(d_l+1)}, \qquad \text{where } t_l\equiv 1-s_i \]

for \(E\sim mc^2\) and \(\mu_l=d_l+i\)

\[ A_l^i= \frac{\alpha(1+\alpha)d_l}{\lambda_i} \left[ \sum_{n=0}^{*}\frac{n+1}{n+\mu_l} \left( x_{i-1}^{\,n+\mu_l}-x_i^{\,n+\mu_l} \right) \right], \]

where \(x_i=\dfrac{t_i}{\alpha+t_i}\), and \(\alpha\) is defined in [1].

For \(mc^2 \ll E < \dfrac{1}{4} E_{\text{crit}}\), the coefficients \(A_l^i\) are expressed in terms of the integral exponential function.

The considerations set forth above make it possible to write the following equation:

\[ I(s_i,x)=\int_{-1}^{1} I(\bar{s}_{i-1},\,x-\lambda_{iB}\cdot \varepsilon)\Phi_i'(\varepsilon)\,d\varepsilon, \tag{5} \]

where \(\varepsilon=\cos\Theta\); \(\Phi_i'=2\pi\Phi_i\); \(I(s_i,x)\) is the electron spectrum at depth \(x\); \(\lambda_{iB}\), the mean projection of the path length \(\lambda_i\) of the electrons onto the direction of their motion, can be calculated according to [6]:

\[ \lambda_{iB}=\lambda_i-\frac{1}{3}A^2(\lambda_i). \]

Direct application of the method of successive collisions to equation (5), after the first two collisions, leads to very cumbersome calculations.

In order to solve equation (5), let us expand the distribution \(I(s,x)\) in a Fourier series, taking into account that \(-1 \le x \le 1\):

\[ I(s,x)=\frac{a_0(s)}{2}+\sum_{k=1}^{\infty} a_k(s)\cos k\pi x+b_k(s)\sin k\pi x. \tag{6} \]

The expansion coefficients are related to \(I(s,x)\) in the usual way. Multiplying equation (4) by \(\cos k\pi x\) or by \(\sin k\pi x\), integrating with respect to \(x\), and taking into account that for \(x>s\) there are no particles having path length \(s\), and also that

\[ a_k(s)=\int_{-1}^{1} I(s,x)\cdot \cos k\pi x\cdot dx,\qquad k=0,1,2\ldots \]

\[ b_k(s)=\int_{-1}^{1} I(s,x)\cdot \sin k\pi x\cdot dx,\qquad k=1,2\ldots \tag{7} \]

we obtain the relation between the expansion coefficients of (5):

\[ a_k(\bar{s}_{i-1}+\lambda_i)=a_k(\bar{s}_{i-1})\cdot C_k^i-b_k(\bar{s}_{i-1})\cdot S_k^i \]

\[ b_k(\bar{s}_{i-1}+\lambda_i)=a_k(\bar{s}_{i-1})\cdot S_k^i+b_k(\bar{s}_{i-1})\cdot C_k^i, \tag{8} \]

where

\[ C_k^i=\int_{-1}^{1}\Phi_i'(\varepsilon)\cos(k\pi\lambda_{iB}\varepsilon)\,d\varepsilon \]

\[ S_k^i=\int_{-1}^{1}\Phi_i'(\varepsilon)\sin(k\pi\lambda_{iB}\varepsilon)\,d\varepsilon. \tag{9} \]

In explicit form, the functions \(C_k^i\) and \(S_k^i\) are represented, taking (4) into account, as follows: for \(\mu\equiv k\pi\lambda_{iB}\),

\[ C_k^i=\sum_{l=0}\frac{2l+1}{2}A_l^i C_{kl}^i,\qquad S_k^i=\sum_{l=1}\frac{2l+1}{2}A_l^i S_{kl}^i \tag{10} \]

where

\[ C_{\kappa l}^{i}=\int_{-1}^{1} P_l(\varepsilon)\cos \nu \varepsilon \cdot d\varepsilon \qquad l=0,\ 2,\ 4,\ldots \]

\[ S_{\kappa l}^{i}=\int_{-1}^{1} P_l(\varepsilon)\sin \nu \varepsilon \cdot d\varepsilon \qquad l=1,\ 3,\ 5,\ldots \tag{11} \]

In the case of a plane unidirectional source, for which the angular distribution function [2] is valid, the initial conditions have the form: \(a_\kappa(0)=1\), \(\varepsilon^\kappa(0)=0\).

Knowledge of the functions \(C_\kappa^i\) and \(S_\kappa^i\) (10) and of the initial conditions makes it possible to compute successively \(a_\kappa^i\) and \(\varepsilon_\kappa^i\) for any \(s\) by means of the recurrence relations (8), and consequently also \(I(s,x)\) from (6). For small \(s\) it is reasonable to carry out the calculations by the method of successive collisions.

The accuracy of the calculation within the framework of the continuous-slowing-down model can be increased if one introduces the range-straggling approximation, according to which one takes the aggregate of distributions \(I(s,x)\), where \(x\) is now expressed in units not of mean but of total ranges, the distribution of the latter

\[ W\left(\frac{\bar R}{R}\right) \]

being given by Blunck’s theory [5] (here \(R\) is the actual range, and \(\bar R\) the mean range in g/cm\(^2\)). Then the transition to the range-straggling approximation is expressed as follows:

\[ I(s,\chi)=\int_{\chi}^{R_m} I(s,\chi/R)\,W\left(\frac{R}{\bar R}\right)\,dR, \tag{12} \]

where \(\chi\) is the depth in g/cm\(^2\).

A direct experimental check of the accuracy of the calculation of \(I(s,x)\) for an infinite medium is at present possible only for \(s=1\). This condition is satisfied by the work of Hørzovsky [4]. Figure 1 gives a comparison of the calculated electron absorption by the present method before and after refinement according to formula (12) (curves 1 and 2, respectively) with the experimental data of [4]. The calculation was carried out for \(\lambda_i\nu\), distinct from \(\lambda_i\) [7], with subdivision into intervals \(\lambda_i\) from condition (2), so that \(\lambda_1=0.55\).

The energy for which the calculation was carried out—1 MeV—is apparently the limiting one for light substances because of the relatively large value of \(\lambda_1\). For heavy substances, calculations may also be carried out at higher energies, especially with refinement according to formula (12).

The analysis shows that criterion (2) for determining the lengths of the intervals \(\lambda_i\) is not universal. The reason is that one cannot make the transition from an energy of 1 MeV, at which \(\lambda_1\) is known and, consequently, \(1/\nu_c\) can be determined for any energy, to small energies, because as the energy decreases the residual range decreases faster than \(\beta^2\), whereas \(\nu\sim\beta^{-2}\). The criterion of constancy of the transport cross section (1) proved to be more universal.

Figure 2 gives a comparison of the distribution of the energy lost by electrons calculated by the present method (curve 1) with Spencer’s calculations (curve 2) and the corresponding experiments cited in [7]. As can be seen, the functions \(I(s,x)\) for large \(s\) can

Fig. 1

Fig. 2

be calculated with acceptable accuracy. In the calculation it was assumed that \(\lambda_{iB}\approx\lambda_i\), while the quantities \(\lambda_i\) were determined from condition (1), where

\[ \overline{<1-\cos\Theta>_c}=1. \tag{13} \]

This same criterion proved suitable for a wide range of energies (for light substances up to 1 MeV) and for various elements at one and the same value of the transport cross section (13).

II. Experimental studies of the distributions of stopped electrons

In order to test the theory of electron transport [8], we undertook experiments to study the distribution of remaining electrons in aluminum and copper. This approach is simpler than comparison with the distribution of lost energy.

Fig. 3

A detector of stopped electrons was developed, built, and tested in this work; its design is shown in Fig. 3. The instrument is based on the thin-plate method [4, 9] and is a further development of it. The device registers the number of electrons stopped in collecting electrode 1, which is located in a brass housing and connected to a dc amplifier through a feedthrough. The dimensions of the collecting electrode overlap the dimensions of the electron beam, independently of the depth in the substance at which the collecting electrode is located. The depth of the latter is regulated by changing the thickness of the absorber in front of the front flange, made of aluminum, whose thickness is 400 microns at the point where the beam falls. The condition of an infinite medium was achieved by placing, after the collecting electrode, a back scatterer 2 made of the material under investigation, with dimensions exceeding the full range of electrons of the initial energy in the given substance. The distances from the collecting electrode to the front flange and to the back scatterer, as a rule, did not exceed a millimeter, thereby ensuring \(4\pi\)-geometry.

The working volume was pumped down to a pressure of the order of \(10^{-5}\) mm. Investigations showed that the operating pressure should be better than \(10^{-2}\) mm Hg; in this case the readings of the instrument do not depend on the pressure inside it.

The thickness of the collecting electrode was \(2.5 \div 5\%\) of the total mean electron range in the given substance. Calculations showed that in this case the error due to the finite thickness of the collecting electrode does not exceed \(1\%\).

Analysis shows that a source of errors in the instrument readings may be the difference in the number of secondary electrons incident on the collecting electrode and leaving it. In order to check for the presence of such an error, grids were placed between the collecting electrode, the front flange, and the backscatterer, to which a negative voltage of up to 10 kV was applied through lead-in 5. No appreciable effect was observed. This is evidently connected with the fact that, as is known, when small thicknesses are penetrated by high-energy electrons, the angular distribution for secondary electrons is close to isotropic.

Using the indicated instrument, distributions of stopped electrons in aluminum and copper were investigated for initial electron energies of \(3.2\), \(4.9\), and \(8.1\) MeV. In all cases they are asymmetric bell-shaped curves; the position of the maximum \(x_n\), the half-width \(\delta_{1/2}\), and the maximum depth \(x_m\), determined by extrapolating the rear front of the curve, expressed in units of the total mean electron range, depend on the initial energy of the particles and on the atomic number of the substance. The results are summarized in Table 1, where the extrapolated range is also indicated, obtained from the curve \(N(x)\), calculated from the distribution \(I(x)\) of stopped electrons by the formula:

\[ N(x)=\int_x^\infty I(x')\,dx' \]

analogously to how this was done in [4].

For an energy of 3 MeV, the results of Gross [9] are given.

Table 1

It is evident from the table that, with decreasing energy and increasing atomic number of the substance, the role of multiple scattering in electron transport increases.

In Figs. 4 and 5, the distributions of stopped electrons in aluminum and copper calculated according to the theory [8] are compared, at energies of 3 MeV and 8 MeV respectively, with the experimental results.

of the present work (open-square symbols) and Gross’s results (crosses). Curves 1 were calculated in the continuous-slowing-down approximation; curves 2 include the spread of total ranges. The comparison shows that already

Fig. 4

Plot labels: \(I(x,l)\), relative units; \(x\) in units of \(R_0\).
In-plot labels: Al; 3 MeV; curves 1 and 2.

Fig. 5

Plot labels: \(I(x,l)\), relative units; \(x\) in units of \(R_0\).
In-plot labels: Cu; 8 MeV; curves 1 and 2.

at energies of 3 MeV for aluminum the continuous-slowing-down approximation proves insufficient, whereas taking into account the spread of ranges ensures good accuracy of the theory.

References

1. Spencer L. V., Phys. Rev., 98, 1597 (1955).
2. Lewis H. W., Phys. Rev., 78, 526 (1950).
3. Adawi I., Phys. Rev., 107, 1476 (1957).
4. Horszowski S. M., J. Sci. Instr., 40, 532 (1963).
5. Blunck O., Z. Physik, 131, 354 (1952).
6. Spencer L. V., Coyne J., Phys. Rev., 128, 2230 (1962).
7. Huffman et al., Phys. Rev., 106, 435 (1957).
8. Vorob’ev A. A., Evdokimov O. B., Kononov B. A. Proceedings of the Second Coordination Conference on High-Dose Dosimetry, p. 93, Publishing House of the Academy of Sciences of the Uzbek SSR, Tashkent, 1966.
9. Gross B., Wright K. A., Phys. Rev., 114, 725 (1959).

Corrections and Noticed Misprints