SOME GENERAL ISSUES IN THE TRANSPORT OF FAST ELECTRONS

A. A. VOROB'EV, O. B. EVDOKIMOV

Scientific and applied problems of radiation physics, space technology, electron therapy, dosimetry, defectoscopy by means of electron beams, and secondary electron emission inevitably encounter the problem of the penetration of fast electrons. In this connection, questions of the theory of transport of these particles in various media and materials acquire great importance. The task set for the theory is to calculate the distribution function of electrons in space, in energy, and in directions of motion. The function is determined from the corresponding kinetic equation under known initial and boundary conditions, or by the Monte Carlo method. The solution of the general Boltzmann kinetic equation for fast electrons is associated with great mathematical difficulties. It is therefore necessary to introduce various kinds of approximations, which leads to the need to write down and solve approximate transport equations in accordance with the nature of the problem. The connection of these equations with the exact Boltzmann equation is usually not discussed, which makes it difficult to estimate the limits of applicability of one approximation or another. In the present article the connection is considered between the kinetic equations used in solving particular problems in the theory of fast-electron transport, with account of the adopted approximations, and the integro-differential Boltzmann equation. The question of the connection with the integral Boltzmann equation will be the subject of another article.

1. Integro-differential equations of electron transport

a. General equation. The transport of fast particles moving with velocity \(v\) can be described by means of the following Boltzmann equation [1]

\[ \frac{\partial N(\bar r, v\Omega, t)}{\partial t} + v\Omega \nabla N(r, v\Omega, t) + v\Sigma(v)N(r, v\Omega,t) = \]

\[ = \int_{v}^{v_m} \Sigma(v')\,v'\,dv'\,c(v') \int_{4\pi} d\Omega'\, f(v'\Omega' \to v\Omega)\, N(r, v\Omega, t) + S(\bar r, v\Omega, t), \tag{1} \]

where the following notation has been used:

\(N(r, v\Omega, t)\) — the distribution function of particles with respect to coordinates \(r\), velocities \(v\), direction of motion \(\Omega\), at the instant \(t\),

\(\Sigma(v)\) is the interaction cross section,

\(f(v'\Omega' \to v\Omega)\) is the probability density that a particle moving with velocity \(v'\Omega'\), after a collision, will have velocity \(v\Omega\).

\(c(v)\) is the mean number of secondary particles of the same kind, produced by a particle with velocity \(v\) in one collision. In equation (1) it is assumed that the interaction cross section does not depend on time, and that there are no delayed particles, while the nonstationarity is due to the source \(S(r, v\Omega, t)\). For such a case, in the theory of electron transport it is convenient to use, instead of time, the length \(R\) of the path traversed by the particle, related to the time and velocity of each particle by the relations:

\[ R=\int_0^t v(t')\,dt', \qquad dR=v(t)\,dt. \tag{2} \]

If the particle appeared at the instant \(t_0\), then one must take \(v(t)=0\) for \(t<t_0\).

We transform equation (1), taking into account the features inherent in electrons; namely, since the velocity of relativistic electrons must be determined with high accuracy, it is convenient to pass from velocities to kinetic energies \(T\) and introduce the function

\[ vN(r, v\Omega, t)=P(r, T, \Omega, R). \tag{3} \]

Then instead of (1) we have:

\[ \frac{\partial P(r,T,\Omega,R)}{\partial R} +\Omega \nabla P(r,T,\Omega,R) +\Sigma(T)P(r,T,\Omega,R)= \]

\[ =\int^{T_0}_{[[unclear: lower limit]]}\Sigma(T')c(T')\,dT' \int_{4\pi} d\Omega'\, q(T',\Omega'\to T,\Omega)\, P(r,T,\Omega,R) +S,(r,T,\Omega)\delta(R). \tag{4} \]

where

\[ dv'\, f(v'\Omega'\to v\Omega) = dT'\, q(T',\Omega'\to T,\Omega). \]

\(\delta(R)\) is the Dirac delta function, taking into account the fact that the source electrons have zero range.

Equation (4) is the most general equation describing the transport of electrons in an isotropic medium in the stationary case without the production of \(\gamma\)-quanta, and giving the distribution of electrons with allowance for their ranges and for the production of secondary electrons. To solve it, it is necessary to know the boundary conditions and the form of the kernel

\[ \Sigma(T')c(T')q(T',\Omega'\to T,\Omega). \]

An exact expression for this function has not been obtained, since an electron moving in a medium interacts simultaneously with a large number of electrons and nuclei. Therefore the question of an exact solution of the problem of electron penetration cannot be posed.

In practice, the absence of an exact expression for the kernel under consideration is not a substantial difficulty in the problem of calculating electron transport. The point is that an electron moving in matter loses energy almost exclusively through interaction with atomic electrons, practically without being deflected. On the other hand, electron deflection occurs due to elastic collisions with nuclei, with energy losses practically equal to zero. These circumstances make it possible

regard as mutually independent the acts associated with energy losses (inelastic collisions that do not lead to deflection of electrons) and collisions with deflections (elastic collisions with nuclei without energy losses). Then, taking into account that the production of secondary electrons in elastic collisions with nuclei is excluded, we have:

\[ c(T')\Sigma(T')q(T',\Omega'\to T,\Omega)\to c(T')\Sigma_i(T')q_i(T'\to T)\delta(\Omega'-\Omega)+ \Sigma_s(T)\cdot q_s(T,\Omega'\to\Omega)\delta(T'-T), \]

where

\[ \begin{aligned} \Sigma_s(T)&\text{ is the cross section of elastic collisions,}\\ \Sigma_i(T)&\text{ is the cross section of inelastic collisions,}\\ q_s(T'\to T)\ \text{and}\ q_i(T,\Omega\to\Omega)&\text{ are the corresponding probability densities.} \end{aligned} \]

Then equation (1) becomes equation (6):

\[ \frac{\partial P(r,T,\Omega,R)}{\partial R} +\Omega\nabla P(r,T,\Omega,R)+\Sigma(T)P(r,T,\Omega,R)= \]

\[ =S(r,T,\Omega)\delta(R)+ \int_T^{T_0} dT'\,c(T')\Sigma_i(T')q_i(T'\to T)P(r,T',\Omega,R)+ \]

\[ +\int_{4\pi} d\Omega'\,\Sigma_s(T)q_s(T,\Omega'\to\Omega)P(r,T,\Omega',R), \]

where

\[ \Sigma(T)=\Sigma_i(T)+\Sigma_s(T). \tag{7} \]

As will be seen from what follows, equation (6) may be taken as the basis of the theory of fast-electron transport if one neglects electron multiplication due to electron–photon cascades and assumes the medium to be isotropic. From it follow all the equations used for solving particular and approximate problems of electron transport.

b. Equation for the distribution function of electrons over angles and energies along the range. If, for example, an electron moves across a thin foil, its range in it is close to its thickness. In this case the problem of determining the distribution function is greatly simplified, since the spatial variable \(r\) can be eliminated by introducing the function

\[ \Phi(T,\Omega,R)=\int_{\infty}dr\,P(r,T,\Omega,R). \tag{8} \]

The equation for \(\Phi\) is identical with equation (6), except for the term with the gradient, since \(P(r,T,\Omega,R)\to0\) as \(r\to\infty\), and is obtained from (6) by integration over all space. We shall denote the equation for \(\Phi\) by (9). The function \(\Phi\) can be measured with the aid of track chambers.

c. Equation for the electron spectrum as a function of the path traversed. In [2] and elsewhere, the problem is considered of determining the energy spectrum of electrons after passage through a foil. If the foil is thin, then the path traversed is approximately equal to the foil thickness, and then the electron spectrum is determined by the function

\[ W(T,R)=\int_{4\pi}\Phi(T,\Omega,R)d\Omega. \tag{10} \]

More precisely, this function characterizes the distribution of electrons by energies as a function of the path traversed and can be measured

by means of track detectors. The equation for \(W(T,R)\) is obtained from (9) by integration over \(\Omega\):

\[ \frac{\partial W(T,R)}{\partial R}+\Sigma(T)W(T,R) = \int_T^{T_0} dT'\, c(T') q_i(T'\to T) W(R,T') \Sigma_i(T') \]
\[ \qquad +\int_{4\pi} \rho\Omega \int_{4\pi} d\Omega'\,\Sigma_s(T)q_s(T,\Omega'\to\Omega)\Phi(T,\Omega',R) +S(T)\delta(R) \tag{11} \]

The double integral in (11) is readily transformed if one takes into account that, for an isotropic medium,
\(q_s(\Omega'\to\Omega)=q_s(\Omega'\cdot\Omega)=q_s(\cos\Theta)\), where \(\Theta\) is the angle between the vectors \(\Omega'\) and \(\Omega\),

\[ 2\pi\int_{-1}^{1} d\cos\Theta\, q_s(T,\cos\Theta)=1. \tag{12} \]

Then the double integral in (11) is equal to

\[ \Sigma_s(T)W(R,T) \]

and, consequently, from (11), taking (7) into account, we obtain:

\[ \partial W/\partial R+\Sigma_i W = \int_T^{T_0} dT'\, c(T')\Sigma_i(T')q_i(T'\to T)W(R,T') +S(T)\delta(R). \tag{13} \]

This is the equation for \(W(R,T)\).

c. The equation in the problem of electron energy dissipation is solved in [3] and can be obtained from equation (13) by integration with respect to \(R\) from 0 to \(\infty\), if one takes into account that \(W(R,T)=0\) at \(R=0,\ R=\infty\):

\[ y(T)\Sigma_i(T) = \int_T^{T_0} dT'\, c(T')\Sigma_i(T')q_i(T'\to T)y(T') +S(T), \tag{14} \]

where it is denoted

\[ y(T)=\int_0^{\infty} W(R,T)\,dR. \]

d. The equation for the function of the energy and angular distribution of electrons in space. Usually, in particle-transport problems, what is of interest is the distribution of particles in energy and angle as a function of position in space, without regard to their path length, i.e., the function:

\[ n(r,T,\Omega)=\int_0^{\infty} P(r,T,\Omega,R)\,dR. \]

The equation for \(n(r,T,\Omega)\) is obtained by integrating equation (4) with respect to \(R\) from 0 to \(\infty\):

\[ \Omega\nabla n(r,T,\Omega)+\Sigma(T)n(r,T,\Omega) = S(r,T,\Omega)+ \tag{15} \]

\[ +\int_T^{T_0} dt'\,\Sigma_i(T')c(T')n(r,T,\Omega)q_i(T'\to T)+ \]

\[ +\int_{4\pi} d\Omega'\,\Sigma_s(T)q_s(T,\Omega'\to\Omega)n(r,T,\Omega). \]

Solving equation (15) presents great difficulties, and evidently for this reason it, like equation (4), is not encountered in the literature.

The difficulties can to a certain extent be avoided if the problem is considered within the framework of the continuous-slowing-down approximation, which is based on the assumption that the energy lost in a collision is small in comparison with the electron’s energy itself. Then equation (15) can be transformed, if one restricts oneself to

\[ n(r,T',\Omega)=n(r,T+\tau,\Omega)\simeq n(r,T,\Omega)+\tau\,\frac{\partial n(r,T,\Omega)}{\partial T}. \tag{16} \]

Then the first integral on the right-hand side of equation (15) is approximately equal to

\[ n(r,T,\Omega)\Sigma_t(T)+\frac{\partial n(r,T,\Omega)}{\partial T}\cdot \zeta(T), \tag{17} \]

\[ n(r,T,\Omega)\Omega_i(T)+\frac{\partial n(r,T,\Omega)}{\partial T}\cdot \zeta(T), \tag{18} \]

where \(\zeta(T)=-\,dT/dR\) is the specific energy loss (18) (we do not consider electron multiplication, i.e., \(c(T)=1\)). Equation (15) in the continuous-slowing-down approximation takes the form:

\[ \frac{dT}{dR}\cdot \frac{\partial n(r,T,\Omega)}{\partial T} +\Omega \nabla n(r,T,\Omega)+\Sigma_s(T)n(r,T,\Omega)= \]

\[ =\int_{4\pi} d\Omega'\, q_s(T,\Omega'\to\Omega)\Sigma_s'(T)n(r,T,\Omega')+S(r,T,\Omega). \tag{19} \]

Since in the continuous-loss approximation there is a one-to-one relation between the electron energy and its range, instead of the energy one introduces into the equation the residual range \(R_0\) and passes to the new function [4] \(I(r,R_0,\Omega)=n(r,T,\Omega)\)

\[ -\frac{\partial I(r,R_0,\Omega)}{\partial R_0} +\Omega\nabla I(r,R_0,\Omega)+\Sigma_s\cdot I(r,R_0,\Omega)= \]

\[ =\int_{4\pi} d\Omega'\, q_s(R_0,\Omega'\to\Omega)\Sigma_s\cdot I(r,R_0,\Omega')+S(r,R_0,\Omega). \tag{20} \]

e. Partial problems in the continuous-slowing-down approximation. In [4] and in a number of other works, the angular distribution of particles is considered as a function of the residual range, i.e., the function

\[ g(R_0,\Omega)=\int dr\, I(r,R_0,\Omega). \]

The equation for \(g\) is obtained from (20) by integration over \(r\) and externally coincides with it, except for the term with the gradient, which becomes zero after integration. In calculating multiple scattering in a foil, one can usually neglect the dependence of \(q_s\) and \(\Sigma_s\) on the energy and, consequently, on \(R_0\). Therefore the equation for the angular distribution function \(U(r,\Omega)\) of electrons after passage through the foil can be obtained by integrating (20) with respect to \(R_0\) from \(0\) to \(\infty\):

\[ \Omega\nabla U(r,\Omega)+\Sigma_s\cdot U(r,\Omega) =\Sigma_s\int_{4\pi} d\Omega'\, q_s(\Omega'\cdot\Omega)\,U(r,\Omega')+S. \tag{21} \]

where

\[ U(r,\Omega)=\int_0^\infty I(r,R_0,\Omega)\,dR_0. \]

i.e., we have obtained the equation usually used for calculating multiple scattering of electrons in foils.

Thus, it has been shown above that all the integrodifferential equations used in calculations of the transport of fast electrons follow from the Boltzmann equation; moreover, more complete equations have been derived, and it has been shown how the equation in the continuous-slowing-down approximation is obtained in a natural way.

REFERENCES

1. B. Davison. Theory of Neutron Transport. Atomizdat, Moscow, 1960.
2. L. D. Landau. J. Phys., USSR, 8, 204, 1944.
3. Spencer L. V., Fano U. Phys. Rev., 93, 1954.
4. Lewis H. W. Phys. Rev., 78, 526, 1950.