Approximate Method for Calculating Fast Electron Transport in Dielectrics in the Presence of a Strong Electric Field
Abstract Generated abstract
An approximate method is proposed for calculating the spatial distribution of energy lost by fast electrons in a dielectric under a strong homogeneous electric field, with boundary effects neglected. The approach uses a segment model in the continuous slowing down approximation, Fourier and Legendre transforms of the particle flux equations, and an angular averaging of field dependent energy losses to reduce the transport problem. Calculations for monoenergetic, monodirectional electrons in nitrogen are compared with exact weak field results and with a nonscattering limiting case, indicating small errors in weak fields and acceptable behavior in very strong accelerating fields, while intermediate fields can produce larger averaging errors. The method is noted to be most applicable over short time intervals, before fluctuations in energy loss become significant.
Full Text
AN APPROXIMATE METHOD FOR CALCULATING FAST ELECTRON TRANSPORT IN DIELECTRICS IN THE PRESENCE OF A STRONG ELECTRIC FIELD
A. P. Yalovets
(Presented by the Joint Scientific Seminar of the Faculty of Physics and Technology and the Electron Defectoscopy Sector of the Scientific Research Institute of EI)
In a number of devices, in particular in the electroionization laser and gas switches, a beam of fast electrons is used to initiate volume charge in overvolted gaps [1], which makes it possible to produce a volume discharge at high gas pressures and thereby increase the power of gas switches and the specific generation of electroionization lasers. The electric field affects the passage of electrons through the gas; in turn, the ionization distribution affects the configuration of the field in the volume, the electrical strength, and thereby the operation of the corresponding devices. A favorable condition for the development of a volume discharge is ionization uniform over the volume, which can be obtained when the gap with a homogeneous electric field is irradiated from the cathode side.
In the present work an approximate method is proposed for calculating the distribution of the energy lost by electrons in a dielectric in the presence of a strong homogeneous electric field, without taking boundaries into account. It should be noted that in the case when energy losses to bremsstrahlung are small, the distributions of lost energy and absorbed energy coincide. In this case the ionization is determined by
\[ N(x)=\frac{d}{e}\frac{D(x)}{\varepsilon} \left[\frac{\text{pair}\cdot\text{ion}}{\text{cm}^{3}}\right], \]
where \(D(x)\) is the distribution of lost energy, \(\varepsilon\) is the specific ionization energy, and \(j\ \mathrm{a/cm^2}\) is the beam current density. The distribution of the energy lost by electrons to ionization was calculated by the formula
\[ D(x)=\int_{0}^{S_{\max}} ds \int_{-1}^{1} du\, I(x,s,u), \tag{1} \]
where \(I(x,s,u)\) is the distribution function of the particle density, related to the differential flux by the relation \(I(x,s,u)\,ds=\psi(x,T,u)\,dT\), \(u=\cos\Theta\), \(\Theta\) is the angle between the \(OX\) axis and the momentum, and \(s\) is the path traversed by the particle.
The equation for the trajectory flux was written according to the segment model [2] in the continuous-slowing-down approximation. This model makes it possible to divide the process of particle transport over a segment into two stages: transport of the particles over the segment and then their scattering at the end of the segment in accordance with the path traversed and with allowance for the focusing action of the electric field.
Thus, it is possible to write two equations:
\[ \tilde I(t,x,u,T)=I[t-\tau,x-v\tau u,u,T+(B(T)-eEu)v\tau], \tag{2} \]
\[ I(t,x,u,T)=\int_{4\pi} d\Omega' g(T,v\tau,u'u)\tilde I(t,x,u,T), \tag{3} \]
where \(t\) is time, \(\tau\) is the time step, \(v\) is the particle velocity, \(B(T)\) is the specific energy loss to ionization, and \(g(T,v\tau,u'u)\) is the probability that a particle with energy \(T\), after traversing a distance \(v\tau\), will be scattered from \(u'\) into \(u\). The initial conditions for equations (2) and (3) are
\[ I(0,x,u,T)=S(x,u,T). \tag{4} \]
For a monoenergetic, unidirectional, and infinitely plane source at the point \(x_0\):
\[ S(x,u,T)=\frac{1}{2\pi}\delta(x-x_0)\cdot \delta(1-u)\cdot \delta(T-T_0). \tag{5} \]
Let us carry out the Fourier transform for equations (2) and (3):
\[ \tilde I_k(t,u,T)=I_k[t-\tau,T+(B-eEu)v\tau,u]\cdot \exp\left(-i\frac{2\pi}{L_0}kv\tau u\right), \tag{6} \]
\[ I_k(t,u,T)=\int_{4\pi} d\Omega' g(v\tau,u'u)\tilde I_k(t,u',T). \tag{7} \]
For electrons moving in an accelerating electric field, the energy losses will depend on the direction of motion of the particle. If the electric-field strength is large, then a large fraction of the particles will be in the process of acceleration and will move in the direction of the field. Therefore, it becomes possible to take the dependence of the energy losses on the angle into account approximately and thereby make the medium isotropic.
\[ \tilde I_k(t,T,u)=I_k[t-\tau,T+(B-eE\bar u_k)v\tau,u]\cdot \exp\left(-i\frac{2\pi}{L_0}kv\tau u\right), \tag{8} \]
where
\[ \bar u_k= \int_{-1}^{1} u I_k(t,T,u)\cdot du \bigg/ \int_{-1}^{1} I_k(t,T,u)\cdot du. \tag{9} \]
The introduction of such isotropy makes it possible to relate the value of the energy uniquely to the step number and thus to eliminate one variable. Let us carry out the Legendre transform for (6) and (7):
\[ \tilde I_{ke}(t)=\sum_{e'}(2e'+1)I_{ke'}(t-\tau)\cdot E_{ee'}(-i\mu v\tau), \tag{10} \]
\[ I_{ke}(t)=\sum_{e'} P_{ee'}\tilde I_{ke'}(t). \tag{11} \]
\(P_{ee'}\) was found from the solution of equation [3] for the function \(g(T,v\tau,u'u)\) and has the form
\[ P_{ee'}=\sum_{m=0}^{\infty}\frac{\alpha_{ee'}^{(m)}}{m!} \tag{12} \]
The \(\alpha_{ee'}^{(m)}\) were calculated by the recurrence formula
\[ \alpha_{ee'}^{(m)} = \varepsilon_e v\tau[\alpha_{e+1,e'}^{(m-1)}-\alpha_{e-1,e'}^{(m-1)}] - S_e v\tau \alpha_{ee'}^{(m-1)}, \tag{13} \]
where
\[ \varepsilon_e=-\frac{eE}{pv}\frac{e(e+1)}{2e+1}, \]
\(p\) is the electron momentum,
\[ S_e=2\pi\int_{-1}^{1} dx\,\sum_s(x)[1-P_e(x)]; \qquad \alpha_{ee'}^{(0)}=\delta_{ee'}. \]
The method for calculating the functions \(E_{cc}(-i\mu v\tau)\) is given in [4]. Thus, by specifying the initial conditions for the flux, one can find the initial conditions for the transforms and then, from relations (10) and (11), find the transforms for any values of \(t\), and the same for any value of the path traversed \(s\).
On the basis of the formulas given above, a program was compiled with which calculations were carried out of the distributions of the energy lost by electrons in nitrogen for a monodirectional, monoenergetic beam. When formulas (10) and (11) were used, a transition was made to relations that included only real quantities [2].
Fig. 1. Distribution of the energy lost in nitrogen by electrons with initial energy \(0.1\) MeV:
\(1\)—\(eE/P=0\); \(2\)—\(eE/P=1.75\) keV/atm·cm (exact solution);
\(3\)—\(eE/P=1.75\) keV/atm·cm (approximate solution)
Figure 1 shows the distributions of the energy lost in nitrogen by electrons with initial energy \(0.1\) MeV without a field (\(1\)) and in the presence of a weak field (\(2, 3\)). Curve 2 was calculated exactly [5] for the value \(eE/P=1.75\) keV/atm·cm; curve 3 was calculated by the approximate method. It is seen from Fig. 1 that the difference occurs at small depths. This is due to the fact that the presence of averaging over energy losses leads to the particles flying in a retarding field receiving the same energy increment as in an accelerating one. Comparison of results (2) and (3) is possible because anisotropy in a weak field is manifested only slightly and therefore the introduction of averaging over energy losses does not lead to a large error. For field values close to the specific ionization losses, the error due to averaging becomes large. In very large fields, i.e., when \(eE \gg (2 \div 3)B(T)\) and higher, averaging over losses will work satisfactorily, since, despite the fact that we have made the medium isotropic, the number of particles not participating in acceleration is small.
The limiting case, when \(eE \gg B\), is shown in Fig. 2 (curve 1). Curve 2 was calculated for a nonscattering medium, i.e., it was obtained from the solution of the nonlinear equation
\[ \frac{dT(x)}{dx} eE - B[T(x)] . \tag{14} \]
In solving equation (14), \(P(x)=B[T(x)]\).
Fig. 2. Distribution of the energy lost in nitrogen by electrons with initial energy 0.1 MeV: 1 — \(eE/P = 30\) kV/atm·cm; 2 — \(eE/P = 30\) kV/atm·cm (non-scattering medium)
From a comparison of curves 1 and 2 one can judge the focusing action of the field.
Figure 3 shows the distribution of the energy lost at a field strength \(eE/p = 15\) kV/atm·cm.
Fig. 3. Distribution of the energy lost in nitrogen.
\(T_0 = 0.1\) MeV, \(eE/p = 15\) kV/atm·cm
It should be noted that, since the results were obtained in the approximation of continuous slowing down, they will be valid when considering the process over short time intervals. In long accelerating intervals, fluctuations in energy losses begin to play a substantial role.
In conclusion, the author expresses gratitude to O. B. Evdokimov for assistance in the work.
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