METHOD FOR CALCULATING FAST ELECTRON TRANSPORT IN DIELECTRICS IN A WEAK ELECTRIC FIELD

O. B. Evdokimov, V. I. Veretel’nik

(Presented by the scientific seminar of the Research Institute of Electronic Introscopy)

In [1], on the basis of the segment model [2], in the approximation of continuous slowing down, a method was developed for calculating the transport of fast electrons in a plate of material with a nonuniform electric field. The kinetic equation of the form [1], with allowance for boundary conditions, is transformed into a system of recurrence relations for the angular moments of the harmonic coefficients of the distribution function. The shortcomings of the work are the complicated special functions arising in the course of the solution, as well as the need, for each electric-field strength, to carry out a separate calculation, which can be avoided in the case of weak fields.

In the present work a variant is presented for solving the kinetic equation on the basis of the segment model for the case of weak electric fields.

The balance equation for the trajectory differential flux \(I(x,T,u)\) in the one-dimensional case is written in the form

\[ I^{j+1}(x,u_{j+1})=\int_{4\pi} I^j(x-\lambda u_j,\; T-\Delta T,\; u_j)\, g_E(\lambda,\; u_j\to u_{j+1})\,du_j, \tag{1} \]

where the notation is the same as in [2], \(\Delta T>0\).

In the case of a weak electric field, the expression for the segment length \(\lambda\) may be written

\[ \lambda=\lambda_0\left(1+\frac{eEu}{B}\right), \]

where \(\lambda_0=\Delta T\cdot B^{-1}\).

The function describing the differential flux will be expanded in a Taylor series at the point \(x-\lambda_0 u_j\).

We represent the trajectory differential flux in both parts of equation (1) and \(g_E(\lambda,u_j\to u_{j+1})\) in the form of an expansion in Legendre polynomials:

\[ I^j(x,u_j)=\sum_l \frac{2l+1}{4\pi}\, I_l^j(x)P_l(u_j), \tag{3} \]

\[ g_E(\lambda,u_j\to u_{j+1})=\sum_l \frac{2l+1}{4\pi}\, g_{El}(\lambda,u_j)P_l(u_{j+1}). \tag{4} \]

The angular moments \(I_l^j(x)\) are expanded on \(-L\le x\le L\) in a Fourier series

\[ I_l^j(x)=\sum_k I_{lk}^j \exp(ik\pi x L^{-1}). \tag{5} \]

The solution for the moments \(g_{El}(\lambda,u_j)\) is given in [3]. Using—

In the case of a weak field, let us expand \(g_l^0\) in a Taylor series in the small parameter \(eEB^{-1}\). Transforming the expressions

\[ \int_{-1}^{1} u_j e^{-\mu u_j} P_l(u_j)P_{l'}(u_j)\,du_j; \]

\[ \int_{-1}^{1} u_j^2 e^{-\mu u_j} P_l(u_j)P_{l'}(u_j)\,du_j, \]

using the recurrence formula for Legendre polynomials

\[ uP_l(u)=\frac{l+1}{2l+1}P_{l+1}(u)+\frac{l}{2l+1}P_{l-1}(u) \]

and denoting

\[ \mu=ik\pi\lambda_0L^{-1};\qquad E_{l,l'}(-\mu)=\frac{1}{2}\int_{-1}^{1} e^{-\mu u_j}P_{l'}(u_j)P_l(u_j)\,du_j; \]

\[ C_{l+1}=\frac{\exp(-S_l\lambda_0)-\exp(S_{l+1}\lambda_0)}{S_{l+1}-S_l};\qquad (2l+1)\exp(-S_l\lambda_0)=G_{l0}; \]

\[ C_{l-1}=\frac{\exp(-S_l\lambda_0)-\exp(S_{l-1}\lambda_0)}{S_{l-1}-S_l};\qquad (2l+1)I_{kl}=I_{kl}, \]

we write equation (1) in the form

\[ \begin{aligned} I_{kl}^{j+1}={}& \sum_{l'} I_{kl}^{j} \left\{ G_{l0}E_{l,l'}(-\mu)+\frac{eE}{B}S_l\lambda_0G_{l0} \right\} \\ &\times \left[ \frac{l+1}{2l+1}E_{l+1,l'}(-\mu) + \frac{l}{2l+1}E_{l-1,l'}(-\mu) \right] \\ &-\frac{eE}{PV}G_{l0}l(l+1) \left[ C_{l+1}E_{l+1,l'}(-\mu)-C_{l-1}E_{l-1,l'}(-\mu) \right] \\ &-\frac{eE}{B}\lambda_0\sum_{l'}I_{kl}^{j} \times G_{l0} \left[ \frac{(l+1)(l+2)}{(2l+1)(2l+3)}E_{l+2,l'}(-\mu) \right. \\ &\qquad\left. +\frac{(l+3)^2}{(2l+1)(2l+3)}E_{l,l'}(-\mu) + \frac{l^2}{(2l)^2-1}E_{l,l'}(-\mu) + \frac{l(l-1)}{(2l)^2-1}E_{l-2,l'}(-\mu) \right]. \end{aligned} \tag{6} \]

For the numerical solution of the problem it is convenient to separate the real and imaginary parts of expression (6), as was done in [4]. Then, taking into account a source with a continuous spectrum, we obtain:

\[ \chi_l^{j+1} = G_{l0}^{j} \left( \sum_{l'} \overset{0}{E}_{l,l'}\chi_{l'}^{j} + \gamma \varepsilon_{l,l'}^{j} z_{l'}^{j} \right) + Q_{\chi l}^{j+1}; \tag{7} \]

\[ z_l^{j+1} = G_{l0}^{j} \left( \sum_{l'} \overset{0}{E}_{l,l'}z_{l'}^{j} - \gamma \varepsilon_{l,l'}^{j}\chi_{l'}^{j} \right) + Q_{zl}^{j+1}; \qquad \gamma=eE; \]

\[ \chi_{kl}^{j}=i^l I_{kl}^{j}+(-i)^l I_{kl}^{j}; \qquad z_{kl}^{j}=i^l I_{kl}^{j}-(-i)^l I_{kl}^{j}; \]

\[ \begin{aligned} \varepsilon_{l,l'}^{j}={}& \left[ \frac{l(l+1)}{PV} \sum_{l'} \left\{ \overset{0}{E}_{l+1,l'}(\mu')C_{l+1} + \overset{0}{E}_{l-1,l'}(\mu')C_{l-1} \right\} \right. \\ &- S_l\lambda_0B^{-1} \sum_{l'} \left\{ \frac{l}{2l+1}\overset{0}{E}_{l-1,l'}(\mu') - \frac{l+1}{2l+1}\overset{0}{E}_{l+1,l'}(\mu') \right\} \\ &+ \lambda_0B^{-1} \sum_{l'} \left\{ \frac{(l+1)(2l+3)}{(2l+1)(2l+3)} \overset{0}{E}_{l+2,l'}(\mu') + \frac{(l+1)}{(2l+1)(2l+3)} \overset{0}{E}_{l,l'}(\mu') \right. \\ &\qquad\left. + \frac{l^2}{(2l)^2-1}\overset{0}{E}_{l,l'}(\mu') + \frac{l(l-1)}{(2l)^2-1}\overset{0}{E}_{l-2,l'}(\mu') \right\} \Bigg]; \end{aligned} \]

\[ Q\text{ is the source;}\qquad \mu'=k\pi\lambda_0L^{-1}. \]

In the case of a weak electric field, \(\chi_l^{j+1}\), \(\varkappa_l^{j+1}\) can be represented in the following form:

\[ \begin{aligned} \chi_l^{j+1} &= \overset{0}{\chi}{}_l^{\,j+1} + \gamma \Delta_\chi^{j+1},\\ \varkappa_l^{j+1} &= \overset{0}{\varkappa}{}_l^{\,j+1} - \gamma \Delta_x^{j+1}. \end{aligned} \tag{8} \]

Now let us obtain recurrence relations for determining \(\overset{0}{\varkappa}\), \(\overset{0}{\chi}\), \(\Delta_\chi\), \(\Delta_x\). For this purpose, in (8) we substitute the expressions for \(\chi_{kl}^{j}\), \(\varkappa_{kl}^{j}\). We obtain a system of recurrence relations:

\[ \begin{aligned} \overset{0}{\chi}{}_{l}^{\,j+1} &= G_{l0}^{j}\sum_{l'} \overset{0}{E}_{l,l'}\,\overset{0}{\chi}{}_{l'}^{\,j} + Q_{l\chi}^{j+1}; \qquad \overset{0}{\varkappa}{}_{l}^{\,j+1} = G_{l0}^{j}\sum_{l'} \overset{0}{E}_{l,l'}\,\overset{0}{\varkappa}{}_{l'}^{\,j} + Q_{lx}^{j+1}; \\[6pt] \Delta_{\chi}^{j+1} &\equiv V_{l}^{j+1} = Q_{l0}^{j}\sum_{l'}\left[\overset{0}{E}_{l,l'}V_{l'}^{j} - e_{l,l'}^{j}\,\varkappa_{l'}^{j}\right]; \\[6pt] \Delta_{x}^{j+1} &\equiv W_{l}^{j+1} = G_{l0}^{j}\left[\overset{0}{E}_{l,l'}W_{l'}^{j} - e_{l,l'}^{j}\chi_{l'}^{j}\right]. \end{aligned} \tag{9} \]

Thus, we have obtained the system of relations (9), the first two of which can be used to calculate the trajectory differential flux \(I_0^j(x,u)\) in the absence of an electric field, while the second two are used to compute the correction \(\Delta I^j(x,u)\), which is independent of the electric field. The general solution of equation (1) in the presence of a weak electric field can in this case be represented in the form

\[ I^j(x,u)=I_0^j(x,u)+eE\Delta I^j(x,u). \]

To calculate the function \(I^j(x,u)\), we use its representation by a Fourier series:

\[ I^j(x,u)=\frac{a_0^j}{2}+\sum_{\kappa} a_\kappa^j\cos \kappa\pi xL_0^{-1} +b_\kappa^j\sin \kappa\pi xL_0^{-1}. \tag{10} \]

Here the coefficients \(a_\kappa^j\) and \(b_\kappa^j\) are determined from the expansion in Legendre polynomials, and the moments \(a_{\kappa l}^{j}\) and \(b_{\kappa l}^{j}\) of the expansion are determined through

\[ \overset{0}{\varkappa}{}_{\kappa l}^{\,j},\quad \overset{0}{\chi}{}_{\kappa l}^{\,j},\quad V_{\kappa l}^{j},\quad W_{\kappa l}^{j}. \]

Having calculated, for example, the distribution of stopped electrons, it is easy to find the electron transmission curve by integrating expression (10) over the limits from \(x\) to \(l\) at the \(T\)-threshold and integrating over angles.

Figure 1 (solid curve) shows the calculated dependence of the quantity

\[ \delta=\frac{I(x)-I_0(x)}{I_0(x)}\,100\% \]

on the dielectric thickness, where \(I(x)\) and \(I_0(x)\) are the numbers of particles that have passed through the dielectric in the presence and absence of an electric field, respectively. The curve was calculated for an isotropic beam of beta particles, a continuous spectrum with limiting energy 765 keV (\(\mathrm{TL}^{204}\)), and a potential difference of 10 kV for a lavsan sample.

The points are experimental results measured for the same parameters of the beta-particle beam, the indicated material, and the potential difference.

Fig. 1. Dependence of the quantity \(\delta\) on the sample thickness \(x\) for an isotropic beam of beta particles from the isotope \(\mathrm{Tl}^{204}\) that have passed through lavsan absorbers at a potential difference of 10 kV; solid curve—calculation; points—experimental results.

REFERENCES

1. O. B. Evdokimov, A. P. Yalovets. Izvestiya vuzov SSSR, “Fizika,” 1973, No. 10.

2. A. A. Vorob’ev, O. B. Evdokimov, B. A. Kononov. In: Dosimetry of Large Doses. Tashkent, Publishing House of the Academy of Sciences, 1966.

3. A. A. Vorob’ev, O. B. Evdokimov, N. K. Ryzhakova. Izvestiya vuzov SSSR, “Fizika,” 1973, No. 1.

4. O. B. Evdokimov. Izvestiya vuzov SSSR, “Fizika,” 1973, No. 1.