CALCULATION OF ELECTRON BEAM CHARACTERISTICS IN A HIGH-CURRENT DIRECT-ACTION ACCELERATOR

G. I. STANEVKO, N. V. TRIKHANOVA

(Presented by the scientific seminar of the Laboratory of Computing Technology and Automation, Scientific Research Institute of Nuclear Physics, Electronics, and Automation)

At present, high-intensity electron beams produced in powerful direct-action accelerators are acquiring great practical and scientific importance. In developing such accelerators, it is necessary to calculate the accelerator optics that provides, at the output, electron beams with prescribed characteristics. The main problem is the calculation of the electric field between the cathode and the anode and the study of the trajectories of the electron beam during acceleration, taking into account the beam’s own electromagnetic field. With a sufficiently complex electrode geometry, such a problem can be solved only by numerical methods.

This article considers the application of the method of successive approximations to the study of the distribution of electric potential and to the construction of electron trajectories during acceleration.

Statement of the Problem

Let the electrode system be represented in the form shown in Fig. 1, where 1 is the anode (a flat plate perpendicular to the \(oz\) axis), 2 is a cylindrical pointed cathode with a hemispherical end, and 3 is an external conductor, which in the general case may be considered as an insulator with a distributed voltage or as part of the anode.

Introduce a cylindrical coordinate system \((r,\theta,z)\), whose axis coincides with the axis of symmetry of the electrode system. Consider the region \(D\), bounded by the contour \(\Gamma\), consisting of the surfaces of the anode, the cathode, and the external conductor.

Fig. 1. Section of the electrode system: 1 — anode, 2 — cathode, 3 — external conductor.

Let a certain distribution of electric potential be prescribed on the surface \(\Gamma\), with its values on the anode equal to \(\varphi\), and on the cathode equal to zero.

It is required to find the distribution of the electric-field potential inside the region \(D\), and also to construct the trajectories of electron motion in the “cathode—anode” space.

For high-intensity electron beams, in which the space charge has a substantial effect on the character of the motion of the particle beam, the potential of the electric field is determined by solving Poisson’s equation:

\[ \Delta \varphi = -4\pi \rho(r,z), \tag{1} \]

where \(\rho(r,z)\) is the space-charge density. The electron trajectories are determined by solving the equation of motion:

\[ \frac{d(m\overline{v})}{dt}=-e\left(\overline{E}+[\overline{v}\cdot\overline{B}]\right), \tag{2} \]

where

\[ \begin{aligned} m&\text{ is the electron mass as a function of velocity;}\\ \overline{v}&\text{ is the electron velocity;}\\ e&\text{ is the electron charge;}\\ \overline{E}&\text{ is the electric-field strength determined by equation (1);}\\ \overline{B}&\text{ is the magnetic induction of the beam’s own magnetic field.} \end{aligned} \]

Solution of equations (1), (2) by the method of successive approximations

Let \(\varphi^{(n)}, \rho^{(n)}, \overline{E}^{(n)}, \overline{B}^{(n)}, m^{(n)}\overline{v}^{(n)}\) be the values of the electric-field potential, the space-charge density, the magnetic induction of the beam’s own magnetic field, and the mass and velocity of the electron in the \(n\)-th approximation. Then problem (1), (2) can be represented as an iterative problem of the form

\[ \Delta\varphi^{(n)}=-4\pi\rho^{(n)}, \tag{3} \]

\[ \frac{d(m\overline{v})^{(n+1)}}{dt} =-e\left(\overline{E}^{(n+1)}+[\overline{v}\cdot\overline{B}]\right). \tag{4} \]

The iterative procedure (3), (4) should be repeated until agreement is established between the electron velocities and their trajectories and the distribution of space charge in the given region \(D\).

Implementation of the iterative problem (3), (4) on an electronic digital computer. Consider a mesh region \(D_s\) approximating the region \(D\). To begin the calculation, it is necessary to specify a zero approximation for the distribution of the electric-field potential inside the region \(D_s\) and a zero approximation for the trajectories of electron motion.

As the zero distribution of the electric-field potential inside the region, we take the solution of Laplace’s equation

\[ \Delta\varphi=0 \]

for prescribed values of the electric potential on the boundary of the region \(D_s\). To construct the zero approximation of the trajectories of electron motion in the region \(D_s\), we divide the cathode into several equal sections—

rons. Let us assume that all trajectories of electrons emitted by one section of the cathode are similar and form an elementary current tube. We shall denote the current in the \(j\)-th tube in the \(n\)-th approximation by \(I_j^{(n)}\). We shall assume that this current is concentrated at the center of the \(j\)-th current tube; consequently, its central trajectory may be regarded as an elementary ray. In the general case the current tubes may intersect.

Fig. 2. Flowchart of the method of successive approximations.

Flowchart labels:

• \(\rho=0,\ \varphi=0,\ B=0\)
• Solution of the equation \(\Delta\varphi=-4\pi\rho\)
• Check for completion
• Construction of trajectories
• Calculation of \(q,\rho,B\)
• Printout of results
• Stop

We construct the trajectories of electron motion in the zeroth approximation by solving equation (2) without taking into account the beam’s own magnetic field and space charge \((\vec B=0,\ \rho=0)\).

To construct subsequent approximations for the potential of the electric field and the trajectories of electron motion, we solve problem (3), (4).

To obtain the \((n+1)\)-st approximation for \(\varphi\), i.e., to solve equation (3), we calculate the space-charge density in the \(k\)-th cell of the region \(D_s\) by the formula

\[ \rho_k^{(n)}(r,z)=\frac{q_k^{(n)}}{d_k}, \tag{5} \]

where

\[ d_k^{(n)} \text{ is the volume of the } k\text{-th cell,} \]

\(q_k\) is the charge introduced by the elementary rays into the \(k\)-th cell in the \(n\)-th approximation.

The charge is determined from the formula

\[ q_k^{(n)}=\sum_j I_j^{(n)}\cdot t_{jk}^{(n)}, \tag{6} \]

where

\(j\) is the number of the elementary ray passing through the \(k\)-th cell;
\(I_j\) is the current of the elementary ray;
\(t_{jk}\) is the residence time of the \(j\)-th ray in the \(k\)-th cell.

To construct trajectories in the \((n+1)\)-st approximation, i.e., to solve equation (4), we calculate the magnetic induction of the beam’s own magnetic field \(\vec B_{(n)}\) by the law of total current [2].

Taking into account the newly constructed trajectories, we calculate the space-charge density of the next approximation by formulas (4), (5) and, having solved equation (3), find the distribution of the electric-field potential, etc.

The iteration process must be continued until the condition

\[ \left|\varphi^{(n+1)}-\varphi^{(n)}\right|\leq \varepsilon \tag{7} \]

is satisfied simultaneously for all points of the region \(D_s\), where \(\varepsilon\) is the prescribed accuracy of the calculation.

The question of convergence of the method was considered by P. Kirstein, G. Kino, and W. Waters [3].

Figure 2 shows the program flowchart. The current density at the cathode was determined on the basis of the “three-halves power” law; in this case, the region located near the cathode was selected as the diode elements.

References

1. R. Feynman, R. Leighton, M. Sands. The Feynman Lectures on Physics. Moscow, “Mir,” 1966.

2. P. Kirstein, G. Kino, W. Waters. Formation of Electron Beams. Moscow, “Mir,” 1970.