ON MODELING THE FIELD IN A DIODE WITH A MULTI-TIP CATHODE

I. Z. Gleizer, A. I. Naletov, B. V. Okulov, G. I. Stanevko

(Presented by the scientific seminar of the Department of Engineering and Computational Mathematics of TPI)

In forming high-current electron beams in direct-action accelerators, the problem arises of calculating the electric field in an electron gun whose cathode is a system of tips and whose anode is a plane. For a given distance between the tips, the maximum tip density is achieved when they are arranged at the vertices of regular triangles. In this case each tip is surrounded by six others, located at equal distances from the first. The diode in this case may be represented as consisting of elementary hexagonal cells (Fig. 1), on whose lateral surfaces the normal component of the electric field is equal to zero. Finding the field distribution for this system of electrodes on a digital computer is difficult because of the complexity of the boundary conditions. Therefore let us assume that the field near a tip—emitter will not change substantially if we replace the hexagonal cell by a cylindrical cell circumscribed about the hexagon, on whose surface the normal component of the field is likewise equal to zero.

Fig. 1. a — layout of emitters in the cathode; b — computational scheme of a diode section.

The surfaces of which have a normal component of the electric field equal to zero. Finding the field distribution for this system of electrodes on a digital computer is difficult because of the complexity of the boundary conditions. Therefore let us assume that the field near a tip—emitter will not change substantially if we replace the hexagonal cell by a cylindrical cell circumscribed about the hexagon, on whose surface the normal component of the field is likewise equal to zero.

For calculating the field, we introduce the cylindrical coordinate system of Fig. 1b. Since the electrode system is axisymmetric, Laplace’s equation has the form

\[ \frac{\partial^2 u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{\partial^2 u}{\partial z^2}=0. \tag{1} \]

To solve equation (1), the finite-difference method was used. The region \(C_1C_2C_3C_4\) was covered by a rectangular grid formed by the straight lines

\[ r=r_i,\quad (i=0,\,1,\,\ldots,\,n), \]

\[ z=z_k,\quad (k=0,\,1,\,\ldots,\,m). \]

If the value of the electric-field potential at the nodes \((i,k)\), \((i,k+1)\), \((i+1,k)\), \((i,k-1)\), \((i-1,k)\) is denoted by \(U_0, U_1, U_2, U_3, U_4\), and the distance of each node from \((i,k)\) by \(h_1, h_2, h_3, h_4\), respectively, then the difference analogue of Laplace’s equation in cylindrical coordinates takes the form

\[ \frac{2}{h_1+h_3}\left(\frac{u_1}{h_1}+\frac{u_3}{h_3}\right) + \frac{2}{h_2+h_4}\left(\alpha\frac{u_2}{h_2}+\beta\frac{u_4}{h_4}\right) - \left(\frac{2}{h_1h_3}+\frac{2}{h_2h_4}+\gamma\right)u_0=0, \tag{2} \]

where

\[ \alpha=1+\frac{h_4}{2r_0},\quad \beta=1-\frac{h_2}{2r_0},\quad \gamma=\frac{1}{h_2r_0}-\frac{1}{h_4r_0}, \]

and \(r_0\) is the distance from the axis of symmetry to the point under consideration. Since at \(z=0\) the field component \(\dfrac{1}{r}\dfrac{\partial u}{\partial r}\) is not defined, from the symmetry conditions, for points lying on the axis, the finite-difference analogue will have the form

\[ \frac{2}{h_1+h_3}\left(\frac{u_1}{h_1}+\frac{u_3}{h_3}\right) + \frac{4u_2}{h_2} - \left(\frac{2}{h_1h_3}-\frac{4}{h_2^2}\right)u_0=0. \tag{3} \]

The values of the potential at the anode are

\[ u_{i,0}=c=\mathrm{const}, \tag{4а} \]

at the cathode,

\[ u_{i,k}=0, \tag{4б} \]

and since the normal derivative of the field on the cylindrical surface is (by condition) equal to zero, on the \((n+1)\)-st layer the potential values satisfy the relation

\[ u_{n+1,k}=u_{n-1,k}. \tag{4в} \]

The system of equations (2–4) was solved on a BESM-4 digital computer by the successive overrelaxation method [1] with acceleration factor \(\omega = 1.6\). The resulting field pattern is shown in Fig. 2 by solid lines. The relative error of the obtained solution does not exceed 1%.

Fig. 2. Field distribution in an elementary cell of the diode.

To verify the correctness of the method for calculating the field on a digital computer, fields were modeled in an electrolytic tank for various values of \(H\), \(h\), \(R\) with \(r_k=\mathrm{const}\). An electrolytic tank was fabricated taking into account the specific arrangement of the tips in the diode. The field pattern was recorded in the most characteristic sections \(OA\) and \(OB\) of the elementary cell (Fig. 1a) in the form of a series of equipotentials, the recording of which was performed automatically with a relative error not exceeding 2%.

The equipotentials of the field in section \(OA\) are shown in Fig. 2 by dashed lines, and in section \(OB\) by points. It is seen from Fig. 2 that in the region of greatest interest—namely, in the vicinity of the emitter tip and between the emitter tip and the anode—the equipotentials practically coincide. It follows from this that, for diodes with multi-tip cathodes and

elementary cell having the structure indicated in Fig. 1, the proposed method for calculating fields on a digital computer gives satisfactory results.

The authors express their gratitude to V. A. Lukutin for assistance in the work.

REFERENCES

1. I. S. Berezin, N. P. Zhidkov. Methods of Computation. Moscow, GIFML, 1959.