On the Calculation of Current Passage Through a Betatron Inflector
Abstract Generated abstract
This paper develops an approximate method for estimating electron current losses in a betatron inflector during high-voltage injection, where space-charge expansion can cause part of the beam to strike the inflector plates. It introduces the inflector current-passage coefficient, defined as the ratio of exit injection current to injector current, and derives calculation relations for a cylindrical electron beam in a flat inflector under simplifying assumptions of laminar paraxial motion and uniform transverse density. The analysis accounts for transverse space-charge forces, the beam’s magnetic field, and the effective accelerating potential, leading to implicit formulas and a nomogram for determining the transmission coefficient from inflector geometry, beam current, initial radius, length, and effective voltage. The same relations are suggested for approximate estimates in cylindrical inflectors when curvature effects are small.
Full Text
ON THE CALCULATION OF CURRENT PASSAGE THROUGH A BETATRON INFLECTOR
B. V. Okulov
(Submitted by the Scientific Research Institute of Nuclear Physics, Electronics, and Automation)
In high-voltage injection into betatrons, the electron current density in the minimum cross section of the beam in the inflector can reach tens and even hundreds of amperes per square centimeter. Therefore, when calculating the motion of an electron beam, it is necessary to take into account the forces of its own space charge, which cause expansion of the beam and may lead to its partial loss on the plates of the inflector [1, 2]. In order to be able, in calculating a high-voltage injection device, to take these losses into account, we introduce the concept of the current-passage coefficient of the inflector. By the current-passage coefficient of the inflector, \(K_d\), we shall mean the ratio of the current at the exit from the inflector, i.e., the injection current \(I_i\), to the current at the entrance to the inflector, i.e., to the current of the injector (gun) \(I_{\text{п}}\) (Fig. 1),
\[ K_d=\frac{I_i}{I_{\text{п}}}. \tag{1} \]
Fig. 1. Scheme of expansion of a cylindrical electron beam in a flat inflector: \(l\) — length of the inflector, \(d\) — width of the inflector, \(O\) — optical axis of the inflector, \(r_0\) — initial radius of the beam, \(r\) — current radius of the beam, \(I_{\text{п}}\) — beam current at the entrance to the inflector, \(I_i\) — beam current at the exit from the inflector, \(\gamma\) — angle of divergence of the beam at the exit from the inflector.
In principle, a divergent, parallel, or convergent beam may be fed to the entrance of the inflector. At the exit from the inflector the beam must be parallel \((\gamma=0)\), or weakly divergent \((\gamma\) small). For a given value of \(I_{\text{п}}\), the current-passage coefficient of the inflec—
tor should be as large as possible. These two requirements on the injected beam at large currents \(I\) are contradictory [3], and satisfying them simultaneously to the greatest possible extent is difficult.
A solution of the problem for the three cases listed above would take up too much space; therefore we shall confine ourselves here to deriving the equation for the current-transmission coefficient of an electron beam that is cylindrical at the entrance to the inflector. In this case, at the exit from the inflector one can obtain a beam with a comparatively small \(\gamma\) and an acceptable value of \(K_d\).
Let us limit the number of forces taken into account in deriving the calculation relations so that the calculation of current transmission is sufficiently simple. At the same time, of course, it will be approximate (of an estimating nature).
Consider the motion of a cylindrical electron beam in a plane inflector of length \(l\) and width \(d\). Let the reduced voltage on the inflector \(U_{di}\), equal to the instantaneous value of the ratio of the voltage on the inflector \(U_d\) to the voltage on the injector \(U_i\), have such a value that the axis of symmetry of the electron beam (the trajectory of the “mean” electron of the beam) coincides with the optical axis of the inflector [1, 7].
We shall take into account the transverse forces of the beam’s own space charge and the forces due to the beam’s own magnetic field. We shall also take into account the increase in the potential of the space on the optical axis of the inflector due to the voltage on the inflector. All other forces will be neglected. Such conditions are equivalent to the assumption that the electron beam moves in the inflector as in an equipotential space. We shall also assume, as is usually done in electron optics when solving similar problems [3], that the electron beam is laminar, paraxial, and has constant density in each transverse section.
Under the stated conditions, the divergence of a relativistic axially symmetric electron beam may be calculated by the formula [4], which in our notation [1] has the form
\[ \frac{l}{r_0} = 1.021\,\frac{U_{\mathrm{e}}^{3/4}}{I_{\mathrm{p}}^{1/2}} \cdot \Phi(U_{\mathrm{e}})\cdot F\!\left(\frac{r}{r_0}\right), \tag{2} \]
where
\[ \Phi(U_{\mathrm{e}})= \left(1+\frac{U_{\mathrm{e}}}{1022}\right)^{3/4}, \tag{3} \]
\[ F\!\left(\frac{r}{r_0}\right) = 2\int_{0}^{[\ln(r/r_0)]^{1/2}} e^{x^2}\,dx, \tag{4} \]
\(r_0\) is the beam radius at the entrance to the inflector, cm;
\(l\) is the inflector length, cm;
\(I_{\mathrm{p}}\) is the beam current at the entrance to the inflector, A;
\(U_{\mathrm{e}}\) is the “effective voltage” of the beam on the optical axis of the inflector [1, 7], kV.
Integral (4) is tabulated in [5].
From equation (2) it follows that
\[ F\!\left(\frac{r}{r_0}\right) = \frac{1}{1.021}\cdot \frac{I_{\mathrm{p}}^{1/2}}{r_0}\cdot \frac{l}{U_{\mathrm{e}}^{3/4}\cdot \Phi(U_{\mathrm{e}})}. \tag{5} \]
The current density at the entrance to the inflector is
\[ j_0=\frac{I_{\mathrm p}}{\pi r_0^2}. \tag{6} \]
Let us denote
\[ \varphi(U_{\mathrm e})=\frac{1}{U_{\mathrm e}^{3/4}\cdot \Phi(U_{\mathrm e})}, \tag{7} \]
then equation (5), taking (6) and (7) into account, is written in the form
\[ F\left(\frac{r}{r_0}\right)=1.735\,j_0^{1/2}\cdot \varphi(U_{\mathrm e})\cdot l. \tag{8} \]
From equalities (4) and (8) it follows that
\[ 2\int_{0}^{[\ln(r/r_0)]^{1/2}} e^{x^2}\,dx = 1.735\,j_0^{1/2}\cdot \varphi(U_{\mathrm e})\cdot l. \tag{9} \]
Fig. 2. Nomogram for calculating the current-transmission coefficient of the inflector. An example of calculating \(K_d\) is shown for: \(I_{\mathrm p}=7\,\mathrm{A}\), \(r_0=0.6\,\mathrm{cm}\), \(U_{\mathrm e}=200\,\mathrm{kV}\), \(l=30\,\mathrm{cm}\), \(d=1.44\,\mathrm{cm}\). In this case \(K_d=0.7\).
From equations (9) and (6), the value of \(r\) is determined as a function of four arguments
\[ r=f_1(r_0,I_{\mathrm p},U_{\mathrm e},l). \]
We shall now find the dependence \(K_d=f_2(r)\).
The electron beam moves in the inflector without losses until (Fig. 1) the equality \(2r=d\) is reached. With further motion
some of the electrons are lost on the plates, and a beam with current \(I_i=K_d \cdot I_{\mathrm{p}}\) emerges from the inflector.
The current-transmission coefficient of the inflector can be determined from Fig. 1 as the ratio of the area of the shaded part of the circle of radius \(r\) to the area of the entire circle.
\[ K_d=\frac{4}{\pi r^2}\cdot(\text{area of sector }PON+\text{area of }\triangle OMN). \]
After the corresponding substitutions and transformations, we obtain
\[ K_d=\frac{2}{\pi}\left[\arcsin a+a(1-a^2)^{1/2}\right], \tag{10} \]
where
\[ a=\frac{d}{2r}. \tag{11} \]
From equations (6), (9), (10), and (11), for various initial conditions one can find the current-transmission coefficient of the inflector as a function of five arguments
\[ K_d=f_3(d,r_0,l,I_{\mathrm{p}},U_{\mathrm{z}}). \tag{12} \]
It is not possible to express \(K_d\) explicitly in terms of these arguments. Therefore, for ease of calculation, the solution of equation (12) is presented graphically in Fig. 2.
The conditions for motion of an electron beam in a cylindrical inflector and in a cylindrical inflector of opposite curvature [1, 6], for sufficiently large mean radii of curvature, differ only slightly from the conditions for motion of an electron beam in a flat inflector; therefore formulas (6), (9), (10), and (11), and Fig. 2, may also be used for approximate (estimated) calculations of the value of \(K_d\) in these inflectors.
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