Selection of Injection Voltage for Compact Betatrons
Abstract Generated abstract
This paper examines how the choice of injection voltage affects the accelerated charge and overall specific characteristics of portable compact betatrons. The authors derive relationships linking captured charge to chamber geometry, injection voltage, injector overshoot, cathode current density, and gun design, showing that increasing voltage also changes injector dimensions and can reduce the useful accelerating volume. Calculations for a PMB-6 type betatron indicate an optimum injection voltage of about 40 to 50 kV for a Kerst injector, while an external injector with an inflector shifts the optimum to about 70 kV and can approximately double the captured charge. The analysis argues that external injection and higher cathode emission density improve utilization of the chamber volume and the charge captured into acceleration.
Full Text
SELECTION OF INJECTION VOLTAGE FOR COMPACT BETATRONS
L. M. Anan’ev, Yu. N. Bel’tyaev, V. L. Chakhlov
(Presented by the scientific seminar of the Department of Industrial Electronics)
A known and generally accepted way of increasing the captured charge in cyclic accelerators is to increase the injection voltage. In considering the question of increasing the injection voltage of portable compact betatrons (PMB), it must be borne in mind that the task of improving the specific dimensional, weight, and economic indicators of the installation as a whole comes to the fore.
Some general questions of improving the specific characteristics of PMB have been considered in [1], and those of elements of the injection system of pulse and filament transformers—in [2].
In the present article the influence of increasing the injection voltage on the magnitude of the accelerated charge is considered, taking into account the change in the structural dimensions of the electron gun caused by the change in injection voltage.
The maximum charge accelerated in the betatron chamber is related to the geometrical dimensions of the installation and to the injection voltage by the relation [3]
\[ Q_0 = (0.1 \div 0.4)\cdot 10^{-11}\cdot \frac{\Delta r^2}{R_0}\sqrt{\frac{1-n}{n}}\cdot U, \tag{1} \]
where \(\Delta r = R - R_0\), \(R\) is the injection radius, \(R_0\) is the radius of the equilibrium orbit, \(n\) is the magnetic-field falloff index, and \(U\) is the injection voltage.
Relation 1 may be written in the form
\[ Q_0 = K_1\cdot S_0\cdot U, \tag{2} \]
where \(S_0 = \pi \Delta r^2 \sqrt{\frac{1-n}{n}}\) is the useful area of the accelerating chamber, determined by the spatial distribution of the potential function of the magnetic field; \(K_1 = (0.1 \div 0.4)\cdot 10^{-11}\frac{1}{R_0}\) is a coefficient independent of the voltage.
In the real case, the radial size of the useful cross section of the accelerating chamber is reduced, owing to the so-called “overshoot” of the injector (Fig. 1), to the value \(S \sim (\Delta r - \Delta)\). In this case the accelerated charge is equal to
\[ Q \sim K\cdot S\cdot U. \tag{3} \]
The relative reduction in the limiting charge capable of being accelerated in the chamber, or the efficiency of utilization of the useful volume of the chamber, is equal to
\[ \frac{Q}{Q_0}=\frac{S}{S_0}=\left(1-\frac{\Delta}{\Delta r}\right)^2 . \tag{4} \]
It follows from (4) that, if the “miss” is of the same order as the half-width of the accelerating chamber, the efficiency of utilization of the useful volume of the accelerating chamber depends substantially on the size of the miss \(\Delta\).
Fig. 1. Cross section of the accelerating chamber
For the Kerst injector (Fig. 1)
\[ \Delta=\frac{b}{2}+d, \tag{5} \]
where \(\alpha\)—the distance from the focusing electrode to the anode box—is related to the injection voltage by the relation (4).
\[ U=\beta d^{\alpha}. \tag{6} \]
Here \(\alpha\) and \(\beta\) are coefficients characterizing the electric strength of the gap \(d\); \(\frac{b}{2}\) is the half-width of the focusing electrode.
If one takes into account that the optimal perveance of the injector beam is
\[ P=\frac{I}{U^{3/2}} \quad [5], \]
where \(I\) is the optimal injection current, and, for simplicity, takes the width of the focusing electrode to be equal to the width of the cathode, then
\[ \frac{b}{2}=\frac{P U^{3/2}}{j l}, \tag{7} \]
where \(j\) is the cathode emission-current density and \(l\) is the cathode length.
Taking into account (5), (6), and (7), expression (3) can be rewritten in the form
\[ Q=K_1\left[\Delta r-\frac{P U^{3/2}}{j l}-\left(\frac{U}{\beta}\right)^{1/\alpha}\right]^2 \cdot U . \tag{8} \]
Dependence (8) has an extremal character; consequently, there is an optimal value of the injection voltage, above which it is inadvisable to design the injector, since the specific characteristics will deteriorate.
Figure 2 shows, in relative units, the dependence of the magnitude of the captured charge, calculated from expression (8), on the injection voltage for a compact betatron of the PMB-6 type. Curve 2—1 corresponds to a cathode emission current density \(j = 5 \frac{a}{cm^2}\), curve 2—2 to a density \(j = 10 \frac{a}{cm^2}\). It follows from Fig. 2 that the value of the optimum injection voltage for the chamber of the PMB-6 betatron with a Kerst injector, at the given cathode emission current densities, lies within the range \(40 \div 50\) kV. In injection by means of an external injector with an inflector, the electrode system of the gun is removed beyond the limits of the useful cross section of the accelerating chamber; consequently, if it is assumed that the beam width at the exit from the inflector is equal to the width of the cathode, then relation (8) may be rewritten in the form
\[ Q \sim \left(\Delta r - \frac{PU^{3/2}}{jl}\right)^2 \cdot U . \tag{9} \]
The values of the intensity, calculated from relation (9) at a density \(j = 5 \frac{a}{cm^2}\), are presented in Fig. 2—3. As is evident from the figure, in this case the optimum value of the injection voltage is approximately 70 kV. The relative magnitude of the captured charge thereby increases by approximately a factor of two in comparison with the Kerst-type injector, also calculated for the optimum value of the injection voltage.
Fig. 2. Dependence of the magnitude of the captured charge on the injection voltage: 1 — at cathode emission current density \(5\,a/cm^2\), 2 — at cathode emission current density \(10\,a/cm^2\), 3 — for an external injector with an inflector at cathode emission current density \(5\,a/cm^2\).
Thus, it follows from the foregoing that the efficiency of use of the useful volume of the accelerating chamber of a betatron depends strongly on the design features of the injector; in particular, the efficiency of use of the chamber volume in compact betatrons is reduced.
Indeed, for the chamber of the PMB-6 betatron with a Kerst injector we have \(\Delta = 4\) mm, \(\Delta r = R - R_0 = 15\) mm, \(\frac{Q}{Q_0} = 0.54\).
It follows from this that it is most expedient to use, in the accelerating chambers of compact betatrons, an external injector with an inflector, since such an injector will make it possible not only to increase the optimum injection voltage, but also to increase the efficiency of use of the volume of the accelerating chamber, which will additionally improve the specific characteristics of the PMB. In addition, as is evident from Fig. 2, increasing the current density from the injector cathode ...
also improves the utilization factor of the region in which focusing forces exist in the working gap of the accelerator and increases the magnitude of the charge captured into acceleration.
REFERENCES
- V. L. Chakhlov. Dissertation, TPI, 1964.
- Ya. S. Pekker. Dissertation, TPI, 1967.
- L. M. Anan'ev et al. Induction electron accelerator—betatron. Atomizdat, 1961.
- I. N. Slivkov et al. Electrical breakdown and discharge in vacuum. Atomizdat, 1966.
- P. A. Cherdantsev. Electron accelerators. Proceedings of the Third Interuniversity Conference, 1961.