On the Question of the Dependence of Betatron Radiation Intensity on the Frequency of the Supply Current

D. A. Noskov

(Presented by the scientific seminar of the Faculty of Physics and Engineering)

The question of choosing the frequency of the supply current for betatrons operating on alternating current is of great practical importance in the design and manufacture of accelerators of this kind.

In some sources there are recommendations to use an increased frequency for feeding the electromagnets, with the aim of increasing the radiation intensity [1, 2]. The radiation intensity, under the condition that in each period the same number of electrons is accelerated at any frequency, will be proportional to the frequency, since the number of accelerating periods is equal to the frequency of the supply current. However, there are still many unclear points in this question. Very little has been published on which alternating-current frequencies are the most acceptable, whether the same number of electrons is accelerated per period at different frequencies, how the rate of increase of the magnetic-field strength affects the capture conditions, and so on. Likewise, little experimental data on these questions has been published.

Considering the theoretical questions of electron motion in the accelerator and the questions of capture of electrons into acceleration, one can distinguish a number of factors, some of which indicate that, with increasing frequency, the number of electrons accelerated during one period increases. Another group of factors says that the number of electrons accelerated during one period does not depend on the frequency. There is a group of reasons of a technical and design character which lead to the fact that, with increasing frequency, the number of electrons accelerated during one period decreases.

Let us consider these factors in somewhat more detail.

With increasing frequency, the increase in the energy of the electron per revolution increases; consequently, the acceleration time is shortened and the path traversed by the electron in the acceleration process decreases, which means that the scattering of electrons on gas molecules decreases.

With increasing frequency, the conditions for bypassing the gun are improved owing to the shortening of the orbit, since the relative shortening of the orbit is directly proportional to the increase in the electron energy

\[ \frac{\Delta r}{r} = \frac{1}{2}\frac{\Delta w}{w} \]

and at the expense of an increase in damping, owing to the fact that the decrease in the amplitude of the electron oscillations about the equilibrium orbit is proportional to the increase in energy

\[ \frac{\Delta x}{x} = -\frac{1}{2}\frac{\Delta w}{w}. \]

At the same time, due to the increase in the damping of the oscillations, the scattering of electrons on the walls of the accelerating chamber decreases. These factors may lead to an increase in the number of electrons accelerated in one period as the frequency is increased.

If the capture process is considered from the point of view of the interaction of instantaneous electron bunches, then one may conclude that an increase in frequency does not change the capture conditions, and that the radiation intensity per period does not depend on frequency.

To the group of causes of a technical and constructional character one should first of all ascribe the difficulty of synchronizing the injection pulse with the magnetic-field voltage, a difficulty that increases with increasing frequency and decreasing duration of the injection pulse, as well as other causes connected with synchronization and stabilization of the voltage on the electromagnet coils and of the injection voltage.

Comparing these factors, it is difficult to say whether the number of electrons accelerated per period at different frequencies will remain constant or will change in the direction of increase or decrease.

One may suppose that the degree of influence of one or another factor on the conditions of capture and acceleration will be different in different frequency ranges. In one range the factors determining an increase in intensity will play the main role, in another—on the contrary, and so forth.

Consequently, there must exist some frequency range in which the capture conditions will be optimal and the radiation intensity maximal. A theoretical prediction of the limits of this range does not appear possible; therefore experimental investigations of this question are necessary.

With the aim of determining experimentally the dependence of the radiation intensity of a betatron on frequency, we carried out a series of experiments on a pulsed betatron without an iron yoke and poles. Instead of changing the frequency of the supply current, we changed the duration of the pulse.

The change in pulse duration is produced by changing the capacitance of the capacitor bank. Such a simple way of changing the pulse duration makes it possible to carry out experiments with pulses whose duration varies over wide limits. This enabled us to carry out a series of experiments to determine the influence of pulse duration (the reduced pulse frequency) on the radiation intensity and to determine the range of optimal frequencies (durations) of the pulses for the given accelerator design. In determining the influence of the pulse frequency on the radiation intensity, the experiment must be arranged so that all other factors affecting the radiation have as little effect as possible, and the dependence of the intensity on frequency is as pure as possible. Direct recording of the dependence of the radiation intensity on frequency by changing the pulse duration and changing the voltage on the capacitors so that the energy of the accelerated electrons is constant at all frequencies is not possible, since in the supply circuit smooth voltage regulation cannot be carried out. As a result of this we adopted another method for measuring this dependence.

For each pulse duration, the dependence of the radiation intensity on the energy of the accelerated electrons was recorded. The energy of the accelerated electrons was varied by changing the voltage on

on the capacitors, when the electrons are dumped onto the target in a phase close to \(85^\circ\). The number of pulses per second for all durations remained constant.

All curves of the dependence of radiation intensity on the energy of the accelerated electrons for different pulse durations are plotted on one graph; moreover, the capacitor voltage for all pulse durations is first recalculated into current, and the graphs may be plotted by laying off on the axis, instead of the electron energy, the values of the current in the electromagnet coils.

Having such a family of curves, it is not difficult to construct the dependence of the radiation intensity on the duration (frequency) of the pulse. For this it is sufficient to take the values of the radiation intensity for pulses of different duration at one value of current (energy) and construct a new graph.

For a betatron without an iron yoke and poles, the curves of the dependence of radiation intensity on the energy of the accelerated electrons were recorded for pulses of duration 500, 670, 1000, and 1430 \(\mu\)sec, which corresponds to frequencies of 1000, 750, 500, and 360 periods/sec.

Graphically these dependences are presented in Fig. 1. The dependence of the radiation intensity on the pulse duration, constructed from the curves of Fig. 1, is shown in Fig. 2. The course of the curve shows that for shorter pulses the radiation intensity is lower. From the form of the dependence it may be concluded that the betatron has a range of optimum frequencies, the upper limit of which is near the frequency 500 periods/sec. The lower limit under our conditions cannot be determined, since the power-supply circuit of the accelerator does not permit operation with pulses of long duration.

Fig. 1

Fig. 2

Thus, as a result of experiments on the investigation of the dependence of radiation intensity on pulse duration (or on pulse frequency), we have established that the given betatron has a definite range of pulse durations in which the number of electrons accelerated in one pulse is independent, or depends very little, on the frequency of the supply current.

This means that increasing the frequency of the supply current in order to raise the radiation intensity is possible only up to a certain limit, so long as the increase in frequency does not lead to a decrease in the number of electrons accelerated in one period. That is, in order to decide the question of choosing the frequency of the supply current, it is necessary to determine experimentally the boundaries of the range of optimum frequencies.

These conclusions are of especially important significance for pulsed betatrons of low energies, where decreasing the pulse duration

is limited by the voltage on the capacitor bank and is not associated with heating of the copper and iron of the electromagnet.

LITERATURE

1. Borob’ev A. A. Accelerators of Charged Particles. Gosenergoizdat, 1949.
2. Grinberg A. P. Methods of Accelerating Charged Particles. GTTI, 1950.