On the Instability of the Electron Beam of a Betatron

V. M. Antonov, V. V. Evstigneev, B. A. Kononov

(Presented by the joint seminar of the Research Institute of Nuclear Physics and the Physics and Technology Faculty of TPI on spectrometry)

Introduction

A betatron with an extracted electron beam is a source of monoenergetic high-energy electrons and is used to study the interaction of electrons with matter.

To obtain quantitative results, extremely important characteristics in this case are the stability of the energy and the small half-width of the energy distribution of the electrons in the beam.

However, as we have shown [1], [2], owing to a number of factors the energy spectrum of the electron beam of an unstabilized betatron has a half-width of about 2%. Investigation of the energy spectrum of the betatron electron beam and of its stability makes it possible to obtain recommendations for the manufacture of highly stable betatrons and to establish the optimum modes of their operation.

The instability of a 10 MeV betatron designed at the Tomsk Polytechnic Institute is investigated; in it, extraction of the electron beam is accomplished by an electrostatic method.

At the end of the acceleration cycle the electrons are displaced by a winding that intensifies the flux in the central core of the betatron pole up to the instability boundary, i.e., to the circle where the electrons leave the region of action of the focusing forces of the magnetic field. The amplitude of the betatron oscillations of the electrons increases sharply; the pitch of their spiral trajectory becomes sufficient to pass around the front surface of the inner plate of the deflecting capacitor. Having passed through the deflecting field, the electrons are freed from the action of the guiding magnetic field and leave the accelerating chamber.

The dumping pulse has a leading edge somewhat shorter than 1 μsec, but this does not produce an instantaneous action on the beam. With the aid of a scintillation counter and an SI-1 oscillograph it was measured that the longest dumping time is 16 μsec [4]. Consequently, the time of displacement and extraction of the electrons has a finite value. This means that some electrons are emitted with nominal energy, while others are delayed and, in connection with this, undergo additional acceleration, acquiring additional energy.

From this one may conclude that the principal energy spread is caused by betatron oscillations, the finite dumping time, the instability of the supply voltage, and the influence of the deflecting capacitor.

Dependence of the Baseline Width of the Energy Spectrum on Betatron Oscillations

Analysis of the radial motion of electrons shows that the motion can be described by a sinusoidal wave with an amplitude decreasing as

\[ \frac{1}{\sqrt[4]{1-n}}\cdot\frac{1}{\sqrt{B}}, \tag{1} \]

where \(n\) is the index of decrease of the magnetic field,
\(B\) is the corresponding induction at the radius of the equilibrium orbit.
The damping of the amplitude of betatron oscillations leads to a decrease in the energy spread of the accelerated electrons.

The density of the accelerating magnetic flux is

\[ \Phi=\Phi_{\mathrm{m}}\cdot \sin \omega t . \tag{2} \]

At a certain value of the magnetic flux, the displacement circuit is triggered and the electrons are emitted, possessing a definite momentum

\[ B\rho=\kappa\Phi . \tag{3} \]

The energy distribution can be represented by the relation

\[ \varepsilon=\frac{\Delta\Phi}{\Phi}=\frac{\Delta B\rho}{B\rho}; \tag{4} \]

or

\[ \frac{\Delta E}{E} = \frac{B_x\rho_x-B_0\rho_0}{B_0\rho_0} = \left(1+\frac{x}{\rho}\right)^{1-n}-1, \tag{5} \]

where \(x\) is the amplitude of the betatron oscillations at a given electron energy,
\(\rho\) is the radius of the equilibrium orbit.

Fig. 1. Dependence of the baseline width of the energy spectrum on the energy of accelerated electrons (the dumping phase is equal to \(75^\circ\)).

Figure 1 presents the calculated dependence of the baseline width of the energy spectrum on the energy of accelerated electrons at a constant dumping phase equal to \(\alpha'_p=75^\circ\). The change in energy is achieved

by regulating the voltage on the betatron magnet. The initial amplitude corresponding to the distance of the injector from the equilibrium orbit is equal to 2.8 cm. It is seen from Fig. 1 that the minimum energy spread of the electrons corresponds to the maximum energy of the accelerated electrons.

Dependence of the Base Width of the Energy Spectrum on the Final Extraction Time

The relation between the energy increment and the change in flux in the betatron is expressed by the relation

\[ \Delta \Phi = k \Delta E . \tag{6} \]

Consequently, to each value of the increment \(\Delta E\) in the betatron there must correspond a definite change in the flux inside the orbit. Since the increase of the magnetic flux takes place according to a sinusoidal law [2], for one and the same extraction time, at different phases, there will correspond different changes in flux and, consequently, different energy increments. The increment of additional energy decreases with increasing phase of extraction of the electrons from the equilibrium orbit. This is given by the following relation, obtained in [3]:

\[ \frac{\Delta E}{E} = \frac{2\pi \cdot \Delta \tau}{T} \cdot \operatorname{ctg}\varphi , \tag{7} \]

where \(\Delta \tau\) is the pulse duration, \(T\) is the period of the extraction phase \(\varphi\).

Fig. 2 presents the calculated dependence of the base width of the energy spectrum on the extraction phase at constant energy of the accelerated electrons. The extraction-pulse duration is equal to 16 \(\mu\)sec. It is seen from Fig. 2 that the minimum energy spread corresponds to an electron-extraction phase equal to \(90^\circ\).

Fig. 2. Dependence of the base width of the electron spectrum on the electron-displacement phase at constant energy.

However, for stable operation of the betatron the extraction phase is always made somewhat less than \(90^\circ\), since extraction at \(90^\circ\) would require good stabilization of the current amplitude of the betatron magnet [3].

Dependence of the Energy Spectrum on Instability of the Supply Voltage

Instability of the betatron supply voltage is caused by a change in the voltage or frequency of the supply network.

1. A change in the voltage leads to a change in the energy of the accelerated electrons if the extraction phase is constant. To eliminate this, betatrons are equipped with stabilization circuits, which ensure a specified energy level by automatic adjustment of the electron-extraction phase. The voltage on the betatron magnet is expressed as

\[ U = U_0 \sin \varphi , \tag{7} \]

where \(U\) is the specified voltage level,
\(U_0\) is the amplitude value of the voltage.

Hence

\[ \varphi=\arcsin \frac{U}{U_0}. \tag{8} \]

Substituting (8) into (6), we obtain

\[ \frac{\Delta E}{E}=\frac{2\pi}{T}\Delta \tau \operatorname{ctg}\arcsin \frac{U}{U_0}. \tag{9} \]

Consequently, a change in the supply voltage leads to a change in the energy spectrum of the electrons at constant energy.

In the plot: vertical axis \(\frac{\Delta E}{E}\%\); horizontal axis \(\frac{\Delta U}{U}\%\); curve labeled \(E=\mathrm{const}\).

Fig. 3. Dependence of the base width of the electron spectrum on the instability of the supply voltage \((E=\mathrm{const},\ \varphi=75^\circ)\).

In the plot: vertical axis \(\frac{\Delta E}{E}\%\); horizontal axis \(\frac{\Delta \omega}{\omega}\%\); curve labeled \(E=\mathrm{const}\).

Fig. 4. Dependence of the base width of the electron spectrum on the instability of the frequency of the supply voltage \((E=\mathrm{const},\ \varphi=75^\circ)\).

Figure 3 shows the calculated dependence of the base width of the betatron energy spectrum on the instability of the supply voltage. It is seen from the figure that a 10% instability of the supply voltage causes the base width of the electron spectrum to change by a factor of 2.

1. A change in the frequency of the supply network leads to detuning of the resonant circuit of the betatron power supply, which results in a sharp change in the current of the secondary winding of the betatron. Therefore, for stable

the level of energy stabilization, the extraction phase must be reduced, which correspondingly increases the width of the energy spread. Fig. 4 shows the calculated dependence of the base width of the energy spectrum of the electron beam on the instability of the frequency of the betatron supply voltage.

Effect of the Deflector on the Electron Beam of a Betatron

The deflecting capacitor for extracting the electrons is fed by a pulsed voltage from a peak transformer [4].

An experimental study was made of the dependence of the intensity of the extracted electron beam on the phase of displacement at a constant extraction phase. The results are given in Fig. 5. The figure

Fig. 5. Dependence of the intensity of the electron beam of a betatron on the phase of displacement at a constant phase of electron extraction.

shows that, in order to obtain a high and stable radiation intensity, the phases of extraction and of displacement of the electrons from the equilibrium orbit must be made identical. It is also necessary to take into account that the deflecting capacitor is a crude electrostatic betaspectrometer, and therefore a change in the voltage on it will affect the energy spectrum of the electron beam of the betatron. And since the phase of displacement of the electrons from the equilibrium orbit is automatically changed by the stabilization circuit, the extraction phase must likewise be changed synchronously with the displacement phase.

Conclusion

Thus it has been shown that the base width of the energy spectrum of a stabilized betatron is tenths of a percent.

The calculations performed prove that, from the standpoint of the energy spectrum, it is best to change the energy of the electron beam of a betatron by changing the voltage on the betatron magnet, while leaving the extraction phase constant. Electron extraction is best carried out at a phase somewhat less than \(90^\circ\), in order to dispense with cumbersome circuits for rigid stabilization of the current-amplitude peak in the betatron magnet. Circuits that provide stabilization of the electron energy by automatic adjustment of the phase of their displacement from the equilibrium or-

beat, at the same time disturb the energy spectrum of the electron beam of the betatron within tenths of a percent.

For rigid stabilization of the energy of electrons extracted from the betatron, the extraction phases and the displacement of the electrons from the equilibrium orbit must be automatically linked.

References

1. B. A. Kononov, V. M. Antonov, V. V. Evstigneev. Investigation of the energy spectrum of the electron beam of a betatron at 7.5 MeV. Izv. AN SSSR, ser. fizika, No. 2, 1964.

2. B. A. Kononov, V. M. Antonov, Yu. N. Beltyaev. Study of the spectrum of electrons extracted from a betatron. Proceedings of the Conference on Electron Accelerators, 1962.

3. T. Westernark, Nuclear Instruments and Methods. 10, 129, 1961.

4. B. A. Kononov. Extraction of electrons from a betatron by an electrostatic method. Izv. TPI, vol. 94, 1958.