THEORETICAL AND EXPERIMENTAL STUDY OF THE ENERGY SPECTRUM OF A BETATRON ELECTRON BEAM AND ITS STABILITY

V. V. EVSTIGNEEV

(Presented by the scientific seminar of the Research Institute of Nuclear Physics)

In studying the spread of electron energies it is necessary to bear in mind that, on the one hand, a betatron is an electronic device in which the electron-optical system plays the determining role. On the other hand, a betatron is an electrical machine with rather complex circuits and automation devices. Consequently, the energy spread of the electrons will be determined, on the one hand, by the process of electron motion itself, and, on the other, by the operating parameters of the betatron.

To obtain the largest charge captured for acceleration, it is necessary to use all the potential wells produced by the electromagnetic field of the given betatron. To accomplish this, electrons must be injected into the betatron chamber at different initial energies, determined by the region of stable orbits for which potential wells exist [1]. In the general case the particles are injected with some angular spread and with momenta not coinciding with the equilibrium value [2].

The energy range \(\Delta E\) of the injected electrons is determined by the position of the injector relative to the radius of the equilibrium orbit and is expressed by the relation

\[ \frac{\Delta E}{E_i}=\frac{E_i-E_0}{E_i}=\left(\frac{r_i^2-r_0^2}{r_i^2}\right), \tag{1} \]

where \(E_i\) is the energy of the electrons injected at the installation radius of the injector \(r_i\);

\(E_0\) is the energy of the electrons injected at the radius of the equilibrium orbit \(r_0\).

From expression (1), the absolute spread of the electrons in energy at the moment of injection increases with increasing injection voltage.

Since electrons are injected not only onto a single equilibrium orbit but into a region of orbits, it is necessary to consider the behavior of an orbit that differs somewhat from the equilibrium one. The expression describing the acceleration regime in a betatron is written as

\[ \frac{dP}{dt}=-\frac{e}{c\Pi}\cdot\frac{\partial \Phi}{\partial t}, \tag{2} \]

where \(P\) is the particle momentum, and
\(\Phi\) is the scalar magnetic flux through its orbit,
\(\Pi\) is the orbit length.

Denoting equilibrium values by the subscript \(S\), and deviations from them by \(\Delta P\), \(\Delta \Pi\), \(\Delta \Phi\), one can obtain [3]

\[ \frac{d\Delta P}{dt} = -\frac{\Delta \Pi}{\Pi_s}\cdot \frac{dP_s}{dt} - \frac{e}{C\Pi_s}\cdot \frac{\partial \Delta \Phi}{\partial t} \equiv 0 \]

\[ \Delta P=\mathrm{const} \tag{3} \]

Thus, in a betatron with a stationary equilibrium orbit, the deviation of the momentum from the equilibrium value is an integral of the motion, independent of the geometry of the orbits. This means that the absolute spread of the momenta of the particles in the beam at the end of acceleration remains the same as at injection.

The electrons injected into the chamber move along instantaneous orbits determined by the position of the injector relative to the equilibrium orbit. As the magnetic field increases, the orbits of all particles asymptotically approach the equilibrium orbit in proportion to \(\langle B\rangle^{-1}\). During acceleration the particles undergo radial oscillations, the amplitudes of which damp as the particle energy increases. The change in electron energy in a betatron proves to be independent of the betatron oscillations [3]. However, the presence of these oscillations leads to a finite energy spread of the electrons in the beam at the moment when they are displaced from the equilibrium orbit. The process of displacing electrons from the equilibrium orbit occurs in a finite time. In this case the energy distribution of the electrons is determined by the expression [4]:

\[ \frac{\Delta E}{E} = \frac{2\Pi\cdot \Delta \tau}{T}\cdot \operatorname{ctg}\varphi, \tag{4} \]

where \(T\) is the period of variation of the magnetic field,
\(\Delta \tau\) is the duration of the radiation pulse,
\(\varphi\) is the phase of electron extraction.

When electrons are extracted from the betatron by the electrostatic method, the electrons pass through the electric field between two plates. The presence of betatron oscillations causes some of the electrons to undergo multiple or grazing collisions with the deflector plates, which leads to additional broadening of the energy spectrum.

The causes of nonuniformity of the electron energies considered above pertain to a single radiation pulse.

A change in the supply voltage of the betatron leads to the fact that the electron energy spectrum will vary from pulse to pulse.

Thus, the principal energy spread of the electron beam is caused by the injection process, betatron oscillations, the finite time of electron extraction, the influence of the deflecting capacitor, and instability of the supply voltage.

The experimental results were obtained with the apparatus described in [5, 6], the scheme of which is shown in Fig. 1.

1. Shape of the energy spectrum of the electron beam

Figure 2 shows a line obtained at an energy of \(7\ \text{MeV}\). It characterizes the energy distribution of betatron electrons that pass through a thin window and an air gap. In a rough approximation the line has a triangular shape with some asymmet-

... toward lower energies, due to energy scattering in the foils and in the air gap. The line width at half-height (half-width) is 2.4% for the stabilized betatron and 3.5% for the unstabilized betatron.

The contribution to the spectral width introduced by the spectrometer, taking its aberrations into account, is 0.14%.

1 — 10-MeV betatron;
2 — magnetic lenses;
3 — correcting magnets;
4 — magnet of the β-spectrometer;
5 — detector;
6 — to the pump;
7 — ionization chamber (monitor);
8 — ionization chamber (monitor).

Fig. 1. Diagram of the experimental setup

The total energy losses in the thickness of the material traversed by the beam amount to 1.3% of the indicated spectral width.

The part of the spectral width due to the accelerator itself is 0.96% for the stabilized betatron and 2.06% for the unstabilized betatron.

Fig. 2. Shape of the energy spectrum of the electron beam:
1 — unstabilized, \(\Delta E/E = 3.5\%\);
2 — stabilized, \(\Delta E/E = 2.5\%\) betatron

Fig. 3. Dependence of the energy spectrum on the extraction phase

2. Dependence of the Energy Spectrum on the Phase of Displacement and Extraction of Electrons from the Accelerator

The phase of electron displacement can be varied by regulating the current in the betatron magnet, while keeping the energy level of the accelerated electrons constant. Figure 3 gives a comparison of the theoretical and experimental dependences of the half-width of the energy spectrum on the extraction phase at a constant electron energy of 7 MeV. The calculation was carried out taking into account that the duration of the radiation pulse is 16 μsec. The character of the variation of \(\Delta E / E\) with the extraction phase corresponds to the theoretical one. It is clear from the figure that, in order to obtain monoenergetic electrons, the displacement of the electrons from the equilibrium orbit and their extraction from the accelerating chamber are best carried out at phases close to \(90^\circ\).

3. Dependence of the Half-Width of the Energy Spectrum of the Electron Beam on Its Energy

The energy of electrons in the betatron can be regulated either by changing the amplitude of the current in the betatron magnet, or by regulating the phase of electron displacement from the equilibrium orbit while keeping the current amplitude in the magnet constant.

Figure 4 shows the dependences of the half-width of the energy spectrum of the electron beam on the energy for different methods of regulating it.

Fig. 4. Dependence of the energy spectrum on energy for different methods of its regulation:
1 — regulation of the extraction phase; 2 — regulation of the voltage on the magnet

It is shown that, in order to obtain a narrow energy distribution, it is better to regulate the energy by changing the amplitude of the current in the betatron magnet.

4. Dependence of the Electron Energy Spectrum on the Voltage on the Deflector

The deflecting capacitor for extracting electrons from the betatron is an electrostatic beta spectrometer; therefore, changing the voltage on it will affect the energy spectrum of the electron ...

of the betatron beam. It has been shown experimentally (Fig. 5) that, at the maximum intensity of the electrons extracted from the betatron, the half-width of the energy spectrum is minimal. When the voltage on the deflector deviates from the optimum value, the maximum shifts toward lower energies. This, as well as the strong dependence of the spectral width on the voltage on the deflector, is due to energy losses of the electrons upon collision with the metal plates of the deflector.

5. Dependence of the energy spectrum of the betatron electron beam on instability of the supply voltage

The system for stabilizing the electron energy in the betatron maintains the specified energy level by automatically adjusting the phase of electron extraction. A change in the extraction phase, in turn, leads to a change in the energy spectrum of the electron beam. Fig. 6 shows the theoretical and experimental dependences of the width of the energy spectrum on instability of the supply voltage.

Fig. 5. Dependence of the energy spectrum and intensity of the extracted electron beam on the voltage in the deflector

Fig. 6. Dependence of the energy spectrum on voltage instability

The experimental rate of change of the spectral half-width with voltage instability on the magnet is higher than expected, which can be explained by a change in the duration of the radiation pulse.

Conclusion

The half-width of the energy spectrum of the extracted electron beam of the betatron in the optimum regime does not exceed 1%.

Deviations from the optimum regime can increase the half-width of the energy distribution of the electrons in the extracted beam up to 2%.

The experimental data obtained are in qualitative agreement with the theoretical data.

REFERENCES

1. B. N. Rodimov. Izv. TPI, vol. 87, 31, 1957.
2. D. Kerst. In: Accelerators, edited by B. K. Yablokov, 1962.
3. A. A. Kolomenskii, A. I. Lebedev. Theory of Cyclic Accelerators. Gosatomizdat, Moscow, 1962.
4. T. Westermark. Nucl. Instr. and Meth. 10, 145, 1961.
5. V. M. Antonov, V. V. Evstigneev, B. A. Kononov. Izv. TPI, vol. 140, 222, 1965.
6. B. A. Kononov, V. M. Antonov, V. V. Evstigneev. Izv. AN SSSR, ser. fiz., No. 2, 1964.