STUDIES ON INJECTION IN A BETATRON

O. V. Sokolov, B. L. Chastokolenko

(Presented by the Scientific-Research Institute of Electron Introscopy)

A general regularity for betatrons and synchrotrons with betatron pre-acceleration is the formation, during injection, of a closed circular current of electrons of large magnitude.

This current, often called the nonstationary current, in its amplitude exceeds by tens and hundreds of times the current of the electrons captured into acceleration, and by several times may exceed, in amplitude, the magnitude of the injection current \([1—7, 11—14]\). In view of the large magnitude of the nonstationary current, it is of interest to investigate the mechanism of its formation, and also to try to create conditions for its capture into acceleration in order to increase the efficiency of betatron operation.

The formation of the nonstationary current is connected with the capture of electrons into acceleration. However, neither in theoretical works considering various mechanisms of electron capture into acceleration, nor in experimental works—with the exception of \([10]\), where the results of the study are considered on the basis of a simplified picture of the motion of a narrow electron beam in the magnetic field of a betatron—has the question of the mechanism of formation of the nonstationary current been considered. The reason for the short-lived existence of the nonstationary current has also not been clarified; the influence of the electron beam emerging from the injector on the character of its formation is not taken into account.

Consideration of the conditions of electron motion in the accelerating chamber during multi-turn injection led us to the conclusion that one of the essential factors in the mechanism of formation of the nonstationary current may be the resonant damping of the amplitudes of oscillations of the electrons forming the nonstationary current. The electron beam emerging from the injector has a definite angle of divergence, owing to angular spread, and also under the action of its own space charge. As a result of the beam divergence, the space-charge density in azimuth will be distributed nonuniformly; this, in turn, should lead to a nonuniform azimuthal distribution of the effective value of the focusing forces of the betatron magnetic field. Thus, conditions are created for resonant damping of the oscillations of the circulating electrons.

Calculations carried out by us of the trajectories of electron motion in a 15 MeV betatron during injection, taking into account the divergence of the injected electron beam, confirmed the resonant character of the pro-

…of the process of formation of the nonstationary current. The calculations of the trajectories of electron motion were based on the solution of the differential equation of radial betatron oscillations, taking into account the azimuthal variation of the magnitude of the focusing forces of the magnetic field:

\[ \frac{d^{2}\rho}{d\theta^{2}}+[1-n(\theta)]\rho=0, \tag{1} \]

where \(\rho\) is the deviation of the electron from the equilibrium orbit;
\(\Theta\) is the independent variable (azimuth);
\(n(\Theta)\) is the effective value of the field index of the betatron magnetic field.

The value \(n(\Theta)\) may be represented as

\[ n(\theta)=n_{\mathrm{cp}}+\Delta n(\theta), \tag{2} \]

where \(n_{\mathrm{cp}}\) is the mean value of the field index of the betatron magnetic field;

Fig. 1. Calculated values of \(\Delta n(\theta)\) for \(\gamma_i=20^\circ\) and \(\gamma_i=13^\circ\)

\(n(\Theta)\) is the change in \(n\) associated with the emission current from the injector.

Figure 1 presents the values of \(\Delta n(\Theta)\) obtained by calculation for a 15 MeV betatron at injector divergence angles \((\gamma_i)\) of \(20^\circ\) and \(13^\circ\) [9]. Equation (1) was solved by the Runge–Kutta method with the aid of a digital computer. Figure 2 shows the change in the amplitude of electron oscillations \((\rho)\) as a function of the number of electron revolutions \((N)\), obtained as a result of calculations of electron trajectories for a 15 MeV betatron (injection voltage \(U_i=20\ \mathrm{kV}\); emission current from the injector \(I_{\mathrm{em}}=150\ \mathrm{mA}\); \(n_{\mathrm{cp}}=0.66\); \(r_0=130\ \mathrm{mm}\), \(r_i-r_0=23\ \mathrm{mm}\), where \(r_i\) is the injection radius). The initial decrease in the amplitudes of the electron oscillations is due to the fact that the conditions for entry of electrons into the inhomogeneity during the first revolutions are favorable for damping the oscillation amplitudes; subsequently, as a result of a change in the phase of entry of the electrons into the inhomogeneity, a resonant increase in the amplitudes of the electron oscillations occurs.

Fig. 2. Change in the amplitude of electron oscillations \((\rho)\) as a function of the azimuth or of the number of electron revolutions \((N)\) in the accelerating chamber

Figure 3 presents the dependences of the number of electron revolutions \((N)\) forming the nonstationary current on the electron emergence angle \((\gamma)\)

from the injector at injector divergence angles \((\gamma_i)\) of \(20^\circ\) and \(13^\circ\). From these dependences one can find the average number of revolutions of the electrons forming the nonstationary current and, assuming that the electron density over the cross section of the injected beam is uniformly distributed, determine the magnitude of the nonstationary current from the formula

\[ I_n=\frac{\gamma_n}{\gamma_i} I_{\mathrm{em}}\cdot \overline{N}, \tag{3} \]

where \(\gamma_n\) is the interval of departure angles of the electrons forming the nonstationary current;

Fig. 3. Dependence of the number of electron revolutions \((N)\) on the electron departure angle \((\gamma)\) from the injector

Fig. 4. Dependences of \(I_n\), \(\overline{N}\), and \(I_\gamma\) on the emission current from the injector \((I_{\mathrm{em}})\) at \(\gamma_i=20^\circ\), \(U_i=20\ \mathrm{kV}\)

\(\gamma_i\) is the divergence angle of the electron beam at the exit from the injector;

\(\overline{N}\) is the average number of revolutions of the electrons forming the nonstationary current;

\(I_{\mathrm{em}}\) is the emission current from the injector.

After substituting all the necessary quantities into (3), it was found that at \(U_i=20\ \mathrm{kV}\), \(I_n=290\ \mathrm{mA}\) for \(\gamma_i=20^\circ\), and \(I_n=520\ \mathrm{mA}\) for \(\gamma_i=13^\circ\). Experimental measurements of the value of the nonstationary current circulating in the 15-MeV betatron at \(U_i=20\ \mathrm{kV}\) gave values of 340 and 600 mA, respectively, for \(\gamma_i=20^\circ\) and \(\gamma_i=13^\circ\).

Figures 4 and 5 show the dependences of the nonstationary current \((I_n)\) on the emission current from the injector \((I_{\mathrm{em}})\), taken at \(\gamma_i=20^\circ\); for comparison, Fig. 4 also gives the dependences of the bremsstrahlung intensity \((I_\gamma)\) and of the average number of revolutions of the electrons forming the nonstationary current \((\overline{N})\) at the same injection parameters, while Fig. 5 includes the dependences of the nonstationary current on the injection voltage \((U_i)\).

It is seen that initially, with an increase in the emission current from the injector, the nonstationary current rises rapidly; then the rate of increase decreases and its value approaches a certain limiting value \((I_{n,\mathrm{lim}})\), determined by the magnitude of the injection voltage \((U_i)\). Comparing the curves \(I_n\) and \(\overline{N}\) (Fig. 4), we see that the rapid rise of the nonstationary current corresponds to an increase in the average number of revolutions-

electrons that form the nonstationary current; moreover, the maximum value \(\overline{N}\) corresponds to the maximum intensity of the betatron bremsstrahlung \((I_\gamma)\). With a further increase in the emission current from the injector \((I_{\mathrm{em}})\), \(\overline{N}\) decreases, and the slow growth of the nonstationary current is due to an increase in the number of electrons per turn as a result of the increase in \(I_{\mathrm{em}}\). Attention is drawn to the fact that the onset of the nonstationary current begins at a certain threshold emission current from the injector, while the greatest values of \(\overline{N}\) and \(I_\gamma\) correspond to another value of the emission current. This is experimental confirmation that the decisive role in the process of formation of the nonstationary current is played by the presence of a space charge of a certain magnitude, produced by the emission current from the injector. The resonant character of the mechanism of formation of the nonstationary current is also confirmed by the extremal character of the curves \(\overline{N}\) and \(I_\gamma\) (Fig. 4) and by the dependences in Fig. 5, which show that, as the energy of the injected electrons is increased, a larger emission current is required for the formation of the nonstationary current.

Fig. 5. Dependences of \(I_n\) on the emission current from the injector \((I_{\mathrm{em}})\) at different \(U_i\).

In studying the nonstationary current, it is of interest to consider the possibility that its decay may be affected by factors whose action could fully or partially eliminate the possibility of its capture into the acceleration regime. In this connection, the effect on the decay of the nonstationary current of resonances of betatron oscillations, as well as of longitudinal and transverse instability of the electron beam during injection, was investigated. Calculations showed that, under the action of the space charge of the nonstationary current, such a change occurs in the relative frequencies of the vertical and radial oscillations that the conditions for the occurrence of resonances of betatron oscillations are satisfied. According to the experiments, the decrease in the magnitude of the nonstationary current due to resonances of betatron oscillations amounted to 10–15% of its maximum value; i.e., resonances of betatron oscillations cannot be the cause of the decay of the nonstationary current. Calculations and experiments carried out by us show that transverse instability of the beam during the existence of the nonstationary current does not occur for the available injection parameters. The longitudinal instability has a time constant of \(\approx 630\ \mu\text{s}\), and therefore its influence may also be excluded.

Modeling of the processes in the chamber, taking into account the inhomogeneity produced by the beam, the defocusing action of the toroidal space charge of the electrons of the nonstationary current, and also the variation of these quantities with time, indicates that the basis of the capture mechanism, as well as the basis of the mechanism of formation of the nonstationary current, is the resonant damping of the amplitudes of electron oscillations.

Modeling of the process of formation of the nonstationary current by means of an electronic computer shows that, for the existing values of the inhomogeneity \(\Delta n(\Theta)\) produced by the emission current from the injector, its very presence ca-

by itself does not lead to an irreversible decrease in the amplitude of the electron oscillations, i.e., to the capture of electrons into acceleration. From the general theoretical point of view, in order to create conditions for the final capture of electrons it is necessary in some way to change, in time, the conditions of electron motion, either by changing the focusing magnetic field or by changing the electric injection field.

Such conditions, as follows from the calculations, arise in two cases:

1. With a rapid decrease in time of the magnitude of the nonstationary current (i.e., as a result of the rapid disappearance of the toroidal space charge that produced defocusing).

2. With a rapid decrease in time of the inhomogeneity \(\Delta n(\Theta)\) produced by the beam over a time of the order of several nanoseconds (i.e., as a result of the disappearance of the cause of the resonant growth of the amplitudes of electron oscillations after damping).

The first mechanism, associated with the existence and rapid disappearance of the electrons of the nonstationary current, is apparently the usual mechanism of capture in a betatron, characterized by low efficiency. Thus, as a result of the accumulation of the nonstationary current and its rapid disappearance, conditions arise for the capture into acceleration of a small fraction of its constituent electrons (3–5%).

Another capture mechanism, not associated with the loss of electrons of the nonstationary current and arising upon the rapid disappearance of the inhomogeneity created by the beam, i.e., with the rapid interruption of the emission current from the injector, should have greater efficiency. With ideal synchronization accuracy and stabilization of the injection pulse with a front steepness of the order of several nanoseconds, as follows from the calculations, it is possible to capture into acceleration up to 30% of the electrons of the nonstationary current, i.e., to increase the efficiency of electron capture into acceleration by \(\sim\)4–5 times.

Fig. 6. Oscillograms of the nonstationary current (first pulse) and the current of electrons captured into acceleration (second pulse): a) with an ordinary injection pulse; b) with an injection pulse with a steep trailing edge

We experimentally checked the possibility of increasing the number of accelerated electrons when injecting pulses with a sharp cutoff of the trailing edge. It should be noted that in producing injection pulses with a trailing-edge duration of \(10^{-9}\) sec and a synchronization accuracy of the order of \(10^{-9}\) sec at a pulse repetition rate of 50 Hz, considerable difficulties arise. In the experimental study of capture of the nonstationary current we used an injection pulse with an amplitude of 20 kV and a trailing-edge steepness of \(2.5 \cdot 10^{-8}\) sec. In this case the captured current was, in individual cases, 23–25 mA, whereas with a sinusoidal pulse of approximately the same duration and at the same voltage the captured current did not exceed 14 mA. An estimate of the instability of the injection phase gave a value of 5.5%, i.e., the synchronization accuracy in carrying out the experiment was within \(1.8 \cdot 10^{-6}\) sec. To reduce the instability, we carried out experiments on capture of the nonstationary current into acceleration while feeding the betatron electromagnet with direct current. The instability in this case was estimated as 0.75%, i.e., the synchronization accuracy was within \(2.4 \cdot 10^{-7}\) sec. The magnitude of the current captured into acceleration was 33 mA. Measurements of the captured current in that

in the other case it was carried out by means of a magnetic-induction sensor in the form of turns arranged on the accelerating chamber along the electron orbits, and by creating an electric field in the accelerating chamber to interrupt the circulating current [8].

Figure 6 shows oscillograms of the measured nonstationary current and of the current captured into acceleration (the first pulse corresponds to the nonstationary current, the second to that captured into acceleration):

a) with the usual sinusoidal injection pulse;
b) with an injection pulse having a steep trailing edge [13–14].

Thus, the possibility of capturing a part of the nonstationary current into the acceleration regime is confirmed experimentally. To achieve greater efficiency it is necessary to create more exact synchronization conditions and to ensure greater steepness of the trailing edge of the injection pulse.

Conclusions

1. The formation of a nonstationary current occurs as a result of the accumulation in the accelerating chamber of electrons injected at different instants of time, having small oscillation amplitudes and making more than one revolution in the accelerating chamber. The appearance of electrons with small oscillation amplitudes and making a sufficiently large number of revolutions occurs as a result of the “resonant” interaction of the circulating electrons with the emission current from the injector.

2. As a result of the rapid disappearance of the nonstationary current, conditions are created for the capture into acceleration of some small fraction of the electrons composing it (3–5%).

3. The possibility has been elucidated of implementing a more effective capture mechanism that is not connected with losses of electrons of the nonstationary current and that arises upon rapid interruption of the injection current.

4. Another method of increasing the number of accelerated electrons may be the abandonment of multi-turn injection with its inherent resonant mechanism and the implementation, for example, of single-turn oscillation-free injection.

REFERENCES

1. V. P. Yashukov. Atomic Energy, 8, 150, 1960.

2. D. P. Ivanov, A. P. Komar, Yu. S. Korobochko. ZhTF, 29, 978, 1959.

3. D. P. Ivanov, A. P. Komar, Yu. S. Korobochko. ZhTF, 29, 1235, 1959.

4. O. E. Zhukov, D. P. Ivanov, Yu. S. Korobochko. ZhTF, 38, 1067, 1968.

5. V. N. Logunov, E. P. Ovchinnikov et al. ZhTF, 27, 1143, 1957.

6. I. P. Chuchalin. Izv. TPI, 87, 256, 1957.

7. I. M. Goloshapov. Izv. TPI, 87, 275, 1957.

8. O. V. Sokolov, B. L. Chastokolenko. Izv. vuzov SSSR, “Fizika,” 9, 106, 1970.

9. O. V. Sokolov. Dissertation. Tomsk, 1964.

10. Yu. N. Lobashov, N. I. Tulipova. ZhTF, 31, 194, 1961.

11. O. V. Sokolov, B. L. Chastokolenko. Izv. vuzov SSSR, “Fizika,” 7, 83, 1971.

12. O. V. Sokolov, B. L. Chastokolenko. Proceedings of the VII Inter-University Conference on Electron Accelerators. Issue 2, 13. Moscow, Atomizdat, 1970.

13. O. V. Sokolov, B. L. Chastokolenko. Materials of the All-Union Scientific-Technical Conference on the Use of Accelerators in the National Economy and Medicine. Report 048, Leningrad, 1971.

14. B. L. Chastokolenko. Dissertation. Tomsk, 1971.