Some Regularities in the Formation of Nonstationary Current in a Betatron

O. V. Sokolov, B. A. Chastokolenko

(Presented by the scientific seminar of the Scientific Research Institute of Electrical Engineering)

During the injection of electrons into the accelerating chamber of a betatron, conditions are created for the short-term circulation of a current of large magnitude, called nonstationary. Its magnitude is many times greater than that of the current captured into the acceleration regime. Thus, for example, it follows from [1, 2] that the magnitude of the current captured into the acceleration regime is 1–10% of the nonstationary current. In the 15 MeV betatron of TPI, at an injection voltage of 20 kV, the nonstationary current is 300 mA, whereas the current captured into acceleration has a magnitude on the order of 15 mA. In this connection, the study of the nonstationary current is of interest both for clarifying the nature of its formation and for creating conditions for capturing it into the acceleration regime.

At present, a number of works [1–7] contain information on the magnitude and lifetime of the nonstationary current, on the dependence of the nonstationary current on the emission current from the injector and on the divergence angle of the electron beam at the injector exit, on the average number of revolutions of the electrons forming the nonstationary current, and on the formation of nonstationary current when the betatron electromagnet is powered by direct current.

However, the data available in the literature do not provide an idea of the regularities in the formation of the nonstationary current.

In works [9, 11–14], a mechanism is given for the formation of the nonstationary current, based on the “resonant” interaction of circulating electrons with the emission current from the injector, and the possibility is shown of capturing into acceleration a portion of the electrons of the nonstationary current and, on this basis, increasing the intensity of the accelerated-electron beam by a factor of 5 or more. The results presented in the indicated works, obtained by computer modeling of the process of nonstationary-current formation in a 15 MeV betatron for various injection and magnetic-field parameters, have been carried through to numerical results and make it possible to compare them with experimental results.

Qualitative and quantitative agreement of the experimental results with the results of modeling the process of nonstationary-current formation can serve as a criterion for the correctness of the “resonant” mechanism of its formation.

In the present work, new data are presented, obtained in an experimental study of nonstationary current in a betatron, confirming the “resonant” mechanism of its formation.

Measurement Method

The study of nonstationary current involves a number of difficulties, associated mainly with its short duration (0.2–1.0 μsec), the small magnitude of the signal it produces in the pickup, and also the presence of noise generated by the betatron control circuits during the measurements. The nonstationary current was measured using a magnetic-induction pickup in the form of 8 turns placed on the surface of the accelerating chamber along the electron orbits. The pickup had a sensitivity of 12 mV/mA and was calibrated against the current of electrons that made one revolution in the accelerating chamber, after which they struck a shutter covering the entire cross section of the accelerating chamber. The characteristics of the pickup and the calibration method are given in [8]. The sensitivity of such a pickup is approximately an order of magnitude higher than the sensitivity of a pickup in the form of a magnetic belt. This makes it possible to feed the signal from the pickup directly to the input of the oscilloscope without additional amplification and thus substantially simplifies the measurement procedure in comparison with that used in [3].

In contrast to [3], where the average number of revolutions of the electrons forming the nonstationary current was determined as the ratio of the amplitude of the nonstationary current to the emission current from the injector, which led to certain distortions of the measurement results, especially in the region of small emission currents from the injector, we introduced the ratio

\[ \overline{N}=\frac{I_{\mathrm{n}}}{I_1}, \]

where \(I_{\mathrm{n}}\) is the amplitude of the nonstationary current measured with the pickup,

and \(I_1\) is the amplitude of the current produced by the electrons at the shutter [8] after they have made one revolution.

Figure 1 presents the dependences of \(I_{\mathrm{n}}\) and \(I_1\) (curves 1 and 4, respectively) on the emission current from the injector (\(I_{\mathrm{em}}\)) and the dependence \(\overline{N}\) on \(I_{\mathrm{em}}\) constructed from them (curve 3).

Fig. 1. \(1\) — \(I_{\mathrm{n}}\); \(2\) — \(I_\gamma\); \(3\) — \(\overline{N}\); \(4\) — \(I_1\)

\[ (U_i = 20\ \text{kV}) \]

Measurement Results and Their Discussion

Figures 1 and 2 present the dependences of the nonstationary current \((I_{\mathrm{n}})\) on the emission current from the injector \((I_{\mathrm{em}})\); moreover, for comparison, Fig. 1 also gives the dependences of the intensity of the bremsstrahlung x-radiation \((I_{\gamma})\) and of the mean number of revolutions of the electrons forming the nonstationary current \((\overline{N})\) (curves 2 and 3, respectively) for the same injection parameters; Fig. 2 includes the dependences of the nonstationary

Fig. 2. 1 — \(U_i = 8\ \mathrm{kV}\); 2 — \(U_i = 12\ \mathrm{kV}\); 3 — \(U_i = 20\ \mathrm{kV}\)

current on the injection voltage. It is seen that at first, as the emission current from the injector increases, the nonstationary current rises rapidly; then the rate of increase decreases and its value approaches a certain limiting value \((I_{\mathrm{n,lim}})\), determined by the magnitude of the injection voltage \(U_i\). Comparing curves 1, 2, and 3 (Fig. 1), we see that the rapid increase of the nonstationary current corresponds to an increase in the mean number of revolutions of the electrons forming the nonstationary current, \(\overline{N}\), while the greatest value of \(\overline{N}\) 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 rise of the nonstationary current is due to an increase in the number of electrons per revolution as a result of the increase in \(I_{\mathrm{em}}\). Attention is drawn to the fact that the onset of the nonstationary current occurs at a certain threshold emission current from the injector \(I_{\mathrm{em,0}}\), whereas the greatest values of \(\overline{N}\) and \(I_{\gamma}\) are reached at another value of the emission current, \(I_{\mathrm{em,opt}}\). This indicates 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, created by the emission current from the injector.

The extremal character of curves 2 and 3 (Fig. 1) testifies to the resonant nature of the interaction of the circulating current with the emission current from the injector. The resonant character of the mechanism of formation of the nonstationary current is confirmed by the dependences in Fig. 2, which show that, with increasing energy of the injected electrons, a larger emission current from the injector is required for the formation of the nonstationary current. From the standpoint of the resonant mechanism [11—14], the formation of the nonstationary current must occur owing to damping of the oscillations of some part of the injected electrons as a result of an azimuthal change in the focusing forces of the magnetic field due to the nonuniform distribution of the density of the space charge. An increase in the energy of the injected electrons must be accompanied by

with an increase in the oscillation energy; therefore, a larger value of the space charge is required for damping the oscillations of the electrons.

The absence of commensurability between the period of revolution of the electrons and the period of their oscillations leads to a change in the conditions for entry of the electrons into the inhomogeneity created by the electron beam emerging from the injector. As a result, resonant damping is replaced by resonant growth of the amplitudes of the electron oscillations, which leads to the loss of a larger fraction of the circulating electrons after several turns. This can explain the short lifetime of the nonstationary current and its dependence on the magnitude, azimuthal position, extent, and lifetime of the inhomogeneity created by the emission current from the injector.

Depending on the divergence angle of the electron beam upon exit from the injector, the extent of the inhomogeneity in the azimuthal distribution of the density of the space charge created by it changes [9, 12]. In Fig. 3, a, the dependence of the nonstationary current \(I_{\mathrm{n}}\) on the injector emission current \(I_{\mathrm{em}}\) is presented for various injector divergence angles \(\gamma_{\mathrm{i}}\). For comparison, the dependence of the intensity of bremsstrahlung radiation \(I_\gamma\) on \(I_{\mathrm{em}}\) is given (Fig. 3, b). It is seen that the largest values of the nonstationary current and of the bremsstrahlung intensity are observed at a divergence angle of \(13^\circ\).

Fig. 3. 1 — \(\gamma_{\mathrm{i}} = 20^\circ\); 2 — \(\gamma_{\mathrm{i}} = 13^\circ\); 3 — \(\gamma_{\mathrm{i}} = 5^\circ\) \((U_{\mathrm{i}} = 20\ \text{kV})\)

The table gives the values of \(I_{\mathrm{em},0}\) (threshold value of the emission current from the injector), \(I_{\mathrm{em,opt}}\) (value of the emission current corresponding to the radiation maximum), \(I_{\mathrm{n,opt}}\) (magnitude of the nonstationary current circulating at \(I_{\mathrm{em,opt}}\)), \(I_{\mathrm{n,lim}}\) (limiting value of the nonstationary current), and \(I_\gamma\) (maximum intensity of the betatron radiation) for different injection voltages \(U_{\mathrm{i}}\) and durations of the injection pulse \(\tau_{\mathrm{i}}\). The shape of the injection pulse is bell-like. The quantities marked with an asterisk are taken for comparison from [10]. From the table it is evident that a decrease in the duration of the injection pulse, which should

Table

leads to a decrease in the lifetime of the nonuniformity created by the emission current from the injector, and to an increase in the rate of its change; it leads to an increase in the threshold value of the emission current from the injector \(I_{\mathrm{em.o}}\), and also to an increase in the nonstationary current.

Fig. 4. \(1\)—\(U_i=12\) kV; \(2\)—\(U_i=22\) kV

In Fig. 4 the dependence of the quantity \(\dfrac{I_{\mathrm{n}}}{I'_{\mathrm{n}}}\) on the amplitude of the first harmonic of the phase nonuniformity of the magnetic field of the betatron \(\Delta H\) is presented. Here \(I'_{\mathrm{n}}\) is the magnitude of the nonstationary current at \(\Delta H=0\) (the betatron electromagnet was supplied with direct current), and \(I_{\mathrm{n}}\) is the magnitude of the nonstationary current measured at the optimal value of the emission current from the injector \(I_{\mathrm{em.opt}}\) for different \(\Delta H\).

It is seen from Fig. 4 that, for \(U_i=12\)—22 kV, a sharp decrease in the magnitude of the nonstationary current occurs at \(\Delta H>7\)—8 gauss. It was noted that capture of electrons into acceleration still takes place if \(I_{\mathrm{n}}/I'_{\mathrm{n}}>0.25\), which corresponds to \(\overline{N}>1.4\). With an increase in the amplitude of the phase nonuniformity of the magnetic field, \(I_{\mathrm{em.o}}\) and \(I_{\mathrm{em.opt}}\) increase; i.e., in this case too an essential role is played by the magnitude of the nonuniformity in the distribution of the space-charge density created by the emission current from the injector.

It is seen from Figs. 1 and 3 that the optimal emission current from the injector, corresponding to the maximum radiation of the betatron at an angle of divergence of the injector beam \(\gamma_i=20^\circ\), corresponds to a nonstationary-current value \(I_{\mathrm{n}}=340\) mA, and an average number of electron revolutions \(\overline{N}\approx 6\); at \(\gamma_i=13^\circ\), \(I_{\mathrm{n}}=600\) mA; at \(\gamma_i=5^\circ\), \(I_{\mathrm{n}}=330\) mA. According to the results—

there, calculations with the aid of an electronic digital computer (for \(U_i=20\ \mathrm{kV}\), \(I_{\mathrm{em}}=200\ \mathrm{mA}\)) give the following values for the nonstationary current \((I_n)\) and \(\overline{N}\): \(I_n=380\ \mathrm{mA}\), \(\overline{N}=6.4\) (for \(\gamma_i=20^\circ\)); \(I_n=630\ \mathrm{mA}\), \(\overline{N}=7.5\) (for \(\gamma_i=13^\circ\)); \(I_n=380\ \mathrm{mA}\), \(\overline{N}=1.9\) (for \(\gamma_i=5^\circ\)) [12–14]. Thus, there is not only qualitative but also quantitative agreement between the calculated and experimental results.

Conclusion

The study of the nature of the regularities in the formation of the nonstationary current leads to the conclusion that an essential role in the mechanism of formation of the nonstationary current is played by the interaction of circulating electrons with the emission current from the injector. The extremal character of the dependences (the latter quantity is proportional to the current captured into the acceleration regime), the short lifetime of the nonstationary current, and also the agreement of the experimental results with the results obtained in calculations on an electronic digital computer, proceeding from the hypothesis of resonant formation of the nonstationary current, confirm the “resonant” mechanism of this interaction.

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