Study of the Process of Electron Capture into Betatron Acceleration for Various Forms of the Injection Voltage Pulse

I. P. Chuchalin

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

At the present time there is no generally accepted theory explaining the mechanism of electron capture into betatron acceleration; there are only separate attempts to explain how electrons bypass the electron gun \[1, 2, 3, 4, 5\].

The process of electron capture into acceleration is influenced by the phase nonuniformity of the magnetic field in azimuth and radius \[6, 7, and 8\].

There are as yet no convincing experimental data confirming one or another theoretical premise. Therefore, the accumulation of experimental material is useful.

It is not our task to consider theoretical explanations of the mechanism of electron capture. Here we shall present only some experimental results that may help in creating a unified theory of capture.

Fig. 1

In most modern betatrons and synchrotrons, to create the voltage pulse on the electrodes of the electron gun, the discharge of a previously charged artificial forming line into a pulse transformer is used.

When the forming line is discharged into a matched load of the pulse transformer, a rectangular voltage pulse is formed in its primary winding. Owing to the presence in the pulse transformer of leakage inductance and distributed capacitance of the windings, a distorted voltage pulse is produced in the secondary winding (Fig. 1). We shall call this pulse shape the “natural shape.”

When electrons are introduced into acceleration, their energy must correspond to the value of the magnetic-field intensity. The required voltage on the gun must satisfy the relation:

\[ U_{\mathrm{req}}=\frac{1}{2}\frac{e}{m_0}\left(r_0 H_a\right)^2, \tag{1} \]

where \(e\) is the charge of the electron, \(m_0\) is the rest mass of the electron, \(r_0\) is the radius of the equilibrium orbit, and \(H_0\) is the magnetic-field strength on the equilibrium orbit.

To ensure the maximum duration of the interval for admitting electrons into the gun, it is necessary to apply an “ideal” voltage pulse (Fig. 2). It was suggested that in this case the x-ray output would also be maximal.

Fig. 2

In practice it is difficult to form a voltage pulse of exactly this shape; however, it is possible to obtain a more or less good approximation. Note [9] mentions an attempt to approximate the ideal form by creating a step-shaped pulse. Pulses with one, two, and three steps were tested. The authors report an increase in radiation intensity of approximately 15% for each step.

Article [10] states that, with an injection voltage pulse having the form of a rising voltage, the radiation intensity increases fivefold in comparison with the intensity for a pulse with a flat top. The shape of the voltage pulse and the method of obtaining it are not indicated in the article.

V. M. Razin [11] studied the dependence of the radiation intensity on the shape of the injection voltage pulse. He formed and applied to the gun a voltage pulse close in shape to the “ideal” one (Fig. 3). Such a pulse was obtained by means of a capacitor of a certain value connected to the secondary winding of a pulse transformer.

Increasing the interval of electron admission when a voltage pulse close in shape to the “ideal” one was applied to the gun, at the optimal gun emission, did not lead to the expected increase in radiation intensity. An increase in radiation intensity was observed only when the gun heating was reduced relative to the optimal value.

Summarizing the results obtained, Razin concluded that the shape of the injection voltage pulse, at optimal gun heating, does not affect the maximum value of the radiation intensity for a given voltage amplitude in the pulse.

Fig. 4.

In investigating the operation of an artificial shaping line, we found another method for changing the shape of the injection-voltage pulse. Figure 4 shows a photograph of the “natural” pulse shape obtained in our experiments. The shaping line consisted of 10 sections (10 coils and 10 capacitors).

When one or several coils of the shaping line are short-circuited, the “natural” pulse shape is distorted. Figure 5 presents pulse shapes when from two to eight coils are short-circuited. It is seen from Fig. 5 that the shape of the injection-voltage pulse can be changed more or less smoothly.

Fig. 3

Fig. 4

Of special interest are “two-peak” pulses (as we shall call them below), in which the second voltage peak is higher than the first. The idea arose as to whether the radiation intensity could be doubled by placing the peaks of the pulse approximately on the same curve of the required injection voltage.

In experimental testing, our supposition was confirmed. Indeed, by applying a “two-peak” voltage pulse to the electron gun of a 15 MeV betatron, we obtained a maximum radiation intensity approximately twice as large as the maximum intensity obtained with the “natural” pulse shape. The gun emission in both cases was equal and optimal, and the amplitudes of the injection-voltage pulses were also equal.

An increase in the amplitude of the injection-voltage pulse was accompanied by a corresponding increase in the radiation intensity. Figure 6 presents the dependences of the maximum radiation intensity on the gun voltage for the “natural” pulse shape and for one of the “two-peak” pulses.

The radiation intensity depends on the ratio of the peak heights, i.e., on the shape of the injection-voltage pulse. Table 1 gives the dependence of the radiation intensity on the pulse shape. During these experiments the injection voltage (amplitude value) was equal to 40 kV, and the average current per layer was 220 μA.

Later, on another 15 MeV betatron, by applying the same “two-peak” voltage pulse to the gun, V. M. Razin also increased the maximum radiation intensity approximately twofold in comparison with the radiation obtained with the “natural” pulse shape.

Table 1

Fig. 5

Graph of the dependence of the maximum radiation intensity on the voltage at the gun.

1 — for the “natural” pulse shape,
2 — when three coils of the forming line are short-circuited.

Having discovered the dependence of the radiation intensity on the shape of the injection-voltage pulse, we could not at first explain the cause and character of this dependence. There could be two reasons here for the increase in radiation intensity: either the radiation increases owing to the presence of two peaks in the pulse, or because, for some ratio of the peak heights, the middle part of the pulse coincides for a longer time interval with the curve of the required voltage.

In order to clarify the cause of the increase in radiation intensity, we decided to investigate the capture of electrons into acceleration for different shapes of the injection-voltage pulse. We needed to know in what part, in time, of the injection-voltage pulse the capture of the maximum number of electrons emitted by the gun filament takes place.

The first to determine the moment of capture of electrons into acceleration was V. M. Razin [11], and somewhat later V. S. Melikhov [5]. The method for determining the moment of capture of electrons into acceleration is based on the following considerations.

At the moment when electrons are captured into acceleration, a circular electron current is formed in the chamber, creating its own magnetic flux \(\Phi_e\). The magnitude of this magnetic flux \(\Phi_e\) is directly proportional to the magnitude of the electron current, i.e., to the magnitude of the charge \(Q\) captured into acceleration:

\[ \Phi_e = k_1 Q, \tag{2} \]

where \(k_1\) is the proportionality coefficient.

When the magnitude of the charge circulating in the chamber changes, the magnitude of \(\Phi_e\) will also change, which will lead to the appearance of a vortex electric field near the poles of the betatron. If one places near the chamber one or several turns of wire encircling the chamber, then at the ends of these turns, in addition to the fundamental fifty-cycle voltage (the electromagnet was powered from a mains supply with frequency 50 cps), a rapidly varying voltage will appear:

\[ E = k_2 \frac{d\Phi_e}{dt} = k_1 k_2 \frac{dQ}{dt}. \tag{3} \]

It is evident that \(E\) will exist during the change in the circulating current, when \(\dfrac{dQ}{dt} \ne 0\), i.e., during the capture or exit of electrons from the acceleration process.

Using a separating \(CR\) circuit, the low-frequency fifty-cycle voltage can be filtered out and the high-frequency voltage \(E\) of interest to us can be isolated. By applying the voltage \(E\) to the vertical amplifier of an oscilloscope and observing the shape of the curve \(E(t)\), one can trace the process of change of the electron current in the chamber.

In the practical determination of the capture moment, a number of difficulties arise. Since the processes occurring during the injection of electrons are very brief (tenths of a microsecond), the oscilloscope amplifier must be broadband (up to 2–3 MHz). With a small passband of the amplifier, the shape of the curve \(E(t)\) may be strongly distorted.

The magnitude of the voltage \(E\), induced in the turns when the electron current changes, is very small; therefore the oscilloscope must have an amplifier with a large amplification factor.

In the turns encircling the chamber, parasitic high-frequency interference from the injection circuit may also be induced. When studying the moment of electron capture, it is necessary, as far as possible, to eliminate parasitic pickup and to ensure good and stable operation of the betatron, since at low radiation intensity the useful signal will also be very weak and unstable.

Figure 7 shows Razin’s circuit, with the aid of which he studied the capture of electrons into acceleration.

On the pole of the betatron, 6–10 turns of insulated wire were placed; the voltage from these, through a coupling capacitor \(C\) (several pF), was applied to the vertical amplifier of an EO-4 oscilloscope.

The voltage pulse from the permalloy sensor PT, located in the magnetic field of the betatron, was applied to the triggered-sweep unit Zh. R. and started the sweep in such a way that the phenomenon under investigation (the instant of electron capture, or the injection-voltage pulse) was reproduced in the middle part of the sweep line on the oscilloscope screen. The phase of generation of the voltage pulse in the winding of the permalloy sensor was varied by the circuit for adjusting the injection phase.

With such an adjustment, the phase of the injection-voltage pulse was correspondingly also changed. However, the relative position in time of the start of the sweep and the instant of generation of the injection-voltage pulse remained unchanged.

Thus, adjustment of the phase of the injection pulse for maximum radiation output did not change the relative position of the image of the injection-voltage pulse on the oscilloscope screen.

The injection-voltage pulse was applied to the vertical input of the EO-4 oscilloscope from a rheostat-capacitive divider \(D\) with switch \(K\) in position 2. The voltage to the divider came directly from the cathode of the electron gun.

Fig. 7

Block diagram for measuring the instant of capture of electrons into acceleration. 1—electron-injection circuit; 2—generator of trigger pulses; 3—converter of the pulses and local oscillator; 4—brightness-modulation and calibration unit; 5—sweep unit.

Razin’s experiments were not sufficiently perfect. He observed only changes in the electron current corresponding to different portions of the injection-pulse shape. A quantitative estimate of the character of the change in the electron current in the chamber at the instant of capture of electrons into acceleration still did not appear possible.

Since the turns were placed on the pole of the betatron, their coupling with the magnetic flux due to the changing electron current was weak; the useful signal was therefore also weak. In addition, the useful signal was distorted, since the amplifier of the EO-4 oscilloscope used in the experiments does not pass frequencies above 200 kHz.

More refined experiments on the study of electron capture into acceleration were those of V. S. Melikhov [5]. He placed the turns not on the pole of the betatron, but in the air gap above and below the chamber. Placing the turns directly near the chamber increased their coupling with the magnetic flux associated with the electron current; as a result, the useful signal also increased. Instead of the EO-4 oscilloscope, Melikhov used an “Orion” oscilloscope, whose amplifier is designed to pass frequencies up to 5 MHz.

To filter out the low-frequency alternating component, a coupling capacitor of capacitance \(0.1\,\mu\mathrm{F}\) was included in the circuit; to eliminate possible distortions, the cable at the end was loaded with a resistance \(R = 100\,\Omega\) (close to the characteristic impedance of the cable, \(\rho = 75\,\Omega\)). Constant-

... the time constant of the circuit, \(RC = 10\ \mu\text{sec}\), ensured almost complete filtration of the alternating voltage at \(50\ \text{cps}\), but did not introduce noticeable distortions into the shape of the pulses under study.

For our study of electron capture into acceleration we used Razin’s circuit with Melikhov’s improvements. Turns of thin insulated wire, from which the useful signal was taken, were attached to the outer wall of the chamber.

The coupling of the turns with the magnetic flux caused by the varying electron current was even stronger than in Melikhov’s experiments, which helped to increase the useful signal. The voltage under study from the turns was fed by a coaxial cable (with a characteristic impedance of \(200\ \Omega\)) through a coupling capacitor of capacitance \(C = 0.15\ \mu\text{F}\) to the terminals of the vertical amplifier of an “Orion” oscillograph. At the end, the cable was loaded by a resistance \(R = 200\ \Omega\). The use of a cable with increased characteristic impedance also helped to increase the signal.

Fig. 8

Figure 8 presents an oscillogram of the disturbances arising in the turns during operation of the injection circuit (the gun filament switched on). Since the energy of the electrons introduced into the chamber does not correspond to the magnetic-field strength on the orbit (the injection-voltage pulse is shifted in phase toward lagging relative to the curve of the required voltage), no capture of electrons into acceleration occurs.

When the phase of the injection-voltage pulse is shifted toward the curve of the required voltage, at some instant electron capture into acceleration begins to occur; a useful signal from the change of current in the chamber is superimposed on the disturbance signal, as a result of which the shape of the curve changes (Fig. 8 б, в, г).

The useful signal is, evidently, easily distinguished by comparing the curves in Fig. 8b and Fig. 8c, d, e.

Figure 8a shows the “natural” form of the injection-voltage pulse used in these experiments. The beginnings of the voltage changes on the oscillograms (point “a” in Fig. 8b, c, d, e) correspond to the beginning of the injection-voltage pulse (point “a” in Fig. 8a).

The curves (Fig. 8b, c, d, e) depict the rate of change of charge. Under the experimental conditions, downward deflection of the oscilloscope beam corresponded to an increase of the current in the chamber, and upward deflection to a decrease of the current. From expression (3) one can find that

\[ Q = \frac{1}{k_1 k_2}\int E(t)\,dt . \tag{4} \]

The integral \(\int E(t)\,dt\) is represented on the graph by the area between the curve \(E(t)\) and the abscissa axis. It follows from this that negative areas are proportional to the charge introduced, and positive areas to the charge thrown off.

From Fig. 8c it is seen that, when the form of the injection pulse is exactly matched to the curve of the required voltage, electron capture occurs once (there is one negative area).

When the injection pulse is shifted in phase in the direction of advance, its intersection with the curve of the required voltage occurs at two points. From Fig. 8d it is seen that in this case electrons are introduced twice (two negative areas): during the rise and during the fall of the injection voltage. With an even larger phase shift of the injection pulse in the direction of advance (Fig. 8e), the points of its intersection with the curve of the required voltage move apart; the moments of electron introduction corresponding to these intersection points likewise move farther apart; electron introduction still occurs at two points (during the rise and during the fall of the injection voltage).

Figure 8f gives an oscillogram of a calibration curve with a frequency of 375 kc/s, by means of which the time scale for the oscillograms (Fig. 8a, b, c, d, e) can be determined.

All the oscillograms described above show the variation of the voltage \(E(t)\), induced in the turns by the magnetic flux caused by the changing electron current. According to (3), the voltage \(E(t)\) is proportional to the rate of change of the charge circulating in the betatron chamber. It is evident that the process of charge variation in the chamber would be more readily visualized if one oscillographed not the rate of change of charge, but directly the change of charge itself. For this it is necessary, evidently, to integrate the voltage \(E(t)\). This can be done with the aid of the simplest electrical integrating circuit \(RC\). However, it is impossible to use only such a simplest integrating circuit, since during integration the useful signal is sharply reduced and the sensitivity of the oscilloscope proves insufficient.

To make it possible to observe directly the change of charge in the chamber, Melikhov [5] used the circuit of a single-stage amplifier with an integrating cell in the anode circuit (Fig. 9).

The 6S2 tube has an internal resistance of about \(7\ \text{k}\Omega\); therefore, the time constant of the integrating circuit at \(R_a = 100\ \text{k}\Omega\) and \(C_a = 0.01\ \mu\text{F}\) is approximately \(65\ \mu\text{s}\), i.e., it exceeds the duration of the processes occurring when electrons are introduced into the chamber by a factor of 20–30. This is quite sufficient to obtain the required accuracy of integration.

For our study of electron capture with different pulse shapes we used the same integration circuit. The only difference was that the voltage from the turns was fed through a coaxial cable with characteristic impedance \(\rho = 200\ \Omega\); the blocking capacitor had capacitance \(C = 0.15\ \mu\mathrm{F}\), and the cable was terminated at the end by a resistance \(R = 200\ \Omega\).

Fig. 9

In Fig. 10 the curves of the charge variation in the chamber for the “natural” pulse shape are presented. Oscillogram \(a\) shows the pulse shape; oscillogram \(b\) represents the interference signal; oscillograms \(v, g, d\) represent the total signals in the presence of electron injection into the chamber for three different values of the injection phase. In photograph \(e\), oscillograms \(b\) and \(v\) are superimposed in order to isolate the useful signal.

Figure 11 shows the useful signals isolated as a result of analysis of the oscillograms (Fig. 10). Curve \(v\) corresponds to exact matching of the injection-voltage pulse with the curve of the required voltage; curve \(g\) corresponds to a leading phase, and curve \(d\) to an even more leading injection phase.

Figure 12 presents the useful signals isolated in the analysis of the oscillograms from a two-peak pulse. Curve \(v\) corresponds to exact matching of the injection-voltage pulse with the curve of the required voltage. As can be seen, capture initially begins from the second, higher peak. Curves \(g, d, e\) correspond to leading injection phases.

Figure 13 represents the useful signals isolated in the analysis of oscillograms from another two-peak pulse.

As a result of the work carried out, the following conclusions can be drawn.

1. By applying to the electrodes of the electron gun, instead of a half-sinusoidal or rectangular voltage pulse, a “two-peak” pulse, one can almost double the number of electrons captured into acceleration, i.e., double the radiation intensity from a betatron or synchrotron with betatron injection.

2. The shaping of a “two-peak” voltage pulse is achieved very simply: for this it is necessary to short-circuit several coils of the artificial pulse-forming line (Fig. 14).

3. Since the injection of electrons occurs not only onto one equilibrium orbit, but into a region of orbits lying inside the vacuum chamber, there exists not one curve of required voltage, but an entire region of required voltages (Fig. 15). Hence it becomes clear that, in order to increase the maximum radiation intensity, it is not at all necessary to arrange

Fig. 10.

Subfigure labels: a), b), c), d), e), f).

Fig. 11.

Vertical axis: \(Q\)

Horizontal axis: \(t\), μsec

Tick labels on the horizontal axis: 0, 3, 6, 9, 12, 15, 18, 21, 24, 27.

Subfigure labels: b), g), d).

Fig. 12.

Vertical axis: \(Q\)

Horizontal axis: \(t\), μsec

Tick labels on the horizontal axis: 0, 3, 6, 9, 12, 15, 18, 21, 24, 27.

Subfigure labels: a), g), d), e).

both peaks of the pulse on a single curve of the required voltage. It is required that both peaks fall within the region of the required voltages.

1. Capture of electrons into acceleration is possible both during the rise and during the fall of the injection-voltage pulse (this is evident from all the capture oscillograms).

Fig. 13

Fig. 14

Fig. 15

1. After each introduction of electrons, they are dumped, and a considerable part of the introduced charge is dumped. The latter circumstance, apparently, is the decisive reason why the remaining electrons bypass the gun.

2. Our results confirm Razin’s conclusions that increasing the interval during which electrons are admitted into the chamber does not lead to an increase in the accelerated charge. The accelerated charge, and consequently also the intensity of the $\gamma$-radiation, can be increased by applying to the gun a voltage pulse that would cross the curve of the required voltage many times. These crossings should not be too frequent; the electrons remaining after the first captures must have time to form into stable bunches, which will not be harmed by the charges subsequently captured into acceleration.

REFERENCES

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