On the Existence of a Closed Beam of Electrons in a Constant Potential Field of Focusing Forces

N. M. Goloshchapov

(Submitted by the scientific seminar of the Faculty of Physics and Technology)

The question of the existence of a closed electron beam in a constant focusing magnetic field has at present acquired considerable interest, since in some theoretical works [1], as well as in proposals for new types of accelerators, it is assumed that under certain conditions such a beam can exist.

The purpose of the present work was to investigate the possibility of the existence of such a beam.

In studying the motion of a beam of electrons of constant velocity in a constant magnetic field, similar in form to that of a betatron, we used an experimental setup that we had employed in another work. The constant magnetic field is produced by an electromagnet with laminated cores—poles from a 15 MeV betatron. Into the chamber, successively from one tube to another, there is placed a partition consisting of a system of Faraday cylinders insulated from one another. From the results of measuring the current from each cylinder, with the current receiver placed successively in all seven tubes, it was possible to find the distribution of the beam current density over the cross section of the chamber, and also to determine the beam trajectory over almost a complete revolution.

Fig. 1. Motion of the beam during the first revolution. The lines connect points of the beam with equal current density in \(\mu\text{A}/\text{cm}^2\).

It turned out that the beam exists throughout the entire first revolution, and the beam current still remains very large in comparison with that which would be captured in acceleration. The distribution of the beam current density for \(U_i = 1000\) V, for the field strength on the equilibrium orbit \(H_{zc} = 7\) oersted, and for the injection current \(I_i^{\mathrm{rav}} \simeq 600\ \mu\text{A}\), which corresponds to the equilibrium charge density of the beam for the given conditions, is shown in Fig. 1. In this case

the current density at the center of the beam at the end of a revolution (at an azimuth of \(309^\circ\) from the injector) is greater than \(20\ \mu\mathrm{A}/\mathrm{cm}^2\). The beam is asymmetric and changes its shape with azimuth, which is due to the initial charge density of the beam, the quality of the field, the geometry of the injector and its position at the moment of injection; and the total beam current during the first revolution decreases by approximately a factor of 5–6 (the current from the conducting layer of the chamber increases by the corresponding amount).

Fig. 2. Variation of the beam current strength with azimuth during the first revolution

Let us note that at the end of the first revolution the beam does not turn into a uniformly distributed electron plasma, but is preserved precisely in the form of a beam, and the rate of decrease of the beam current strength is significantly reduced (Fig. 2).

Not considering the experiments described above sufficient to speak of the possibility of the existence of a closed electron beam (which, in fact, we could not have observed with the given experimental method), we conducted other experiments under conditions in which a pulsed voltage was applied to the injector. This facilitates observation of a closed electron beam.

Fig. 3. Injection circuit

The electromagnet used was the same one; the current receiver was removed from the chamber, and the injector was supplied by a pulsed voltage from the injection circuit (Fig. 3).

If it is assumed that the existence of a closed beam in a constant magnetic field is possible, then during the interval of time when the injection voltage corresponds to the magnetic field (i.e., to the required voltage), a beam will be created with a density varying from 0 to

maximum and again to 0 as the beam disperses. Thus a pulse of the beam current and a pulse of the magnetic flux of this beam will be produced.

If a turn (or several turns) is placed along the outer or inner diameter of the chamber, an emf will be induced in it proportional to the change of this flux, i.e.,

\[ E=\frac{d\Phi_e}{dt}=\frac{dQ}{dt}, \]

where \(Q\) is the quantity of charge in the beam.

Such a method, first proposed at the Tomsk Polytechnic Institute by V. M. Razin, was used by V. S. Melikhov and I. P. Chuchalin to study the capture of electrons into acceleration.

Fig. 4. Formation of a closed electron beam when electrons are introduced at the crest of the injection pulse

Fig. 5. Character of the change in the emf \(E\) taken from the turns, and of the quantity of charge \(Q\) in the beam (general case)

We placed two turns on the chamber along the inner diameter, since it turned out that 6 turns outside the chamber, as Melikhov and Chuchalin had done, have a considerably larger inductance. In order to eliminate interference, all filament transformers and autotransformers were blocked, the pulse transformer was shielded, and a common shield was placed between the injection circuit and the electromagnet of the installation (Fig. 3).

The injection pulse (Fig. 4) had a duration \(\tau = 17\ \mu\text{sec}\) and a flat top of \(6\ \mu\text{sec}\) (at \(U_i > 10\ \text{kV}\)) and could be varied from 0 to 30 kV. The signal from the turns was fed to a 25I pulse oscilloscope by means of a coaxial cable terminated at the end by a resistance of 70 ohms. In terms of sweep duration and frequency passband, the oscilloscope is quite suitable.

In addition, in order finally to filter out the high-frequency interference arising in the turns, these turns were shunted by a capacitance of 2000 pF. The signal shape was not distorted thereby. Thus it was possible to obtain on the oscilloscope screen the emf signal induced in the turns, practically in pure form without interference (Fig. 5). Additional experiments also confirm that the observed signals are indeed due to the formation and existence of an electron beam in the chamber.

When the injection pulse, for example, has only just begun to enter with its crest into the region of the required voltage, the emf signal \(E\) from the turns will have the form shown in Fig. 5. But in this case it may be assumed that in each—

at a given instant of the time interval \(\Delta \tau\), one and the same number of electrons is emitted from the cathode of the injector (since the top of the injection pulse is flat, and the filament voltage does not change).

Analysis of the emf curve \(E\) shows that in the time interval from point \(K_1\) to \(K_2\) (Fig. 5) an increase occurs in the number of charges in the beam (this is easily verified by integrating the curve \(E\)). But since during this time interval the number of electrons leaving the injector cathode and their velocity remain constant, the increase in the beam current is possible due to accumulation of electrons in the beam, i.e., due to the fact that the electrons make a considerable number of revolutions. Moreover, in order to make certain that this was the signal of a beam making more than one revolution, a partition was placed behind the injector over the whole cross section of the chamber, with the aim of intercepting the beam that had made only one revolution. In this case the amplitude of the signal from the turns (the signal of the beam that had made only one revolution) proved to be more than 20 times smaller than without the partition.

Thus, if the beam made only one or two revolutions, then without the partition the signal from the turns, first, should not have differed appreciably in amplitude from the signal obtained with the partition present, since the difference in the number of revolutions would have been small; and second, the signal should have been many times shorter than what we observe (\(\Delta \tau \simeq 2—6\ \mu\text{sec}\), depending on where the introduction of electrons into the beam occurs—on the front, the fall, or the top of the injection pulse), since the transit time of the beam through two revolutions is very small.

By integrating (graphically or electrically) the signal \(E\), we obtain the curve of the change of charge \(Q\) in the beam. In the general case (when introduced on the front or fall of the injection pulse), the curves \(E\) and \(Q\) have the form shown in Fig. 5. Here \(A\) is the region of accumulation of electrons in the beam, and \(B\) is the region of ejection of electrons from the beam.

In the case when the introduction occurs at the top of the injection pulse (the top being flat), the curve \(Q\) has another form (Fig. 4), i.e., for some time the beam has a constant charge density.\(^1\)

On the basis of the experiments carried out, the following conclusions may be drawn.

1. In a constant focusing magnetic field of the betatron type, the existence of a closed beam of electrons is possible during the time of electron introduction, i.e., during many revolutions of the beam. The charge density of the beam in this case increases from zero to some maximum value and then slowly decreases almost to zero. It may be considered that the initial beam exists at least within the limits from point \(K_1\) to \(K_2\) (Fig. 4), i.e., up to the moment when the beam charge density increases. This time was usually \(1 \div 3\ \mu\text{sec}\). If one takes \(1\ \mu\text{sec}\), then at an injection energy \(U_i = 20\ \text{kV}\) the beam makes more than 100 revolutions. After point \(K_2\), enhanced spreading of the beam begins and electrons are ejected (we shall not speak here yet about the causes of the ejection).

Thus, for our given conditions (equilibrium-orbit radius \(r_0 = 13\ \text{cm}\), \(n = 0.55\), and a definite form of the injection pulse), the average “lifetime” of the beam varying in density is

\[ \tau_{\text{av}} \simeq \frac{1}{2}\Delta \tau \simeq 1 \div 3\ \mu\text{sec}, \]

and the minimum number of revolutions is

\[ N = 0.732\sqrt{U_i}, \]

where \(U_i\) is the injection voltage in volts.

\(^1\) The intensity of the glow and the stability of the signals on the oscilloscope screen were available for observation, but it was not possible to photograph the signals from the screen.

With a change in the indicated conditions, the coefficient preceding the root will change correspondingly.

1. Under certain conditions (injection at the peak of the injection pulse, with the peak of the injection pulse being flat), the existence of a beam of constant density is possible. The “lifetime” of such a beam is \(\Delta \tau \simeq 2 \div 3\ \mu\text{sec}\).

REFERENCES

1. Rodimov, B. N. On the mechanism of capture of electrons into acceleration in a betatron. Izvestiya LPI, vol. 87, 1957.