On Certain Factors Influencing the Process of Electron Capture into Acceleration in a Betatron

N. M. Goloshchapov

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

Up to now, a sufficiently well-founded theoretical explanation has not yet been created for what chiefly determines the capture of electrons into acceleration in a betatron. From the point of view of the theory of motion of a single electron in a betatron, for example, it is impossible to explain the basic phenomena, such as the dependence of radiation intensity on the emission current and the injection voltage. According to this theory, the electron must strike the injector already in the first revolutions [1, 2]. In view of this, the problem arose of electrons bypassing the injector, and on this basis—experimental attempts to facilitate the bypassing of the injector by electrons (various contractors, compression of the orbit at the moment of injection, etc.) [3—7]. However, most betatrons that have been built operate successfully, while all possible improvements in the form of contractors, short-time local disturbances of the magnetic field, etc., can have a positive effect only in the case when the betatron gives a small radiation output.

Some authors have attempted to explain the possibility of capturing electrons into acceleration by various factors, namely:

1. With a certain distribution of azimuthal-phase inhomogeneities of the magnetic field (arising naturally or created artificially), favorable conditions are created for electrons to bypass the injector [8, 9].

2. The presence, under certain conditions, of resonance between the free betatron oscillations (in \(r\) and \(z\)) and the circulation of electrons around the orbit can lead either to rapid damping of the oscillations and to electrons bypassing the injector or, conversely, to an increase in the amplitude of the oscillations and a sharp decrease in the radiation output [10—12].

3. Since the electrons during injection lose part of their energy to the creation of the electromagnetic field, compression of the instantaneous equilibrium orbit occurs and, thus, the possibility arises of bypassing the injector [13].

4. The presence of a space charge in the chamber, moreover a decreasing one, also favors the bypassing of the injector and creates the possibility of capturing electrons into acceleration (capture on the falling edge of the injection pulse) [14]. All these proposals have to some extent received experimental confirmation. Let us also note that here the matter concerns the bypassing of the injector by electrons as the main task of the injection process. Of course, all the indicated factors occur and in some way affect the capture of electro-

... in acceleration, but, as it seems to us, none of them is the principal one. There exist experimental data where the action of these factors should have led to the disappearance or to a considerable decrease of the radiation, which in fact does not occur. Consequently, the principal determining factor for the capture of electrons into acceleration is not taken into account in the existing theories.

At one time Howard [15] and other authors pointed out that, in order to explain the process of capture of electrons into acceleration, it is necessary to take into account the interaction of electrons and electron beams. Taking into account the (electrostatic) interaction of the electrons in the beam and of the beams with one another during the injection process, we come closer to the real picture.

This problem was solved in first approximation in the work of B. N. Rodimov [16]. It was shown that, as a consequence of the electrostatic interaction of the electrons in the beam and of the beams during the injection process, there occurs an alternating displacement of the equilibrium orbit to one side and the other in the radial direction, and the period and amplitude of the oscillations of the beams change. In this case it is found that the problem of bypassing the injector does not play the important role that is now ascribed to it.

Under normal conditions (optimal field distribution, optimal injections, emission, position of the gun, etc.), the electron beams, owing to the electrostatic interaction between them, themselves ensure their capture into acceleration. Of course, some fraction of the electrons also strikes the injector and the walls of the chamber; however, considerably more electrons strike the walls than the injector. It has been shown that, when the charge density of the beam is increased up to certain limits, the capture conditions should improve, and the radiation intensity should increase.

It should be noted that some authors dispute this proposition, asserting that the electrostatic interaction of the beams not only does not favor the capture of electrons into acceleration but, on the contrary, hinders capture, and they consider the magnetic interaction of the beams to be the main factor assisting capture. In the work of B. N. Rodimov it is shown that the magnetic interaction of the beams at the moment of injection proves to be insignificant and practically does not affect the capture of electrons into acceleration; moreover, the magnetic interaction of the beams does not hinder but promotes capture, only its effect is considerably weaker than the electrostatic interaction.

We carried out experiments with the aim of determining whether, and in what way, the electrostatic interaction of the beams during the injection process affects the capture of electrons into acceleration. The experimental arrangement is presented in Fig. 1. The work was carried out on a 15 MeV betatron. The ordinary and transverse injectors were typical Kerst injectors, except that the transverse injector injected electrons along the radius. The injection pulses were close to sinusoidal. For the ordinary injector \(U_i = 30\) kV, \(\tau_i = 10\) μsec, and for the transverse injector \(U_i\) — up to 12 kV and \(\tau_i \approx 16\) μsec. At the optimum radiation value, the current to the foil from the ordinary injector was constant and equal to 65 μA. Then, switching on the transverse injector, we found the dependence of the radiation intensity \(J\) on the injection voltage and on the emission current of the transverse injector, i.e.,

\[ J = f(U_i) \quad \text{at } U_n = \text{const} \]

\[ J = f(I_{\mathrm{sl}}) \quad \text{at } U_i = \text{const} \]

\[ J = f(U_i, I_{\mathrm{sl}}), \]

where \(U_n\) is the filament voltage of the transverse injector, \(I_{\mathrm{sl}}\) is the current to the foil from the transverse injector (i.e., the current of the transverse beam).

The injection pulse to the transverse injector was applied in phase with the injection pulse of the ordinary injector. Since it was necessary for us to determine the influence of the electrostatic interaction of the beams, which depends on the charge density \(\rho\) of the transverse beam, we presented the measurement results—

Fig. 1. Experimental scheme

1 — equilibrium orbit, 2 — ordinary injector,
3 — transverse injector, \(\varphi \simeq (25^\circ \div 30^\circ)\)

—in the form of a family of curves, as in Fig. 2. From this it was not difficult to find the dependences

\[ J = f(\rho) \quad \text{at } I_{c.l} = \text{const} \]

\[ J = f(I_{c.l}) \quad \text{at } \rho = \text{const}. \]

For the curves of Figs. 2 and 3, \(\rho\) was determined in the following way (all arguments are given for the beam from the transverse injector). The current density of the beam can be represented as

\[ i = e \rho V \tag{1} \]

where \(e\) is the electron charge,

\[ \rho \;-\; \frac{\text{el}}{\text{cm}^3} \text{ — charge density,} \]

\(V\) — cm/sec — electron velocity,

\(i\) — A/cm\(^2\) — beam current density.

The electron velocity can be expressed as:

\[ V = 5.93 \cdot 10^7 \sqrt{U_i}, \tag{2} \]

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

Fig. 2

Dependence of the radiation intensity \(J\) on the current strength
and the charge density of the transverse beam

Fig. 3

Dependence of the radiation intensity on the charge density
of the transverse beam at a constant current strength of it (curve \(a\),
where \(I_{\mathrm{cl}} = 210\,\mu\mathrm{A} = \mathrm{const}\)) and on the current strength at a constant
charge density (curve \(b\), where \(\rho = 3 \cdot 10^{-6}\,\mathrm{A} = \mathrm{const}\))

The beam current, which we determine from the current to the collector, i.e. \(I_{c.l}\), will be

\[ I_{c.l}=iS, \tag{3} \]

where \(S\) is the beam cross section, \(\mathrm{cm}^2\),
\(I_{c.l}\) is in amperes.

Thus one can find

\[ \rho=A\frac{I_{c.l}}{\sqrt{U_i}}, \tag{4} \]

where

\[ A=\frac{50.59}{Se}=\mathrm{const}. \]

For each pair of values \(I_{c.l}\) and \(U_i\), \(\rho\) can be calculated. Let us note that \(S\) is taken as the initial beam cross section, i.e. the area of the anode aperture. Then the charge density \(\rho\) will also be the initial density.

Taking into account expression (4) for \(\rho\), the experiment can be carried out in such a way that the following are obtained directly:

\[ J=f(\rho)\quad \text{for } I_{c.l}=\mathrm{const}, \]

\[ J=f(I_{c.l})\quad \text{for } \rho=\mathrm{const}. \]

By appropriate selection of the injector filament heating and the injection voltage, these conditions, i.e. \(I_{c.l}=\mathrm{const}\) and \(\rho=\mathrm{const}\), can be fulfilled.

Curve \(a\) (Fig. 3), showing the dependence of the radiation intensity on the charge density of the beam, confirms the theoretical consideration of this phenomenon given by B. N. Rodimov; namely, electrostatic interaction takes place, and an increase in the charge density of the beam (up to certain limits) leads to an increase in the radiation intensity.

Concerning curve \(b\) (Fig. 3), the following may be said. Since the magnetic interaction of transverse beams is practically negligible, we assume that the magnetic field of the transverse beam produces a considerable local perturbation of the guiding magnetic field, as a result of which the capture conditions deteriorate and the radiation intensity decreases.

References

1. D. W. Kerst, R. Serber—Phys. Rev. 60, 53, 1941.
2. J. A. Raichman, W. H. Cherry—Journ. Franklin Inst. 243, 261, 345, 1947.
3. I. Davis—Phys. Rev. 81, 656, 1951.
4. W. B. Jones et al.—Phys. Rev. 78, 60, 1950.
5. G. D. Adams—Rev. Sci. Instr. 19, 607, 1948.
6. G. D. Adams, D. W. Kerst, D. T. Scag—Rev. Sci. Instr. 18, 783, 1947.
7. J. Davis—Rev. Sci. Instr. 21, 971, 1950.
8. F. K. Goward—Phys. Soc. 61, 315, 284, 1948.
9. V. S. Melikhov, Dissertation, Tomsk Polytechnic Institute, 1954.
10. S. E. Barden—Proc. Phys. Soc. 64, 373, 85, 1951.
11. S. E. Barden—Proc. Phys. Soc. 64, 379, 579, 1951.
12. E. D. Courant—Journ. Appl. Phys. 20, 6, 611, 1949.
13. D. W. Kerst—Phys. Rev. 74, 503, 1948.
14. R. Wideröe—Journ. Appl. Phys. 22, 3, 362, 1951.
15. F. K. Goward et al.—Proc. Inst. E. E. 97, 108, 320, 1950.
16. B. N. Rodimov, “On the Mechanism of Electron Capture into Acceleration in a Betatron,” Proceedings of Tomsk Polytechnic Institute, vol. 87, 1957.