ON THE POSSIBILITY OF INCREASING THE PARTICLE CAPTURE COEFFICIENT IN A BETATRON USING ADDITIONAL FOCUSING MICROWAVE FIELDS

Yu. A. Nepokrytykh, V. I. Zelentsov

(Presented by the scientific seminar of the sector of ultrahigh frequencies and accelerator theory of the Nuclear Physics Research Institute)

In connection with the use of betatrons in various fields of science and technology, considerable attention is being paid to improving the parameters and reliability of these accelerators, in particular to increasing the captured charge. There is a whole series of theories explaining the mechanism of electron capture into acceleration [1—6]. It has been established that the main electron losses occur at the moment of injection and in the comparatively short interval of time following it. The intensity can be increased by:

1) increasing the injection energy;
2) increasing the cross section of the working part of the vacuum chamber;
3) applying a nonoscillatory mechanism of electron capture into acceleration.

Along with this, a number of works have been published on the creation of installations with additional focusing fields [7, 8, 9], which make it possible to increase the volume of the phase space of the circulating beam. Apparently, this should lead to an increase in the captured charge. In [10] the possibility was first considered of obtaining additional focusing of particles through the action of microwave fields excited in the accelerator chamber, which should lead to a reduction in the losses of injected electrons.

Additional focusing of the beam is strong when the inequality \(h \gg H_z(t)\) is satisfied, where \(h\) is the field strength of the wave; \(H_z^0(t)\) is the field strength of the guiding field.

Since \(H^0(t)\) is a function of time, as the guiding field increases the additional focusing decreases, while the focusing by the guiding field increases; i.e., it is advisable to use microwave fields when \(H_z(t)\) is small and intensive loss of particles occurs at the moment of injection.

In [10] it was shown that, in order to satisfy the condition for stability of the motion, it is necessary that the inequality \((\chi R)^2 \gg 1\) be fulfilled (explanations are given below).

The authors of [10] proposed experimentally studying the influence of microwave fields on the process of electron capture in a betatron.

According to [5], the magnetic field of the betatron changes only slightly during the injection process and does not affect the capture process. Proceeding from this, an electromagnet was made of ordinary St-3 steel (Fig. 1), with pole diameters, profiled analogously to betatron poles, equal to

400 mm, with a magnetic-field falloff coefficient at the radius of the equilibrium orbit \(n = 0.65\). The magnetic field in the interpole space, where the chamber was placed, was produced by a coil fed with direct current and mounted on the central core. At a field strength of 45 oersteds (\(IW \approx 900\) ampere-turns), free circulation of an electron beam with an injection energy of 35 keV is ensured at an equilibrium-orbit radius \(R = 11.5\) cm.

Fig. 1. Chamber          Fig. 2. Wave antinodes

As the focusing wave, a wave of type \(H_{110}\) was chosen (Fig. 2), since calculation of the conditions under which focusing is realized is considerably simplified because there are no variants in \(\varphi\) [11]. In addition, the surface currents have only a longitudinal component, which facilitated the design of the chamber and made it possible to obtain a high \(Q\). Finally, this wave is readily excited when the waveguide is bent in the \(E\)-plane, when the bent axis lies in the same plane as the electric-field vector \(E\) of the exciting wave of type \(H_{10}\) [12, 13].

The components of the electromagnetic-wave field are

\[ E_\varphi = jD'\varkappa \nu \sin \frac{\pi}{a} z\, \Phi_0'(\varkappa r)e^{j\omega t}; \]

\[ H_z = D'\varkappa^2 \sin \frac{\pi}{a} z\, \Phi_0(\varkappa r)e^{j\omega t}; \]

\[ H_r = D'\varkappa \frac{\pi}{a} \cos \frac{\pi}{a} z\, \Phi_0'(\varkappa r)e^{j\omega t}; \]

\[ E_z = E_r = H_\varphi = 0; \]

\[ \Phi(\varkappa r) = I_0(\varkappa r) + P' N_0(\varkappa r); \qquad P' = -\frac{I_0'(\varkappa r_{1,2})}{N_0'(\varkappa r_{1,2})} = -\frac{I_1(\varkappa r_{1,2})}{N'(\varkappa r_{1,2})}. \]

A wave of this type has one variant each of the \(H\) component along the \(r\) and \(z\) axes. \(H_\varphi = 0\), i.e., the magnetic lines of force are curves closed around the equilibrium orbit, and the planes in which they lie are perpendicular to the direction of particle motion. Such a field configuration is the most convenient, since it provides radial and vertical focusing.

Of the \(E\) components, there is only a longitudinal one, which has no variants in \(\varphi\), i.e., the lines of force are closed circumferentially—

… In the course of its motion the electron experiences the influence of \(E_{\varphi}\), i.e., it is alternately accelerated and decelerated, as if oscillating relative to a uniformly moving point.

The phase velocity of wave propagation is \(v_{\phi}=\infty\), and therefore there can be no synchronization in the motion of the particles and the field.

The method for calculating the chamber, which is a resonator, is set out in detail in [12]. The chamber is a rectangular uniformly bent waveguide, inside which an electromagnetic field is excited (Fig. 1).

Labels in the figure: injector; flag; ferrite; magnetron; TsVL-100.

Fig. 3. Assembly of the setup

The dispersion equation \(I'_0(\varkappa r_1)N'_0(\varkappa r_2)-I'_0(\varkappa r_2)N'_0(\varkappa r_1)=0\) is transformed into \(I_1(\varkappa r_2)N_1(\varkappa r_2)-I_1(\varkappa r_2)N_1(\varkappa r_1)=0\), where \(I_1\) and \(N_1\) are Bessel and Neumann functions of the first order.

\[ \varkappa=\sqrt{\left(\frac{2\pi}{\lambda}\right)^2-\left(\frac{\pi}{a}\right)^2} \]

is the root of the equation,

where \(\lambda\) is the wavelength at which the resonator is excited,

\(a\) is the height of the chamber.

If \(x=\varkappa r_1\) and \(kx=\varkappa r_2\), where \(k=\dfrac{r_2}{r_1}\), the equation takes the form:

\[ I_1(x)N_1\left(\frac{r_2}{r_1}x\right)=I_1\left(\frac{r_2}{r_1}x\right)N_1(x) \]

and is solved numerically.

The calculation and fabrication were carried out comparatively easily in connection with the above and with the use of a microwave generator with a wide range of smooth frequency tuning.

Two coaxial copper cylinders were used as the side walls of the chamber, of height \(a=83\ \mathrm{mm}\) and radii \(r_1=80\ \mathrm{mm}\), \(r_2=150\ \mathrm{mm}\). The lower cover was vacuum-soldered to them with PSR-45 solder. From above, the chamber was closed with a cover \(20\ \mathrm{mm}\) thick and tightened with 33 M6 bolts. Rubber rings, placed in grooves machined on the end faces of the cylinders, were used as seals.

Three openings were cut in the outer side wall (Fig. 3): 1) a window for introducing the injector, which was an ordinary betatron gun; 2) a window for evacuating the chamber by a TsVL-100 pump; 3) a resonant slot for exciting the chamber, consisting of a diaphragm with dimensions \(6.8\times1.5\ \mathrm{cm}^2\). With a side-wall thickness of \(0.5\ \mathrm{mm}\) and power supplied to the chamber through a standard \(44\times72\ \mathrm{mm}^2\) waveguide, the VSWR was \(1.6\). Measurements of the field distribution showed that, substantially…

there were no distortions due to the holes; the loaded quality factor at resonance was \(Q = 2 \cdot 10^3\) and the shunt resistance \(R_{\mathrm{sh}} = 1.3\) MΩ [15]; the resonant frequency was 2732 MHz. Conductors used for beam indication were passed through the bottom of the chamber. The seals were Teflon and epoxy resin.

To excite the field in the chamber, a generator with smooth frequency tuning, operating in the decimeter range, was built. The duration of the generated pulse was 3 μs. The high-frequency section was evacuated, and pumping was carried out through the coupling window and the chamber. It was assembled from the following elements: a magnetron, a transition waveguide, a ferrite valve, and a connecting waveguide.

Fig. 4. Injection pulse

In [14] a theoretical calculation is given of the focusing fields and of the change in the number of betatron oscillations of particles per revolution for various amplitudes of microwave fields, for parameters close to the installation described above. Thus, for \(n = 0.5\) and \(Q = 3 \cdot 10^3\), a generator power \(P = 10^5\) W is required in order to increase the number of betatron oscillations per revolution from \(\nu = 0.7\) to \(\nu = 2\). And since the phase volume of the beam’s betatron oscillations \(V_\perp \sim \nu^2\), it increases by a factor of 8. This should lead to a sharp increase in the captured charge.

The experimental procedure was analogous to that described in [16]. The displacement winding, consisting of two turns, was placed inside the chamber. A current of 80 A with a duration of 2 μs was passed through it. Installation of the winding outside the chamber is impossible (the chamber walls shield the displacement field).

A copper plate \(40 \times 10 \times 1\), moved 3 mm closer to the equilibrium orbit than the end of the injector, was used as the target. The injection pulse is shown in Fig. 4. The operating region is the peak, with a duration of about 1.5 μs and an amplitude of 20 mA. In this case, during the spiralization process, about \(10^{10}\) electrons could be captured. The following figure shows the dumping of the beam onto the target during free circulation; the dashed line shows the extrapolatedיזֹ

curve, drawn according to the maximum current dumped onto the target in the time interval up to 15 μsec. Figure 6 shows the loss of electrons when the magnetron generator is operating at the moment of injection. The dashed line shows the extrapolated curve of the dumped current. A comparison of the behavior of the curves (Figs. 5, 6) is shown in Fig. 7. Thus it is evident that during operation of the magnetron generator the loss of particles is sharply reduced, and at the moment of switching off a peak appears. This may be explained as follows:

1. The additional focusing field retains particles with a greater spread of parameters than the guiding field can retain, and after the generator ceases operating they are lost.

Fig. 5. Beam circulation without an RF field

1. Since the injection pulse is somewhat longer than the RF pulse, after the latter ends the beam circulation is disturbed by electrons injected at that moment.

Fig. 6. Beam circulation under the action of an RF field

injected at that moment. According to [1], along the equilibrium orbit there are formed concentrically the orbits of beams with energies greater and smaller than the energy of the electrons at the equilibrium radius. These beams interact with one another in such a way that, at the equilibrium radius and near it, favorable conditions are created for the existence of the beam, and electrons entering the chamber after the focusing has ended disturb the motion.

Fig. 7. Comparison of the behavior of the curves in Figs. 5, 6

The experimental results are also consistent with [5], in which the statistical character of the capture mechanism was considered, where it was shown that in the process of circulation the beam is cooled due to the loss of some electrons with large transverse momentum. Under the action of focusing, the cooling process should proceed more intensively in connection with the increase ...

by an increase in the beam density, and when its action ends part of the beam is also lost.

Table 1 gives a comparison between the intensity of particle spill-out during particle spiralization without the action of the microwave field and with it, over a time up to 20 μsec after the end of injection. The comparison shows that the spill-out in the absence of the microwave field occurred more intensively.

Table 1

Table 2

Table 2 presents a comparison of the amplitudes of the current dumped onto the flag, likewise in the case of action of microwave fields and without them. The experiment was carried out at injection currents from 30 to 150 mA and electron energies from 20 to 50 keV, while varying the applied power over a wide range. The regularities obtained are in full agreement with the data in [1, 5, 14]. At a vacuum higher than \(2 \cdot 10^{-6}\) mm Hg, under the action of a high microwave field, a discharge arose in the chamber.

The experimental data confirmed the theoretical calculation [14] concerning the possibility of increasing the particle capture coefficient in a betatron when additional focusing microwave fields are used.

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