ON THE POSSIBILITY OF CREATING ACCELERATING SYSTEMS WITH LARGE APERTURES FOR ELECTRON SYNCHROTRONS

A. N. Didenko, R. G. Kaminskaya, G. P. Fomenko, Yu. G. Yushkov

In high-energy electron synchrotrons, very high values of the accelerating-field voltage per turn are required because it is necessary to compensate the energy losses of the electrons in the form of intense electromagnetic radiation. Increasing the voltage per turn by increasing the number of accelerating resonators is, in many cases, not feasible.

Another way of obtaining a high voltage per turn is to use the accelerating fields of sections of linear accelerators placed in straight sections. In this case the radio-frequency harmonic number usually increases. As the harmonic number grows, it becomes possible, for a given length of the straight sections, to design resonant systems with large values of shunt impedance. It must be noted, however, that increasing the harmonic number will have a twofold effect on the intensity of the synchrotron. On the one hand, increasing the harmonic number increases the average current of particles captured from the injector; on the other hand, it increases particle losses caused by the effect of quantum fluctuations of radiation on phase oscillations. In high-energy synchrotrons, the predominant effect will be exerted by the factor due to the electromagnetic radiation of relativistic electrons at high energies.

The present article is devoted to a theoretical and experimental examination of the possibility of using sections of diaphragm-loaded waveguides with cutoff waveguides at the ends as accelerating systems for high-energy electron synchrotrons.

At high energies, owing to the quantum character of the radiation, betatron and synchrotron oscillations are excited. This leads to the loss of a significant fraction of the accelerated electrons. To reduce electron losses it is necessary to increase the voltage amplitude across the accelerating gap. In synchrotrons with weak focusing, the requirements on the magnitude of the accelerating field increase sharply at energies above \(10^9\) eV.

The number of electrons remaining in the accelerator chamber at time \(t\) can be determined from the formula [1, 2]

\[ N = N_0 \exp\left(-\int_0^t \frac{dt}{\tau}\right), \tag{1} \]

where

\(\tau\) is the “lifetime” of the electron beam.

\(N_0\) is the number of electrons at the beginning of acceleration.

To determine \(\tau\), one may use the approximate formula obtained with the aid of the diffusion equation for electrons in phase space

\[ \tau=\frac{1}{a_s}\left(\frac{\Delta U}{U_{\mathrm{в}}}\right)'\left(e^{\frac{\Delta U}{U_{\mathrm{в}}}}-1-\frac{\Delta U}{U_{\mathrm{в}}}\right), \tag{2} \]

where \(\Delta U\) is the depth of the potential well. In the case of synchrotron oscillations it is determined by the formula

\[ \Delta U=\frac{2\mathcal{E}^2}{\sin \varphi_s}\left(\sin \varphi_s-\varphi_s\cos \varphi_s\right), \tag{3} \]

where \(U_{\mathrm{в}}=\frac{1}{2}\bar A_s^2\Omega^2\) is the energy of the synchrotron oscillations excited by quantum fluctuations of the radiation, \(a_s\) is the damping coefficient of the synchrotron oscillations, \(\Omega\) is the frequency of the synchrotron oscillations, \(\varphi_s\) is the equilibrium phase, and \(\bar A_s^2\) is the mean-square amplitude of the synchrotron oscillations.

As is seen from the formulas given, the magnitude of electron losses in a given accelerator depends on the ratio \(\Delta U/U_{\mathrm{в}}\). Then, in order for the electron losses for harmonic numbers \(q_1\) and \(q_2\) to be identical, it is necessary that the condition

\[ \left(\frac{\Delta U}{U_{\mathrm{в}}}\right)_{q_1} = \left(\frac{\Delta U}{U_{\mathrm{в}}}\right)_{q_2} \tag{4} \]

be satisfied, or

\[ \frac{\sin\varphi_{s1}-\varphi_{s1}\cos\varphi_{s1}}{q_1\cos\varphi_{s1}} = \frac{\sin\varphi_{s2}-\varphi_{s2}\cos\varphi_{s2}}{q_2\cos\varphi_{s2}}. \tag{5} \]

A calculation carried out for the NIIYaF electron synchrotron at an energy of \(1.5\ \mathrm{GeV}\) showed that, for \(q=4\), for losses not exceeding \(10\%\), this ratio is equal to \(0.25\). Then it follows from (4) and (5) that, for example, for \(q=100\) (wavelength of the accelerating high-frequency field \(\lambda=33\ \mathrm{cm}\)), the phase \(\varphi\) will be such that the required amplitude of the accelerating voltage at the end of the acceleration cycle must be \(2.5\ \mathrm{MV}\).

It is therefore clear that the amplitude of the accelerating voltage increases with increasing harmonic number. However, the transition to high harmonic numbers makes it possible to use, as the accelerating system, sections of iris-loaded waveguides whose shunt impedance may be 1–2 orders of magnitude higher than the shunt impedance of resonators operating at low harmonic number. Therefore, cases are possible in which it is simpler to create a higher voltage in an accelerating system with a large shunt impedance than a lower voltage in a system with a smaller shunt impedance.

As is known, the shunt impedance of an iris-loaded waveguide depends on the loading parameter \(g/\lambda\) and decreases sharply as \(g/\lambda\) increases. Apparently, in order to obtain a large shunt impedance, it makes sense to use a rectangular iris-loaded waveguide and to place it in the straight sections so as not to reduce the aperture of the chamber in the radial direction. Since the irises must not reduce the aperture of the chamber in height either,

then for each wave there will be its own minimum value of the parameter \(g/\lambda\). For example, for the electron synchrotron of the Nuclear Physics Research Institute, the minimum value of the parameter \(g/\lambda\) will be 0.254 at \(q=100\).

The beam-passage holes in the end walls of the diaphragm-loaded waveguide cannot be chosen large, since they will reduce the aperture of the cavity. To eliminate the influence of the passage holes, it was proposed in [3] to use cutoff waveguides at the ends of the diaphragm-loaded waveguide.

For an experimental investigation of the possibility of creating such accelerating systems for a \(1.5\ \mathrm{GeV}\) synchrotron, a diaphragm-loaded resonator with cutoff waveguides at the ends, shown in Fig. 1, was fabricated.

Fig. 1. Diaphragm-loaded resonator with cutoff waveguides

Fig. 2. Calculated and experimental dispersion characteristics of the waveguide

The calculation of the diaphragm-loaded waveguide was carried out using the formula:

\[ \frac{v_{\phi}}{c} = \frac{2\pi L}{\lambda_{\mathrm{cr}}\psi} \left[ 1+ \frac{\pi a^{2}(1-\cos\psi)} {2aL\left(1+\dfrac{a^{2}}{b^{2}}\right)} \right], \tag{6} \]

where

\[ \cos\psi = \cos\gamma_{0}L - \frac{\beta_{0}a}{\pi} \left(\frac{a}{\pi g}\right)^{2} \sin\beta_{0}L \]

\[ \gamma_{0}^{2} = \frac{\omega_{0}^{2}-\omega_{\mathrm{cr}}^{2}}{c^{2}}. \]

\(\omega_{\mathrm{cr}}\) is the critical frequency, determined as for a rectangular waveguide;
\(L\) is the period of the structure;
\(g\) is the gap between diaphragms;
\(a\) is the width of the resonator;
\(b\) is the height of the resonator.

The experiments were carried out on a waveguide in the 10-cm band. The dimensions of the diaphragm-loaded waveguide were chosen from the condition of the greatest frequency separation between different modes of oscillation (\(a=7.8\ \mathrm{cm}\), \(g=2.6\ \mathrm{cm}\), \(b=7.2\ \mathrm{cm}\), \(L=2.5\ \mathrm{cm}\)). The calculated and experimental dispersion characteristics of the waveguide are given in Fig. 2, from which it is seen that formula (6) gives a comparatively large discrepancy only in the long-wavelength region.

The determination of the oscillation modes and of the wave type was carried out with the aid of magnetic and electric probes and test bodies [4], moved along the \(z\)-axis of the waveguide.

In the general case, in an iris-loaded waveguide of rectangular cross section there exist in-phase and out-of-phase waves of the \(LE\) and \(LM\) types. For acceleration of particles, however, as shown in [5], only the in-phase wave \(LE_{11}\) can be used. The remaining waves are parasitic; among them the chief danger is posed by the \(LE_{01}\) wave, which is the lowest mode of the iris-loaded waveguide. To suppress this wave in the resonator under study, special cuts were made in the side walls; these did not disturb the field of the main \(LE_{11}\) wave.

An experimental study of the distribution of the field components of the \(LE_{11}\) wave along the length of the resonator for various transverse dimensions of the cutoff waveguides showed that the cutoff waveguides have only a slight effect on the field structure in the iris-loaded waveguide. The length of the cutoff waveguides was not changed during the experiment, and the attenuation in them was not less than \(40\ \mathrm{dB}\).

Fig. 3. Dependence of \(\dfrac{R_{\mathrm{sh}}}{Q}\) for the oscillation \(\dfrac{\pi}{2}\) on the width of one of the walls of the cutoff waveguide for \(L' = L\) and \(L' = \dfrac{L}{2}\).

A study was made of the influence of the cutoff waveguides on the value of the shunt resistance of the iris-loaded waveguide. Figure 3 gives the curves of \(\dfrac{R_{\mathrm{sh}}}{Q}\) for an oscillation of the \(\dfrac{\pi}{2}\) type as a function of the width of one of the walls of the cutoff waveguide for cases in which the iris-loaded waveguide is connected to the cutoff waveguide when the distance \(L'\) from the beginning of the cutoff waveguide to the first iris is equal to the structure period \(L\) and to \(\dfrac{L}{2}\). The second wall of the cutoff waveguide remained constant and equal to the gap between the irises. It is seen from Fig. 3 that, for large transverse dimensions of the cutoff waveguide, the distance \(L'\) must be chosen close to the structure period. In this case the shunt resistance of the iris-loaded resonator approaches the value of the shunt resistance of a resonator without holes in the end walls.

It is known that the shunt resistance of an iris-loaded waveguide of constant length is proportional to \(\dfrac{1}{\sqrt{\lambda}}\). Therefore, on the basis of the experimental data obtained in the \(10\)-cm range, one can estimate the shunt resistance of the synchrotron accelerating system for a wavelength \(\lambda = 33\ \mathrm{cm}\) and a waveguide length of \(\sim 1\ \mathrm{m}\).

This wavelength agrees well with the aperture dimensions of the synchrotron chamber of the Research Institute of Nuclear Physics. For the example given above, when the harmonic number is \(q = 100\) and the loading parameter is \(q/\lambda = 0.254\), the shunt resistance of the iris-loaded accelerating system of the synchrotron will be equal to \(30\ \mathrm{M\Omega}\). It is not difficult to calculate that, in order to create a voltage of \(2.5\ \mathrm{MV}\) in a system with such a shunt resistance, a high-frequency generator with a power of \(\sim 100\ \mathrm{kW}\) is required. We note that when using a resonator operating at a frequency of \(365\ \mathrm{MHz}\), for co-

...to create a voltage of approximately 250 kV, which is necessary for accelerating particles with 10% losses from the final energy at harmonic number \(q = 4\), the 1.5 GeV synchrotron of the Nuclear Physics Research Institute uses, among other things, a generator with a power of 125 kW.

Thus, as a result of the investigations carried out, it has been shown that, by using the properties of evanescent waveguides, it is possible to create effective accelerating systems for high-energy electron synchrotrons that require less power even when operated at a high radio-frequency harmonic number. When such accelerating systems are placed in the straight sections of a synchrotron, sections of the vacuum porcelain chamber may be used as evanescent waveguides, provided that a thin layer of silver is deposited on its walls.

References

1. H. A. Kramers. Physik, 7, 285, 1940.
2. M. Sands. Proceedings of the Intern. Conf. on High-Energy Accelerators and Instrument, 298, CERN, Geneva, 1959.
3. A. N. Didenko, R. G. Kaminskaya. ZhTF, 11, 1969.
4. E. L. Ginzton. Measurements on Centimeter Waves. IL Publishing House, 1960.
5. E. S. Kovalenko, V. I. Shimanskii. Izv. vuzov SSSR, Radiotekhnika, 2, 153, 1960.