WAVEGUIDE ELECTRON CYCLIC ACCELERATOR

A. A. VOROB'EV, A. N. DIDENKO, E. S. KOVALENKO

Introduction

The waveguide cyclic electron accelerator proposed by A. A. Vorob'ev, in our opinion, combines the principal advantages of linear and cyclic accelerators.

On the one hand, being a cyclic accelerator, it can accelerate particles to very high energies with a comparatively small waveguide length. On the other hand, being a waveguide accelerator, it can in principle provide very large energy gains per revolution. At the present stage of development of accelerator technology, these qualities make such an installation highly promising.

Such accelerators may be of two variants: with an external control field and without it. In the latter case, particle control is effected by the high-frequency field of the wave itself. This question has been considered by us, but we shall not touch upon it here; instead, we shall confine ourselves to presenting only the basic propositions relating to a waveguide cyclic accelerator with a control field.

In principle, such an accelerator will differ from an ordinary synchrotron only in the character of the rf field and in the absence of straight sections. Indeed, if the rf field of a synchrotron has only one field component, \(E_{\varphi}\), decreasing with radius as \(\frac{1}{r}\), then the rf field of a waveguide has all components, depending in a complicated way on \(r\) and \(z\). From this point of view, a waveguide cyclic accelerator with a control field may be called a synchrotron with a generalized rf field. As for the absence of straight sections, this property of waveguide accelerators, while immaterial at comparatively small energies and long acceleration times, may prove very important in those cases when radiation losses increase greatly, or when it is necessary to ensure a large energy gain per revolution. This is explained by the fact that in such cases ordinary synchrotrons must be provided with so many resonators that the total length of the straight sections may exceed that at which the particle motion will be stable. Such a violation of stability will not occur in waveguide accelerators.

A schematic accelerating system of a waveguide cyclic accelerator with an external control field is shown in Fig. 1.

The remaining units of the accelerator do not differ especially strongly from analogous units of an ordinary synchrotron. From Fig. 1 it is seen that the difference consists only in the fact that the accelerating system of this accelerator is not a set of resonators, but is made in the form of a bent iris-loaded waveguide, which must at the same time also serve as the accelerator chamber.

The use of waveguides makes it possible to create conditions under which the particles will be accelerated not over short sections of the path, as in synchrotrons, but at every point of the trajectory. This is achieved by using waves slowed down to the velocity of the particles, i.e., in the present case, to the velocity of light. Any of the known systems may be used as slowing systems. Since the slowing required in the present case is small, and there are in fact no requirements on the broadbandness of the system, simple and double ridges are suitable for our purposes.

Fig. 1.

Simple and double ridges are suitable for the purpose. Round iris-loaded waveguides may also be used; however, in this case part of the interpole space in height is occupied by the diaphragms. This leads to an increase in the interpole distance and, consequently, to an increase in the cost of the accelerator’s magnetic system. Therefore almost all calculations connected with this accelerator were carried out for a double ridge forming the side walls of the chamber (Fig. 1).

The technical realization of a waveguide cyclic accelerator requires consideration of a whole series of closely interrelated questions. These were considered by us. Thus, the question of particle motion in a generalized high-frequency field was considered; the behavior of the waveguide in a variable magnetic field was investigated; the electrodynamics of bent iris-loaded waveguides was studied; a technique was developed for measuring phase velocities and field distributions inside waveguides; and the question of the techno-economic properties of the system was considered. All the questions considered make it possible to conclude that this accelerating system can be used in accelerator technology alongside the resonator system. Moreover, the waveguide system may prove preferable in those cases where the use of resonators is associated with fundamental difficulties. We shall briefly consider the questions listed above.

Particle dynamics in a waveguide accelerator

As already indicated, a waveguide accelerator differs from a synchrotron only in the nature of the high-frequency field. This field has, generally speaking, all components, depending on the coordinates in a complex way.

Thus. At the time when work on the waveguide accelerator was only beginning, it was not clear what this circumstance would lead to. Moreover, it could be assumed that precisely for this reason the waveguide accelerator would be unable to operate. In a number of works A. A. Kolomenskii and A. N. Lebedev, and earlier K. W. Robinson, found that radial variations of the high-frequency field lead to the appearance of an exponential growth of the amplitudes of the axial oscillations of particles and, to the same extent, to damping of the radial betatron oscillations. In particular, it was proposed to use this effect to damp oscillations induced by radiation in strong-focusing installations.

The calculations we carried out showed that the indicated point of view is erroneous and that the high-frequency field introduces no changes into the damping decrements. The presence of a complicated field structure leads only to a redistribution of the frequencies of betatron and synchrotron oscillations, but in general this redistribution is small in most cases. The stability conditions are therefore close to the stability conditions of an ordinary synchrotron and differ from them only in very strong high-frequency fields. The conditions for particle capture into the synchronous acceleration regime are likewise analogous to the capture conditions in a synchrotron, and only in the case when the separatrix is approximately equal to the transverse dimensions of the chamber is a substantial difference possible. We note that in a waveguide it is not at all necessary to use the traveling-wave regime. Standing waves may be used with equal success. It has been shown that this has almost no effect on the dynamics of the particles.

In the region of high energies, as in a synchrotron, radiation is significant in a waveguide cyclic accelerator. Since in the given system propagation of waves with a velocity less than \(c\) is possible, in addition to ordinary radiation an additional, Cherenkov, radiation is possible here. Calculations showed that mainly a redistribution of the radiation over the spectrum occurs, but its total magnitude changes hardly at all. On the whole, the questions of dynamics considered show that the waveguide cyclic accelerator is quite analogous to the synchrotron, and that no complications of any kind should be expected here.

Behavior of the Waveguide in an Alternating Magnetic Field

During operation the waveguide must be placed in an alternating magnetic field. This will lead to the induction in it of Foucault currents, which distort the controlling magnetic field inside the waveguide. With a large waveguide thickness and a rapid time variation of the magnetic field, the distortion may be large and may lead to a disruption of the acceleration regime. Strictly speaking, this question must also be considered for an ordinary betatron, since there too the chamber is covered with a thin layer of silver. However, for our case it acquires greater importance, and it is clear that the distortion will be strong unless special measures are taken.

This question was considered both theoretically and experimentally. It was shown that the difficulties can be avoided if the upper walls of the waveguide are made of glass or porcelain and coated with a layer of silver. The thickness of the coating layer should be equal to 5–6 skin layers for a wavelength in the 10-centimeter range. In this case, as measurements showed, when such a waveguide is placed in the alternating field of a betatron at 15 MeV, the distortion does not exceed fractions of a percent.

Electrodynamics of a Diaphragm-Loaded Waveguide

It should be noted that before the beginning of our work the electrodynamics of bent diaphragm-loaded waveguides had in practice not been studied at all. When beginning the work, we had at our disposal only two papers by D. V. Voskresenskii, in which questions of wave propagation in bent, non-diaphragm-loaded systems were considered. We investigated, from the electrodynamic point of view, a system consisting of a closed uniformly bent waveguide of rectangular cross section, diaphragm-loaded along the side walls. It was necessary to consider what types of waves exist in such systems, which wave is the most suitable for our purposes. In addition, having chosen a definite type of wave, it was necessary to determine under what conditions its phase velocity would be equal to the velocity of light, and also how its components vary with height and radius.

It was shown that the waves existing in diaphragm-loaded waveguides are most conveniently classified along the axis parallel to the ribs of the diaphragms, i.e., in our case along the \(z\) axis. With such a classification all waves are divided into two types:

\[ LE\,(E_z = 0) \quad \text{and} \quad LM\,(H_z = 0)\text{-waves}. \]

In addition, waves of each type may be in phase and out of phase, depending on whether they are symmetric or antisymmetric with respect to the mean radius. It was shown that the best for our purposes is the symmetric \(LE_{11}\)-wave. The dispersion properties of this wave in a bent waveguide were studied in detail by B. N. Morozov. In particular, he showed that bending of the waveguide leads to acceleration of the wave and, consequently, the diaphragm height required for slowing the \(LE_{11}\) wave in a bent waveguide is greater than in a straight one. The calculated curves make it possible to determine the dimensions of the waveguide for which the phase velocity of the \(LE_{11}\) wave on the circumference of the mean radius will be equal to the particle velocity. For large bending radii the influence of bending may be neglected and the waveguide may be regarded as straight. The electrodynamics of a straight waveguide has also been studied by us in detail, both theoretically and experimentally.

Besides dispersion in the waveguide, electrodynamics includes determination of the dependence, on the frequency and dimensions of the waveguide, of such quantities as the \(Q\)-factor, coupling resistance, shunt resistance, etc. We shall discuss these quantities in detail below.

Generator Power and Technical-Economic Characteristics of the System

In determining the power of the high-frequency generator it is necessary to proceed from the electric-field strength required for normal operation of the waveguide accelerator.

The field strength must, first, ensure the transfer of the greater part of the accelerated particles from the betatron regime to the synchronous one and, second, ensure such an increase in particle energy per revolution as would correspond, for a given equilibrium phase, to the growth of the magnetic field. Both of these requirements are important, but depending on the specific conditions one of them becomes essential. In small accelerators with a chamber aperture comparable with the radius of the insta-

…of the installation, the first requirement is essential. Conversely, for large installations at the end of the acceleration cycle, because of the presence of intense radiation, the second requirement becomes decisive.

Let us first consider accelerators for low energies. For them the first requirement is essential, and in order to satisfy it, it is necessary that, at the moment the high-frequency field is switched on, the radial dimensions of the separatrix be larger than the spread of the equilibrium orbits of the beam in the radial direction, which also determines the beam dimensions. Depending on the preliminary acceleration regime, these dimensions may be different. Since the dimensions of the separatrix are proportional to the field strength, it is clear that this condition determines the minimum value of the field required for effective particle capture.

Specific calculations carried out for an accelerator with a betatron acceleration regime and with the following parameters:

\[ \begin{aligned} r_s &= 13.4\ \text{cm}; & g &= 3.25\ \text{cm}; & \lambda &= 9.35\ \text{cm};\\ a &= 6\ \text{cm}; & E_u &= 0.53\ \text{MeV}; & E_s &= 3\ \text{MeV} \end{aligned} \]

(\(E_u\) is the injection energy, \(E_s\) is the energy of transition to the synchrotron regime) showed that the minimum electric-field strength is equal to \(56\ \text{V/cm}\)

\[ \left( V_0 = 4.7\ \frac{\text{kV}}{\text{revolution}} \right). \]

As for the second requirement, its fulfillment for small accelerators is not associated with any difficulties. Indeed, if the acceleration time is \(3\text{–}5\ \text{msec}\) (as in an ordinary betatron), then the particle must increase its energy by \(20\text{–}30\ \text{eV}\) per revolution. In betatrons this is provided by the emf of induction arising when the magnetic flux changes. In our case the increase per revolution is

\[ \Delta E = e V_0 \cos \varphi_s . \]

(\(\varphi_s\) is the equilibrium phase).

It follows from this that \(\varphi_s\) must be chosen very close to \(90^\circ\). This is very advantageous from the standpoint of capturing particles into the synchrotron acceleration regime, since in this case the dimensions of the separatrix will be very large. At the same time, this is disadvantageous from the standpoint of using high-frequency power. Practically all the high-frequency power is needed not for acceleration, but for capturing particles into the synchrotron regime. We note that this is characteristic not only of the waveguide accelerator, but of any synchrotron in which the ratio of the chamber width to the radius is large and the multiplicity of the accelerating harmonic is large. When radiation becomes significant, the equilibrium phase, according to the principle of autophasing, decreases and the accelerating system is used more efficiently.

Let us now determine the power required to create, inside the waveguide, a high-frequency field of such strength. As is known, the characteristic of the efficiency of any high-frequency system is its shunt resistance:

\[ R_{\text{sh}} = \frac{V_0^2}{P}, \]

where \(P\) is the power of the generator. It can be shown that, for an iris-loaded waveguide operating in the traveling-wave regime,

\[ R_{\mathrm{sh}} = 4\pi m R_{\mathrm{c}} Q \frac{v_{\mathrm{gr}}}{c}, \tag{2} \]

where \(m\) is the harmonic number of the rf field, \(R_{\mathrm{c}}\) is the coupling resistance, which depends on the field distribution, \(Q\) is the quality factor, and \(v_{\mathrm{gr}}\) is the group velocity of the wave under consideration. If we assume that the waveguide has

\[ Q = 5000; \qquad v_{\mathrm{gr}} = 0.1\,c; \qquad R_{\mathrm{c}} = 25\ \Omega \]

(these values do not exceed the limits of technical feasibility; this is discussed in more detail below), then, for \(m = 9\), we find that for such an installation \(R_{\mathrm{sh}} = 1.4\ \mathrm{M}\Omega\), i.e., to create the required field inside the waveguide an rf generator with a power of \(300\ \mathrm{W}\) is sufficient. For comparison, let us note that the resonator of the FIAN synchrotron has \(R_{\mathrm{sh}} = 1.2\ \mathrm{k}\Omega\). For efficient resonant systems \(R_{\mathrm{sh}}\) may reach several \(\mathrm{M}\Omega\), but nevertheless it can be shown that in all cases the waveguide accelerating system will be at least an order of magnitude more effective than the resonant one.

The shunt resistance of the system is the most essential parameter determining the efficiency and techno-economic properties of the system. In general, the question of techno-economic properties is very complicated, since systems must be compared both in terms of fabrication costs and in terms of costs during operation of the entire system. It is precisely these costs that are determined by the shunt resistance. We believe, nevertheless, that in accelerators for high energies, when any increase in power exceeds the limits of technical feasibility, the shunt resistance is an indicator of the efficiency of the system, and it must be studied in greater detail. From the formula (2) given above it follows that, for a waveguide, \(R_{\mathrm{sh}}\) depends on the product of a number of electrodynamic characteristics, which it is therefore expedient to study in greater detail. The calculation performed shows that all these characteristics depend strongly on the width of the interaction space. This is shown in Fig. 2, in which, for a waveguide with rigorously calculated characteristics, the graph of the shunt resistance \(r_{\mathrm{sh}}\) per unit length is given. As can be seen, it varies from \(0.1\) to \(0.5\ \frac{\mathrm{M}\Omega}{\mathrm{cm}}\). The graph makes it possible to determine quickly the total shunt resistance of any waveguide system. Thus, it is not difficult to show that if even a waveguide with \(r_{\mathrm{sh}} = 0.1\ \frac{\mathrm{M}\Omega}{\mathrm{cm}}\) is used as

Fig. 2.

of the accelerating system of the German synchrotron at 6 Bev, the total shunt impedance of the system would be 2000 megohms. To obtain such a shunt impedance in a resonator system, it would be necessary for each of the 80 resonators used in this synchrotron to have a shunt impedance of 25 megohms. In reality, however, the resonators have a shunt impedance equal to 2 megohms, i.e., an order of magnitude lower. This means that, using a waveguide system, a generator with a power not of 400 but of about 40 kW would be sufficient. It should also be noted that the accelerating system of the German synchrotron itself is very complicated: 80 resonators are fed from an X-602 klystron amplifier, manufactured by the firm “Eimak,” through a bent waveguide surrounding the entire installation.

Thus, from the standpoint of attainable shunt impedances, the waveguide system leaves far behind even the most modern and highly efficient resonator systems.

Analysis of Existing Accelerating Systems of Synchrotrons and Their Comparison with the Waveguide Accelerating System

The accelerating systems of synchrotrons, and especially synchrotrons for higher energies, are usually made in the form of a large number of resonators. This is due to the need to obtain extremely high voltages per revolution in order to compensate radiation losses. As the energy increases, the required number of resonators must grow rapidly, since the energy losses are proportional to \(E^4\). This leads to a rapid increase in the total length of the straight sections, as a result of which, at certain energies, loss of stability of particle motion is possible. From this point of view, the use of resonators with high \(R_{\mathrm{sh}}\), requiring the presence of straight sections, is possible only over a limited energy range. As for the use of resonators that do not require straight sections, it is excluded because of the small shunt impedances of these systems. The waveguide accelerating system, apparently, is free of the above-listed shortcomings: precisely because it is a waveguide system, it makes it possible to obtain shunt impedances at least an order of magnitude higher than those of the best resonator systems and at the same time does not require the presence of straight sections.

The only question that we have not yet considered in comparing the waveguide system with resonators is the question of the influence of increasing the harmonic number of the accelerating harmonic on the operation of a waveguide accelerator. Indeed, if in the Cambridge and German synchrotrons the wavelength is 60 cm, then in our case it is approximately 10 cm (using longer wavelengths in waveguide cyclic accelerators apparently makes no sense). On the one hand, an increase in the harmonic number leads to an increase in the current of particles captured into acceleration, but on the other hand—to an increase in the rms phase spread because of the quantum nature of the radiation, which is, of course, very undesirable. This leads to the fact that the amplitude of the rf voltage must be increased if we want the particle losses due to fluctuations of the radiation to remain approximately the same. Analyzing the Christy and Robinson formula for particle losses due to radiation fluctuations, one can show that the particle losses depend on a certain expression that is a complicated function of the equilibrium phase and the harmonic number. The argument of this function will be the expression

\[ F(\varphi_s, m) = \frac{1}{m}\,(\operatorname{tg}\varphi_s - \varphi_s). \tag{3} \]

Therefore, the particle losses for two accelerators will be equal if

\[ \frac{1}{m_1}(\operatorname{tg}\varphi_{s1} - \varphi_{s1}) = \frac{1}{m_2}(\operatorname{tg}\varphi_{s2} - \varphi_{s2}). \tag{4} \]

It follows from this equality that, as the multiplicity is increased, the equilibrium phase must increase sufficiently so that the equality is not violated. Since in both cases we assume the radiation losses to be the same, this must lead to an increase in the voltage according to the relation:

\[ \frac{V_{02}}{V_{01}} = \frac{\cos \varphi_{s1}}{\cos \varphi_{s2}} . \tag{5} \]

However, with increasing multiplicity the \(R_{\text{ш}}\) of the waveguide system increases; therefore it is by no means obvious that, taking into account fluctuations of the radiation, it is disadvantageous to go over to higher multiplicities. Indeed,

\[ P = \frac{V_0^2}{R_{\text{ш}}}, \]

and therefore, since with increasing multiplicity by a factor \(m\), \(R_{\text{ш}}\) increases, an increase in multiplicity is even advantageous in the region where

\[ \frac{V_{02}}{V_{01}} < \sqrt{m}. \]

Calculations show that such a region exists and includes the most acceptable range of equilibrium phases \((0^\circ < \varphi_s < 50^\circ)\).

Conclusions

The analysis carried out permits the conclusion that the waveguide accelerating system for electron cyclic accelerators is highly efficient. The analysis shows that this system can be used on an equal footing with a resonator system. Moreover, in those cases where the use of a resonator system is associated with great difficulties, the use of waveguide systems proves to be even more preferable. In particular, in our opinion, such a system may prove indispensable in iron-free electron accelerators for high energies. In this case, with a resonant system it is generally impossible to compensate radiation losses. In addition, in iron-free installations the acceleration time can be greatly reduced, and this makes it possible to make better use of the advantages of the waveguide accelerating system, which in principle permits large energy increments. Finally, the use of short acceleration times (tens of microseconds, as is the case in plasma accelerators) will greatly reduce the cost of the high-frequency generator, which is very important.