ACCELERATING SYSTEM OF AN UNDULATOR SYNCHROTRON

R. G. Ziyakaev

(Presented by the seminar of the High-Energy Laboratory of the Nuclear Physics Research Institute at TPU)

In [1] a method was proposed for accelerating an electron beam that satisfies stringent requirements on the energy spread and geometrical characteristics of the beam. Such requirements are usually imposed on injectors for high-energy accelerators. The basic scheme of the proposed accelerator—an undulator synchrotron—is shown in Fig. 1. Acceleration of the electrons takes place in the high-frequency field of resonator 2, in which standing waves of the type \(H_{10p}\) are excited. Electrons emerging from the injector (1) enter the resonator at a maximum of the standing wave of the electric field, acquire the energy \(eU\cos\varphi\), and after leaving the resonator move in the field of the bending magnets (3). The magnetic-field strength is chosen so that an electron which has acquired the energy \(eU\cos\varphi\) in the resonator, after being deflected through an angle \(\pi\), enters the resonator at a maximum of the electric field and again acquires the energy \(eU\cos\varphi\). The calculations given in [1] show that the beam parameters of an undulator synchrotron are no worse than the beam parameters of a microtron; however, the undulator synchrotron has a larger capture region for electrons into acceleration, which will make it possible to obtain larger accelerated electron currents. The combination of high intensity and strict beam characteristics makes this accelerator promising for use as an electron injector in high-energy accelerators, whose intensity depends significantly on the intensity and parameters of the beam injected into the accelerator.

Fig. 1. Schematic diagram of an undulator accelerator. 1 — injector, 2 — accelerating resonator, 3 — bending magnets.

1. Study of an accelerating resonator with capacitive pins

The condition of synchronism between the time of motion of the beam outside the resonator and the period of the microwave oscillations can be satisfied either by changing the distance between the magnets or by changing the radius of turn-

of electrons in a magnetic field. In the first case the accelerating resonator may be made with dimensions constant along the length of the resonator; in the second case it is necessary to change the distance between the drift holes in the resonator. This can be accomplished either by changing the wavelength along the resonator, or by introducing capacitive pins at the locations of the electric-field maxima. In this case, by changing the distance between the pins along the length of the waveguide one can achieve the required synchronism, while by changing the transverse distance between the pins one can keep the wavelength constant along the accelerating resonator.

To determine the parameters of an accelerating system in the form of a waveguide with capacitive pins, a resonator was fabricated with five accelerating gaps located at different distances from one another. The depth of insertion of the pins could be varied, thereby tuning the accelerating system to a single frequency. A resonator of this type can be calculated as a system of individual resonator cells having different geometrical dimensions and tuned to one frequency. The method for calculating an individual resonator cell is based on treating it as a connection of a lumped capacitance and a “radial line” [2]. On the fabricated resonator, measurements were made of the relative field distribution along the length of the resonator, the frequency, and the shunt impedance. Because of a small increase in shunt impedance compared with a resonator based on a smooth waveguide (since the required depth of insertion of the pins varied from 2 to 6 mm), and because of the difficulty of manufacture, tuning, and operation of such a system, preference was given to a resonator based on a smooth waveguide (a prismatic resonator).

2. Calculation of the Parameters of a Prismatic Resonator

From the design standpoint, the simplest accelerating system is a prismatic resonator excited in the \(H_{10n}\) mode. Having a lower shunt impedance than a resonator with pins, this resonator makes it possible to obtain the required energy gain per pass with only a slight increase in the power of the microwave generator. Let us consider the characteristics of a prismatic resonator needed for calculating an undulator synchrotron.

Fig. 2. Accelerating resonator. Electron acceleration is produced in the direction of the \(y\)-axis.

a. Power Loss in the Walls of a Prismatic Resonator

The calculation of the power loss in the walls of a rectangular resonator excited in the \(H_{101}\) mode is given in [3]. For a resonator excited in the \(H_{10n}\) mode, whose dimensions are shown in Fig. 2, the power can be calculated analogously.

\[ P=\frac{R_s \lambda^2 E^2}{8\eta_1^2} \left[ \frac{b\cdot c}{a^2} +n^2\left(\frac{a\cdot b}{c^2}+\frac{a}{2c}\right) +\frac{c}{2a} \right], \tag{1} \]

where \(\lambda\) is the resonant wavelength of the generator,

\(R_s\) is the specific surface resistance of the resonator walls, taking account of the skin effect;

\(E\) is the electric-field strength in the resonator.

The graph of the dependence of the loss power on the resonator dimensions for various electric-field strengths is shown in Fig. 3. The calculations were carried out for a resonator made of copper; the constant coefficients of Eq. (1) were chosen as

\[ R=12\cdot 10^{-2}\ \Omega,\qquad \eta_1=377\ \Omega,\qquad a=7.2\ \text{cm}. \]

The dimensions \(c\) and \(a\) of the resonator are related to the resonant wavelength \(\lambda\) by the relation

\[ \lambda=2\sqrt{\left[\left(\frac{1}{a}\right)^2+\left(\frac{n}{c}\right)^2\right]}. \]

Fig. 3. Loss power in the resonator walls as a function of its dimensions and of the field strength
\((a=7.2\ \text{cm},\ R=1.2\cdot 10^{-2}\ \Omega,\ \tau=377\ \Omega)\).

The loss power calculated from Eq. (1) does not coincide with its true value, since in the practical construction of the resonator additional losses are introduced owing to imperfect contacts and to the radiation of energy through the openings present in the resonator. The real value of the loss power can be estimated as follows. By definition, the quality factor of the resonator \(\Theta\) is equal to

\[ \Theta=2\pi W/P, \tag{2} \]

where \(W\) is the total stored energy,

\(P\) is the energy lost per period.

For a prismatic resonator the value of the quality factor calculated from formula (2) is

\[ \Theta= \frac{\lambda}{\delta}\cdot \frac{abc}{2}\cdot \frac{ \left[\left(\frac{1}{a}\right)^2+\left(\frac{n}{c}\right)^2\right]^{3/2} }{ \left(\frac{1}{a}\right)^2\cdot c(a+2b) + \left(\frac{n}{c}\right)^2\cdot a(c+2b) }. \tag{3} \]

The unloaded quality factor of the manufactured resonator \(\Theta_u\), obtained from measurements, can be written in the form

\[ \Theta_u=2\pi W_u/P_u. \]

Let us form the ratio

\[ k=\Theta/\Theta_u=(W/W_u)\cdot(P/P_u). \]

Since the quantities \(W/W_u>1\) and \(P/P_u>1\), a pessimistic estimate for the loss power of a real resonator can be written in the form

\[ P^{*}=k\cdot P . \tag{4} \]

The prismatic resonators manufactured by us usually have \(k=1.5\div1.6\). Cylindrical resonators used in the microtron, whose manufacturing technology has been developed more thoroughly, have \(k=1.3\div1.4\). Using formulas (1), (4), and taking the power required for beam acceleration in the form

\[ P_{\text{acc}}=N\cdot I\cdot \Delta U , \]

where \(N\) is the number of passages of the beam through the resonator,
\(I\) is the beam current,
\(\Delta U\) is the increase in electron energy per one passage through the resonator,

one can easily estimate the power of the microwave generator required for an undulator synchrotron with specified parameters.

Thus, in order to obtain an accelerated current of \(0.5\ \text{A}\) in a pulse with an energy of \(10\ \text{MeV}\), using a resonator with transverse dimensions \(40\times72\ \text{mm}\) at \(N=13\), it is sufficient to have a generator providing an output power of \(10\ \text{MW}\).

6. Choice of the width of the accelerating resonator.
In [1] it was shown that betatron oscillations in an undulator synchrotron are stable if the condition

\[ |\cos \mu|<1, \]

\[ \cos \mu=-\cos \varkappa\pi+\frac{\varkappa L}{2R}\sin \varkappa\pi , \]

is satisfied, where \(\varkappa=\sqrt{1-n}\) for radial and \(\varkappa=\sqrt{n}\) for vertical oscillations. This condition imposes restrictions on the magnitude of the permissible values of \(L/R\); however, calculations performed by the matrix method show that the value of \(L/R\) selected from the condition of obtaining the maximum energy gain per passage permits stable betatron oscillations. Let us consider how the resonator width affects the increase in electron energy per passage. The usual representation of the energy gain in the form \(\Delta T=eU\cos\varphi_s\) is not expedient in the present case, since the time of motion of electrons in the resonator is close to \(\pi\); therefore it is more convenient to write the force acting on an electron in the resonator in the form \(F=eE_0\sin\omega t\), since when it is written in the form \(F=eE_0\cos\omega t\) one has to operate either with large positive or with negative phases. For estimating the energy gain it is sufficient to consider the motion of the electron only in the electric field of the resonator; the equation of motion is then written in the form

\[ \frac{d}{dt}m\dot{y}=eE_0\sin\omega t . \]

The solution of this equation in the relativistic case makes it possible to obtain an expression for the increase in electron energy in passing through the resonator,

\[ \Delta T=\frac{eU}{\pi}\cdot\frac{\lambda}{b}\sin\pi\cdot\frac{b}{\lambda}\sin\left(\varphi_0+\pi\frac{b}{\lambda}\right), \tag{5} \]

where \(U\) is the “voltage” between the resonator walls,
\(\lambda\) is the generator wavelength,
\(\varphi_0=\omega t_0\) is the phase of entry of the electron into the resonator.

Since changing the dimensions of the resonator leads to a change in the power of losses in the walls, we determine \(U=E\cdot b\) from equation (1) and substitute it into equation (5). In this case expression (5) is written in the form

\[ \Delta T=\frac{e\eta}{\pi} \sqrt{ \frac{8P}{ R\left[ b\left(\frac{c}{a^2}+\frac{n^2 a}{c^2}\right) +\frac{1}{2}\left(\frac{n^2a}{c}+\frac{c}{a}\right) \right] } } \sin \pi\frac{b}{\lambda}\, \sin\left(\varphi_0+\pi\frac{b}{\lambda}\right) \tag{6} \]

A family of characteristics for \(\Delta T\) as a function of the phase of entry of electrons into the resonator for different values of the ratio \(b/\lambda\) of the resonator width to the wavelength is given in Fig. 4.

Fig. 4. Dependence of the energy increment in the accelerating gap on the input phase and the relative width of the resonator \((n=5,\ P=0.6\ \text{MW},\ a=7.2)\).

In the calculations it was assumed that \(P=0.6\ \text{MW}\), \(n=5\), \(a=7.2\ \text{cm}\); the remaining constants are the same as in equation (1). It follows from Fig. 4 that:

1. Increasing the resonator width up to values \(b/\lambda=0.5\) leads to an increase in the energy increment of the electrons, despite the increase in the power of losses in the resonator walls.

2. Depending on the ratio \(b/\lambda\), one can preliminarily choose the phase of entry of the equilibrium electron and estimate the region of phase stability of the accelerator. In order for the phase oscillations of the electrons to be stable, for a magnetic system with an orbit spatial-compression coefficient \(k>0\), which is intended to be used in this accelerator, the phase of entry of the equilibrium electron must be chosen on the falling section of the characteristic, located to the right of the maximum value of \(\Delta T\). It is evident from Fig. 4 that the phase of entry of the equilibrium

of the electron must vary as a function of \(b/\lambda\) and must increase as \(b/\lambda\) decreases.

1. As usual, the region of phase stability depends on the choice of the equilibrium phase. As the injection phase of the equilibrium electron is increased, the region of phase stability increases; however, in this case the magnetic system of the accelerator must ensure the passage of electrons with large values of \(\Delta T/T\). If the magnetic system ensures the passage of electrons with \(\Delta T/T=0.1\), then the maximum range of permissible values of the input phases is practically independent of the ratio \(b/\lambda\) and is approximately equal to \(62^\circ\).

2. Choosing \(b/\lambda>0.4\) is inadvisable, since in this case particles injected in a decelerating field lie in the capture region, and this will require the creation of an injector for the undulator synchrotron at high energies.

c. On the possibility of the occurrence of a high-frequency discharge (HFD).

The mechanism of HFD is as follows [4]. Electrons emitted for some reason from one electrode, being accelerated by the electric field, strike the opposite electrode. If the energy of the electrons upon impact is sufficient to knock out secondary electrons, then a new group of electrons is formed, which in turn is accelerated by the field of opposite sign toward the first electrode and knocks electrons out of it. If the coefficient of secondary emission is greater than unity, then repeated repetition of this process leads to the appearance in the accelerating gap of a large current, which loads the microwave generator and interferes with the normal operation of the resonator. For a copper resonator the coefficient of secondary emission is greater than unity for incident electrons with energies lying in the range from 0.2 to 3 keV; therefore the region of existence of HFD is usually narrow. At low voltages on the resonator, HFD can be avoided if the steepness of the voltage rise on the resonator is increased so that HFD does not have time to develop; and at a steady voltage on the resonator, HFD is prevented by choosing the width of the resonator. The regions of existence of HFD for different voltages and sizes of the accelerating gap, as well as for different types of HFD, are given in works [4], [5], [6]. From the data of work [6] it can be seen that at voltages \(\ge 5\cdot 10^5\) V HFD does not arise if the generator frequency \(f\) and the resonator size \(b\) are related by the relation \(f\cdot b \ge 10^4\ \text{MHz}\cdot\text{cm}\). For a frequency of the order of \(3000\) MHz we obtain that the resonator width \(b\) must be \(b \ge 3\) cm.

d. On matching the resonator with the waveguide line.

In work [7] the question of matching a microtron resonator with a waveguide line was considered. It is shown in the work that the value of the coupling of the resonator with the waveguide line must be chosen as a function of the magnitude of the accelerated current. In this case, for stable operation of the accelerator it is necessary to choose the coupling value greater than the critical one; then a random change in the resonator load is automatically accompanied by such a change in the input power that the acceleration process is not disturbed. The work gives the dependences of the coupling value on the ratio of the beam power to the loss power, obtained experimentally and theoretically. Since the equivalent circuits of the rectangular resonator of a microtron and of the accelerating resonator of an undulator synchrotron are identical, and the difference consists only in the magnitude of the power losses in the resonator walls, the curves given in work [7] can be used to determine the coupling value.

In conclusion, the author expresses his gratitude for the discussion of the work and for the comments made by the staff members of the Research Institute of Nuclear Physics: I. P. Chuchalin, Yu. G. Yushkov, and G. P. Fomenko.

References

1. G. P. Fomenko, Yu. G. Yushkov. ZhTF 39, 1959, 1965.

2. V. F. Kovalenko. Introduction to Microwave Electrical Engineering. Sovetskoe Radio Publishing House, Moscow, 1950.

3. S. Ramo and J. Whinnery. Fields and Waves in Modern Radio Engineering. Translated from English. Edited by Yu. B. Kobzarev. Moscow, OGIZ, 1948.

4. H. Tamagawa. Electr. I. Japan. 3. 3. 93, (1957).

5. B. A. Zager, V. G. Tishin. ZhTF, vol. 34, issue 2, 1964.

6. V. G. Mudrolyubov, V. I. Petrushin. Proceedings of the All-Union Conference on Charged-Particle Accelerators. Vol. 2, Moscow, 1968, p. 254.

7. I. M. Matora, R. V. Khar’yuzov. ZhTF, vol. 39, issue 11, 1969.