On the Calculation of Optimal Dimensions of Smooth Waveguide Accelerating Systems for Cyclic Accelerators
Abstract Generated abstract
The paper examines whether smooth circular waveguide sections can serve as accelerating systems for cyclic accelerators at high energies, where diaphragm-loaded structures are technologically more complex. It derives relations for the length, radius, phase slip, and shunt resistance of resonant straight waveguide sections operating in the E01 family of modes, and compares the dependence of efficiency on longitudinal mode number and allowed phase shift. The analysis indicates that the commonly used E010 resonator is less effective than modes with higher longitudinal number, with the E012 mode giving the largest shunt impedance under the assumptions used. For maximum phase shift of pi, the corresponding optimal dimensions are found to be L = 1.5 lambda and R = 0.513 lambda, although the resulting efficiency of short smooth-waveguide sections remains insufficient for high-energy accelerator design.
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ON THE CALCULATION OF THE OPTIMAL DIMENSIONS OF SMOOTH WAVEGUIDE ACCELERATING SYSTEMS FOR CYCLIC ACCELERATORS
N. V. PISKOV
(Presented by the seminar of the Microwave and TU sector of the Nuclear Physics Research Institute)
Recently, increasing attention has been paid to waveguide accelerating systems in the development of cyclic accelerators. Thus, in [1] it is proposed to use closed curved waveguides and sections of waveguides for accelerating electrons. The studies carried out in [1, 2] showed that at high energies it is most expedient to use diaphragm-loaded waveguides and various pin structures, while at energies of the order of 10–50 MeV smooth curved waveguides may be used as accelerating systems. In [3] it was proposed to use accelerating systems based on sections of diaphragm-loaded waveguides for accelerating protons in a cybernetic accelerator. Calculations showed that in this case the accelerating system proposed by Schnell [4] is the most promising. It should be noted, however, that the technology for manufacturing diaphragm-loaded waveguides is rather complicated. In this connection, it is of interest to investigate the question of using accelerating systems based on smooth waveguides in accelerators not only for low, but also for ultrahigh, energies. This article is devoted to the study of this question.
The use of smooth curved waveguides as accelerating systems of low-energy electron synchrotrons is possible because inside the waveguide one can choose such a radius at which the phase velocity of one of the waves will be equal to the velocity of the particles. If one strives to satisfy such conditions in accelerators for high energies, then, as the radius is increased, it is also necessary to increase the width of the waveguide (the width of the waveguide must always be approximately \(1/3\) of the mean radius of the waveguide). In this case the efficiency of such accelerating systems will decrease. This leads to the fact that, starting from certain radii, the use of smooth curved waveguides as accelerating systems will be inexpedient.
The situation changes if the smooth curved waveguides are replaced by sections of straight waveguides arranged along rectilinear segments. In this case the phase velocities of all waves will be greater than the velocities of the accelerated particles, and therefore the condition of synchronism between the particles and the waves will not occur. However, if the transverse dimensions of the waveguide are chosen large in comparison with the wavelength, then the phase velocities of the lower modes of waves will approach the speed of light. If the length of the waveguide is such that the phase shift between
of the wave and the accelerated particles \(\Delta\varphi \leq \pi\), then the particles will experience only an accelerating action of the waves; therefore such sections can be used for acceleration.
When the phase velocity of the wave is not equal to the speed of light, the phase shift will be determined by
\[ \Delta \varphi=\frac{2\pi L}{\lambda}\left(1-\frac{1}{\beta_{\phi}}\right), \tag{1} \]
where \(L\) is the length of the waveguide section,
\(\lambda\) is the wavelength in free space,
\[ \beta_{\phi}=\frac{1}{\sqrt{1-\left(\frac{\lambda}{\lambda_{\mathrm{cr}}}\right)^2}}, \tag{2} \]
It was shown in [5] that the most preferable for creating accelerating systems for high energies are circular-section waveguides operating on a wave of type \(E_{01}\). Therefore all subsequent calculations were carried out for the \(E_{01}\) wave.
Let us consider a resonant accelerating system. For it
\[ \lambda_0=\frac{1}{\sqrt{\left(\frac{1}{2.62\,R}\right)^2+\left(\frac{p}{2L}\right)^2}}, \tag{3} \]
where \(p=0,1,2\ldots\) is the longitudinal wave number.
From (1), (2), (3) we obtain expressions for the quantities \(\dfrac{L}{\lambda}=f(p_1\Delta\varphi)\) and \(\dfrac{R}{\lambda}=f(p_1\Delta\varphi)\),
\[ \frac{L}{\lambda}=\frac{p\pi+\Delta\varphi}{2\pi}, \tag{4} \]
\[ \frac{R}{\lambda}=\frac{p\pi+\Delta\varphi}{\sqrt{2p\pi\Delta\varphi+(\Delta\varphi)^2}}. \tag{5} \]
Finally, it is necessary to determine the shunt resistance of the system
\[ R_{\mathrm{sh}}=\left.\frac{(LE_z)^2}{2P_{\mathrm{g}}}\right|_{R=0}, \]
where \(P_{\mathrm{g}}\) is the power fed into the resonator from the high-frequency generator, which in the steady-state regime is equal to the power dissipated in the resonator walls \(P\)
\[ P=\frac{c}{8\pi}\oint Re\!\left[E_t H_t^{*}\right]\,df =\frac{c}{8\pi}\sqrt{\frac{\omega\mu}{8\pi\sigma}}\oint |H_t|^2\,df, \]
where \(\omega\) is the angular frequency,
\(\mu\) is the magnetic permeability of the material,
\(\sigma\) is the conductivity of the material.
It is not difficult to show that for the \(E_{01}\) wave
\[ R_{\mathrm{sh}}= \frac{\nu_{01}^{2}\sqrt{\lambda}\,\lambda^{2}L^{2}} {c\sqrt{\dfrac{\mu c}{\sigma}}\,\pi^{2}I_{1}^{2}(\nu_{01})R^{3}(L+2R)} . \tag{6} \]
Expression (6) can be brought to the form
\[ R_{\mathrm{sh}}= k\sqrt{\lambda}\left(\frac{\lambda}{R}\right)^{3} \frac{\Delta\varphi}{\pi}\, \frac{1} {\left[ 1+4\frac{\pi}{\Delta\varphi}\frac{R}{\lambda} \left(1-\sqrt{1-0.146\frac{\lambda^{2}}{R^{2}}}\right) \right] \left(1-\sqrt{1-0.146\frac{\lambda^{2}}{R^{2}}}\right)} \]
Fig. 1. Dependence of the quantity \(\dfrac{R_{\mathrm{sh}}}{\sqrt{\lambda}}\) on the longitudinal wave number \(p\):
\(1\)—\(\Delta\varphi=\pi\);
\(2\)—\(\Delta\varphi=\dfrac{2}{3}\pi\);
\(3\)—\(\Delta\varphi=\dfrac{\pi}{2}\).
Fig. 2. Dependence of the change in the relative dimensions of the accelerating section on
\(1\)—\(\dfrac{R}{\lambda}\left(\dfrac{\pi}{2}=\pi\Delta\right)\);
\(2\)—\(\dfrac{R}{\lambda}\left(\Delta\varphi=\dfrac{2}{3}\pi\right)\);
\(3\)—\(\dfrac{R}{\lambda}(\Delta\varphi=\pi)\);
\(4\)—\(\dfrac{L}{\lambda}(\Delta\varphi=\pi)\);
\(5\)—\(\dfrac{L}{\lambda}\left(\Delta\varphi=\dfrac{2}{3}\pi\right)\);
\(6\)—\(\dfrac{L}{\lambda}\left(\Delta\varphi=\dfrac{\pi}{2}\right)\).
where
\[ k=0.5\, \frac{\nu_{01}^{2}} {c\sqrt{\dfrac{\omega c}{\sigma}}\,\pi^{2}I_{1}^{2}(\nu_{01})}. \]
or, taking (5) into account,
\[ R_{\mathrm{sh}}= \frac{2.62^{4}k\,[2p\pi\Delta\varphi+(\Delta\varphi)^{2}]} {\pi(p\pi+\Delta\varphi)^{2} \left[4\pi+2.62\sqrt{2p\pi\Delta\varphi+(\Delta\varphi)^{2}}\right]} . \tag{7} \]
Figures 1 and 2 show the dependences \(\dfrac{R_{\mathrm{sh}}}{k\sqrt{\lambda}}=f(p)\), \(L/\lambda=f(p)\), \(R/\lambda=f(p)\) for various values of \(\Delta\varphi\). From Fig. 1 it is seen that resonators operating in \(E_{010}\) oscillations, which have become widespread in accelerator technology, are less effective than resonators operating on waves with increased longitudinal wave numbers \(p\). The resonator with the \(E_{012}\) type of oscillation has the greatest shunt impedance. With a further increase in \(p\), the efficiency of the accelerating systems decreases, tending in the limit to zero. This is explained by the fact that, as \(p\) increases, the transverse dimensions of the waveguide increase (Fig. 2), the longitudinal component of the electric field \(E_z\) decreases, and the shunt impedance falls.
It should be noted that real accelerating systems must have rather large holes in the end walls for the passage of particles. Beyond-cutoff waveguides, which change the \(Q\) of the resonator only insignificantly, can be used as effective short circuits [6].
Thus, in constructing accelerating systems based on smooth circular waveguides for high-energy accelerators, the most effective systems are those operating in the \(E_{012}\) mode of oscillation. At the maximum \(\Delta\varphi=\pi\), the dimensions of the system are \(L=1.5\lambda\) and \(R=0.513\lambda\). As \(\Delta\varphi\) is decreased, the length of the system decreases, while the transverse cross section increases.
Thus, when accelerating sections based on smooth waveguides are used, it turns out that long sections are less effective than short ones. The efficiency of short sections, however, is insufficient in the design of high-energy accelerators.
In conclusion, the author expresses gratitude to A. N. Didenko for his interest in the work and for a number of useful suggestions.
References
-
A. A. Vorob’ev, A. N. Didenko [et al.]. Waveguide Synchrotrons. Atomizdat, Moscow, 1966.
-
A. A. Vorob’ev, A. N. Didenko, Yu. N. Potekhin, V. L. Kaminskii. PTE No. 3, 20, 1967.
-
O. A. Val’dner, A. V. Shal’nov, Yu. A. Khlestkov. Proceedings of the First All-Union Conference on Charged-Particle Accelerators. VINITI, vol. 2, 149, 1970.
-
Schnell. Proc. V. International Conference on High Energy Accelerators, Frascati, 1965, Rome, 1966.
-
N. P. Didenko, N. V. Piskov. Proceedings of the Research Institute of Nuclear Physics, Electronics, and Automation (in press).
-
A. N. Didenko, R. G. Kaminskaya. ZhTF, 39, No. 11, 2029, 1969.