THE INFLUENCE OF THE DISPERSION OF THE MAGNETS OF AN UNDULATOR ACCELERATOR ON THE MAGNITUDE OF THE ACCELERATED CURRENT

R. I. Ziyakaev, I. P. Chuchalin

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

The calculation of the region of permissible values of the input phases for an undulator accelerator in [1] was carried out for voltage-amplitude values at the resonator of 200, 300, and 400 kV, for which it was shown that the size of the region of input phases and the optimal equilibrium phase depend on the voltage at the resonator.

The powers of the available microwave generators make it possible to obtain in the resonator of an undulator accelerator field strengths limited by breakdowns in the resonator, i.e., not less than 200–250 kV/cm; in this case the voltages at the resonator reach values of \(\sim 1\) MV. Calculations of the region of permissible values of the input phases for such voltages show the presence of limitations on the magnitude of the accelerated current due to the dispersion of the deflecting magnets of the undulator accelerator. The principal limitation of the accelerated current occurs at the initial stage of acceleration, since at the exit from the first accelerating gap the electrons have maximum values of the ratio of the momentum difference of the equilibrium and nonequilibrium electrons to the momentum of the equilibrium electron, i.e., \(\Delta p/p\). With a finite width of the apertures in the dispersive magnetic system, the principal electron losses are observed on the walls of the second aperture. To obtain a region of input phases \(\geq 1\) rad, it is necessary either to increase the dimensions of the apertures in the resonator or to use low-dispersion magnetic systems.

Since increasing the diameters of the apertures in resonators of the 10-cm range beyond 2 cm leads to a significant decrease in the quality factor of the resonators, the most realistic way of increasing the permissible region of input phases, and consequently the accelerated current, is the use of low-dispersion magnetic systems. Below an estimate is given of the region of permissible input phases, in terms of the size of the input-phase region that allows electrons to pass through the second aperture, for three variants of magnetic systems. The influence of stray magnetic fields is not taken into account, since by changing the shape of real magnets one can achieve a sufficiently accurate approximation of electron trajectories in real magnets to the trajectories of electrons in idealized magnets. We shall assume that the injector of the undulator accelerator is a point source of electrons, and that at the exit from the first aperture of the resonator the electrons with different momenta have the same direction of velocities.

1. When “ordinary” weak-focusing bending magnets, considered in [1], are used, the displacement of a nonequilibrium-particle trajectory from the trajectory of an equilibrium particle can be written in the form [2]

\[ x=\frac{R_0}{1-n}\cdot\frac{\Delta p}{p}\left(1-\cos\sqrt{1-n}\,\alpha\right), \tag{1} \]

where \(x\) is the displacement of the trajectory of a nonequilibrium particle from the trajectory of an equilibrium particle after bending in a magnetic field through an angle \(\alpha\);

\(R_0\) is the radius of the trajectory of the equilibrium particle;

\(n\) is the magnetic-field falloff index;

\(p\) is the momentum of the equilibrium particle.

The resonator of a \(2\ \mathrm{MeV}\) ondulator accelerator has a beam-aperture size in the radial plane of \(1.6\ \mathrm{cm}\) and a distance between the centers of the beam apertures \(2R_0=8.5\ \mathrm{cm}\). If the magnetic-field falloff index is \(n=0.5\), then electrons with \(|\Delta p/p|\leq 0.06\) will pass through the second beam aperture.

Figure 1 gives the dependence of the electron energy at the exit from the first beam aperture of the resonator on the phase at which the electrons enter the resonator for the case when the voltage on the resonator is \(1\ \mathrm{MV}\), the resonator width is \(3.4\ \mathrm{cm}\), and the energy of electron injection into the resonator is \(200\ \mathrm{keV}\). From Fig. 1 one can determine that, for the given magnetic system, the maximum range of admissible values of the input phases for the obtained value \(\Delta p/p\) is \(\Delta\theta_0\leq 0.90\ \mathrm{rad}\); in this case, at the exit of the second beam aperture the current will be \((0.9/2\pi)I_0\), where \(I_0\) is the injector current.

Fig. 1. Dependence of the electron energy at the exit of the first beam aperture on the phase of entry into the resonator. Voltage on the resonator \(1\ \mathrm{MV}\), resonator width \(3.7\ \mathrm{cm}\), electron injection energy \(200\ \mathrm{keV}\), microwave-generator wavelength \(10.9\ \mathrm{cm}\).

1. Let us consider the use of bending magnets of the “electron mirror” type, presented in [3]. Neglecting the stray field, such magnets create a uniform magnetic field, for which the displacement can be written in the form

\[ x=2\frac{R_0\Delta p}{p}. \]

For the resonator parameters given in item 1, the value of the admissible values of \(|\Delta p/p|\) and the maximum range of admissible values of the input phases \(\Delta\theta_0\) are, respectively, \(0.094\) and \(1.22\ \mathrm{rad}\), which gives for the current at the exit of the second beam aperture the value \((1.22/2\pi)I_0\).

1. Let us consider bending magnets whose shape in the radial plane is shown in Fig. 2. In this figure, the angle \(\beta\) denotes the angle formed by the normal to the exit edge of the first magnet and the trajectory of the equilibrium particle \(ABB'D'\). \(ACC'D'\) is the trajectory of a nonequilibrium particle. The magnetic system is symmetric with respect to the plane passing through the straight line \(SS'\) perpendicular to the surface of the drawing. The fall-off index of the magnetic field in the magnets is different from zero. It can be shown that such a system is nondispersive if the condition

Fig. 2. Nondispersive magnetic system.

\[ \operatorname{tg}\beta = -\frac{\sqrt{1-n}\,\sin\sqrt{1-n}\,\alpha} {1-\cos\sqrt{1-n}\,\alpha}. \tag{2} \]

is satisfied.

Condition (2) means that the trajectories of particles with different momenta in the field-free gap between the magnets are parallel to the trajectory of the equilibrium particle.

The structural realization of magnets of such a shape for an undulator accelerator makes it possible, relatively simply, to achieve nondispersiveness for electrons with \(|\Delta p/p| = 0.2\), which, for the size of the admissible region of input phases, gives a value of \(1.75\) rad; in this case the current is equal to \((1.75/2\pi)I_0\). A further increase in the region of input phases is apparently no longer expedient because of the insignificant energy gain of the equilibrium electron.

The estimates given for the magnitude of the accelerated current show that the use of nondispersive magnets for bending the electron beam in an undulator accelerator makes it possible to obtain the magnitude of the region of admissible input phases given in [1] and equal to \(1.3\) rad. As the last estimate shows, the magnitude of the region of admissible input phases may be even larger.

REFERENCES

1. G. P. Fomenko, Yu. G. Yushkov, ZhTF, XXXIX, 1959, 1969.
2. J. Livingood. Principles of Cyclic Accelerator Operation. Moscow, IL, 1963.
3. V. M. Kelman and S. Ya. Yavor. Electron Optics. Publishing House of the Academy of Sciences of the USSR, Moscow–Leningrad, 1959.