CALCULATION OF THE EXTRACTION OF A BEAM OF ACCELERATED ELECTRONS FROM A 300 MeV ELECTRON SYNCHROTRON

V. P. ANOKHIN

Introduction

The methods proposed for extracting accelerated particles from cyclic accelerators can be divided, according to the principle of operation of the extraction device, into three main groups:

1. Slow extraction of accelerated particles.
2. Regenerative extraction of accelerated particles.
3. Pulsed single-turn extraction of accelerated particles.

In the slow extraction method, a gradual expansion of the equilibrium orbit of the accelerated particles is carried out by means of pulsed magnetic fields [1—5]. Near the boundary of the operating region of the accelerator’s magnetic field, the particles enter the deflecting device, receive an additional impulse, and emerge into the region of radial instability in the form of a focused beam. Further transport of the beam, depending on the requirements imposed on its parameters, is carried out with the aid of magnetic channels or without them. According to the method of action on the accelerated particles, the deflecting devices are divided into two groups:

1. Deflection of particles by an electrostatic field.
2. Deflection of particles by a magnetic or electromagnetic field.

In extracting accelerated particles by the regenerative method, magnetic fields are created which cause excitation of radial oscillations of the particles being extracted [6—13]. The accelerated particles are released from the action of the focusing forces of the accelerator’s magnetic field in a narrow azimuthal region in the form of a sufficiently focused beam.

In pulsed single-turn extraction, the deflecting device—a deflector—deflects the accelerated particles directly from the equilibrium orbit by an amount sufficient to direct the beam of the particles being extracted into the extraction channel [3], [4], [14—16]. In this case, the rise of the deflecting field to its steady value must occur in a time shorter than the period of revolution of the particles in the accelerator. Extraction is carried out during one revolution of the particles in the accelerator.

In the present work, the possibility is considered of extracting a beam of accelerated electrons from a synchrotron with weak focusing, at an energy of 300 MeV, by exciting the parametric resonance of radial betatron oscillations of the electrons [2], [10].

The electron extraction method under consideration was developed for ring cyclic accelerators and therefore cannot be used directly in our case. To study the stability of the free oscillations of electrons, the “racetrack”-type synchrotron must be replaced by a circular synchrotron having the very same periodicity of radial oscillations. In weak-focusing accelerators with a split magnet, the unperturbed free oscillations of accelerated particles are described by Hill’s equation. The solution of the equation of free oscillations has the form:

\[ x = C(\sigma)\cos[\mu K+\varphi(z)], \tag{1} \]

\[ \cos\mu=\cos\chi\,\frac{\pi}{2}-\frac{\chi l}{2r_0}\sin\chi\,\frac{\pi}{2}, \]

where

\[ \begin{aligned} r_0&\text{ is the radius of the equilibrium orbit,}\\ l&\text{ is the length of the straight section,}\\ \chi=\sqrt{1-n}&\text{ is for the radial deviation,}\\ n&\text{ is the index of the radial decrease of the magnetic field in the sectors,}\\ C(\sigma)&\text{ is the envelope of the oscillations.} \end{aligned} \]

For a sufficiently small value of \(l/r_0\) (in our case \(l/r_0=0.533\)), the ratio

\[ \eta=\frac{C(\sigma)_{\max}}{C(\sigma)_{\min}} \]

differs little from unity (\(\eta=1.0527\)), and the particle trajectory is close to sinusoidal. In this case the fundamental harmonic component of the free oscillations has the form [18]:

\[ x=C\cdot\cos\left(\frac{2\mu}{\pi}\omega_0 t+\varphi\right), \tag{2} \]

where \(\omega_0\) is the angular frequency of revolution of the particle.

For a weak-focusing synchrotron with 4 straight sections, the concept of the dimensionless frequency of free oscillations, equal to \(4\mu\), is introduced, where \(\mu\) is the quantity determined from (1). The angular frequency of radial oscillations can then be determined from the expression

\[ \omega_{r(\mu)}=\frac{4\mu}{2\pi}\omega_0. \tag{3} \]

The angular frequency of radial oscillations for an accelerator with a ring magnet is determined by the expression

\[ \omega_r=\sqrt{1-n}\,\omega_0. \tag{4} \]

To replace a “racetrack”-type synchrotron by a synchrotron with a ring magnet having the same periodicity of radial oscillations, we find the magnetic-field falloff index \(n_{\mathrm{cr}}\) of the equivalent circular synchrotron from expressions (3), (4).

§ 1. Determination of the liberation circumference

The liberation circumference is the circumference on which the focusing forces of the accelerator magnetic field are equal to zero. To determine it, we use the data of magnetic measurements of the radial field distribution in the magnet quadrants at the end of the acceleration cycle:

\(H_z=\varphi(r)\) and the dependence \(H'_z=H_0\dfrac{r_0}{r}\), which determines the limiting gra-

gradient of the field with respect to radius, at which the focusing action of the field is still preserved (Fig. 1). The scatter in the values of \(r_{\mathrm{osc}}\) is explained by the imperfection of the accelerator magnetic field, as well as by errors in the magnetic measurements and in the graphical construction. We take

\[ r_{\mathrm{osc,av}}=\frac{\sum_m r_{\mathrm{osc}\,m}}{m}=102.4\ \mathrm{cm} \]

where \(m\) is the number of azimuths for which \(r_{\mathrm{osc}}\) was determined.

Fig. 1. Radial distribution of the magnetic field in a 300 MeV synchrotron at the end of the acceleration cycle

The outer radius of the aperture of the synchrotron vacuum chamber is

\[ R_{\mathrm{out}}=(106.0\pm0.5)\ \mathrm{cm}. \]

Consequently, the electrons will enter the region of instability of the radial motion within the aperture of the accelerator vacuum chamber.

§ 2. Perturbing magnetic field. Equations of motion

To implement the extraction mechanism we use the azimuthal magnetic field shown in Fig. 2. This function can be represented in the form of a Fourier series:

\[ f(\Theta)=\frac{2\Delta H_0}{\pi} \left[ \left(\frac{\Theta_1}{4}-\frac{\Theta_{\mathrm{p.p}}}{2}\right) +\sum_{n=1}^{\infty}(-1)^n \frac{\sin\left(n\frac{\Theta_1}{2}\right)-\sin\left(n\Theta_{\mathrm{p.p}}\right)}{n}\cos\Theta \right], \tag{5} \]

where \(\Theta_{\mathrm{p.p}}\) is the azimuthal extent of the straight section

\[ (\Theta_{\mathrm{p.p}}=22^\circ45'), \]

\(\Theta_1\) is the azimuthal size of the perturbing magnetic field

\[ (90^\circ \leq \Theta_1 \leq 157^\circ15'), \]

\(\Theta\) is the current coordinate, \(0\leq \Theta \leq 2\pi\).

In the calculations we use the constant component \(f(\Theta)_0\) and the first harmonic \(f(\Theta)_1\) of this function.

The required change in the magnetic field in the region \(r_0\) for shifting the equilibrium orbit by \(\Delta r = r_{\mathrm{eq.\,avg}} - r_0\) is found from the expression

\[ \Delta H_0 = H_0(1-n)\frac{\Delta r}{r_0} \frac{\sin \pi \sqrt{1-n}} {\sin\left(\dfrac{\Theta_{1\mathrm{calc.}}\sqrt{1-n}}{2}\right)}, \tag{6} \]

where \(\Theta_{1\mathrm{calc.}}=\Theta_1-\Theta_{\mathrm{s.s.}}\).

The deviation from the equilibrium orbit \(x\) is found in the form

\[ x=x_0+h_0\cos t+y, \tag{7} \]

Fig. 2. Perturbing magnetic field used for calculating the extraction of an electron beam

where \(x_0\) is the symmetric expansion of the equilibrium orbit under the action of \(f(\Theta)_0\):

\[ x_0=\frac{r_0}{H_0(n-1)}\,f(\Theta)_0; \tag{8} \]

\(h_0\) is the amplitude of the asymmetric displacement of the equilibrium orbit under the action of the force \(f(\Theta)_1\):

\[ h_0=\frac{r_0}{H_0 n}\,f(\Theta)_1; \tag{9} \]

\(y\) is the radial oscillation of the electrons with frequency \(0.5\omega_0\) about the displaced orbit.

The equation of motion, written with respect to \(y\), has the form:

\[ \frac{d^2y}{dt^2}+\omega^2y = \varepsilon\left[3x_0\left(y^2+2hy\cos t+h^2\cos^2 t\right)+y^3+3y^2h\cos t+\right. \]

\[ \left. +3yh^2\cos^2 t+h^3\cos^3 t \right] + 2\frac{dh}{dt}\sin t, \tag{10} \]

where \(\varepsilon\) is an empirical coefficient \((\varepsilon\ll 1)\).

The solution of equation (10) is found, according to [19], in the form:

\[ y=a\cos\left(\frac{t}{2}+\vartheta\right), \tag{11} \]

where \(a\) is the amplitude of the electron oscillations,

\(\vartheta\) is the phase of the electron oscillations.

\(a\) and \(\vartheta\) are found from the equations:

\[ \frac{da}{dt}=aA\sin 2\vartheta-aB, \qquad \text{(a)} \]

\[ \tag{12} \]

\[ \frac{d\vartheta}{dt}=C-Da^2+A\cos 2\vartheta, \qquad \text{(b)} \]

where \(A, B, C, D\) are coefficients depending on \(\omega_r\) and varying slowly with time:

\[ \begin{gathered} A=\frac{3x_0 h\varepsilon}{2\omega_r}, \qquad B=\frac{\varepsilon}{2\omega_r}\frac{\partial \omega_r}{\partial \tau},\\[4pt] C=\omega_r-\frac{1}{2}-\frac{3h^2\varepsilon}{4\omega_r}, \qquad D=\frac{3\varepsilon}{8\omega_r}. \end{gathered} \tag{13} \]

The magnitude of the stationary value of the amplitude \(a_{\mathrm{stat}}\) is determined by the expression

\[ a_{\mathrm{stat}}^{2}= \frac{C\pm\sqrt{A^{2}+B^{2}}}{D} \sim \frac{C\pm A}{D}. \]

Equations (12) admit hard excitation of oscillations, described by equation (11), when the slowly varying coefficients of equations (12) satisfy the condition \(A\ge C\).

Let us consider the moment of resonance, when \(A=C\). This occurs at

\[ \omega_r=\omega_{r\mathrm{res}}. \]

In this case extraction of electrons is possible if the amplitude of the oscillations excited at resonance, \(a_{\mathrm{stat}}\), is not less than \(y=|\Delta r-(x_0+h_0)|\). Calculations have shown that the parameters of the synchrotron under consideration satisfy this condition, \(a_{\mathrm{stat}}>y\).

§ 3. Dynamics of the amplitude and phase of radial free oscillations of electrons during extraction. Radial velocity of electrons on the liberation circumference

In the normal acceleration process, up to the moment of displacement of the equilibrium orbit, the electrons are distributed uniformly over the phases of the radial free oscillations in the range \((0 \div 2\pi)\). Preliminary calculations showed that it is sufficient to compute the dynamics for the initial phase values \(\vartheta_0=0^\circ \div 180^\circ\).

In a synchrotron, owing to the presence of a rising magnetic field, the betatron oscillations of the electrons are damped proportionally to \(\sqrt{H}\), where \(H\) is the magnetic-field strength at the equilibrium radius \(r_0\). In addition, at energies of accelerated electrons of the order of hundreds of MeV, damping of betatron oscillations occurs due to radiation friction. During the acceleration process the electrons interact with one another and with molecules of the residual gas, and because of this there is some increase in the dimensions of the beam. The distribution of electrons over oscillation amplitudes may be described with the aid of the function \(C\cdot e^{-x^{2}}\). Taking the above into account, we find that the initial value of the amplitude of the free radial oscillations at the end of the acceleration cycle may be taken equal to \(a_0=0.1\ \mathrm{cm}\). The solution of equations (11, 12) is carried out for the initial conditions: \(a_0=0.1\ \mathrm{cm}\); \(\vartheta_0=(0^\circ \div 180^\circ)\).

Equations (12) are solved by the grapho-analytical method [20]. In equation (12a) the term \(aB=0\), since we consider the case \(\omega=\omega_{\mathrm{res}}=\mathrm{const}\).

With an appropriate choice of scale for the quantities

\[ \frac{da}{dt}=\varphi(t), \qquad \frac{d\vartheta}{dt}=f(t), \]

\(a=\psi(t)\) and \(\vartheta=f(t)\), the graphical construction can be replaced by numerical calculations. Along with simplifying the solution, this increases the accuracy of the calculations. Equations (12) were tabulated and solved on a computer for all initial conditions. In Figs. 3 and 4 the solution graphs are presented:

\[ a=\psi(t), \qquad \vartheta=f(t). \]

Fig. 3. Dependence of the amplitude of parametric oscillations of electrons for \(a_0 = 0.1\ \text{cm}\) and various initial values of the phase \(v_0 = (0^\circ \div 180^\circ)\)

Fig. 4. Dependence of the phase of parametric oscillations of electrons for \(a_0 = 0.1\ \text{cm}\) and various initial values of the phase \(v_0 = (0^\circ \div 180^\circ)\)

Equation (11) was also tabulated for solution on computers. The solutions \(y=f(t)\) were found for values of \(t\) close to \(t_{\text{rel}}\) (\(t_{\text{rel}}\) is the instant at which the electrons reach the release circumference, in radians).

The results of the calculations show that:

1. The time from the beginning of the resonant buildup of the free radial oscillations of the electrons to their arrival at the release circumference depends on the initial value of the oscillation phase \(\vartheta_0\).

2. At the release circumference the electrons are grouped, according to their oscillation phases, into two bunches: \(\varphi_{s_1}\sim 70^\circ\) and \(\varphi_{s_2}\sim 250^\circ\).

Fig. 5. Spectrum of trajectories of a beam of electrons moving in the stray field of the accelerator after the instant of release (on the abscissa axis: I — straight section, II — magnet quadrant)

1. The release of electrons from the action of the focusing forces of the accelerator magnetic field occurs in a narrow azimuthal region of the orbit (\(\sim 15^\circ\)). The increase in the oscillation amplitude of an electron per revolution by the instant of release is \(\sim 1\ \text{cm}\).

2. The divergence of the electron beam at the release circumference is \(\sim 1.2\cdot 10^{-2}\ \text{rad}\).

The dynamics of the amplitude and phase of the radial free oscillations of the electrons was also calculated on an IPT-5 electronic analog computer. The results of the solution were recorded on a digital voltmeter and photographed from the indicating device. Qualitatively, the results of the calculations and the solution obtained on the IPT-5 agree well.

Construction of the trajectories of the motion of the electron beam in the stray magnetic field of the synchrotron, under the assumption that the electrons are monoenergetic and that their energy does not change during extraction, shows that the beam is strongly blurred in the horizontal plane (Fig. 5).

The initial divergence of the beam at the release circumference was \(\sim 1.2\cdot 10^{-2}\ \text{rad}\). If no additional extraction device is used, then it is impossible to obtain practically acceptable beam parameters in the region of the extraction window (divergence, extraction efficiency). This observation applies to an even greater extent to accelerators of higher energies.

To pass the electron beam through the stray field of the synchrotron magnet, pulsed magnetic focusing can be used.

beam by magnetic fields with a front steepness on the order of \(10^{-8}\) sec (air-core magnetic lenses, or lenses with cores made of high-frequency ferrites), a magnetic deflector with a sharply decreasing magnetic field [21], or an electromagnetic deflector.

The use of the latter appears to be the most promising and, in practice, the most readily feasible.

REFERENCES

1. B. A. Kononov. Dissertation, TPI, Tomsk, 1957.
2. L. S. Sokolov. Dissertation, TPI, Tomsk, 1957.
3. V. V. Vladimirsky et al. PTE, No. 3, 1956.
4. Intern. conf. on high energy acceler. and instr., CERN, 1959.
5. CERN, CEA, p. 30, 1960.
6. K. J. Le Couteur, Proc. Phys. Soc., B 64, 1073, 1951.
7. S. Cohen, A. Crewe, Proc. CERN Symp., 1, 140, 1956.
8. Yu. Ya. Lembra. ZhTF, 29, 992, 1959.
9. Yu. Ya. Lembra. ZhTF, 30, 405, 1960.
10. A. A. Vorob’ev, L. S. Sokolov. Collection “Accelerators.” Gosatomizdat, 1960.
11. A. Turren. «Nuovo Cimento», X, 8, 511, 1958.
12. F. W. Brasse, G. E. Fischer, M. Fotino, K. W. Robinson, Intern. conference on high energy accelerators, Dubna, 1963.
13. U. Bizzari, A. Turren. «Nuovo Cimento», 37, 2, 751, 1965.
14. Kh. P. Simonyan. Report EKU-49, E, 1958.
15. A. Ts. Amatuni. Report EKU-57, E, 1958.
16. L. N. Bondarenko. Dissertation, IYaF SO AN SSSR, 1963.
17. M. S. Rabinovich. Proceedings of FIAN USSR, vol. 10, Publishing House of the USSR Academy of Sciences, 1958.
18. G. I. Dimov, Yu. K. Petrov. Izvestiya Vuzov, “Fizika,” No. 5, 1959.
19. N. N. Bogolyubov, I. A. Mitropolsky. Asymptotic Methods in the Theory of Nonlinear Oscillations. Fizmat, 1955.
20. V. I. Smirnov. Course of Higher Mathematics. Vol. 2, p. 25, 1956.
21. A. V. Popov et al. ZhTF, vol. 32, 9, 1037, 1962.