Particle Motion in Waveguide Synchrotrons

(Accounting for Nonlinear Terms)

A. N. Didenko, V. K. Kononov

Accelerating systems based on bent closed waveguides may be of definite interest for accelerator technology [1, 2]. A distinctive feature of accelerators with such accelerating systems is that in them the accelerating field has all components, rapidly varying along the radius and height of the waveguide.

In works [3, 4], particle motion in waveguide synchrotrons was considered in the linear approximation. It was shown that, in such an approximation, the high-frequency field does not lead to a redistribution of the damping decrements. However, it is necessary to note the following: since in waveguide synchrotrons the quantity in which the expansion is carried out is \(\chi\), the linear theory is valid only for those \(\rho\) for which the relation \(\chi \ll 1\) is satisfied. The quantity

\[ \chi \sim \frac{1}{r_2-r_1} \]

(\(r_1\) and \(r_2\) are the inner and outer radii of the chamber), and therefore the linear theory is valid for deviations considerably smaller than those for which the linear theory of particle oscillations in ordinary synchrotrons is valid, where the expansion is carried out in

\[ \frac{\rho}{r_0}. \]

It follows from this that nonlinear terms in waveguide synchrotrons must have a more substantial influence on particle motion in comparison with their influence in ordinary synchrotrons.

The present work is devoted to an investigation of the influence of nonlinear terms on particle motion in waveguide synchrotrons. For simplicity of calculation, we shall assume that waves of type \(LE\) of bent closed waveguides of rectangular cross section \((E_z=0)\), with diaphragms arranged along the bent walls, are used for acceleration.

Writing the equations of motion, one can verify that in this case the oscillations in \(z\) are not coupled with the radial oscillations. Therefore it is more convenient to begin the consideration precisely with the axial oscillations. In expanding the high-frequency field in a series, we shall restrict ourselves only to quadratic terms. As for the expansion in a series of the components of the external magnetic field, we shall assume that for it it is still sufficient to restrict ourselves to retaining the linear terms. Without accounting for

of radiation for particles moving in an external magnetic field in the presence of the high-frequency field of an \(LE\)-wave of a curved closed waveguide of rectangular cross section, there exists the following integral of motion:

\[ E\left(\frac{r^2\dot{\theta}}{c^2}-\frac{q}{\omega}\right) +\frac{er}{c}\left(A_\theta^{\mathrm{rf}}+A_\theta^{\mathrm{ext}}\right) =\mathrm{const}, \tag{1} \]

where \(A_\theta^{\mathrm{ext}}\) and \(A_\theta^{\mathrm{rf}}\) are the \(\theta\)-component of the vector potential of the external magnetic and high-frequency fields,
\(q\) is the multiplicity.

The existence of this integral greatly simplifies the solution of our problem. Restricting ourselves to first-order terms in expanding \(\dot{\theta}\) in a series, one can show that

\[ \dot{\theta}=\dot{\theta}_s\left[ 1-\frac{\rho}{r_s} -\frac{E_{\theta,s}^{\mathrm{rf}}}{\kappa r_s H_z^{\mathrm{ext}}}\psi \right], \tag{2} \]

where \(E_{\theta,s}^{\mathrm{rf}}=E_{\theta,\max}^{\mathrm{rf}}\cos\varphi_s\).

Using this expression for \(\dot{\theta}\), the equation of motion of particles in a waveguide synchrotron, with allowance for second-order small terms, can be written in the following form:

\[ \frac{d}{dt}\left(\frac{E}{c^2}\dot{z}\right) +\frac{E}{c^2}\dot{\theta}_s^{\,2} n_0 A z = -\frac{E}{c^2}\dot{\theta}_s^{\,2} n_0 z \left\{-B\frac{\rho}{r_s}+D\psi\right\}, \tag{3} \]

where

\[ A=1+\frac{\kappa_z^2 r_s^2}{\kappa r_s}\, \frac{1}{n_0}\, \frac{E_{\theta,s}^{\mathrm{rf}}}{H_z^{\mathrm{ext}}}\tan\varphi_s, \]

\[ B=\frac{\kappa_z^2 r_s^2}{n_0\kappa r_s}\, \frac{E_{\theta,s}^{\mathrm{rf}}}{H_z^{\mathrm{ext}}}\tan\varphi_s, \tag{4} \]

\[ D=A\,\frac{E_{\theta,s}^{\mathrm{rf}}}{\kappa r_s H_z^{\mathrm{ext}}} +\frac{\kappa_z^2 r_s^2}{n_0\kappa r_s}\, \frac{E_{\theta,s}^{\mathrm{rf}}}{H_z^{\mathrm{ext}}}, \]

\(n_0\) is the magnetic-field index,

\[ \kappa_z=\frac{\pi}{b}, \]

\(b\) is the height of the waveguide.

Since the quantities on the right are of second order, instead of \(\rho\) and \(\psi\) one may substitute their values found from the solution of the linear equations.

The presence of a term \(\sim \rho\) on the right-hand side of the equation leads only to a transfer of the energy of radial oscillations into axial ones and therefore does not pose a great danger.

A much greater danger is posed by the term \(\sim \psi\), which leads to coupling of the transverse and longitudinal oscillations and, in the case of resonance, may lead to an unbounded increase of the amplitudes of the transverse oscillations. We shall therefore examine in detail the influence of this term on the axial betatron oscillations of the particles. Equation (3) will then have the form:

\[ \frac{d}{dt}(m\dot{z}) +\dot{\theta}_s^{\,2}m \left[ n_0-\left(\frac{\kappa_z}{\kappa}\right)^2 \frac{qeV_{\max}\sin\varphi_s}{2\pi E} \right]z = \]

\[ = m\dot{\theta}_s^{\,2} \left(\frac{\kappa_z}{\kappa}\right)^2 \frac{qeV_{\max}\sin\varphi_s}{2\pi E}\,z\psi. \tag{5} \]

As is known [5, 6], equations of this type describe the action of external periodic forces on nonlinear oscillatory systems with slowly varying parameters. In the general case, which also occurs in our problem, the frequencies and amplitudes of the external forces will vary slowly with time.

Using the solution method described in detail in [5], one can show that, in the first approximation,

\[ Z=a\cos\left(\frac{1}{2}\int \nu\,d\psi\right), \tag{6} \]

where \(a\) and \(\psi\) must satisfy the equations

\[ \frac{da}{dt} = -\frac{a}{2m\omega}\frac{d(m\omega)}{dt} + \dot{\Theta}_s^{\,2}\left(\frac{\kappa_z}{\kappa}\right)^2 \frac{qeV_{\max}\cos\varphi_s}{2\pi E}\, a\tilde{\psi}_0\frac{\sin 2\psi}{2\nu}, \]

\[ \frac{d\psi}{dt} = \omega-\frac{1}{2}\nu + \dot{\Theta}_s^{\,2}\left(\frac{\kappa_z}{\kappa}\right)^2 \frac{qeV_{\max}\cos\varphi_s}{2\pi E}\, \tilde{\psi}_0\frac{\cos 2\psi}{2\nu}, \tag{7} \]

\(\omega\) and \(\nu\) are the frequencies of betatron and phase oscillations, respectively:

\[ \omega = \dot{\Theta}_s \sqrt{ n_0-\left(\frac{\kappa_z}{\kappa}\right)^2 \frac{qeV_{\max}\sin\varphi_s}{2\pi E} }, \]

\[ \nu = \dot{\Theta}_s \sqrt{ \frac{qeV_{\max}\sin\varphi_s}{2\pi(1-n_0)E} }. \]

Introducing the new variable

\[ a'=ae^{\frac{1}{2}\ln\left(\frac{m\omega}{m_H^{\omega}}\right)}, \]

the system of equations (7) can be transformed to the following form:

\[ \frac{da'}{d\tau} = \dot{\Theta}_s \left(\frac{\kappa_z}{\kappa}\right)^2 \tilde{\psi}_0 \frac{qeV_{\max}\cos\varphi_s}{2\pi E}\, a'\frac{\sin 2\psi}{2\nu}, \]

\[ \frac{d\psi}{d\tau} = \omega-\frac{1}{2}\nu + \dot{\Theta}_s^{\,2} \left(\frac{\kappa_z}{\kappa}\right)^2 \tilde{\psi}_0 \frac{qeV_{\max}\cos\varphi_s}{2\pi E}\, \frac{\cos 2\psi}{2\nu}, \tag{8} \]

where \(\tau=\dot{\Theta}_s t\).

It follows from expressions (8) that if, at some instant of time,

\[ \omega(t)=\frac{1}{2}\nu(t), \]

then at that instant the resonance conditions between the betatron and phase oscillations will be satisfied. However, since the frequencies vary with time, the increase in the amplitude of the betatron oscillations during passage through resonance will depend on the rate of change of the parameters. For a certain rate of change of the system parameters, the presence of such resonances will not exert a significant influence on the motion of particles in waveguide synchrotrons.

Figures 1–3 present the results of a numerical solution of system (8) for an accelerator with the following parameters:

\[ n_0=0.6;\qquad \left(\frac{\kappa_z}{\kappa}\right)^2=0.6;\qquad r_s=33\ \text{cm};\qquad t_{\text{rev}}=10^{-6}\ \text{sec}; \]

\[ q=500;\qquad \lambda=40\ \text{cm};\qquad eV_{\max}=5\cdot10^5\ \text{eV}. \]

These parameters are close to those that may be of practical interest. The high intensity of the high-frequency field

at the initial instant of time is necessary in order that the dimensions of the separatrix be equal to the radial dimensions of the chamber at a high harmonic number. The dependence of the particle energy on time was chosen in the form

\[ E = E_0 \left( 1 + \frac{\beta}{T_{\text{acc}}}\, t \right), \qquad \beta = 100, \qquad E_0 = 5 \cdot 10^7 \ \text{eV}. \]

\[ \xi = \frac{\omega}{\tfrac{1}{2}\nu} - 0.9857 \]

Fig. 1

\[ \xi = \frac{\omega}{\tfrac{1}{2}\nu} - 0.9985 \]

Fig. 2

and the calculations were performed for \(T_{\text{acc}} = 0.01\ \text{s}, 0.1\ \text{s}\), and \(1\ \text{s}\) (Figs. 1, 2, and 3, respectively).

It is evident from these figures that, when passing through resonance, the amplitude of betatron oscillations changes only slightly if \(T_{\text{acc}} = 1\) sec, and will increase strongly when the acceleration time is decreased. This result, unexpected at first glance, is a consequence of the fact that, when we change the acceleration time, we simultaneously change the strength of the high-frequency field.

\[ \xi=\frac{\omega}{\frac12\nu}-0.9994 \]

Fig. 3

If, however, the voltage is not changed as the acceleration time is changed, then the coefficients of \(\sin 2\psi\) and \(\cos 2\psi\) in equations (8) will remain constant, and the results obtained in this case will indicate that, as the speed of passage through resonance increases, the increase in the oscillation amplitudes will decrease.

In an analogous way, nonlinear terms can also be taken into account for radial betatron oscillations. The equation obtained can likewise be solved by numerical methods, and for it one can determine such speeds of passage through resonance at which the increase in amplitude will be insignificant.

Thus, from what has been considered it is clear that, in waveguide synchrotrons, the appearance of additional resonances is possible. At certain speeds of passage through the resonances, their influence may become insignificant.

REFERENCES

1. A. A. Vorob'ev, A. N. Didenko, E. S. Kovalenko, B. N. Morozov. Proc. Intern. Conf. CERN, p. 680, 1959.

2. A. A. Vorob'ev et al. Waveguide synchrotron at 10 MeV. Atomic Energy, 18, 6 (1965).

3. E. S. Kovalenko. Proceedings of Higher Educational Institutions, Physics, No. 6, 85 (1959).

4. E. S. Kovalenko. Ibid., No. 3, 175 (1960).

5. N. N. Bogolyubov, Yu. A. Mitropol'skii. Asymptotic Methods in the Theory of Nonlinear Oscillations. Moscow, 1958.

6. Yu. A. Mitropol'skii. Problems of the Asymptotic Theory of Nonstationary Oscillations. Moscow, 1964.