FEATURES OF SOLVING ACCELERATOR ENGINEERING PROBLEMS ON ANALOG COMPUTERS

V. A. KOCHEGUROV, B. A. RYZHKOV, N. V. TRIKHANOVA, V. P. IVANCHENKOV

(Presented by the Department of Computer Engineering)

The calculation of accelerators is reduced to the calculation of trajectories of motion of charged particles in electromagnetic fields that are inhomogeneous in space and vary in time. In most cases these trajectories are described by nonlinear differential equations that have no analytical solution. The use of numerical and graphical-analytical methods leads to cumbersome computations. Solving the problem on analog computers, however, is carried out quite simply.

The features of constructing mathematical models and programming are shown using examples of calculating trajectories of charged particles in a microtron and analyzing the extraction of accelerated particles from the chamber of a cyclic accelerator. A device is described for optimizing beam-transport systems that operates jointly with an analog computer.

Modeling the Motion of Particles in a Microtron

A microtron belongs to particle accelerators with a constant controlling magnetic field [1]. In the acceleration process, the motion of particles in it occurs along orbits of increasing radius. In this case the particles are accelerated under the action of a high-frequency electric voltage over a small section of the trajectory in the resonator, while outside the accelerating gap the particles move along cyclic orbits in a constant magnetic field. To determine the parameters of the accelerating gap and the best position of the sources of charged particles, it is necessary to know the trajectories of the particles on the first turns. These problems can be solved on an analog computer with accuracy sufficient for engineering practice. A special feature of solving such problems is the need to change, during the solution according to a specified program, the structural diagram of the mathematical model.

Fig. 1. Electron trajectories in a microtron

Figure 1 shows the trajectories of motion of charged particles in a microtron on the first turns. Taking the relativistic effect into account, the equations of motion of electrons have the form [2]: in the resonator, for \(|x| \le h,\ y < b\):

\[ \begin{aligned} \frac{d^{2}y}{dt^{2}} &= \frac{e}{m_{0}}\sqrt{1-\frac{1}{c^{2}}\left(\frac{dx}{dt}\right)^{2}-\frac{1}{c^{2}}\left(\frac{dy}{dt}\right)^{2}} \left\{-\left(\frac{dx}{dt}\right)\left(\frac{dy}{dt}\right)\times\right.\\ &\quad \times \frac{E_{0}}{c^{2}}J_{0}\left(y_{01},\frac{y}{y_{0}}\right)\sin(\omega t+\varphi_{0})-\\ &\quad \left.-\frac{dx}{dt}\left[\frac{E_{0}}{c}J_{1}\left(y_{01},\frac{y}{y_{0}}\right)\cos(\omega t+\varphi_{0})+B_{z}\right]\right\}, \end{aligned} \]

\[ \begin{aligned} \frac{d^{2}x}{dt^{2}} &= \frac{e}{m_{0}}\sqrt{1-\frac{1}{c^{2}}\left(\frac{dx}{dt}\right)^{2}-\frac{1}{c^{2}}\left(\frac{dy}{dt}\right)^{2}} \left\{\left[1-\frac{1}{c^{2}}\left(\frac{dx}{dt}\right)^{2}\right]\times\right.\\ &\quad \times E_{0}J_{0}\left(y_{01},\frac{y}{y_{0}}\right)\cdot\sin(\omega t+\varphi_{0})+\\ &\quad \left.+\frac{dy}{dt}\left[\frac{E_{0}}{c}J_{1}\left(y_{01},\frac{y}{y_{0}}\right)\cos(\omega t+\varphi_{0})+B_{z}\right]\right\} \end{aligned} \tag{1} \]

and outside the resonator \((|x|>h,\ y>y_{0})\):

\[ \frac{d^{2}y}{dt^{2}}= \frac{e}{m_{0}}\sqrt{1-\frac{1}{c^{2}}\left(\frac{dx}{dt}\right)^{2}-\frac{1}{c^{2}}\left(\frac{dy}{dt}\right)^{2}}\cdot \left(-\frac{dx}{dt}B_{z}\right), \]

\[ \frac{d^{2}x}{dt^{2}}= \frac{e}{m_{0}}\sqrt{1-\frac{1}{c^{2}}\left(\frac{dx}{dt}\right)^{2}-\frac{1}{c^{2}}\left(\frac{dy}{dt}\right)^{2}}\cdot \left(\frac{dy}{dt}B_{z}\right), \tag{2} \]

where

\[ e,\ m_{0} \]
are the charge and rest mass of the electron;

\[ E_{0} \]
is the amplitude of the electric-field intensity on the resonator axis;

\[ J_{0}\left(y_{0},\frac{y}{y_{0}}\right),\quad J_{1}\left(y_{01},\frac{y}{y_{0}}\right) \]
are Bessel functions of zero and first order;

\[ B_{z} \]
is the induction of the constant magnetic field;

\[ \varphi_{0} \]
is the initial phase of the electron motion relative to the accelerating voltage.

An analytical solution of these equations is rather labor-intensive, since the calculation must be carried out for a large number of variants of the initial values and of the parameters of the accelerating gap. Owing to the simplicity of programming and sufficient speed, analog computers make it possible to obtain a large number of solutions in a short time.

The continuity of the solution process, i.e., the transition from system (1) to system (2), is provided by means of a specially developed switching device. The use of relays for this purpose is impractical because of the long contact switching time, comparable with the time required to solve the trajectory equations in the accelerating gap. Electronic switching devices based on vacuum diodes and operational amplifiers have been developed and used; they are assembled from standard analog-computer elements. Figure 2 shows the circuit of a programmed switching device that provides selection of the region bounded by three straight lines.

Diodes \(D_{3}, D_{4}, D_{5}, D_{6}\) and amplifier 4 constitute the circuit of a key that switches voltages of either polarity. Diodes \(D_{1}, D_{2}\) and amplifiers 2, 3 generate a control signal selecting the region \(|x|\le h\). Diodes \(D_{7}, D_{8}\) and amplifiers 5, 6 generate a control signal selecting the region \(y\le b\), where \(b\) and \(h\) are the height and half-width of the resonator.

Connecting a switch between the input resistance of the amplifier and the summing point, whose potential is close to zero, makes it possible to significantly reduce the influence of the control signals on the controlled voltage. The number of control signals applied to the switch circuit is determined depending on the configuration of the constraint region.

Fig. 2. Schematic of the programming device

Study of the Extraction of Accelerated Particles in Cyclic Accelerators

For conducting physical experiments, beams of charged particles are extracted from accelerator chambers. In doing so, it is necessary to ensure the displacement of particles from the equilibrium orbit to the extraction channel and to carry out the extraction. To obtain efficient extraction of accelerated particles, one must know the trajectories of all particles under an external perturbation and accordingly choose the required perturbation. In known methods of investigation it is impossible to evaluate the behavior of the entire beam of extracted particles during the extraction process. Such an evaluation can be obtained by representing the beam characteristics on the phase plane. Because of the nonlinearity of the focusing forces, in this case it is impossible to evaluate the beam behavior from the envelope, since the nonlinearity leads to distortion of the phase ellipse [1]. The solution was carried out for the case when the displacement of particles from the equilibrium orbit was achieved by means of an asymmetric displacement of the equilibrium orbit. The motion of the particles is then described by the equation

\[ \frac{d^2 x}{d\Theta^2} + F(x) = r_0 \delta(\Theta), \tag{3} \]

where
\(x\) — deviation from the equilibrium orbit;
\(\Theta\) — particle azimuth;
\(r_0\) — radius of the equilibrium orbit;
\(\delta(\Theta)\) — perturbation;
\(F(x)\) — focusing force.

The perturbations causing the asymmetry of the equilibrium orbit may be located at different azimuths. Depending on the position of the perturbation, it is necessary to provide for a switching device ensuring continuity of the solution process. The circuit for solving the equation describing particle extraction is presented in Fig. 3. In this case the perturbation is applied at azimuths from \(\pi\) to \(2\pi\), and the magnitude of the perturbation increases linearly from 0 to \(\delta\).

In the case of placing the perturbation at different azimuths, it is necessary to provide additional switching devices.

Automatic Optimization of Charged-Particle Beam Transport Systems

The transport system for charged-particle beams in accelerators is, in the general case, a complex electron-optical system consisting of some set of focusing and steering elements. In the development and study of charged-particle beam transport systems, one of the main problems is the task of finding such a set of parameters of the electron-optical system that would most fully meet the specified requirements. Analysis and synthesis of such systems for the purpose of selecting optimal parameters is a complex task and requires the use of computing equipment and devices implementing one or another search method. The search for optimal parameters of an electron-optical system for specific conditions of its use in accelerators of various types can be carried out according to the following criteria:

Fig. 3. Block diagram for solving equation (3)

1. Selection of the optimal parameters of the system from the condition of matching the characteristics (emittance) of the beam at the output of the channel with characteristics satisfying the specified requirements.

2. Selection of the optimal parameters from the condition of ensuring transport of the charged-particle beam without losses at the minimum possible cost of the system.

3. Selection of the optimal parameters of the system from the conditions of satisfying the first and second criteria simultaneously.

The selection of optimal parameters of the transport channel can be carried out comparatively easily and simply on an analog computer using the correlation-extremal system developed by us [4]. Let us consider the solution of the problem of matching the emittance of the injected beam with the admittance of the accelerator. A similar problem is solved for external injection in powerful cyclic accelerators [5].

The search for optimal parameters of the forming system is carried out as follows: on the analog computer, the equation for the amplitude function $\beta$ is solved, which characterizes the change in the beam parameters on the phase plane as it moves along the channel [3]. The voltage characterizing, in the chosen scale, the amplitude function $\beta$ and its derivative at the output of the channel is fed to the device for transforming (generating) the phase ellipse. Construction of the phase ellipse, for example for the horizontal plane, from the obtained values of $\beta$ and $\beta'$ is performed by solving the equation

\[ x'_{1,2}=\frac{1}{\beta}\left[\pm \sqrt{\varepsilon\beta - x^{2}}+\frac{1}{2}\beta' x\right], \tag{10} \]

where $x$ is the transverse displacement of particles from the axis,
$x'$ is their angular divergence.

The voltage proportional to the variable \(x\) in equation (10) is set by a linearly increasing voltage generator, whose level is varied by the discriminator \(x_{\max}\) so that, as \(\beta\) changes during the solution process, the condition is always satisfied

\[ x_{\max}=\sqrt{\varepsilon\beta}. \tag{11} \]

To reproduce the beam emittance in the form of an optical image on the screen of a cathode-ray tube, a special photoelectronic scanning device has been developed, by means of which the area of the phase ellipse is illuminated; its position and shape change depending on the parameters \(K_i\) of the focusing lenses [6]. In the optical correlator, the beam emittance at the output of the channel, obtained for specified values of the coefficients \(K_i\), is compared with the admittance of the accelerator. A Berger optical correlator [7] is used as the optical-type computing device. With the aid of the optimizer OP, the parameters of the electron-optical system are sought for which maximum matching of the beam emittance at the channel output with the admittance of the main accelerator is obtained. The search for the optimal parameters \(K_i\) is carried out by one of the stochastic-search algorithms, for example, by the coordinatewise self-learning algorithm with an arbitrary law of variation of the probability [8]. Switching in the structural circuit for solving the equation of the amplitude function \(\beta\) and synchronization of the operation of the extremal system are carried out by a special programming device.

If it is necessary to minimize the cost of the system for transporting a beam of charged particles, then the volume of the air gap for all magnets of the system is chosen as the function to be minimized,

\[ V=\sum_{i=1}^{n} l_i \sqrt{\varepsilon_x\cdot \varepsilon_z \cdot \beta_i^{(x)} \cdot \beta_i^{(z)}}, \tag{12} \]

where

\(n\) is the number of forming elements of the transport system;
\(l_i\) is the length of the \(i\)-th element.

In this case, to search for the optimal parameters of the focusing system, an additional device for computing the volume according to expression (12) is introduced into the extremal system. The device for computing the volume is built from standard analog-computer blocks.

Fig. 4. Block diagram of a device for optimizing the parameters of a system for transporting charged-particle beams

Fig. 4 shows a block diagram of the device for optimizing the parameters of a system for transporting beams of charged particles.

With the aid of the developed analog computing device, calculations were performed and the optimal parameters of the injection channel of a synchrotron for \(1.5\ \text{GeV}\) at the Scientific Research Institute of Nuclear Physics, Electronics, and Automation (NII YaFEA) of Tomsk Polytechnic Institute were found [5].

References

1. J. Livingston. Principles of Operation of Cyclic Accelerators. Translated from English, IL, 1963.

2. H. Rogenhagen and F. Haberstok. Zeitschrift für Angewandten Phyjsik, 19, Band, Heft 1, Berlin (1965), S. 5—10.

3. V. A. Kocheturov, N. V. Trikhanova. Elektronnaya tekhnika, I, 1, No. 1, 1967.

4. V. P. Ivanchenkov, V. A. Kocheturov, G. A. Zheltovskii. Application of an optical correlator for calculating the injection path of accelerators. Abstracts of a report at the XXI Scientific-Technical Conference of LIAP. L., 1967.

5. A. A. Vorob’ev et al. The TPI Synchrotron at 1.5 GeV. Moscow, Atomizdat, 1968.

6. V. P. Ivanchenkov, G. A. Zheltovskii. A photoelectronic scanning device for analyzing charged-particle beam transport systems. Abstracts of a report at the scientific-technical conference devoted to Radio Day. Tomsk, 1968.

7. F. B. Berger. Optical Cross-Correlator, Patent of United States of America, No. 27 87188, 1957.

8. A. A. Rastrigin. Statistical Search Methods. Moscow, Nauka Publishing House, 1968.