ANALOG COMPUTER STUDIES OF THE MOTION OF CHARGED PARTICLES IN A CYCLOTRON

A. A. KOMOV, B. A. RYZHKOV, N. V. TRIKHANOVA, S. N. YUZHAKOVA

(Presented by the Department of Computer Engineering)

The large spread of ion energies in the beam extracted from a cyclotron makes it difficult to carry out physical experiments on resolving closely spaced energy levels of atomic nuclei. The monochromaticity of the extracted beam of accelerated ions can be improved by applying magnetic analysis, which, however, is associated with a large loss of intensity. Other methods of improving the monochromaticity of the extracted beam are possible: the use of focusing diaphragms on the cyclotron dees, displacement of the source from the geometric center, and rotation of it through some angle [1, 2].

Fig. 1. Schematic representation of the cyclotron dees

To obtain complete information on the possibility of placing slits and determining their location, it is necessary to calculate the ion orbits for various initial conditions. The motion of ions in the central region of the cyclotron (Fig. 1) in a Cartesian coordinate system is described by the system of differential equations:

\[ \frac{d^{2}x}{dt^{2}}=\omega \frac{dy}{dt}+\frac{q}{m_{0}}\frac{V_{m}}{\Delta}\cos(\omega t+\varphi_{0}), \]

\[ \frac{d^{2}y}{dt^{2}}=-\omega \frac{dx}{dt} \tag{1} \]

in the accelerating gap, and by the system of equations

\[ \frac{d^{2}x}{dt^{2}}=\omega \frac{dy}{dt}, \]

\[ \frac{d^{2}y}{dt^{2}}=-\omega \frac{dx}{dt} \tag{2} \]

outside the accelerating gap.

Here \(\omega\) is the ion revolution frequency,

\(q, m_0\) — charge and mass of the ion,

\(V_m\) — amplitude of the accelerating high-frequency electric field,

\(\Delta\) — distance between the dees.

The solution of the systems of differential equations (1), (2) was carried out in [1] by the graphical-analytical method in analyzing the operation of a cyclotron in the regime with unregulated energy of the extracted ions. In this case the action of diaphragms, which shield the space between the dees from the sagging of the electric field, is not taken into account. With the introduction of shielding, the pattern of the electric field relative to the geometrical axis passing along the dees becomes asymmetric. The use of the graphical-analytical method for calculating the orbits of a cyclotron operating in the regime with regulated energy requires excessively large expenditures of time, since it is necessary to compute orbits for cases of ion acceleration with different initial conditions and with different charge-to-mass ratios.

For the calculation, analog computing machines (ACM) were used, providing sufficient accuracy of the results in computing the first several orbits, which also determine the formation of the beam. ACMs provide a visual character to the solution, which makes it possible immediately to see unsuitable variants, the convenience of specifying initial conditions and changing the values of parameters, which facilitates the rapid carrying out of calculations in a form convenient for analysis. The structural diagram of the model, compiled on the basis of systems of equations (1), (2), is presented in Fig. 2, where 1—6 are integrators, 7 is an inverter, \(K_1\)—\(K_4\) are constant-coefficient units. Integrators 1, 2 are intended for solving the first equation, and 3, 4 for the second equation of system (1). Blocks 5—7 are intended for reproducing the accelerating sinusoidal high-frequency field. With the switch open, system (2) is solved; with the switch closed, system (1) is solved. Control of the solution process, i.e., the transition from solving the system of equations (1) to system (2), is performed by a programming device that makes it possible to use the values of the variables obtained at the end of the preceding section as the initial values for the following section. The use of electromagnetic relays for constructing the programming device is inadmissible, since, for a favorable time scale, the flight time of the relay contacts is comparable with the solution time on the section of ion motion in the accelerating gap; diode switches fully meet the necessary requirements and do not introduce any appreciable error.

Fig. 2. Structural block diagram for solving systems of equations (1) and (2)

The circuit of the programming device is shown in Fig. 3. The control voltage is the voltage from the output of integrator 2, representing the coordinate \(x\). Diodes \(D_1, D_2\) and operational amplifier 1 select the modulus of the control voltage, which is compared with the voltage \(E_0\), corresponding to half the distance between the dees. To reduce the actuation time of the switch, diode elements should be included in the feedback circuit of operational amplifier 2, ensuring a steep leading edge of the signal controlling the switch circuit. The circuit

a switch, consisting of diodes \(D_3\)—\(D_6\) and amplifier 5, can operate with signals of either polarity. Resistors \(R_1\) and \(R_2\) are the input resistances of operational amplifier 5; diodes \(D_4\) and \(D_5\) are connected directly to the summing point, whose potential is close to zero. This connection of the diodes increases the sensitivity with respect to the control voltage. The error in triggering the switch circuit when the control voltage changes does not exceed \(0.1\ \mathrm{V}\). The asymmetry of the electric-field pattern with respect to the \(y\) coordinate, with shielding taken into account, is implemented by a simple additional circuit controlled by a voltage proportional to the first derivative

\[ \frac{dx}{dt}. \]

Fig. 3. Diagram of the programming device

The results of the calculation were observed on an I-4M oscilloscope. Figure 4 shows oscillograms of the trajectories of charged particles for different initial phases, constant amplitude, and different frequencies of the accelerating high-frequency field. As can be seen from Fig. 4, the particle trajectories intersect one another, and the placement of the diaphragms

Fig. 4. Solution results—oscillograms of ion trajectories in a cyclotron at different initial phases

of the accelerating high-frequency field. As can be seen from Fig. 4, the particle trajectories intersect one another, and the placement of the diaphragms

would lead to the loss of a considerable fraction of the ions. Figure 5 shows oscillograms of ion trajectories for various frequencies and various amplitudes of the accelerating high-frequency field. It is evident from Fig. 5 that by varying the amplitude of the high-frequency field one can bring closer together the orbits of ions accelerated at different frequencies of the high-frequency field.

Fig. 5. Results of the solution—oscillograms of ion trajectories at various amplitudes and frequencies of the accelerating voltage

field. Analysis of the calculation results showed that the focusing diaphragms for an accelerator operating in a regulated-energy mode can be used provided that the accelerating voltage is regulated in the range of 100–150 kV.

The calculation of ion orbits when the source is turned through some angle was also carried out on an analog computer. In this case the motion of the ion during the first half-turn is described by the system of equations:

\[ \frac{d^{2}x}{dt^{2}}=\omega \frac{dy}{dt}-\frac{q}{m_{0}}\frac{V_{m}}{\Delta}\cos \alpha \cos(\omega t+\varphi_{0}), \]

\[ \frac{d^{2}y}{dt^{2}}=-\omega \frac{dx}{dt}-\frac{q}{m_{0}}\frac{V_{m}}{\Delta}\sin \alpha \cos(\omega t+\varphi_{0}), \tag{3} \]

where

\(\alpha\) is the angle of rotation of the ion source. The subsequent orbits are described by the systems of equations (1) and (2). The use of an analog computer makes it much simpler to determine the required angle of rotation and the magnitude of the displacement of the ion source in comparison with the graphical-analytical method used in [2].

REFERENCES

1. N. I. Venikov, N. I. Chumakov. Zh., “Atomic Energy,” vol. II, no. 3, 1961.

2. S. I. Kozlov, R. Ts. Oganesyan, Pek Den Gi, I. A. Shchelaev. Calculation of the motion of ions in the central region of a cyclotron when the source is turned. JINR Preprint, 1966.