ELECTRONIC MODELING OF THE PROCESS OF PARTICLE MOTION IN THE INJECTION DEVICE OF CYCLIC ACCELERATORS

N. V. TRIKHANOVA, P. M. SHCHANIN

(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)

In external injection in cyclic accelerators, particles traverse part of their path in a stray magnetic field, which bends their trajectory and reduces the efficiency of injection. Therefore, in order to place the particles onto a prescribed orbit and with the required parameters, it is necessary to compensate the action of the magnetic field. One of the methods of compensation is to create, in the injection device, an electric field whose action is opposite to that of the magnetic field. If the injection channel is set exactly tangent to the orbit of the accelerated particles, the electric field is created in an inflector made in the form of a plane capacitor. If the injection channel is arranged at an angle to the orbit, it is necessary to use an inflector that is a cylindrical capacitor. In practice, complete compensation of the action of the stray magnetic field on the particles being injected cannot be obtained, since the inflector electrodes would have to be of a very complex design in order to provide an electric-field distribution identical to the magnetic one. Usually the electrodes of the injection device are divided into several parts, and different voltages are applied to them, creating a stepwise distribution of the electric field.

Let us consider the calculation of electron trajectories in inflectors of two types: in a plane capacitor and in a cylindrical capacitor, divided into three parts.

For the case of a cylindrical capacitor (Fig. 1), the dependence of the magnetic field \(H\), directed perpendicular to the plane of the drawing, on the angle \(\varphi\) is shown in Fig. 2. The electron-optical properties of such a system, but with homogeneous electric and magnetic fields, are described in the literature [1]. In deriving the equation of the electron trajectory in the inflector, the method set forth in that work was used.

Since only trajectories that differ little from the circle \(r=a\) are considered, one may put

\[ r = a(1+z) \]

and neglect both \(z^2\) and the products of \(z\) with other small quantities. We also assume that the energy of the electrons changes only insignificantly during their motion in the injection device.

Using the equations of motion in polar coordinates,

\[ m\frac{d}{d\psi}\left(r^2\dot{\psi}\right)=r\frac{dr}{dt}eH, \tag{2} \]

\[ m\frac{d^2r}{dt^2}=mr\dot{\psi}^{\,2}-\frac{eU}{r\ln\frac{R_1}{R_2}}-r\dot{\psi}eH . \tag{3} \]

Fig. 1. Cylindrical capacitor.

Fig. 2. Dependence of the magnetic field on the angle.

and expression (1), we write the equation of the electron trajectories

\[ \frac{d^2 z}{d\psi^2} = 1+z - \frac{ eU\left[1-\dfrac{2aeHz}{mv_0}+3z\right] }{ a^2m\left(\dfrac{v_0}{a}\right)^2\ln\dfrac{R_1}{R_2} } - \frac{ aeH\left(1-\dfrac{aeHz}{mv_0}+3z\right) }{ v_0 }, \tag{4} \]

where
\(e, m\) are the charge and mass of the electrons;
\(v_0\) is the initial velocity of the electrons;
\(H\) is the magnetic-field intensity;
\(U\) is the potential difference on the capacitor;
\(R_1\) and \(R_2\) are the radii of curvature of its plates.

Introduce the notation:

\[ H=H_{\max}b(\psi), \]

\[ U=U_{\max}a(\psi), \]

where \(e(\psi)\) and \(a(\psi)\) are functions whose maximum value is equal to unity,

\[ \frac{eU_{\max}}{mv_0^2\ln\dfrac{R_1}{R_2}}=A; \qquad \frac{eH_{\max}}{\dfrac{v_0}{a}m}=B. \]

The quantity

\[ \left(\frac{eH}{\dfrac{v_0}{a}m}\right)^2 \]

in equation (4) may be neglected. Equation (4) takes the form

\[ \frac{d^2z}{d\psi^2} = 1+z - Aa(\psi)\,[1-2Bb(\psi)z+3z] - Bb(\psi)\,[1+3z]. \tag{5} \]

Numerical solution of this equation leads to cumbersome calculations. Therefore, the motion of particles was modeled on the MNB-1 analog computer.

The initial data are the electron energy and the beam parameters at the inflector exit: the dimensions and the angle of convergence. It is necessary to choose the voltages on the capacitor plates in such a way that the beam has the smallest deviation from an arc of radius \(a\). For selected plate voltages, it is necessary to determine the deviation of the beam from an arc of radius \(a\), the beam dimensions, and the angle of convergence at the entrance to the inflector.

To simplify the structural block diagram of the model, we shall adopt the following assumption: in the product \(2Aa(\psi)Bb(\psi)z\) we shall take \(a(\psi)=1\). In reality \(a(\psi)\) differs little from 1, and the multiplier is the small quantity \(z\); by adopting this assumption, we introduce an insignificant error. We write equation (5) in a form convenient for solution on an analog computer:

\[ \frac{d^{2}z}{d\psi^{2}} = 1+z - Aa(\psi)(1+3z) + Bb(\psi) - 3Bb(\psi)z + 2ABb(\psi)z . \tag{6} \]

Replacing \(\psi\) in equation (6) by the machine time \(\tau\), and \(z\) by the voltage \(U\) with the corresponding scale factors \(M_\tau\) and \(M_z\), we obtain the equations in machine variables,

Fig. 3. Structural diagram for a cylindrical capacitor.

\[ -\frac{d^{2}U}{d\tau^{2}} = M_{\tau}^{2} \left[ \frac{1}{M_z} + U - Aa\!\left(\frac{\tau}{M_\tau}\right)(1+3U) + Bb\!\left(\frac{\tau}{M_\tau}\right) - \right. \]

\[ \left. - 3Bb\!\left(\frac{\tau}{M_\tau}\right)U + 2ABb\!\left(\frac{\tau}{M_\tau}\right)U \right]. \tag{7} \]

The structural block diagram for solving this equation is presented in Fig. 3, where 1, 2 are operational amplifiers in the integration mode; \(\alpha_1, \alpha_2, A_1, B_1\) are blocks of constant coefficients; 4, 5, 6, 7 are blocks of variable coefficients reproducing the functions \(a\left(\dfrac{\tau}{M_\tau}\right)\) and \(b\left(\dfrac{\tau}{M_\tau}\right)\).

The equation describing the behavior of the structural circuit has the form

\[ \frac{d^2U}{dz^2} = \alpha_1 k_{21} \left[ -k_{12}U_3 a(\tau) + k_{13}b(\tau)U_4 + k_{11}U_5 + \alpha_2 k_{14}U - \right. \]

\[ \left. - k_{16}A_1Ua(\tau) - k_{15}B_1Ub(\tau) \right]. \]

Here

\[ k_{1i}=\frac{1}{R_{1i}C}, \qquad (i=1,2,\ldots 6); \]

\[ k_{21}=\frac{1}{R_{21}C}; \qquad k_{31}=\frac{R_0}{R_{31}}; \]

\(U_3, U_4\) are the voltages at the inputs of blocks 6 and 7, which reproduce the perturbing forces. Comparing equations (7) and (8), we obtain relations for determining the scale factors and the transfer coefficients of the computing elements:

\[ \alpha_1\frac{M_\tau^2}{M_z}k_{21}k_{12}U_3=A; \]

\[ \alpha_1\frac{M_\tau^2}{M_z}k_{13}k_{21}U_4=B; \]

\[ \alpha_1k_{21}k_{11}\frac{M_\tau^2}{M_z}U_5=1; \]

\[ \alpha_1\alpha_2k_{14}M_\tau^2k_{21}=1; \]

\[ \alpha_1k_{21}k_{16}A_1M_\tau^2=3A; \]

\[ \alpha_1k_{21}k_{15}B_1M_\tau^2=3B-2AB. \]

In solving the problem the following method was used: making use of the reversibility property of the electron-optical system (cylindrical capacitor), we assume that the electrons move in the reverse direction, i.e., the equation is solved from the “end.” As initial conditions we specify the value of the coordinate and of the convergence angle that must be obtained at the injector exit. As a result of the solution we obtain the beam parameters at the injector entrance.

The configuration of the electric field must be chosen in such a way that a particle with zero initial conditions at the end of the solution has a coordinate and a trajectory inclination angle as close to zero as possible. The maximum value of the voltage applied to the plates is taken as 1. Relative to this level we choose the voltages on the plates.

The results of the solution for \(A=1.062,\ B=0.076\), for a particle with various initial conditions and for various distributions of the electric field, are summarized in Table 1.

Table 1

It is seen from Table 1 that it is not possible to compensate completely for the action of the magnetic field; the best results were obtained with the following distribution of the electric field on plates 1, 2, 3: 1, 0.89, 0.95, respectively. The particle trajectories obtained for the above voltage distribution are shown in Fig. 4.

For the case of a plane capacitor, the plots of the variation of the magnetic and electric fields are presented in Fig. 5. The equation of the electron trajectory in the $xy$ plane (the direction of particle motion coincides with the $y$ axis) has the form:

\[ x''+\frac{eH}{W\beta}\sqrt{1+x'^2+z'^2}\,x'^2+ \frac{e}{W\beta}\sqrt{1+x'^2+z'^2}\times \]

\[ \times\left[\frac{E}{\beta}\sqrt{1+x'^2+z'^2}-H\right]=0. \]

Here $E$ and $H$ are the intensities of the electric and magnetic fields,

$W$ is the total energy of the particle, $\beta=\dfrac{v}{c}$ is the ratio of the particle velocity to the velocity of light. The equation can be simplified. If it is assumed that the energy of the particle and its direction change only insignificantly during the motion in the injection device, then the quantities $x'^2$ and $z'^2$

Fig. 4. Particle trajectories in a cylindrical capacitor.

Fig. 5. Magnetic field in a flat capacitor.

because of their smallness in comparison with unity, they may be neglected, and \(W\) and \(\beta\) may be regarded as constant. Denoting

\[ \frac{H}{H_{\max}}=a(y),\qquad \frac{E}{E_{\max}}=b(y),\qquad \frac{eH_{\max}}{W\beta}=A;\qquad \frac{eE_{\max}}{W\beta^{2}}=B \]

and taking into account the assumptions stated above, we finally obtain

\[ x''+Aa(y)x'^2+[Bb(y)-Aa(y)]=0. \]

Fig. 6. Block diagram for a plane capacitor.

The structural block diagram for solving this equation is shown in Fig. 6, where 1 and 2 are operational amplifiers; 3, 4, 5 are blocks of variable coefficients reproducing the functions \(a(y)\) and \(b(y)\); 6, 7 are blocks of constant coefficients; BP is the product block.

Fig. 7. Particle trajectories in a plane capacitor.

Table 2

Table 3

Table 2 gives the final values of the angle and coordinate for particles with different initial conditions after passing through the injection device at \(A = 0.234\), \(B = 0.234\).

As a result of selecting the voltages on the plates, the output data of the particles improved somewhat. The new final values of the angle and coordinate, for the same initial conditions as those given in Table 2, are presented in Table 3.

The trajectories of particles with the initial conditions given in Table 2 are shown in Fig. 7. The model was also used to determine tolerances for the stability of the electric field and particle energy, which, in the case of a flat inflector, were found to be \(\pm 0.5\%\) for the energy and \(\pm 0.1\%\) for the electric field. It was assumed here that the angular direction of the particle at the exit, due to changes in the particle energy or in the electric field, must not change by an amount greater than \(0.1\) of the allowable angle.

In conclusion, it should be noted that, despite the errors introduced into the solution by instability in the operation of the elements of the modeling machine, the model makes it possible to obtain rapidly data both on the particle parameters and on the optimal parameters of the injection device.

REFERENCES

1. B. Ya. Kogan. Electronic Modeling Devices and Their Application to the Investigation of Automatic Control Systems. Fizmatgiz, 1959.

2. A. M. Strashkevich. Electronic Optics of Electrostatic Fields That Do Not Possess Axial Symmetry. Fizmatgiz, 1959.

3. V. M. Kelman and S. Ya. Yavor. Electron Optics. Publishing House of the Academy of Sciences of the USSR, 1959.