INVESTIGATION ON AN ANALOG COMPUTER OF THE ROTATION OF A MICROTRON BEAM ABOUT THE OPTICAL AXIS OF THE FORMING SYSTEM

V. P. Ivanchenkov, P. A. Lyugai, L. V. Shilyagina

(Presented by the scientific seminar of AVTA)

When transporting a beam from the injector to the main accelerator, the task is to guide the beam without losses and to form, at the output of the injection channel, beam characteristics that most fully satisfy the conditions for capturing particles into acceleration. A characteristic feature of the TPI synchrotron injector at 1.5 GeV is that the horizontal emittance \(\varepsilon_x\) of the beam extracted from the microtron is larger than the emittance of the beam in the vertical plane \(\varepsilon_y\), while at the same time the horizontal admittance of the synchrotron is considerably smaller than the admittance in the vertical plane. In this connection, in order to increase the coefficient of electron capture into acceleration at the first stage of injection, it was proposed to carry out a “transfer” of particles from the horizontal plane to the vertical plane and vice versa, thereby, as it were, replacing the horizontal emittance by the vertical one, and the vertical by the horizontal one [1].

Such an “exchange” of emittances can be carried out in two ways: by rotating the plane of the microtron magnet through \(90^\circ\), or by rotating the beam at the microtron output through the same angle. The first variant in our case is associated with great technical difficulties; therefore we considered the problem of rotating the beam about the optical axis of the system by means of the inhomogeneous axially symmetric magnetic field of a short lens. It is known from [2, 3] that a lens will be short if the extent of its field is small in comparison with the focal length.

In the cylindrical coordinate system \(\{r,\ \varphi,\ z\}\), the motion of an electron in the field of a short lens is described by the following system of equations:

\[ \left\{ \begin{aligned} \frac{d^2 r}{dz^2} &= -\frac{e}{8m_0\Phi_r} B_z^2 r, \\ \frac{d\varphi}{dz} &= \sqrt{\frac{e}{8m_0\Phi_r}}\, B_z, \end{aligned} \right. \tag{1} \]

\[ \tag{2} \]

where \(\frac{e}{m_0}\) is the specific charge of the electron,

\(\Phi_r\) is the relativistic potential, determined in terms of the accelerating potential \(\Phi\) by the relation

\[ \Phi_r = \Phi\left(1+\frac{e\Phi}{2m_0c^2}\right), \tag{3} \]

\(B_z\) is the distribution of the magnetic-field induction along the \(z\) axis.

\(B_z\) was chosen in the form

\[ B_z=\frac{B_0}{1+\left(\frac{z}{d}\right)^2}, \tag{4} \]

which well approximates the field of a short lens [2]. Here \(B_0\) is the maximum value of \(B_z\), and \(d\) is the half-width characterizing the extent of the field. Equation (1) describes the trajectory of electron motion in a plane rotating together with the electron about the optical axis of the system with an angular velocity determined by expression (2).

The problem was solved on an MN-14 analog computer. The solution reduced to finding such a value of \(B_0\) for which the beam, on passing through the lens, would be rotated by \(90^\circ\). The solution was carried out in a cylindrical coordinate system, with parallel transformation of the results into a Cartesian system according to the relations

\[ \left\{ \begin{aligned} x&=r\cos\varphi;\\ y&=r\sin\varphi;\\ x'&=r'\cos\varphi-r\sin\varphi\,\varphi';\\ y'&=r'\sin\varphi+r\cos\varphi\,\varphi'. \end{aligned} \right. \tag{5} \]

The block diagram for solving the problem is shown in Fig. 1. Integrator 1, functional block 2, and scaling block 3 are used to reproduce the distribution of the magnetic induction \(B_z\); multiplier blocks 17, 22, integrators 18, 20, and inverters 19, 21 solve equation (1); integrator 6 solves equation (2); and the circuit assembled on the basis of trigonometric blocks 7 and 8, multiplier blocks 9, 10, 11, 13, 14, 15, and summators 12 and 16 performs the transformation of the solution results from the cylindrical coordinate system into the Cartesian system.

Fig. 1. Block diagram of the model

In the course of the solution, the trajectory of motion of the extreme (boundary) particles was followed in the phase planes \(xx'\), \(yy'\), and in the plane \(xy\) when the beam was rotated by \(90^\circ\) relative to the optical axis.

In a cylindrical coordinate system, particles in the horizontal plane satisfy the condition \(\varphi_0=0\). If the electron source is located outside the field of the lens, then \(\varphi_0'=0\). This condition is fulfilled in our case; therefore, from (5) it follows that

\[ x_0=r_0,\quad x_0'=r_0',\quad \text{and}\quad y_0=0\quad \text{and}\quad y_0'=0. \]

The initial conditions \(r_0=x_0\) at block 20, and \(r_0'=x_0'\) at block 18, were specified for concrete values of \(x_0,\ x_0'\) of the horizontal emittance \(\varepsilon_x\) (Fig. 2).

Fig. 2. Horizontal emittances: \(\varepsilon_x\)—initial, \(\varepsilon_x^*\)—final

Fig. 3. Vertical emittances: \(\varepsilon_y\)—initial, \(\varepsilon_y^*\)—final

The results of the solution are shown in Figs. 4, 5, and 6. From these figures it is seen that the trajectories of particles that initially had a displacement \(x_0\) and divergence \(x_0'\), as a result of passing through the magnetic lens rotating the beam by \(90^\circ\), converge in the phase plane \(xx'\) to the origin (Fig. 5), while in the plane \(yy'\) (Fig. 6) these same particles, after leaving the origin, acquire certain values of \(y\) and \(y'\).

Fig. 4. Projection of the trajectory of motion onto the \(xy\) plane for a rotation by \(90^\circ\) for particles of the horizontal plane

Fig. 5. Projection of the trajectory of motion onto the \(xx'\) plane for a rotation by \(90^\circ\) for particles of the horizontal plane

Thus, a “transfer” of particles from the horizontal plane \(xx'\) into the vertical plane \(yy'\) takes place. Projections of the trajectories of particle motion onto the \(xy\) plane are shown in Fig. 4.

Points 1, 2, ..., 6 on the horizontal emittance \(\varepsilon_x\) (Fig. 2) correspond to curves 1, 2, ..., 6 in Figs. 4, 5, and 6.

Particles in the vertical plane satisfy the condition \(\varphi_0=\dfrac{\pi}{2}\); therefore for them, from (5), \(x_0=0,\ x_0'=0,\ y_0=r_0,\ y_0'=r_0'\). By specifying the initial conditions \(r_0=y_0,\ r_0'=y_0'\) for concrete values of \(y_0,\ y_0'\) of the vertical emittance \(\varepsilon_y\) (Fig. 3), trajectories of motion were obtained for particles initially located in the vertical plane (Figs. 7, 8, 9).

Fig. 6. Projection of the trajectory of motion onto the plane \(yy'\) for a rotation through \(90^\circ\) for particles of the horizontal plane

Fig. 7. Projection of the trajectory of motion onto the plane \(xy\) for a rotation through \(90^\circ\) for particles of the vertical plane

From Figs. 8 and 9 it is seen that the same “transfer” of particles occurs from the vertical plane \(yy'\) into the horizontal plane \(xx'\).

From a comparison of the phase characteristics \(\varepsilon_x,\ \varepsilon_y\) at the entrance and \(\varepsilon_x^*,\ \varepsilon_y^*\) at the exit of the lens it is seen that the shape and position of \(\varepsilon_x^*\) and \(\varepsilon_y^*\) changed somewhat. In this case the emittance \(\varepsilon_x^*\) became equal to the value \(\varepsilon_y\), and \(\varepsilon_y^*\) became equal to \(\varepsilon_x\).

Fig. 8. Projection of the trajectory of motion onto the plane \(xx'\) for a rotation through \(90^\circ\) for particles of the vertical plane.

The investigation showed that inside the lens there is a substantial increase in the radius \(r\). The maximum value \(r_{\max}\) exceeds the value of the radius at the lens entrance by several times. The value of \(r_{\max}\) determines the choice of aperture of the rotating lens. The value of \(r_{\max}\) decreases when the field is compressed along the \(z\) axis. But since the rotation of the beam about the optical axis must remain the same \(\left(\dfrac{\pi}{2}\right)\), the value of the integral \(\displaystyle \int_{z_0}^{z} B_z\,dz\) does not

must change. Therefore, compression of the field along the \(z\) axis leads to an increase in its maximum value \(B_0\). Calculations showed that, for a field half-width \(d=2\) cm and a maximum induction value \(B_0=1.328\) T, the inner diameter of the coil must be equal to 20 cm. Such a concentrated field can be produced by using a shielded magnetic lens with a pole tip.

Fig. 9. Projection of the trajectory of motion onto the \(yy'\) plane for a rotation by \(90^\circ\) for particles of the vertical plane

The investigations carried out showed that the “exchange” of the beam emittances at the entrance to the channel can be accomplished by means of a simple magnetic lens having a nonuniform field distribution along the axis of rotation.

It should be noted that the use of a uniform magnetic field of a solenoid [1] for rotating a microtron beam may prove ineffective, since in a uniform field each individual electron has its own axis of rotation, which in the general case does not coincide with the optical axis of the forming system. The projection of the trajectory of motion onto a plane perpendicular to the lines of force is a circle with radius

\[ \rho=\frac{mV_n}{eB_z}, \tag{6} \]

where \(V_n\) is the transverse component of the particle’s velocity. From (6) it is evident that a uniform magnetic field has no effect at all on electrons moving parallel to the optical axis. In this connection it is not possible to rotate the beam completely through \(90^\circ\) and to carry out a complete “transfer” of particles from one plane to another, for example, from the horizontal to the vertical.

Calculations performed for the selection of the optimal parameters of the forming elements of the injection channel showed [4] that rotation of the microtron beam through \(90^\circ\) makes it possible to increase the capture coefficient at the first stage of injection by a factor of \(1.5\)–\(2\).

REFERENCES

1. R. G. Ziyakas, L. D. Opisko, I. P. Chuchalin, M. G. Shvyrtalov, P. M. Shanin. Rotation of the emittance of a microtron injector. Proceedings of the VII Interuniversity Conference on Accelerators. Atomizdat, 1970.

2. V. Glazer. Fundamentals of Electron Optics. Moscow, Gostekhizdat, 1957.

3. V. M. Kelman, S. Ya. Yavor. Electron Optics. Leningrad, “Nauka,” 1968.

4. V. P. Ivanchenkov, P. A. Dyugai, V. A. Kocheturov. Development and investigation of a correlation-extremal system for optimizing the parameters of charged-particle beam transport channels. Scientific report, NII YaFEA No. 03825. Tomsk, 1970.