ANALOG-COMPUTER STUDY OF THE FOCUSING OF AN INTENSE ELECTRON BEAM BY QUADRUPOLE FIELDS

N. V. TRIKHANOVA, L. V. PILGER, P. M. SHCHANIN

(Submitted by the Department of Computer Engineering)

In the focusing of particles of high energies, quadrupole lenses are widely used, in which the magnetic fields are perpendicular to the motion of the particles. In this case the focusing forces are proportional to the longitudinal velocity of the particles, and therefore such lenses act more effectively than lenses with a longitudinal field. However, in a space free of charges or currents it is impossible to form fields that would create radial forces directed toward the axis simultaneously at all angles. In particular, the fields in quadrupole lenses create forces that focus in one plane and defocus particles in the perpendicular plane. Therefore, in order to produce focusing in two planes, it is necessary to use a system of quadrupole lenses in which the subsequent lenses are rotated by \(90^\circ\) relative to the preceding ones. The equations of motion in a strong-focusing channel formed from magnetic quadrupole lenses are well known, and in the single-particle approximation they have analytic solutions. If the beam intensity is high (i.e., it is necessary to take Coulomb repulsion into account), then the variables of the equations for the beam envelope are not separated and have no tabulated solutions.

In the present work an attempt is made to determine the possibility of using quadrupole lenses for focusing high-intensity beams. Owing to the specific features of electron sources, it is difficult to use ordinary explicit-pole lenses; therefore it was decided to create quadrupole fields by means of linear conductors with current. Such a focusing system will have low inductance, and it can be supplied by short current pulses.

In general form, the equations of the beam envelope in a focusing system have the form [1]:

\[ \frac{d^{2}x}{d\tau^{2}}+Q_x\cdot a(\tau)\cdot x-\frac{F_0^{2}}{x^{3}}-\frac{2r_a^{2}}{x+y}=0, \tag{1} \]

\[ \frac{d^{2}y}{d\tau^{2}}-Q_y\cdot a(\tau)\cdot y-\frac{F_0^{2}}{y^{3}}-\frac{2r_a^{2}}{x+y}=0, \]

where

\[ r_a=S\sqrt{\frac{2I_{\text{beam}}}{\beta^{3}\gamma^{3}I_0}}; \]

\[ Q_x=\frac{eS^2}{m_0\gamma\beta c}\cdot\frac{\partial B_y}{\partial x} =\frac{S^2\,2\mu_0 e I_{\ell}}{m_0\gamma\beta c\pi r_{\mathrm{m}}^2}; \]

\[ Q_x=Q_y=Q; \]

\[ I_0=4\pi\varepsilon_0\frac{m_0c^3}{e}; \]

\[ \gamma=\frac{1}{\sqrt{1-\beta^2}}; \]

\(r_{\mathrm{m}}\) is the distance from the optical axis to the center of the current-carrying conductor,

\(S\) is the length of the system period,

\[ \tau=\frac{\beta c}{S}\,t \]
is a dimensionless variable expressed in units of periodicity,

\(I_{\text{beam}}\) is the beam current,

\(I_{\ell}\) is the current in the lens,

\(\mu_0\) is the magnetic permeability,

\(\varepsilon_0\) is the dielectric constant,

\(e, m_0\) are the charge and mass of the electron,

\(\beta\) is the relative velocity of the electron,

\(F_0\) is the beam emittance.

If \(\gamma\) and \(\beta\) are expressed through the accelerating voltage, then finally we obtain the expressions for the constant coefficients \(Q\) and \(r_a\) of the system of equations (1):

\[ Q=\frac{S^2\mu_0\sqrt{\dfrac{2e}{m_0}}\,I_{\ell}} {\pi r_{\mathrm{m}}^2 U^{1/2}\left(1+\dfrac{eU}{2m_0c^2}\right)^{1/2}}, \]

\[ r_a^2= \frac{S^2 I_{\text{beam}}} {\dfrac{10^7}{c^2}\sqrt{\dfrac{2e}{m_0}}\,U^{3/2} \left(1+\dfrac{eU}{2m_0c^2}\right)^{3/2}}. \]

We solve the problem under the assumption that the beam at the source exit is parallel. In this case the emittance \(F_0=0\), and equations (1) are simplified:

\[ \frac{d^2x}{dz^2}+Q_x\,a(z)\,x-\frac{2r_a^2}{x+y}=0, \]

\[ \frac{d^2y}{dz^2}+Q_y\,a(z)\,y-\frac{2r_a^2}{x+y}=0. \tag{2} \]

If the beam is intense, then \(r_a^2\gg F_0\), and this assumption will be satisfied more accurately. In order to select the focusing system, equations (2) were solved on analog computing machines (ACMs) of the MPT-9 and MN-7 types. The current in the lenses and the periodicity of the system were determined for various values of the beam current and particle energy. In order to make it possible to increase the beam transport distance, the following condition was imposed: the beam parameters (its dimensions and divergence) at the exit must be equal to the beam parameters at the entrance to the system. In our case the beam must be parallel and must have a radius equal to the initial one.

Previously, using the method described in [2], it was calculated that, without a focusing system, the radius of the electron beam

(\(r_0 = 1\) cm, \(I_{\text{beam}} = 1000\) A) will increase by a factor of 2 at a distance of 5.8 cm, by a factor of 3 at a distance of 8.7 cm, and by a factor of 20 at a distance of 50 cm. These data were then used to determine the error in solving the equations on the analog computer.

In the first variant, a focusing system was studied consisting of quadrupole lenses of identical length, formed by straight current-carrying conductors. In this case \(a(\tau)\) is a periodic function taking the value \(+1\) during focusing and \(-1\) during defocusing. It should be noted that if focusing occurs in the \(xz\) plane, then defocusing occurs in the \(yz\) plane.

The successive switchings that realize the periodic function \(a(\tau)\) are performed by the step finder from block SB-5. No divider is used in this case; the slider of the step finder, successively, every ten steps, supplies the voltage \(U_x\) or \(-U_x\) to the block output.

The solution was carried out for various values of the period length and for various values of the focusing force. However, for this focusing system it was not possible to obtain even one variant satisfying the requirements stated above, since, for a small channel diameter, the beam is lost on the channel walls in the defocusing lens. It was also not possible to obtain a satisfactory result with lenses of different lengths, although a focusing system consisting of lenses of different lengths has some advantage.

Next, focusing of the beam by a helical quadrupole lens was investigated. A helical quadrupole lens can be obtained from an ordinary symmetric quadrupole lens if, with variation of the longitudinal coordinate, the poles (electrodes) of the ordinary lens are uniformly rotated about the lens axis [3]. In the present case, we consider the variant in which the current-carrying conductors forming the lens describe a left-hand helix, i.e., the rotation is counterclockwise.

The equations of the beam envelopes in a helical quadrupole lens have the form

\[ \frac{d^2 x}{d\tau^2} + Q(x\cdot \cos 4\pi z - y\sin 4\pi z) - \frac{2 r_a^2}{x+y} = 0, \]

\[ \frac{d^2 y}{d\tau^2} - Q(y\cdot \cos 4\pi z + x\sin 4\pi z) - \frac{2 r_a^2}{x+y} = 0. \tag{2} \]

The block diagram for solving equations (2) is shown in Fig. 1, where 1, 2, 5, 6, 7, 8 are integrators; 3, 9, 10, 11 are inverters; 14 is an adder; \(a_1\)—\(a_{14}\) are constant-coefficient blocks; 4 is a function generator; 12, 13, 15, 16 are multiplication blocks. The independent variable \(\tau\) is represented in the circuit by machine time \(t\); the variable \(x\) is represented by the voltage at the output of the second integrator, and the variable \(y\) by the voltage at the output of the sixth integrator. The upper half of the circuit (blocks 1, 2, 3, 4, 14, 15, 16) serves to reproduce the trajectory equation in the \(xz\) plane, and the lower half to reproduce the trajectory equation in the \(yz\) plane. \(z\) is the longitudinal coordinate, coinciding with the optical axis of the system. To reproduce the functions \(\sin 4\pi t\) and \(\cos 4\pi t\), we use the method of a defining equation.

The block diagram for reproducing \(\cos 4\pi t\) consists of two integrators, designated in the diagram by the numbers 7, 8, and the inverter 9. In order to obtain a sinusoidal voltage with an amplitude of 100 V, one more amplifier, 10, was added to the circuit.

In studying the focusing system with helical quadrupole lenses, a solution was obtained that is close to the required one.

mum. The result that best satisfied the requirements was obtained at \(I_\ell = 1370\) A, \(\tau = 3\). Figure 2 gives oscillograms of the solution for this case.

The capabilities of the given focusing system were investigated, i.e., the behavior of a beam of charged particles for various values

Fig. 1. Structural diagram for solving equations (2)

Fig. 2. Oscillograms of the solution at
\(\tau = 3,\ I_\ell = 1370\) A

Fig. 3. Dependence of the lens current on the beam current, \(\tau = 3\)

of the beam energy and current. The solution of equations (2) was carried out for the following values of the accelerating voltage: 100, 200, 300, 400, 500, 1000 kV and for a beam current \(I_{\text{beam}} = 1000\) A and \(r_m = 3\) cm.

At small values of the accelerating voltage (\(U = 100\) kV, \(U = 200\) kV), no satisfactory result was obtained for any value of \(\tau\). At values of the accelerating voltage \(U = 300\) kV and higher, satisfactory results were obtained for \(\tau = 3\) and \(\tau = 4\). From consideration of the results of the solution it may be concluded that focusing intense beams of low energy by quadrupole fields is ineffective. As was to be expected, with increasing particle energy the lens current decreases.

For an accelerating voltage \(U = 300\) kV, values of the lens current were found for various values of the beam current. The graph of the dependence of the lens current \(I_\ell\) on the beam current \(I_{\text{beam}}\) is shown in Fig. 3. It is seen from the graph that at large values of the beam current the lens current practically does not change, i.e., the system enters saturation. For small changes in the beam current, the results of the solution satisfy the requirements.

Investigating the focusing system at the initial divergence angles of the beam, it was established that at small initial divergence angles \((2, 5, 10^\circ)\) the beam parameters at the system output are close to the required ones. With increasing initial angle, the beam divergence angle at

at the exit of the system increases, compared with the initial value, by a factor of 1.5 in the worst case.

Thus, it has been established that focusing of an intense electron beam is possible by means of a helical strong-focusing channel. The lens current required to confine the beam is comparable with the current in the beam. For a beam current \(I_{\text{beam}} = 1000\ \text{A}\), the best result is obtained if three periods fit into a length of \(0.5\ \text{m}\) (the wire makes three turns).

REFERENCES

1. I. M. Kapchinsky, Dynamics of Particles in Linear Resonance Accelerators. Moscow, Atomizdat, 1966.
2. D. Ivey. Problems of Physics. No. 6, 1956.
3. C. Yavor. Focusing of Charged Particles by Quadrupole Lenses. Moscow, Atomizdat, 1969.