INVESTIGATION OF THE FORMATION OF AN INTENSE ELECTRON BEAM BY MAGNETIC HELICAL QUADRUPOLE LENSES

G. I. Stanevko, N. V. Trikhanova, P. M. Shchanin

(Submitted by the scientific seminar of the Department of Computing Technology)

In connection with the development of sources of high-intensity electron beams, interest has recently increased in the problem of transport and focusing of intense electron beams [1]. In the present work, the possibility of focusing an intense electron beam by helical quadrupole lenses (HQL) is considered; the behavior of the beam is investigated when the beam current and the lens current are varied, and also when the amplitude and shape of the accelerating-voltage pulse are varied. Since the accelerating voltage is a sequence of pulses of short duration (on the order of several microseconds) and low repetition rate, in order to reduce power it is expedient to use pulsed power for the lens. It is therefore desirable to choose a system that would have, as far as possible, low inductance.

The investigations were carried out in two variants: without taking into account the longitudinal magnetic field and with allowance for the longitudinal field of the HQL. In the investigation, a system with the following parameters was considered: cathode diameter — 100 mm, electron-guide diameter — 140 mm, electron-guide length — 150 cm, beam current — 50 kA, electron energy — 1 MeV. It is assumed that the beam emittance at the entrance to the electron guide is equal to zero. Such an assumption is valid for a high-intensity beam, since the behavior of the beam is substantially influenced by the action of the space charge, and not by the initial distribution of divergence angles [4].

At the first stage, a parallel beam at the entrance was considered, and the problem was posed: to choose the parameters of the focusing system ensuring passage of the beam through the electron guide without losses; at the exit it is necessary to obtain a parallel beam (or one close to parallel) with a diameter of 100 mm.

The investigation was carried out by solving the equations for the beam envelopes on an analog computer (AVM) of the MPT-9 type in combination with an MN-7 machine and a set of nonlinear blocks NBN-1. For the variants selected as a result of modeling, a more accurate solution was performed on a BESM-4 digital computer (TsVM).

A helical quadrupole lens can be obtained from an ordinary symmetric quadrupole lens if, with variation of the longitudinal co-

the coordinates of the pole of an ordinary lens will rotate uniformly about the lens axis with a certain pitch determining the period of the system. In the present case, a non-equipolar lens formed by conductors carrying a current describing a left-handed helix was considered; that is, the rotation was counterclockwise.

The equations for the beam envelopes in the MHQL, without allowance for the longitudinal magnetic field, have the form [2]:

\[ \begin{aligned} \frac{d^{2}x}{d\tau^{2}}+Q(x\cos 4\pi\tau-z\sin 4\pi\tau)-\frac{2r_a^2}{x+z}&=0;\\ \frac{d^{2}z}{d\tau^{2}}+Q(z\cos 4\pi\tau+x\sin 4\pi\tau)-\frac{2r_a^2}{x+z}&=0, \end{aligned} \tag{1} \]

where

\[ Q= \frac{ 2S^{2}\mu_{0}\sqrt{\dfrac{e}{m_{0}}}\,I_{\ell} }{ \pi r_{\mathrm{m}}^{2} \left[ U\left(1+\dfrac{eU}{2m_{0}c^{2}}\right) \right]^{1/2} }, \]

\[ r_a^{2}= \frac{ S^{2}I_{n} }{ \sqrt{\dfrac{2\cdot10^{7}}{c^{2}}}\cdot \sqrt{\dfrac{e}{m_{0}} \left[ U\left(1+\dfrac{eU}{2m_{0}c^{2}}\right) \right]^{3/2}} }; \]

\(r_a^{2}\) is the parameter with space charge;

\(\tau\) is a dimensionless variable expressed in units of the periodicity, \(\tau=y/S\), where \(y\) is the longitudinal coordinate and \(s\) is the length of the period of the system;

\(I_{n}\) is the beam current;

\(I_{\ell}\) is the lens current;

\(U\) is the accelerating voltage;

\(r_{\mathrm{m}}\) is the maximum possible beam radius;

\(\mu_{0}\) is the magnetic permeability of vacuum;

\(e, m_{0}\) are the charge and mass of the electron.

The block diagram for solving system (1) on an analog computer and the modeling procedure are described in [3]. The system of nonlinear equations was solved on a digital computer by the Runge–Kutta method.

In the course of the investigation, the focusing strength and the periodicity of the system were varied. At each fixed value of \(\tau\), by varying the focusing strength, it was ensured that the transverse dimensions of the beam did not exceed the radius of the electron-beam guide, and that the beam parameters at the system output were equal to the beam parameters at the system input. The results of solving system (1) on a digital computer for the most optimal value \(\tau=10\) and \(I_{n}=50\ \mathrm{kA}\), for various values of the lens currents, are summarized in Table 1.

It is seen from Table 1 that, when the lens current is varied, the beam dimensions change slowly. No results satisfying the requirements of the posed problem were obtained: the maximum beam dimensions in the \(xy\) and \(zy\) planes exceed by 50–60 mm, on average, the tube radius \(r_{\mathrm{m}}=70\) mm. In this case it is necessary to increase \(r_{\mathrm{m}}\) and to select the value of \(\tau\) and the magnitude of the current in the lens. The most satisfactory result was obtained for \(I_{\ell}=18.5\ \mathrm{kA}\) and \(\tau=10\). For this case the beam envelope was constructed in the \(xy\) plane and in the \(zy\) plane (Fig. 1). From an analysis of Fig. 1 it may be noted that the beam in the \(xy\) and \(zy\) planes does not

symmetric, that is, simultaneous reduction of the beam dimensions in the \(xy\) and \(zy\) planes does not occur.

Studies were carried out to establish tolerances for variations in the lens current. It was found that the beam radius increases by 20% when the lens current is decreased by 10% and when it is increased by 30%. When the accelerating voltage is varied by 2% from the nominal value,

Table 1

the beam parameters change only slightly. A decrease in the accelerating voltage by 14% causes a change of 10% in the beam dimensions in the channel and at the channel exit. From these data one can determine the requirements on the stability of the supply voltage.

Fig. 1. Beam envelopes in the \(xy\) and \(zy\) planes at \(I_l = 18.5\ \mathrm{kA}\) and \(\tau = 10\).

To clarify the focusing capability of the system, it is of interest to know the dependence of the lens current on the beam current. This dependence was established by solving the system of equations (1) for various values of the beam current. For each fixed value of the beam current, a value of the lens current was selected at which the beam at the exit of the sys-

system meets the specified requirements. Figure 2 presents plots of the dependence \(I_l=f(I_p)\) and of the number of coherent oscillations of the beam envelope, fitting within a length of \(1.5\ m\), on the beam current: \(\nu=F(I_p)\). From Fig. 2 it is evident that the dependence of the lens current on the beam current is nonlinear. At a certain value of the beam current, “saturation” occurs, i.e., the increase in the lens current is not proportional to the increase in the beam current.

It is not difficult to see (Fig. 1) that the beam envelope undergoes complex oscillations. First of all, there are incoherent oscillations, whose frequency coincides with the frequency of the focusing field (solid line in Fig. 1). The frequency of the coherent oscillations (dashed line in Fig. 1) is several times lower than the frequency of the focusing field and varies depending on the beam-current density, which is consistent with

Fig. 2. Plots of the dependences of the lens current and the number of oscillations on the beam current.

the theory [4]. At beam-current values below \(30\ ka\), the frequency of the coherent oscillations increases linearly with increasing beam current. At beam-current values greater than \(30\ ka\), the increase in the frequency of the coherent oscillations is insignificant (Fig. 2). From Fig. 2 it is evident that the dependence of the number of coherent oscillations fitting within a length of \(1.5\ m\) on the beam current, \(\nu=F(I_p)\), has a “saturation” region (at \(I_p=30\ ka\)), analogous to the region on the curve \(I_l=f(I_p)\). Such behavior of the envelope is characteristic of a “mismatched” beam, which is caused by the action of the space charge.

For \(\tau=10\), \(I_p=50\ ka\), and \(I_l=18.5\ ka\), a solution was obtained with divergence angles different from zero. With a converging beam at the entrance, namely with initial divergence angles equal to \(-10^\circ\) and \(-20^\circ\), the focusing of the beam improved: the beam passes through the beam pipe without losses, and its parameters at the exit are close to the required ones.

In the HQL, in addition to quadrupole fields, a longitudinal magnetic field is produced, which exerts an additional focusing action on the beam. The equations for the envelopes of the electron beam in an HQL, taking account of the longitudinal magnetic field, may be written in the form

\[ \begin{aligned} \frac{d^{2}x}{d\tau^{2}} &+ Q\left\{(z\sin 4\pi\tau + x\cos 4\pi\tau) -\frac{dz}{d\tau}\cdot\frac{2\pi}{S^{2}}\times\right.\\ &\left.\times\left[(z^{2}-x^{2})\cos 4\pi\tau + 2xz\sin 4\pi\tau\right]\right\} -\frac{2r_{a}^{2}}{x+z}=0; \\[6pt] \frac{d^{2}z}{d\tau^{2}} &+ Q\left\{(z\cos 4\pi\tau + x\sin 4\pi\tau) +\frac{dx}{d\tau}\cdot\frac{2\pi}{S^{2}}\times\right.\\ &\left.\times\left[(z^{2}-x^{2})\cos 4\pi\tau + 2xz\sin 4\pi\tau\right]\right\} -\frac{2r_{a}^{2}}{x+z}=0. \end{aligned} \tag{2} \]

The values of the parameters \(Q\) and \(r_a^{2}\), and the initial conditions, are the same as in the system of equations (1). The solution of system (2) was carried out on a BESM-4 digital computer.

In the course of the investigation the focusing force (the lens current) and the periodicity of the system were varied, beginning with \(\tau=5\) up to \(\tau=12\). Results satisfying the requirements were obtained for the following values of \(\tau\): 8, 9, 10, 11, 12. The beam dimensions do not exceed the tube dimensions for \(\tau=9\) at lens currents \(I_l=5\ \mathrm{kA}\) and \(I_l=6\ \mathrm{kA}\); for \(\tau=10\) at \(I_l=5\ \mathrm{kA}\); for \(\tau=11\) at \(I_l=7\ \mathrm{kA}\) and \(I_l=5\ \mathrm{kA}\); \(\tau=12\) at \(I_l=10\ \mathrm{kA}\) and \(I_l=5\ \mathrm{kA}\).

Fig. 3. Beam envelopes in the \(xy\) and \(zy\) planes, obtained taking account of the action of the longitudinal magnetic field of the HQL.

For \(\tau=8\), a result close to the requirements was obtained for \(I_l=5\ \mathrm{kA}\). In this case the maximum value is \(z=61\ \mathrm{mm}\), \(x=70\ \mathrm{mm}\), i.e., in the \(xy\) plane the beam size is equal to the radius of the tube. For \(\tau=7\), at \(I_l=5\ \mathrm{kA}\) and \(I_l=4\ \mathrm{kA}\), the maximum beam dimensions exceed the tube radius by \(4\text{–}5\ \mathrm{mm}\), i.e., at \(\tau=7\) some fraction of the electrons will be lost during transport. At \(\tau=5\) and \(\tau=6\), no satisfactory results were obtained; the beam dimensions exceeded the tube radius for all values of the lens current.

In the investigation it was observed that for $\tau = 5$ and $\tau = 6$ the values of the coordinates $x, z$ as functions of the lens current change “more rapidly” than for $\tau = 9 \div 12$; i.e., as $\tau$ is increased, the sensitivity of the system to the lens currents becomes smaller. Thus, the larger $\tau$ is, the less the instability of the lens power supply will affect the focusing system.

The possibility of focusing the beam was investigated for initial angles of divergence equal to $10$, $20$, $-10$, and $-20^\circ$. If the results obtained with the indicated initial angles are compared with the results for a parallel beam at the entrance, it is seen that in the worst case the focusing deteriorated by 18%, while in the best case, for an initial divergence angle of $+10$, it deteriorated by 3%.

From an analysis of the beam envelopes in the $xy$ and $zy$ planes, shown in Fig. 3, it may be concluded that the amplitude of the oscillations decreased in comparison with the case in which the action of the longitudinal magnetic field of the HQL is not taken into account. The character of the oscillations remained the same.

Comparing the results of solving systems of equations (1) and (2), it may be concluded that the longitudinal magnetic field of a helical quadrupole lens has a substantial effect on the beam dimensions, improving its focusing. As a result of solving system (2), values of the lens currents and of $\tau$ were obtained for which the transverse dimensions of the beam do not exceed the tube radius, i.e., the beam parameters satisfy the specified requirements. The values of the currents in the lenses in this case are 3 times smaller than in the best variants obtained by solving system (1). Analysis of the results of solving system (2) shows that the number of periods $\tau$, i.e., the number of turns of the helical lens, can be reduced to 8 or 7, thereby simplifying the focusing system. A smaller number of periods gives a smaller value of the HQL inductance, which is very important for pulsed powering of the lenses.

REFERENCES

1. A. A. Rukhadze, V. G. Rukhlin. Injection of a relativistic electron beam into plasma. ZhETF, vol. 61, issue 1(7), 1971.

2. S. Ya. Yavor. Focusing of Charged Particles by Quadrupole Fields. Moscow, Atomizdat, 1969.

3. N. V. Trikhanova, L. V. Pilger, P. M. Shanin. Computer study of the focusing of an intense electron beam by quadrupole fields. Proceedings of TPI, vol. 258.

4. I. M. Kopchinskii. Particle Dynamics in Linear Resonance Accelerators. Moscow, Atomizdat, 1966.