Generation of magneto-immersed electron beams

Generating laminar electron beams has a long and rich history of ideas and approaches. Some applications require magnetic compression of the electron beam with the goal to increase its current density, like the Electron Beam Ion Sources (EBIS)¹ and electron lenses for the proton colliders.^2,3 Considering that in the process of electron beam compression a fraction of the longitudinal energy is transferred into the transverse plane resulting in increased electron trajectory angle and, that the magnetic mirror entrance cone of main solenoid has a limited value, it would be beneficial for the beam transmission to have as laminar initial electron beams as possible.

If some beam electrons have trajectory angles outside of the entrance cone of the magnetic mirror, these electrons will reflect from the mirror back to the cathode and will oscillate between the cathode and the magnetic mirror while slowly drifting outwards.⁴ Due to energy exchange within the beam, some of these oscillating electrons can acquire enough energy to hit the surface of the cathode or Wehnelt electrode and produce secondary electrons with multiplicity larger than 1 providing a positive feedback mechanism for a process of accumulating oscillating electrons.⁵ When the accumulated space charge of these electrons exceeds a certain threshold, the oscillated electrons can bunch, affecting the primary electron beam, which also becomes bunched with a frequency determined by the period of these axial oscillations. Such beam excitations have been observed in real systems.^6,7

While electron beams for electron coolers are not subjected to high magnetic compression, and sometimes undergo magnetic expansion to achieve additional transverse cooling, highly laminar electron beams with low initial transverse electron temperature are still needed for effective cooling.

The largest contributor to the transverse electron energy is the electron gun optics with its combination of electric and magnetic fields in a cathode-anode gap.

An attempt to form an optimum electrostatic field in a magneto-immersed gun with 5 anodes was tried at CERN,⁸ with the main purpose of generating an electron beam for an electron cooler with a transverse energy lower than 1 eV. An analysis of the beam electrons generated with this gun reveals a highly non-uniform dependence of the transverse energy component on the magnetic field with several minima and maxima. The magnetic field corresponding to the most pronounced minimum at 0.70 kG was selected as a working point for this gun with an electron current of 8.3 A and energy of 60 keV. A sharp dependence of the transverse electron energy on the magnetic field was noted, which limited the effective magnetic field range.

As a trend, for most magneto-immersed electron guns, the radius of the Larmor electron motion decreases with increasing magnetic field in a gun region. This effect can be used to make the electron beam more laminar simply by increasing the magnetic field in the gun because the ratio of the Larmor radius to the beam radius is conserved at all magnetic fields for such guns. However, increasing the magnetic field on the cathode limits the achievable magnetic beam compression into the final solenoid, where the magnetic field is limited by technical issues. This limitation prompts the search for a solution which allows using lower magnetic field in the gun region and still produces a laminar electron beam.

For some time, an adiabatic electron gun with an almost uniform axial electric field in a cathode-anode gap⁹ has been considered being the best approach to generate a laminar electron beam. This electron gun has an anode, which overlaps the cathode, a feature providing the needed uniform electrostatic field distribution. The next development step for this gun was equipping it with a spherical convex cathode and with a Wehnelt electrode having a similar convex shape and acting as an extension of the cathode surface.^10,11 This convex spherical shape of the cathode surface looks counterintuitive since the near-cathode electric field has visible angles with the guiding magnetic field. However, this magneto-immersed gun employing an electrode geometry with a convex cathode shape geometry provides smaller radius of the Larmor motion at all magnetic fields than a gun with a flat cathode. Simulations of guns with convex cathodes¹¹ show a non-linear dependence of the transverse electron energy with the radial coordinate of its emission. A similar dependency of the transverse electron energy on the radius of emission from a cathode with an ellipsoid shape has been shown, with several maxima and minima within the full beam radius with a general trend of increasing the transverse energy with the radius of emission.

In Ref. 12, an analysis of some electron guns with non-adiabatic fields has been done and some dependencies of the beam laminarity on the gun geometry have been established. In this paper, we have performed further investigations of electron guns immersed in a magnetic field and the generation of electron beams using magnetic confinement. As before, for our simulations, we used a 2D software package from Field Precision LLC.¹³ 

We analyzed different electron guns studying the dependence of the maximum value of the electron trajectory angle of all simulated particles emitted from the cathode as a function of the magnetic field at the cathode and the electron current. The simulations have been done for uniform magnetic and electrostatic fields inside the anode tube downstream of the cathode-anode gap, where the anode tube radius is constant and at an energy determined by the cathode-anode potential difference in a way that the Larmor radius remains constant in the area of analysis. For easy comparison of different guns, all angles are normalized to an electron energy of 25 keV. The result of a series of simulation for a specific electron gun is presented in a table where the maximum trajectory angles are listed as a function of the magnetic field and the beam current. The color-coded table of trajectory angles helps to visualize areas of magnetic field with small angles, where the gun performs best and areas with large angles to avoid. All electron gun simulations have been done for a cathode diameter of 9.2 mm, which is used in the BNL EBIS electron gun.

The schematic of an adiabatic gun¹⁰ built by Budker Institute of Physics for BNL EBIS is presented in Fig. 1. The electron gun has very smooth distributions of both axial and radial electrostatic fields. The geometry is quite popular because of this quality.

Figure 2 presents the laminarity map for this adiabatic electron gun. Figure 3 presents a sub-selection of the data, i.e., the maximum angle for varying magnetic fields, but with a fixed electron current I_el of 12.0 A.

The scan of this map at electron current I_il = 12.0 A is presented in Fig. 3.

This dependence is smooth and does not have extremes. The common practice shows that such an electron gun is used with a magnetic field on the anode close to 2.0 kG and sometimes with magnetic fields as low as 1.5 kG.

The criteria for acceptable maximum angle are determined by the specific application. If the electron gun is to be used for generating a high-current, high-density electron beam using further magnetic compression with a superconducting solenoid having a maximum magnetic field of 6 T and the magnetic field on the gun as 1.5 kG, the acceptance cone is 9.06°. One can see that the operation of this gun seems safe at 1.4 kG, corresponding to a maximum angle of 7.2°, but with limited margin considering other contributors to the trajectory angle, like non-symmetrical gun electrodes, non-uniform electron emission, misalignment of the gun, and main solenoid.

At Brookhaven National Laboratory, we modified Budker’s gun. A cylindrical anode overlapping the cathode that we replaced with a conical funnel-like anode opened to the anode with angle of a cone 63°.¹² We simulated a number of different configurations of the electron guns with electrostatic fields close to adiabatic. Most of these guns had a conical funnel-like anode and different cathode-anode distances, cone angles, and shapes. The dependence of maximum trajectory angle on the magnetic field for these electron guns is shown in Fig. 4.

This plot demonstrates that these guns have individual dependencies of angle on the magnetic field similar to the one in Fig. 3 and none of them can provide 12 A electron beam with maximum angles smaller than 5° at magnetic field lower than 1.5 kG.

The Larmor motion of the electrons is determined by the external electric and magnetic fields in the cathode-anode gap of the gun in combination with the space charge field of the electron beam. An attempt was made to modify the external fields in a way that at some operating conditions the integral of all transverse actions on the beam electrons is smaller than that in an adiabatic electron gun. These field variations should be local, confined to the cathode-anode gap. Since the Larmor motion period is comparable to the length of these variations, we call such electron guns with sharply varying fields as non-adiabatic guns. The overall goal of this attempt is to generate a laminar electron beam (small trajectory angles) also for an electron gun with a magnetic field lower than that used at an adiabatic gun so that a higher magnetic compression of the electron beam is attainable.

A strong variation of the magnetic field in the cathode-anode gap can be produced with a magnetic shim made of soft iron located close to the cathode-anode gap and producing a local dip in the magnetic field distribution (Fig. 5).

The electrostatic geometry of this electron gun is the same as used at the BNL EBIS and is a result of morphing from the adiabatic gun in Fig. 1. The laminarity map of this gun with the non-adiabatic magnetic field is shown in Fig. 6.

One can clearly see regions with small trajectory angles, and one of them is located in a low-magnetic field area. For an electron beam current I_el = 12.0, the optimum magnetic field is 1.4 kG and the maximum trajectory angle at this field is 3.45°, which is significantly smaller than that for the adiabatic gun at this magnetic field. However, the price for the small angle at the optimum magnetic field is larger angles at lower electron currents, which means that during the current ramp the beam crosses areas with larger trajectory angles. This effect has to be taken into consideration when a pulsed electron gun is designed.

The magnetic field for a small-angle region on this map at a given electron beam current depends on the distance between the magnetically shimmed anode and the cathode (Fig. 7). This is a quite convenient dependence as the optimum magnetic field is reduced for shorter cathode-anode distances. In other words, the smaller the gap—and therefore the higher the gun perveance—the lower the optimum magnetic field.

The effect of the magnetic field alternation in the cathode-anode gap depends on a thickness of the magnetic shim because it determines the value of absorbed magnetic flux. As one would expect, a thinner shim produces less pronounced effect than a thicker shim (Fig. 8). It is therefore possible to control the effect of the magnetic field variation on the beam trajectory angle by changing the thickness of the shim. Our simulation also demonstrated that if the magnetic shim is replaced with a magnetic coil in the same volume, the effect on the magnetic field distribution and on the gun performance can be made similar. The magnet coil has the convenience of controllable modifications of the gun performance in situ.

The transverse electric field in a cathode-anode gap can be modified by enhancing the transverse electric field with the gun electrodes, which means either the Wehnelt electrode or the anode or both.¹² The geometry of such a gun is shown in Fig. 9. The transverse electric field has a bump in the cathode-anode gap. Such modification of electrodes and their effect on the electron beam is controllable on the design stage. Our simulations show that the laminarity map of such a gun resembles the map of a magnetically non-adiabatic electron gun, but the areas of small trajectory angles extend to even lower magnetic field. As with magnetically non-adiabatic electron guns, the effect of the field modification depends on the strength of this modification. The stronger the modification, the more pronounced the contrast of the map, with smaller minimums and larger maximums in trajectory angles.

The value of the optimum magnetic field for such a gun also depends on the cathode-anode gap (Fig. 10), but this dependence is opposite to the magnetically non-adiabatic gun.

The optimum magnetic field for electron guns with the non-adiabatic electrostatic field for the geometry presented in Fig. 9 decreases with increasing cathode-anode distance. This effect may be explained by an increase in the period of the Larmor motion at lower magnetic field so that it matches the extended modified electrostatic field. For this kind of electron gun, the benefit of reducing the optimum magnetic field comes at the cost of a reduction of the gun perveance.

To estimate the contribution of the electron space charge to the pattern of the gun laminarity map, we simulated electron beam transmission with space charge and without it. The simulations have been done with identical magnetic and electrostatic fields for the non-adiabatic electron gun shown in Fig. 9. The electrostatic field in this simulation corresponds to the extraction of the electron beam with current 12.0 A. Figure 11 presents dependencies of the maximum relative Larmor radius as a ratio of the Larmor radius to the beam radius (dr/r_beam) on the magnetic field for both cases.

The trajectory tracks at the magnetic fields corresponding minimums on curves Fig. 11 are shown in Figs. 12(a) and 12(b).

These plots demonstrate that with some difference in the shape of curves in Fig. 11 for cases with and without space charge, the general pattern of this dependence seems to be determined primarily by the external fields with the electron space charge playing a secondary role.

Even the distributions of the electron Larmor radius across the beam radius for cases with and without the space charge in Figs. 12(a) and 12(b) show that it is mostly affected by the external fields as well.

The introduction of non-adiabatic fields in the cathode-anode gap of an electron gun has strong effect on the performance of the gun. It changes the dependence of the electron trajectory angle on the magnetic field for a fixed electron current compared to an adiabatic gun. The main benefit of such modification is the possibility to operate the gun at a lower magnetic field than for an adiabatic gun, and for some applications, this benefit is important. The downside of non-adiabatic guns is increased beam oscillations in areas outside the optimum conditions: at different magnetic fields and during the ramping up and down of the electron current. Knowing the effect of field modification, one can optimize the gun to satisfy both the requirements of the minimum magnetic field and acceptable level of oscillations in other important operational regions.

Our simulations show that the observed pattern of the Larmor motion is affected primarily by the external fields with the electron space making much smaller modification to this pattern.

The authors are very grateful to Fredrik Wenander for productive discussions and indispensable help in preparation of this manuscript.

Work was supported by Brookhaven Science Associates, LLC, under Contract No. DE-SC0012704 with the U.S. Department of Energy.