Two types of electron guns are used in the EBIS: the Brillouin type and the magnetically imaged type.
The cathode of the Brillouin electron gun is placed where the magnetic field is close to zero, generally through a magnetic shield. When the electron beam is first emitted from the cathode, it is electrostatically focused by the electrode. After that, as the beam transmits, it enters the area where the magnetic field is, and the magnetic compression occurs in the high magnetic field in the breeding region. There is an advantage of obtaining a high current density through these two types of compression. At this time, the radius of the beam, in which the temperature of electrons is not considered, in the magnetic field is expressed as the Brillouin radius,rB, in Eq. (2.2),
rB= s
2meIe
π ε0vzeB2, (2.2)
where me, Ie, vz, and B are the mass of electron, the electron current, the electron velocity, and the magnetic field in the breeding region, respectively. However, the electron beam is accelerated by the emission of thermal electrons at the cathode, so the electron temperature is not zero. Therefore, the accurate current density of the electron beam can be calculated only when the temperature is considered.
So the radius of the electron beam considering the temperature is calculated as the Hermann radius [43],
rH, by Hermann theory. The Hermann radius containing 80% of the beam is represented in Eq. (2.3),
rH=rB
v u u t1
2+1 2
s 1+4
8mekbTcrc2
e2B22rB4 +B2cr2c B2rB4
, (2.3)
wherekb,Tc,Bc, andrc are the Boltzmann constant, the temperature of the cathode, the magnetic field at the cathode, and the cathode radius, respectively. The last term of Eq. (2.3) can be neglected when the magnetic field is zero at the cathode position, and the first term in the bracket is much larger than 1, so it is the most dominant in the inner root. Therefore, Eq. (2.3) can be approximated like Eq. (2.4) [37].
rH≈ 4 r8me
e2 kbTc rrc
B ≈2.59×10−3Tc1/4
rrc
B. (2.4)
This way, a high current density may be obtained through the electron beam with the Hermann radius.
However, since the Brillouin electron gun has an electron beam emitted from the zero magnetic fields and transmitted to a high magnetic field, mutual alignment of the electrical potential and magnetic field is essential. If there is a misalignment between the two types of field, the oscillation in the radial direction may occur, degrading the quality of the electron beam and causing problems in the performance of the EBIS [44].
Conversely, the magnetic-immersed electron gun has no significant risk for the alignment problem because the cathode is in the magnetic field, and the electron beam is emitted from there. Unlike the Brillouin type, the transverse size of the magnetically immersed electron beam is independent of the current or energy of the beam. It depends only on the size of the cathode and the magnetic field strength, as shown in Eq. (2.5),
r(z) =rc s
Bc
B(z), (2.5)
wherezis the axial coordinate. The Brillouin type used in the CERN Light Ion Therapy (LIT) facility obtained a current density of higher than 1.5 kA/cm2 [45]. In contrast, the RHIC EBIS of BNL [34]
and the REXEBIS of REX-ISOLDE/CERN [46] use a magnetic imaged gun to use a current density of several hundred A/cm2. Brillouin electron guns can obtain a higher current density. But, since a current density is enough below 1 kA/cm2based on the CBSIM calculation in the previous section, the RAON EBIS uses the magnetic-immersed type considering the advantages of alignment, etc.
The most widely used cathode is LaB6, but the IrCe cathode manufactured by BINP (Budker Institute of Nuclear Physics, Russia) [47] can emit an electron beam with twice or more current density compared to the monocrystalline-LaB6. In addition, the evaporation rate of the IrCe cathode is more than 20 times lower the LaB6at the same temperature at the cathode. And, it has an advantage in the lifetime, up to 40000 hr generating 15∼17 A/cm2, so the RAON EBIS uses the IrCe cathode [48–50]. The magnetic- immersed electron gun is characterized by radial oscillation because the cathode is in the magnetic field during the emission and transmission. The envelope equation of this oscillation are described by Humphries [51], as shown in Eq. (2.6),
d2R dZ2 =
ωp
γ ωc
2
1 2R−R
4
1− 1 R4
, (2.6)
whereR, the oscillation amplitude, which is the ratio between the beam radius and the initial radius, and Z, the normalized axial position used, are represented following,
R= r r0, Z=zωc
βc.
(2.7)
The first frequency used Eq. (2.6),ωp, is the plasma frequency, expressed as Eq. (2.8), ω2p= n0e2
γ ε0me, (2.8)
and, the second frequency,ωc, is the cyclotron frequency being Eq. (2.9), ωc= eB
γme. (2.9)
This equation can be applied under the condition of the paraxial beam with the space-charge limit. In Eq. (2.6), whenωp is zero, the oscillation of the beam disappears, or the oscillating effect is reduced by increasingωc. That is, the amplitude and period of the beam oscillation decrease when the magnetic field increases at the position of the cathode. This effect can be confirmed, as shown in Fig. 2.5, by simulating the beam trajectories by changing the magnetic field of the cathode [47] using the TRAK code by Field Precision LLC [52].
Figure 2.5: Variation of beam oscillation with different magnetic fields of e-gun solenoid [47].
The magnet configuration for using the RAON EBIS electron gun consists of the gun coil, the main solenoid using the superconducting magnet, and the collector coil, as shown in Fig. 2.1. The magnetic field configuration generated by these three magnets is shown in Fig. 2.6. Figure 2.6a shows the magnetic field in the area where the electron beam enters the magnetic field of the superconducting magnet from the cathode. And Fig. 2.6b shows the strength of the axial magnetic field on the axis. To reduce the radial oscillation of the electron beam, the magnetic field at the cathode position uses about 0.25 T, and the electron beam is transmitted to a magnetic field of 6 T in the breeding region. when transmitting charged particles to a trap using the magnetic field like the Penning trap, they always pass through the region, where the magnetic field gradient exist, as illustrated in Fig. 2.6a. In that case, the magnetic mirror effect [53, 54] is a physical phenomenon to be considered. For the axisymmetry condition of the
(a) Magnetic field line from cathode to main solenoid.
- 0 . 5 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5
- 1
01234567
C o l l e c t o r Bz [T]
z [ m ] C a t h o d e
(b) Magnetic field strength on axis.
Figure 2.6: Magnetic field configuration of RAON EBIS calculating by TRAK.
magnetic field, the equation of the radial magnetic fieldBrin the cylindrical coordinate can be obtained from Gauss’ law for the magnetism (∇·B=0), as Eq. (2.10),
1 r
∂
∂r(rBr) +∂Bz
∂z =0. (2.10)
The radial magnetic field is obtained as Eq. (2.11) under the condition that the axial magnetic field Bz is almost unchanged compared to the radial motion of the particle.
Br=−r 2
∂
∂zBz. (2.11)
And the component of the Lorenz force for the axial direction usesBr, so the axial force for theBθ =0 is following,
Fz=−m 2B
∂Bz
∂z , (2.12)
wherev⊥is the perpendicular velocity of the particle as the source of the magnetic moment, µ≡1
2 mv2⊥
B . (2.13)
The magnetic moment has a property of invariance even though the magnetic field changes. Thus, the perpendicular velocity changes by the relation in Eq. (2.14),
1 2
mv2⊥i Bi
=1 2
mv2⊥f Bf
, (2.14)
according to the magnetic field that changes when the particle passes through the magnetic field gradient region. In a particular magnetic field, when the parallel component of the particle velocity becomes zero, and only the component remains in the perpendicular direction, a reflection of the particle occurs at this point called the magnetic mirroring effect. In this case, when reflected, the amplitude of the perpendicular velocity is equal to the particle’s initial velocity by energy conservation,v2⊥f =v2i. The relational expression of Eq. (2.14) can be converted to Eq. (2.15) as the ratio of the magnetic fields at the place where the particle starts and is reflected, respectively.
Bi Bf
= v2⊥i v2⊥f =v2⊥i
v2i =sin2θ. (2.15)
Assuming that reflection occurs in the maximum magnetic field in the trap device, a condition in which the beam is reflected may be confirmed, as shown in Eq. (2.16),
Bi
Bmax
<sin2θm, (2.16)
whereθmis the pitch angle where the particle is injected. With this ratio, the boundary on which particles are not reflected can be expressed as a cone shape in the velocity coordinate system, as shown in Fig. 2.7, a loss cone [54]. From the standpoint of the magnetic mirror, if the starting particle is inside the loss
Figure 2.7: Loss cone [54].
cone, it is not reflected but lost, however for using a trap device, it is the condition that the particle can be loaded into the trap. In the electron beam transmission of the EBIS, the conditions under which electrons can be transmitted without reflection by the magnetic mirror effect need to be confirmed. And, it can be derived by the ratio of the magnetic field at the cathode position and the superconducting magnet.
The TRAK code simulated the electron beam transmission with 1 A using the magnetic immersed gun of the EBIS. Figure 2.8a shows the trajectory of the electron beam emitting the cathode into the
main solenoid area. The beam’s envelope from the cathode position increases and decreases again in the part where the magnetic field intensity is slightly reduced and then strengthens again, as illustrated in Fig. 2.6a. In addition, the radial oscillation of the electron beam, a characteristic of the magnetic immersed type, is also confirmed. It can be seen that the magnitude of the oscillation is small as the magnetic field of the gun coil is set at 0.25 T. Figure 2.8b shows the trajectory until the electron beam passes through the breeding region and dumps into the collector. It was confirmed that the beam radius
(a) Trajectory of electron beam from cathode to SC magnet.
(b) Trajectory of electron beam from cathode to col- lector.
Figure 2.8: Electron beam transmission of 1 A of RAON EBIS calculating by TRAK.
changes according to the magnetic field strength and that it was stably transmitted to the collector and dumped without the reflection due to the magnetic mirror effect. After analyzing the simulation results, the characteristics of the electron beam considering the space charge effect in the breeding region may be checked, and the conditions for the injection of the ion beam are confirmed.