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.
5.6 mm Diameter 4.2 mm Diameter
Figure 2.9: IrCe cathodes of 5.6 and 4.2 mm in diameter [47].
This calculation analyzed the simulation results to confirm. From the results of the TRAK, the distribution of the radial current density in the breeding region with a 6 T magnetic field was confirmed, as shown in Fig. 2.10. The current density is plotted for the 1200 mm to 1500 mm part contained in
- 0 . 1 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0
- 5 0
0
5 0 1 0 0 1 5 0 2 0 0 2 5 0
Current Density [A/cm2 ]
r [ m m ]
z = 1 2 0 0 m m z = 1 3 0 0 m m z = 1 4 0 0 m m z = 1 5 0 0 m m
Figure 2.10: Current density distribution of 1 A electron beam in breeding region with 6 T simulated by TRAK.
the breeding region in the axial direction. The size of the electron beam can be confirmed through this distribution, as the electron beam does not exist at the part where the current density is zero. Figure 2.10 illustrates that the position where an electron beam disappears is slightly different based on 0.44 mm, which seems to be caused by the radial oscillation and the cyclotron motion of the electron beam. Hence, the average radius of the electron beam is about 0.44 mm, which is consistent with the calculated value above. In addition, most values between 150 and 200 A/cm2 exist for the current density, and when averaged, about 170 A/cm2, which is also consistent with the calculation.
While the electron beam is transmitted in this way, the ion beam is injected and trapped to perform the charge breeding. At this time, the injection and energy conditions of the ion beam are determined by the electrical potential in the breeding region, which should consider the electron beam’s space charge effect along with the voltage applied to the drift tubes. The potential of the electron beam due to the space charge effect can be calculated from the Poisson equation, Eq. (2.17),
∇·⃗E= ρ ε0
, (2.17)
whereρ andε0 are the volume charge density and the vacuum permittivity, respectively. The volume charge density required in the Poisson equation is obtained as shown in Eq. (2.18),
ρ= −I
veS, (2.18)
whereI,ve, andSare the electron beam current, the velocity of electrons, and the transverse area, respec- tively. The negative sign of Eq. (2.18) expresses that electrons have a negative charge. Equation (2.20), the electric field in the radial direction, can be obtained by calculating from Eq. (2.19), an integral form of the Poisson equation, using the volume charge density of Eq. (2.18).
I
⃗E·d⃗A= 1 ε0
Z Z Z
ρdV. (2.19)
→⃗Er·2π⃗rL= 1 ε0
−I
veπr2b·πr2L, (2.20)
whereLandrbare the length of the breeding region and the radius of electron beam, respectively. Inside the electron beam, the total charge calculated depends on the value ofr, and outside, the total charge becomes a constant value calculated asr=rb. Therefore, the radial electric field is expressed by dividing it into the inside and the outside electron beam, as shown in Eqs. (2.21) and (2.22),
⃗Erin(r) =− I
2π ε0ver2brr,ˆ r<rb, (2.21)
⃗Erout(r) =− I 2π ε0ve
1
rr,ˆ r>rb. (2.22)
Since the electric field and the potential have a relation like in Eq. (2.23), they can be obtained by integrating the radial electric field with respect tor, as shown in Eq. (2.24).
⃗E=−∇V, →⃗Er(r) =−∂
∂rV(r)r,ˆ (2.23)
V(r) =− Z
⃗Er·d⃗r. (2.24)
After Eqs. (2.21) and (2.22) are integrated, the electrical potentials of Eqs. (2.25) and (2.26) are calcu- lated, and the integration constants ofCinandCout may be obtained by boundary conditions.
Vin(r) = I
4π ε0ver2br2+Cin, r<rb, (2.25) Vout(r) = I
2π ε0veln(r) +Cout, r>rb. (2.26)
The first boundary condition is determined from the voltage applied to the drift tube. When the voltage ofVDT is applied to the drift tube at the position (rDT) of the inner diameter of the drift tube,Cout is calculated from Eq. (2.26).
Cout=VDT− I 2π ε0ve
ln(rDT), (V(rDT) =VDT atr=rDT). (2.27) The electric potential outside the electron beam,Vout, by usingCout of Eq. (2.27), is arranged as shown in Eq. (2.28).
Vout(r) = I 2π ε0ve
ln r
rDT
+VDT, r>rb. (2.28)
After that, it becomes possible to calculateCinusing Eqs. (2.25) and (2.28) at the electron beam’s radial boundary (r=rb).
I
4π ε0ver2br2b+Cin= I 2π ε0veln
rb rDT
+VDT, (by B.C. atr=rb). (2.29) If Eq. (2.25) is organized using theCin value obtained by Eq. (2.29), the electrical potential inside the electron beam is obtained, as shown in Eq. (2.30).
Vin(r) = I
4π ε0ver2b r2−r2b + I
2π ε0veln rb
rDT
+VDT
= I 2π ε0ve
1 2
r2 rb2−1
+ln
rb rDT
+VDT, r<rb.
(2.30)
This way, the radial potential well is calculated through the electrical potential inside and outside the electron beam, Eqs. (2.28) and (2.30), including the space charge effect. By using the value of the
- 1 0 - 5 0 5 1 0
1 8 . 7 5 1 9 . 0 0 1 9 . 2 5 1 9 . 5 0 1 9 . 7 5 2 0 . 0 0 2 0 . 2 5
Electric Potential [kV]
r [ m m ]
T h e o r e t i c a l C a l c u l a t i o n S i m u l a t i o n R e s u l t b y T R A K
Figure 2.11: Radial electric potential in breeding region from theoretical calculation (blue) and simula- tion by TRAK (red).
electron beam radius 0.437 mm mentioned above, along with transmitting the electron beam to 1 A
with an energy of 13 keV, the potential distribution can be obtained as represented by the blue line in Fig. 2.11. In this plot,the voltage of the drift tube,VDT, is 20 kV with an inner radius of 12.25 mm.
The red line in Fig. 2.11 shows the radial potential distribution at a position corresponding to z=1400 mm, which is the breeding region, from the simulation result in Fig. 2.8. Compared to the theoretical calculation, there is a difference of about 100 V, but this is because the radial electron distribution is not entirely uniform in the simulation, and the beam size is also slightly different. Both results show that the space charge effect of the electron beam decreases by about 1 kV from 20 kV, which is the voltage applied to the drift tube. Considering this lowered electrical potential, the breeding region’s potential and the voltage applied to the gate DT #08 should be determined. The energy of the incoming beam should be slightly higher than the potential of the breeding region, and the gate should be set higher than the beam’s energy. Furthermore, the shape of this radial potential well determines the conditions under which the ion beam can overlap the electron beam in the breeding region, which is discussed in the following sections. Since the space charge effect occurs not only in the breeding region but also in all regions where the electron beam exists, a potential lower than the voltage applied to each drift tube is distributed. Figure 2.12 shows the axial distribution of electrical potential according to the presence or absence of the electron beam on the axis. It can be seen that the potential is significantly lowered in
0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5
05
1 0 1 5 2 0 2 5 3 0
D T # 1 0 D T # 0 9
D T # 0 8
D T # 0 4 ~ 0 7 D T # 0 3
D T # 0 2 D T # 0 1
A n o d e
Electric Potential on axis [kV]
z [ m ]
W i t h o u t E - B e a m
W i t h E - B e a m
C a t h o d e
Figure 2.12: Axial electric potential from cathode to DT #10 from simulation result by TRAK.
the breeding region where the magnetic field is the highest and the electron beam size is the smallest, and a little less so in the other areas. However, the electric potential on the right side of DT #03 and the left side of DT #08∼09 are relatively much lower. It is caused by a change in the magnitude of the magnetic field in one drift tube region with the same voltage applied. In particular, in the case of DT
#09, this results in a very shallow additional trap region in a space other than the breeding region, called a parasitic trap [55]. In this parasitic trap, if the gate timing of DT #08 is correct, the ion beam is not trapped, but the residual gas ions ionized by the electron beam can be trapped. Since the space charge effect of the electrons makes this trap, the trapped ions are extracted simultaneously as the electron
beam is turned off. Therefore, this should be considered in the experimental sequence of producing and ejecting the highly charged ions from the EBIS. It is also necessary to predict the total amount of charge that can stably trap in this breeding region. The trap capacity can be calculated using Eq. (2.31) [56],
CTrap=1.05×1013 IeL
√Ee
f, (2.31)
whereIe, L, Ee, and f are the electron beam current, the length of the breeding region, the energy of electrons, and the charge compensation, respectively. The length of the RAON EBIS breeding region is 0.76 m, and when an electron beam 1 A is transmitted with an energy of 13 keV, the trap capacity becomes∼7.00×1010charges when the charge compensation is 1. That is, based on132Sn25+, about 2.8×109 particles can be trapped. However, the number of trappable particles is reduced when there is a charge compensation effect, including the capacity loss caused by the residual gas inside the EBIS.
Considering the bunch capacity, which is one of the specifications of RFQ-CB that sends the bunched ion beam to the EBIS, it seems that the trap capacity could sufficiently trap it.