1. Preliminary results for electron lens with beam current of 20 A
A. Barnyakov, D. Nikiforov, A. Levichev
BINP-CERN,

2. Equilibrium beam radius
Equilibrium radius is the constant radius for the beam, which it will has in the uniform magnetic field. It depends on the cathode radius Rc, magnetic field on the cathode Bc, beam current I, beam voltages U and magnetic field in the channel B (Appendix 1)
To obtain the beam diameter of 2 mm in the magnetic field of 5 T the cathode radius has to be 5 mm with cathode magnetic field of about 0.2 T and beam potential of kV
Cathode field 0.2 T

3. Estimation of the maximum of the vacuum tube perveance for thin beam Rb<<Rt
The main problem for propagation through the vacuum chamber of high current beam with low energy is virtual cathode inside the vacuum chamber. This effect can be estimated with help of maximum tube perveance. For thin beam the beam and tube can be presented as cylindrical capacitor and the tube perveance can be estimated as (see Appendix 2)

4. Vacuum tube and beam perveance
I=20, U=10 kV microperveance is 20, Rt/Rb=1
I=20, U=25 kV microperveance is 5, Rt/Rb<12.5
I=20, U=35 kV microperveance is 3, Rt/Rb<40
With tube aperture of 80 mm to obtain the beam diameter of about 2 mm the beam voltage must be about 35 kV

5. Magnetic field in the solenoids
Equilibrium diameter of the beam with I=20 and V=10-35 kV for magnetic field of 5 T is about 2 mm for the cathode radius of 5 mm and cathode magnetic field of 0.2 T

6. Beam dynamics I=20 A, V=25 kV
In high magnetic field of the bending solenoid the beam is compressed and when the ratio Rt/Rb>12.5 the virtual cathode is arisen. To remove this effect without changing the aperture of vacuum chamber and beam current the beam potential has be increased (~35 kV). For the cone type chamber the ratio Rt/Rb>12.5 becomes earlier and such design is worse than tube with constant radius.

7. Possible design of electron gun with 35 kV and 20 A
Beam current is I=20 A
Voltage is 35 kV
Magnetic field is 0.2 T
Cathode radius is 5 mm

8. Electron gun with 35 kV and 20 A
Gun parameters
Cathode diam.
10 mm
Cathode material
Ir-Ce
Energy
Up to 35 kV
Current
21 A
Tr. Profile
Flat top
Emit (gun exit)
7.8 mm mrad

9. All current (20 A) propagates through the electron lens
Beam dynamics I=20 A, V=35 kV
All current (20 A) propagates through the electron lens

10. Beam dynamics I=20 A, V=35 kV
Beam potential is decreased when the beam is compressed and inputted in the vacuum chamber with high aperture of the bending solenoid. Due to asymmetric vacuum chamber the beam potential becomes also asymmetric and particles get different velocities.

11. Beam dynamics I=20 A, V=35 kV: transverse beam space
Due to asymmetric vacuum chamber the beam profile is also asymmetric. For particles with different potential the Lorenz radius are differed. Besides the individual rotation for every particle the beam is rotated as a whole due to magnetic field on the cathode therefore the potential distribution along the radius is changed during the time.

12. Beam dynamics I=20 A, V=35 kV: energy of the particles

13. Beam dynamics I=20 A, V=35 kV: conclusion
The initial beam potential of 25 kV is not enough for beam current of 20 A and vacuum tube aperture of 80 mm.
For vacuum tube aperture of 80 mm and beam current of 20 A the initial beam potential has to be about 35 kV.
To compress beam down to 2 mm by magnetic field of 5 T the cathode diameter has to be about 10 mm with magnetic field of 0.2 T. Reducing the cathode magnetic field allows the cathode diameter to be increased.
Increasing of the ratio between transverse size of the vacuum chamber and beam size decreases the beam potential.
Asymmetric vacuum chamber leads to asymmetric distribution of the potential of the particles and profile of the beam.
For single particles the potential can be reduced down to 9 kV in the vacuum tube with aperture of 80 mm and beam diameter of about 3 mm. In this case the average beam potential is about 20 kV.
Due to nonuniform beam potential the velocities of the particles are differed, the front of the beam can be increased and particles motion is not laminar.
The typical beam size in the main solenoid is about 3 mm. Most likely, to decrease this size the new design of the bend can be needed.
Needs to investigate of 10 mm cathode lifetime (See Appendix 3).

14. Beam dynamics I=20 A, V=35 kV: vacuum chamber symmetrization
Symmetrization of the vacuum chamber can improve the beam potential

15. Beam dynamics I=20 A, V=35 kV: vacuum chamber symmetrization
More particles have potential less than 20 kV
10 kV
30 kV
Potential distribution is more uniform
12 kV

16. Beam dynamics I=20 A, V=35 kV: vacuum chamber symmetrization
Transverse beam space

17. Beam dynamics I=20 A, V=35 kV: vacuum chamber symmetrization
Conclusion:
Symmetrical vacuum chamber can improve the beam potential distribution
More uniform beam potential can reduce transverse beam size
For this solution the design of the vacuum chamber has to be changed and, may be, bend design too
Needs more investigation to choice and optimize new design

18. Additional researches: beam propagation not along solenoid axis
Taking into account high magnetic field the beam will go along the magnetic field force lines. In the center of the solenoid, the beam will move rectilinearly due to high radiuses of the solenoid. In the end of the magnet the beam will have some angle which is excited by magnetic field force lines. Additional corrector can compensate this angle.

19. Proton beam with electron beam
Additional researches: compensation of the electron beam field by proton field
Proton beam with electron beam
There is no influence of the proton field on the electron field due to ultra relativistic energy of proton (relativistic factor is about 7000)
Electric field from the electron beam is up to 1.6 MV/m
Electric field from the proton beam is up to 22 kV/m

20. Additional researches: collimator with wires
Current is 230 A with the same direction, maximum magnetic field is 0.01 T

21. Thank you for the attention

22. Appendix 1: equilibrium beam radius in magnetic field
I – beam current
U – beam potential
Bc – magnetic field on the cathode
Rc – radius of the cathode
B – magnetic field in the transport channel

23. Appendix 2: maximum of the tube perveance

24. Appendix 3: IrCe Current density vs. Temperature