# Presentation: 056174

##      Electron Beam Optics For Magnetized Electron Cooling

** ****J****ö****rg Kewisch, Xiangyun Chang**

## Magnetized Electron Cooling

- The transverse velocity of the electrons is much larger than the longitudinal velocity.
- If the electron moves on small Larmour circles the average transverse velocity is zero: Cooling is improved.
- Parkhomchuk’s Friction Force approximation:

## Coulomb Log

## Beam Requirements in the Cooling Section

- Bunch charge:	  		20 nC
- Beam Size:				1 mm max
- Bunch length:			5 cm rms
- Emittance (normalized)
-        inside the solenoid:		50 mm mrad
- Solenoid Field:			5 Tesla

## What is a "Magnetized Beam", Busch's Theorem

**When a non-magnetized beam enters a solenoid, the fringe field increases the normalized emittance:**
**                                                                                       ****with**
**A magnetized beam rotates around the longitudinal axis      (x ~ y’, y ~ -x’), the effect of the fringe field cancels that motion.**
**Busch’s Theorem: If only axial symmetric fields are applied then:**
**	****	****	****	****                             ****with**
**A magnetized beam can only be made using a magnetic field on the cathode!**
**The beam transport matrix from the cathode to the cooling section must be axial symmetric.**
**We try to subtract two large numbers and hope to get zero. Non-linearities disturb the balance.**
** **

## Layout

**Layout**

## Optics elements

## Super-conducting gun

**11⁄2-cell gun**
**30 MeV/m on the cathode**
**1 MW RF power**
**Beam energy 4.75 MeV **
**             ****at the gun exit**
**Cathode solenoid inside the gun**
**360 Gauss on the cathode**
**400 Gauss on the wall**
**Radius on the cathode 1.2 cm**
**Laser pulse 63 picoseconds**

## Radial Dependence of the longitudinal Field

- Cathode position

**The 4D emittance of a beam created with the above solenoid was compared to that of an artificial radius-independent field distribution (obeying Maxwell’s laws), using a 12 mm cathode spot. No significant degradation was observed.**

## Emittance compensation

**Emittance compensation**

**On the cathode the emittance is (close to) zero. **
**Each longitudinal slice maintains the low emittance, but has due to space charge a different phase advance.The over-all emittance at the gun exit is larger.**
** ****We use the space charge to reverse this effect. **

## Emittance compensation

**Emittance compensation**

**A solenoid focuses the beam. Slices with small radius converge more and experience more space in the following drift.**
**By optimizing the solenoid strength and drift length we achieve that all slices have a waist in approximately at the same distance. **
**The accelerating cavity is placed here to increase the beam energy and reduce the space charge.**

## Emittance compensation (magnetized beam)

**Magnetization hampers emittance compensation**
**Variations in the beam radius cause additional emittance growth. **
**A second solenoid is used to optimize the emittance. **
**Focusing of the z-merge dipoles must be taken into account.**

## Elliptical beam

**The effects of space charge are most important near the cathode**
**The charge distribution inside the bunch is critical**
**Two distributions have been tested:**
- Beer Can: Uniform density cylinder
- Longitudinal shaping of the laser pulse is proven technology
- Elliptical Distribution: Uniform density ellipsoid
- Space Charge is linear in all directions inside the ellipsoid
- Two dimensional shaping  of the laser pulse must be tested
**Other distributions will be tested in future**

## Merge of Low Energy and High Energy Beam(from MAC 2004)

**Longitudinal phase space at the ****first dipole and at second dipole **

## Merging system and Emittance Compensation

**Z-merge (Litvinenko,Kayran) defines 4 integrals to be zero**
**Optimize drifts and angles to make all integrals zero**
**Quadrupoles place horizontal and vertical beam waist in the same position**

## Linac

**Acceleration on top of the RF wave. Head and tail of the bunch are accelerated less than the center.**
**De-cellerating 3rd harmonic cavities compensate this effect:**
**	****Gun 4.7 MeV + Linac 55 MeV – 3****rd**** harmonic 5 MeV = 55 MeV**
**3****rd**** harmonic cavities have small aperture. Strong focusing is necessary.**
**3****rd**** harmonic cavities must be in the middle of the linac to avoid negative beam energies.**

## Stretcher, Merge with Ions, Compressor

**Stretching the beam to 5 cm inside a 3 inch diameter beam pipe requires 450 degrees of bending magnets.**
**The last cavity in the linac is miss-phased to increase the energy spread from 4•10****-4**** to 2•10****-3****. The tail has more energy.**
**A 200 MHz cavity at the end of the stretcher reduces the energy spread to 8•10****-5****.**
**The cavity must be in a dispersion free region.**
**The Kayran-Litvinenko integrals must be zero for the stretcher/merger beam line. MAD and a post-processing program is used to find an approximate solution. This solution is then optimized. **
**A 200 MHz cavity before the compressor introduces the opposite energy spread. **
**The compressor optics must inject the beam into the linac.**

## Chromaticity

**Chromaticity can destroy the correlation between horizontal and vertical motion: x’ ~ y and y’ ~ -x**
**Sextupoles can not be used:**
- It would require 8 sextupole families
- The betatron beam size is very large. Therefore it can not be done with fewer families
- It is impossible to find sextupole locations that are sufficiently othogonal
**The best strategy is to keep the beam radius (beta function) small**

## Multipoles

**Multipoles must be kept small**
**A perfect dipole has an effective sextupole component. It is a geometric effect (like weak focusing).**
**This sextupole must be compensated locally.**
**In the simulations a sextupole element is included at both ends of each dipole. The strength is determined by optimization**

## Solenoid Gap

**For technical reasons the cooling solenoid will be split into two sections. Extra focusing is necessary to maintain magnetization. **
**We will use quadrupoles to obtain 180****o****/360****o**** phase advance. This allows opposing field direction in the solenoid halfs.**

## Tracking Codes with space charge

- PARMELA
- closed source, runs only in WINDOWS
- proven
- Impact-T
- Fortran 90
- source available
- cryptic input
- MPI capable
- ASTRA
- closed source, LINUX and WINOWS
- can’t do dipoles (they are working on it)

## Optimizing

**New program developed:**
- Uses CYGWIN inside WINDOWS and PARMELA
- Uses POWELL method (Numerical Recipes)
- Optimizing directives included as comment lines in PARMELA input
- May call MAD for pre-optimization,  output translated to PARMELA
- Prepares PARMELA input file
- Any number can be modified
- Calculates cavity phase shift from shift in position
- Calculates magnetic field on cathode from spot size
- Spawns PARMELA
- Reads PARMELA binary file for results
**Typical run with 2000 particles:** 3 days
- Mistakes waste much time
- Using POWELL requires a start value
-       not too far from optimum

## Better optimizing

**Genetic algorithm using the PISA system (ETH Zürich) spawns Impact-T **
**Runs on computer clusters (BLC and NERSC Seaborg system)**
**Close to completion**
**Will be used to optimize the longitudinal and transversal bunch shaping.**

## Beam Size from Gun to Cooler Solenoid

## Achieved Beam Quality

- The required emittance was achieved using **elliptical** beam distrubution on the cathode:
- Normalized emittance: 50 mm mrad.
- Bunch length 5 cm rms
- Energy spread 8*10-5
- Using a **beer-can** distribution a normalized emittance it was not possible to get better than 65 mm mrad.

## 4D Emittance

**4D Emittance**

## Conclusion

## Conclusion

**The optics for magnetized electron cooling is an extremely complicated system **
**The Space charge makes the use of linear design tools ineffective. Tracking is time consuming and tedious**
**The required beam quality can be achieved **
**with an elliptical distribution**
**with appropriate diagnostics**
**Using non-magnetized cooling seems to be simpler and cheaper**
**Experience and tools will be useful **