Electron Beam Optics For Magnetized Electron Cooling Jrg

Electron Beam Optics For Magnetized Electron Cooling Jörg Kewisch, Xiangyun Chang Jörg Kewisch, January 24, 2006

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: Jörg Kewisch, January 24, 2006

Coulomb Log Jörg Kewisch, January 24, 2006

Beam Requirements in the Cooling Section • Bunch charge: • Beam Size: • Bunch length: • Emittance (normalized) inside the solenoid: • Solenoid Field: 20 n. C 1 mm max 5 cm rms 50 mm mrad 5 Tesla Jörg Kewisch, January 24, 2006

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 effect of the fringe field cancels that motion. (x ~ y’, y ~ -x’), the 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. Jörg Kewisch, January 24, 2006

Layout Jörg Kewisch, January 24, 2006

Optics elements Jörg Kewisch, January 24, 2006

Super-conducting gun • 1½-cell gun • 30 Me. V/m on the cathode • 1 MW RF power • Beam energy 4. 75 Me. V 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 Jörg Kewisch, January 24, 2006

Radial Dependence of the longitudinal Field Cathode position The 4 D 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. Jörg Kewisch, January 24, 2006

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. Jörg Kewisch, January 24, 2006

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. Jörg Kewisch, January 24, 2006

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. Jörg Kewisch, January 24, 2006

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 Jörg Kewisch, January 24, 2006

Merge of Low Energy and High Energy Beam (from MAC 2004) Longitudinal phase space at the first dipole and at second dipole Jörg Kewisch, January 24, 2006

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 Jörg Kewisch, January 24, 2006

Linac • Acceleration on top of the RF wave. Head and tail of the bunch are accelerated less than the center. • De-cellerating 3 rd harmonic cavities compensate this effect: Gun 4. 7 Me. V + Linac 55 Me. V – 3 rd harmonic 5 Me. V = 55 Me. V • 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. Jörg Kewisch, January 24, 2006

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. Jörg Kewisch, January 24, 2006

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 Jörg Kewisch, January 24, 2006

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 Jörg Kewisch, January 24, 2006

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. Jörg Kewisch, January 24, 2006

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) Jörg Kewisch, January 24, 2006

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 Jörg Kewisch, January 24, 2006

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. Jörg Kewisch, January 24, 2006

Beam Size from Gun to Cooler Solenoid Jörg Kewisch, January 24, 2006

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. Jörg Kewisch, January 24, 2006

4 D Emittance Jörg Kewisch, January 24, 2006

Conclusion Jörg Kewisch, January 24, 2006

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 Jörg Kewisch, January 24, 2006