Adiabatic Thermal Beams in a Periodic Focusing Field

1. Adiabatic Thermal Beams in a Periodic Focusing Field Chiping Chen, Massachusetts Institute of Technology Presented at Space Charge 2013 Workshop CERN April 16-19, 2013 *Research supported by DOE Grant No. DE-FG02-13ER41966, Grant No. DE-FG02-95ER40919, Grant No. DE-FG02-05ER54836 and MIT Undergraduate Research Opportunity (UROP) Program.

2. MIT faculty, staff and graduate students T.R. Akylas T.M. Bemis (now at Beam Power Technology) R.J. Bhatt (now at McKinsee) R.V. Mok K.R. Samokhvalova (now at MathWorks) J. Zhou (now at Beam Power Technology) MIT Undergraduate Researchers T.J. Barton K. Burdge D.M Field A. Jimenez-Galindo K.M. Lang T. Phan T. Rabga H. Wei (now at Cornell U.) IF-UFRGS (Brazil) R. Pakter F.B. Rizzato Contributors

3. Beam losses and emittance growth are important issues related to the dynamics of particle beams in non-equilibrium It is important to find and study beam equilibrium states to maintain beam quality preserve beam emittance prevent beam losses provide operational stability control chaotic particle motion Control halo formation Thermal equilibrium maximum entropy Maxwell-Boltzmann (“thermal”) distribution most likely state of a laboratory beam smooth beam edge Why is thermal beam equilibrium important? Phase space for a KV beam Qian, Davidson and Chen (1994) Pakter, Chen and Davidson (1999) Zhou, Chen, Qian (2003)

4. International Linear Collider (ILC) Large Hadron Collider (LHC) Muon Collider Free Electron Lasers (LCLS, NGLS, JLAMP) Energy Recovery Linac (ERLs) Light Sources Spallation Neutron Source (SNS/ESS)) Accelerator Driven Systems (ADS) High Energy Density Physics (HEDP) Applications of high-brightness charged-particle beams • RF and Thermionic Photoinjectors • Thermionic DC Injectors • High Power Microwave Sources

5. Adiabatic thermal beam theory* (Periodic solenoidal focusing) = - = P xP yP const q y x ( ) ( ) º @ 2 E w s H x , y , P , P , s const ^ x y 1 K K [ ] ( ) ( ) ) ( ) ( = + + + + f + + 2 2 2 2 2 2 2 self H x , y , P , P , s P P x y x , y , s w s x y ( ) ( ) ^ x y x y 2 2 2 w s 2 qN 4 r s b brms 2 d w s K 1 ( ) + k - = ( ) ( ) ( ) s w s w s z 2 2 3 ( ) ds 2 r s w s ( ) brms ( ) ] { [ } = - b - w f x , y , P , P , s C exp E P q b x y b b w , C , are constants b Angular momentum (exact): Scaled transverse Hamiltonian (approximate): Thermal distribution: * K. R. Samokhvalova, J. Zhou, and C. Chen, Phys. Plasmas 14, 103102 (2007); K. R. Samokhvalova, Ph.D Thesis, MIT (2008); J. Zhou, K. R. Samokhvalova, and C. Chen, Phys. Plasmas 15, 023102 (2008).

6. Self-consistent equations* ( ) ì ü é ù e f 2 self 2 ï ï 4 q r , s C r K ( ) = - + - th n r , s exp í ý ê ú ( ) ( ) ( ) b e g 2 2 2 2 ï ï 2 r s 4 r s k T s ë û î þ ^ brms th brms b B 2 self Ñ f = - p q n 4 b w 2 r 1 ( ) ( ) W = - W + b b 0 s s ( ) b c 2 2 r s brms ( ) ( ) ( ) 2 2 é ù k T s r s W s ( ) e = = ^ 2 B brms const k = c s ê ú th g b 2 2 2 m c z b 2 c ë û b b b Beam density Poisson’s equation Beam rotation Envelope equation perveance rms beam radius focusing parameter thermal rms emittance

7. UMER edge imaging experiment* • 5 keV electron beam focused by a short solenoid. • Bell-shaped beam density profiles • Not KV-like distributions *S. Bernal, B. Quinn, M. Reiser, and P.G. O’Shea, PRST-AB, 5, 064202 (2002)

8. Comparison between theory and experiment for 5 keV, 6.5 mA electron beam* Experimental data z=6.4cm z=11.2cm z=17.2cm *S. Bernal, B. Quinn, M. Reiser, and P.G. O’Shea, PRST-AB 5, 064202 (2002); J. Zhou, K. R. Samokhvalova, and C. Chen, Phys. Plasmas 15, 023102 (2008)

9. Beam density and self-electric field* • Density • Thermal beam has a higher density than the KV-like beam. • Thermal beam has a smooth profile. • Self-electric field • Thermal beam has a smooth field. • Thermal beam has a weaker field near the beam edge. *H. Wei and C. Chen, PRST-AB 14, 024201 (2011).

10. Focusing lattice and beam envelope

11. Poincare map in phase space KV-like beam Adiabatic thermal beam Normalized momentum = Normalized radius =

12. Poincare map in phase space KV-like beam Adiabatic thermal beam No chaotic motion in adiabatic thermal Beam! Chaotic seas outside KV beam envelop.

13. Parameters of 2D PIC simulation* *Barton, Field, Lang & Chen, IPAC 2012 paper WEPPR032; Barton, Field, Lang & Chen, PRST-AB 15, 124201 (2012).

14. Results of PIC simulation rms envelope from PIC simulation Focusing parameter Same as the solution of the rms envelope equation for the adiabatic thermal beam.

15. Results of PIC simulation (continued) Horizontal Momentum distribution Vertical momentum distribution Both are Gaussian distributions.

16. Results of 2D PIC simulation Beam density distribution Horizontal and vertical emittances Proof of emittance conservation with nonlinear space charge!

17. Issues in beam generation Current state of the art 1 A, 500 kV 1.1 mm-mrad for 1.5 mm radius cathode (Spring-8 injector - Tagawa, et al., PRST-AB, 2007) Is the intrinsic emittance achievable? 0.27 mm-mrad per mm cathode radius Low beam quality limited Spring-8 XFEL performance 0.25 mJ 10% of design goal (Tanaka, paper WEYB01, IPAC 2012)

18. Innovative thermionic gun design* CATHODEELECTRODE CATHODEELECTRODE ANODE ELECTRODE ANODE ELECTRODE BEAMTUNNEL BEAMTUNNEL ANODE APERTURE ANODE APERTURE ELECTRONBEAM ELECTRONBEAM CATHODE CATHODE Initial simulation without magnetic field Second iteration of simulation with magnetic field *C. Chen, T.M. Bemis, R.J. Bhatt, and J. Zhou, US. Patent No. 7,619,224 B2 (2009).

19. OMNITRAK3D simulation for a cold beam Particle distribution at z = 10 mm Axial magnetic field profile

20. 1D adiabatic thermal C-L flow 1D adiabatic warm-fluid equations Schematic of 1D C-L flow Emitter Collector Chen, Pakter & Rizzato, IPAC proceedings, 2011.

21. Theoretical predictions Conservation laws: At the emitting surface: Poisson equation:

22. Examples of 1D adiabatic thermal C-L flow Parameters: Parameters:

23. First results of 1D self-consistent simulation High-temperature example • Model • Charged sheets • Parameters: Chen, Pakter & Rizzato, IPAC proceedings, 2011.

24. Results of refined 1D self-consistent simulation R.V. Mok, T.R. Akylas & C. Chen

25. 2D adiabatic thermal C-L flow theory and issues Need: Some generalized adiabatic equations of state

26. PIC simulation of thermionic gun • Explore beam physics • Longitudinal (Chen, Rizzato, Pakter, paper Proc. IPAC 2011, p. 694) • Transverse • Longitudinal-transverse coupling • Re-examine the merit of intrinsic transverse and longitudinal emittances • 0.27 mm-mrad per mm cathode radius (transverse) • Develop emittance-preserving techniques • Approach • PIC simulation & Theory Work in progress by R.V. Mok, T.R. Akylas & C. Chen

27. Discovery of adiabatic thermal beams is an important advance in beam physics relevant to present and future accelerator applications. Warm-fluid theory Kinetic theory Equivalence between warm-fluid and kinetic theories Integrable charged-particle orbits Verification by 2D PIC simulation Some advances have been made in beam generation. Innovative thermionic gun design 1D adiabatic thermal Child-Langmuir flow Initiated investigation of 2D adiabatic thermal Child-Langmuir flow Future R&D opportunities Engineering design of thermionic gun, beam matching, and beam transport Experimental demonstration Apply and generalize the concept of adiabatic thermal beam in high-brightness electron and ion beam design Conclusion