Thermal Beam Equilibria in Periodic Focusing Fields C

Thermal Beam Equilibria in Periodic Focusing Fields* C. Chen Massachusetts Institute of Technology Presented at Workshop on The Physics and Applications of High-Brightness Electron Beams Maui, Hawaii November 16 -19, 2009 Collaborators: T. R. Akylas, T. M. Bemis, R. J. Bhatt, K. R. Samokhvalova, J. Taylor, H. Wei and J. Zhou Thanks to the UMER group, especially S. Bernal. *Research supported by DOE Grant No. DE-FG 02 -95 ER 40919, Grant No. DE-FG 02 -05 ER 54836 and MIT Undergraduate Research Opportunity (UROP) Program.

Outline • Background Ø Importance of thermal beams Ø Historical perspective Ø Issues • Beams in Periodic Solenoidal Focusing Ø Warm-fluid and kinetic theories Ø Comparison between theory & experiment Ø Control of chaotic particle motion • Beams in Alternating-Gradient Focusing Ø Warm-fluid theory Ø Comparison between theory & experiment • Research Opportunities in Thermionic DC Beam Approach to High. Brightness, High-Average Power Injectors • Conclusions • Future Directions HBEB 09 2/33

Why is thermal beam equilibrium important? • Beam losses and emittance growth are important issues Ø related to the dynamics of particle beams in non-equilibrium • It is important to find and study beam equilibrium states to Ø maintain beam quality Phase space for a KV beam Ø preserve beam emittance Ø prevent beam losses Ø provide operational stability Ø control chaotic particle motion Ø Control halo formation • Thermal equilibrium Ø maximum entropy Ø Maxwell-Boltzmann (“thermal”) distribution Ø most likely state of a laboratory beam Ø smooth beam edge HBEB 09 Qian, Davidson and Chen (1994) Pakter, Chen and Davidson (1999) Zhou, Chen, Qian (2003) 3/33

Applications of high-brightness chargedparticle beams Ø International Linear Collider (ILC) Ø Free Electron Lasers (FELs) Ø Energy Recovery Linac (ERLs) Ø Light Sources Ø Large Hadron Collider (LHC) Ø Spallation Neutron Source (SNS) Ø High Energy Density Physics (HEDP) Ø RF and Thermionic Photoinjectors Ø Thermionic DC Injectors Ø High Power Microwave Sources HBEB 09 4/33

University of Maryland Electron Ring (UMER) • UMER Ø Circumference = 11. 52 m Ø Scaled low-energy e- beam Ø Space-charge-dominated regime • Linear beam experiments Ø Solenoidal and quadrupole focusing experiments Ø Density profile measurements S. Bernal, B. Quinn, M. Reiser, and P. G. O’Shea, PRST-AB 5, 064202 (2002) S. Bernal, R. A. Kishek, M. Reiser, and I. Haber, Phys. Rev. Lett. 82, 4002 (1999) HBEB 09 5/33

Linear focusing channel y z x jq Beam Alternating-Gradient Quadrupoles Solenoid Weak Focusing HBEB 09 Strong Focusing 6/33

Rigid-rotor equilibrium in a uniform magnetic field Brillouin Density dc Beam (non-neutral plasma column) *R. C. Davidson and N. A. Krall, Phys. Rev. Lett. 22, 833 (1969); A. J. Theiss, R. A. Mahaffey, and A. W. Trivelpiece, Phys. Rev. Lett. 35, 1436 (1975); L. Brillouin, Phys. Rev. 67, 260 (1945). HBEB 09 7/33

Thermal rigid-rotor equilibrium in a uniform magnetic field Davidson and Krall, 1971 Distribution function HBEB 09 Trivelpiece, et al. , 1975 8/33

Periodic Focusing • Solenoid (weak focusing) S/2 s + I 1 - I 1 + I 1 • Single particle orbits x ¢¢(s) + k (s) x(s)= 0 x(s) = Ax wx (s) cos[y x (s ) + y x 0 ] • Quadrupole (strong focusing) s S Bq S x N HBEB 09 N S y N S σv=60 o S N N S/2 S/2 9/33

Previous equilibrium theories Equilibria Thermal Beam Equilibria Other Beam Equilibria Focusing Uniform Rigid-rotor kinetic Cold-fluid beam R. C. Davidson, Physics of nonneutral plasmas (Addison-Wesley, Reading, MA, 1990). M. Reiser and N. Brown, Phys. Rev. Lett. 71, 2911 (1993). R. C. Davidson, Physics of nonneutral plasmas (Addison-Wesley, Reading, MA, 1990). Warm-fluid beam S. M. Lund and R. C. Davidson, Phys. Plasmas 5, 3028 (1998). Approximate (small σv) Periodic Solenoidal R. C. Davidson, H. Qin, and P. J. Channell, Phys. Rev. Special Topics-Accel. Beams 2, 074401 (1999). Rigid-rotor kinetic C. Chen, R. Pakter and R. C. Davidson, Phys. Rev. Lett. 79, 225 (1997). Cold-fluid beam R. C. Davidson, P. Stoltz, and C. Chen, Phys. Plasmas 4, 3710 (1997). Periodic Quadrupole HBEB 09 Approximate (small σv) Kapchinskij-Vladimirskij (KV) R. C. Davidson, H. Qin, and P. J. Channell, Phys. Rev. Special Topics-Accel. Beams 2, 074401 (1999). I. M. Kapchinskij, and V. V. Vladimirskij, in Proc. of the International Conf. on High Energy Accel. (CERN, Geneva, 1959), p. 274. 10/33

Issues of previous theories • There was a lack of a fundamental understanding of beam equilbria beyond cold fluid Ø KV-type equilbria are mathematical and cannot be realized or seen experimentally. Ø Smooth-beam approximations were not accurate at high vacuum phase advance. Self-similar density distribution Constant-density contours are ellipses of the same aspect ratio • RMS envelope equations (Sacherer, 1971; Lapostolle; 1971) Ø Assumption of a self-similar density distribution Ø No self-consistent description of emittance evolution Ø No self-consistent description of density evolution HBEB 09 0 11/33

Warm-fluid equilibrium theory* (Solenoidal focusing) • Continuity equation • Force balance equation • Poisson’s equation • Pressure tensor • Ideal gas law is ignored in paraxial treatment HBEB 09 *K. R. Samokhvalova, J. Zhou, and C. Chen, Phys. Plasmas 14, 103102 (2007) 12/33

Warm-fluid equilibrium theory* (Solenodial focusing) • Adiabatic equation of state æp ö V × Ñç ^2 ÷ = 0 èn ø 2 T ^ ( s ) rbrms ( s ) = const • RMS beam radius • Transverse beam velocity HBEB 09 2 2 = rbrms s r () ¢ (s) rbrms V^ = r Vz eˆ r + r Wb (s)eˆ q rbrms (s) *K. R. Samokhvalova, J. Zhou, and C. Chen, Phys. Plasmas 14, 103102 (2007) 13/33

Warm-fluid equilibrium theoretical results* (Solenoidal focusing) éK 4 e th 2 ù q f self (r , s )üï ý ê + 2 ú- 2 ( ) g 2 rbrms s û ë b k B T^ s ï þ Beam density ìï r 2 nb (r , s ) = 2 expí- 2 rbrms (s ) ïî 4 e th Poisson’s equation Ñ 2 f self = - 4 p q n b Beam rotation w b rb 20 1 b (s ) = - c (s )+ 2 2 rbrms (s ) C Envelope equation rms beam radius focusing parameter é c (s )ù ( ) kz s = ê ú ë 2 b b c û HBEB 09 2 perveance thermal rms emittance 2 (s ) k B T^ (s )rbrms e = = const 2 2 g b 2 m b b c 2 th *K. R. Samokhvalova, J. Zhou, and C. Chen, Phys. Plasmas 14, 103102 (2007) 14/33

Kinetic equilibrium theory* (Solenoidal focsuing) • Vlasov equation • Single-particle Hamiltonian • Paraxial approximation Cartesian Coordinates Þ ( x, y , Px , Py ) y Larmor Frame (~ x, ~y , P~ ) x c y (s) Þ Courant-Snyder transformation ( x , y , Px , Py ) 2 x HBEB 09 *J. Zhou, K. R. Samokhvalova, and C. Chen, Phys. Plasmas 15, 023102 (2008) 15/33

Constants of motion and thermal distribution Angular momentum (exact): Pq = x. Py - y. Px = const Scaled transverse Hamiltonian (approximate): E º w 2 (s)H ^(x , y , Px , Py , s) @ const H ^ (x , y , Px , Py , s) = ( ) 1 K K 2 2 2 2 self + + ) [ ] P P x y x , y , s w s x y ( ) ( x y 2 2 w 2 (s) 2 q. N b 4 rbrms (s) d 2 w (s) K 1 + k = s w s ( ) ( ) z 2 ds 2 2 rbrms w 3 (s) (s ) Thermal distribution: f b (x , y , Px , Py , s) = C exp{- b [E - w b Pq ]} b , C , wb are constants HBEB 09 J. Zhou, K. R. Samokhvalova, and C. Chen, Phys. Plasmas 15, 023102 (2008) 16/33

Beam envelope and density warm beam cold beam HBEB 09 17/33

UMER edge imaging experiment* • 5 ke. V 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) HBEB 09 18/33

Comparison between theory and experiment for 5 ke. V, 6. 5 m. A electron beam* Experimental data z=6. 4 cm HBEB 09 z=11. 2 cm *S. Bernal, B. Quinn, M. Reiser, and P. G. O’Shea, PRST-AB 5, 064202 (2002); K. R. Samokhvalova, J. Zhou, and C. Chen, Phys. Plasmas 14, 103102 (2007); J. Zhou, K. R. Samokhvalova, and C. Chen, Phys. Plasmas 15, 023102 (2008) z=17. 2 cm 19/33

Chaotic phase space for a KV beam Qian, Davidson and Chen (1994) Pakter, Chen and Davidson (1999) Zhou, Chen, Qian (2003) HBEB 09 20/33

KV Beam Self-electric Field Map HBEB 09 Normalized Momentum Control of chaos in thermal beams (preliminary results) Wei & Chen, paper presented at DPP 09 Thermal Beam KV Beam Normalized Radius 21/33

Warm-fluid equilibrium theory (AG focusing) Solenoidal Lattice Quadrupole Lattice Force-balance equation æ Ñf self V ´ B ext + nb m(V × Ñ )g b V = nb qçç 2 gb c è ö ÷ - Ñp ^ ÷ ø Equation of state (adiabatic process) 2 ( s ) = const T ^ ( s ) rbrms T^ (s )xbrms (s )ybrms (s ) = const Transverse flow velocity r ¢ (s ) V^ = r brms b b ceˆ r + e. W b (s )eˆ q rbrms (s ) x (s ) y (s ) V^ (x, y , s ) = x brms b bceˆ x + y brms b bceˆ y xbrms (s ) ybrms (s ) ¢ ¢ Beam density profile æ g b m b b 2 c 2 r 2 çç 2 k BT^ (s ) C nb = 2 expç rbrms (s ) ç q f self (r , s ) ç - g 2 k T (s ) è b B ^ HBEB 09 ì rbrms ¢¢ (s ) Wb (s )× [Wb (s )+ Wc (s )]ü ö í ý÷ 2 2 ( ) b î rbrms s þ÷ bc ÷ ÷ ÷ ø æ g m b 2 c 2 é x¢¢ (s ) ù ö b brms ç- b + k q (s )ú x 2 ÷ ê ç 2 k BT^ (s ) ë xbrms (s ) û ÷ ç ÷ ù 2÷ ¢¢ (s ) C ç g b m b b 2 c 2 é ybrms - k q (s )ú y ÷ nb = exp ê xbrms (s) y brms(s) ç 2 k BT^ (s ) ë ybrms (s ) û ÷ ç self ç q f (x, y , s ) ÷ çç ÷÷ 2 g ( ) k T s ^ b B è ø 22/33

Thermal beam equilibrium theoretical results (AG focusing) Beam density Poisson’s equation Envelope equations focusing parameter HBEB 09 perveance 4 D thermal rms emittance 23/33

Beam equilibrium properties - Temperature effects • Transverse beam temperature is constant across the cross section of the beam. • 4 D rms emittance is conserved. e 42 Dth = k B T^ (s) x brms (s) y brms (s) mg bb c 2 b 2 = const • Rms beam envelope increases with temperature. HBEB 09 KS =4 Kˆ = e 4 4 Dth 24/33

Beam equilibrium properties - Density profile on x-axis HBEB 09 Density profile on y-axis 25/33

Beam equilibrium properties - Equipotential and density contours • Equipotential contours are ellipses. • Constant density contours are also ellipses. HBEB 09 26/33

• The density is not self-similar é ab ù ú 1ú ´ 100 % ë xbrms ybrms û Elliptical symmetry but not self-similar • Numerical proof of self-field averages HBEB 09 27/33

UMER 6 -quadrupole experiment* 10. 48 HBEB 09 • 4 ke. V electron beam focused by 6 quadrupoles • 2/3 of the beam is chopped by round aperture • Beam density profiles are bell-shaped in the x-direction and hollow in the y-direction • Cannot be explained by KV distribution 13. 43 17. 13 26. 83 35. 28 42. 43 49. 88 57. 98 66. 08 73. 98 *S. Bernal, R. A. Kishek, M. Reiser, and I. Haber, Phys. Rev. Lett. 82, 4002 (1999) 28/33

Comparison between theory and experiment Z=13. 43 cm HBEB 09 Z=17. 13 cm Z=26. 83 cm Z=35. 28 cm 29/33

Research opportunities in thermionic dc gun approach to high-average-power beams • Current state of the art Ø 1 A, 500 k. V Ø 1. 1 mm-mrad for 1. 5 mm radius cathode (Spring-8 injector - Tagawa, et al. , PRST-AB, 2007) • Is the intrinsic emittance achievable? Ø 0. 25 mm-mrad per mm cathode radius • How can we control beam halo? • Need gun and beam matching theory including thermal effects Ø Current research at MIT (Taylor, Akylas & Chen) HBEB 09 30/33

Experimental opportunities • Periodic solenoidal focusing channel Ø New design based on a patented highbrightness circular electron beam system (C. Chen, T. Bemis, R. J. Bhatt and J. Zhou, US Patent Pending, 2009). Ø Minimize beam mismatch. Ø Demonstrate adiabatic thermal beams in a long channel. T. M. Bemis, R. Bhatt, C Chen & J. . Zhou, APL (2007) • AG focusing channel Ø New design a patented high-brightness elliptic electron gun (R. J. Bhatt, C. Chen and J, Zhou, US patent No. 7, 318, 967, 2008) Ø Minimize beam mismatch. Ø Demonstrate adiabatic thermal beams in a long channel. R. Bhatt, T. M. Bemis & C. Chen, IEEE Trans PS (2006) HBEB 09 31/33

Conclusions • Adiabatic thermal beam equilibria shown to exist in Ø Periodic solenoidal focusing Ø AG Focusing • Adiabatic equation of states assures the conservation of normalized rms emittance with space charge Ø 2 D normalized rms emittance in periodic solenoidal focusing Ø 4 D normalized rms emittance in AG focusing • Gaussian density distribution for emittance-dominated beams • Flat density in the center with a characteristic Debye fall off at the edge for space-charge-dominated beams • Predictions for AG focusing Ø Conservation of 4 D normalized rms emittance Ø Elliptical constant density and potential contours Ø Non-self-similar density distribution HBEB 09 32/33

Future directions • Perform high-precision experiments to further test the adiabatic thermal beam equilibrium in periodic solenoidal focusing. • Perform high-precision experiments to test the adiabatic thermal beam equilibrium in AG focusing. • Develop a better understanding of thermal effects in thermionic electron guns and beam matching. • Apply the concept of adiabatic thermal beams in the research, development and commercialization of high-brightness, highaverage-power electron sources and beams. HBEB 09 33/33