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Thermal Beam Equilibria in Periodic Focusing Fields*

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

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Thermal Beam Equilibria in Periodic Focusing Fields*

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  1. 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-FG02-95ER40919, Grant No. DE-FG02-05ER54836 and MIT Undergraduate Research Opportunity (UROP) Program.

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

  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) Free Electron Lasers (FELs) Energy Recovery Linac (ERLs) Light Sources Large Hadron Collider (LHC) Spallation Neutron Source (SNS) High Energy Density Physics (HEDP) Applications of high-brightness charged-particle beams • RF and Thermionic Photoinjectors • Thermionic DC Injectors • High Power Microwave Sources

  5. University of Maryland Electron Ring (UMER) • UMER • Circumference = 11.52 m • Scaled low-energy e- beam • Space-charge-dominated regime • Linear beam experiments • Solenoidal and quadrupole focusing experiments • 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)

  6. Linear focusing channel y z x jq Alternating-Gradient Quadrupoles Solenoid Beam Weak Focusing Strong Focusing

  7. Rigid-rotor equilibrium in a uniform magnetic field BrillouinDensity 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).

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

  9. Periodic Focusing ( ) ( ) ( ) ¢ ¢ + k = S/2 S/2 S/2 S/2 x s s x s 0 S/2 S/2 S/2 S/2 s + + + + + + I I I I I I 1 1 1 1 1 1 - - - - - - I I I I I I 1 1 1 1 1 1 + + + + + + I I I I I I 1 1 1 1 1 1 ( ) ( ) ( ) [ ] = y + y x s A w s cos s 0 x x x x s S S S N N N N N N B B B N N N S S S q q q S S S S S S y y S S S x x N N N S/2 S/2 N N N N N N S S S S/2 S/2 • Solenoid (weak focusing) • Single particle orbits • Quadrupole (strong focusing) σv=60o

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

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

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

  13. Warm-fluid equilibrium theory*(Solenodial focusing) æ ö p Ñ = V × ^ ç ÷ 0 2 n è ø ( ) ( ) = 2 T s r s const ^ brms ( ) = 2 2 r s r brms ( ) ¢ r s ( ) = + W ˆ ˆ brms V r V e r s e ( ) ^ q z r b r s brms • Adiabatic equation of state • RMS beam radius • Transverse beam velocity *K. R. Samokhvalova, J. Zhou, and C. Chen, Phys. Plasmas 14, 103102 (2007)

  14. Warm-fluid equilibrium theoretical results*(Solenoidal focusing) ( ) 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 *K. R. Samokhvalova, J. Zhou, and C. Chen, Phys. Plasmas 14, 103102 (2007)

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

  16. Constants of motion and thermal distribution = - = 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: J. Zhou, K. R. Samokhvalova, and C. Chen, Phys. Plasmas 15, 023102 (2008)

  17. Beam envelope and density warm beam cold beam

  18. 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)

  19. 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); 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)

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

  21. Control of chaos in thermal beams (preliminary results) Thermal Beam Normalized Momentum KV Beam Normalized Momentum Normalized Radius Wei & Chen, paper presented at DPP09 Normalized Radius KV Beam Self-electric Field Map

  22. Warm-fluid equilibrium theory(AG focusing) æ ö Ñ f ´ self ext V B ( ) ç ÷ × g = - + - Ñ n m V Ñ V n q p ç ÷ ^ b b b g 2 c è ø b ( ) ( ) ( ) ( ) ( ) = = T s r 2 s const T s x s y s const ^ ^ brms brms brms ( ) ¢ r s ( ) ( ) ( ) ¢ ¢ x s y s = b + W ˆ ˆ ( ) brms V r c e e s e = b + b ( ) ˆ ˆ V x , y , s x brms c e y brms c e ^ q b r b ( ) ( ) r s ^ b x b y x s y s brms brms brms ( ) æ ö é ù ¢ ¢ g b 2 2 m c x s ( ) ç ÷ - + k 2 b b brms s x ê ú ( ) ( ) [ ] ( ) ( ) ( ) ( ) q ç ÷ æ ö 2 k T s x s ì ü ¢ ¢ g b W × W + W ë û 2 2 2 m c r r s s s s ^ B brms ç ÷ - - ç ÷ b b brms b b c í ý ( ) ( ) ( ) b ç ÷ 2 2 é ù ¢ ¢ 2 k T s r s c g b 2 2 C m c y s C ç ÷ î þ ( ) ^ B brms b = = - - k 2 b b brms n exp n exp s y ç ÷ ê ú ç ÷ ( ) ( ) ( ) ( ) ( ) ( ) b b q 2 x s y s 2 k T s y s r s f self ë û q r , s ç ÷ ç ÷ ^ brms brms B brms brms - ç ÷ ( ) ( ) ç ÷ g 2 f k T s self q x , y , s è ø ^ - b B ç ÷ ( ) ç ÷ g 2 k T s è ø ^ b B Equation of state (adiabatic process) Transverse flow velocity Beam density profile

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

  24. Beam equilibrium properties - Temperature effects ( ) ( ) ( ) k T s x s y s e = ^ = 2 B brms brms const 4 Dth g b 2 2 m c b b KS ˆ = = K 4 e 4 4 Dth • Transverse beam temperature is constant across the cross section of the beam. • Rms beam envelope increases with temperature. • 4D rms emittance is conserved.

  25. Beam equilibrium properties - Density profile Density profile on x-axis Density profile on y-axis

  26. Beam equilibrium properties - Equipotential anddensity contours • Equipotential contours are ellipses. • Constant density contours are also ellipses.

  27. Elliptical symmetry but not self-similar • The density is not self-similar % 100 ´ ù ú û brms y b a brms x - 1 é ë ú • Numerical proof of self-field averages

  28. UMER 6-quadrupole experiment* • 4 keV 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 10.48 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)

  29. Comparison between theory and experiment Z=13.43cm Z=17.13cm Z=26.83cm Z=35.28cm

  30. Research opportunities in thermionic dc gun approach to high-average-power beams • 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? • 0.25 mm-mrad per mm cathode radius • How can we control beam halo? • Need gun and beam matching theory including thermal effects • Current research at MIT (Taylor, Akylas & Chen)

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

  32. Adiabatic thermal beam equilibria shown to exist in Periodic solenoidal focusing AG Focusing Adiabatic equation of states assures the conservation of normalized rms emittance with space charge 2D normalized rms emittance in periodic solenoidal focusing 4D 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 Conservation of 4D normalized rms emittance Elliptical constant density and potential contours Non-self-similar density distribution Conclusions

  33. 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, high-average-power electron sources and beams. Future directions

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