Slide 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.

Slide 2

HBEB092/33 Outline 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

Slide 3

HBEB093/33 Why is thermal beam equilibrium important? 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 Qian, Davidson and Chen (1994) Pakter, Chen and Davidson (1999) Zhou, Chen, Qian (2003) Phase space for a KV beam

Slide 4

HBEB094/33 Applications of high-brightness charged- particle beams 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) RF and Thermionic Photoinjectors Thermionic DC Injectors High Power Microwave Sources

Slide 5

HBEB095/33 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. OShea, PRST-AB 5, 064202 (2002) S. Bernal, R. A. Kishek, M. Reiser, and I. Haber, Phys. Rev. Lett. 82, 4002 (1999)

Slide 6

HBEB096/33 Linear focusing channel z x y qq Alternating-Gradient Quadrupoles Solenoid Beam Weak FocusingStrong Focusing

Slide 7

HBEB097/33 Rigid-rotor equilibrium in a uniform magnetic field *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). dc Beam (non-neutral plasma column) Brillouin Density

Slide 8

HBEB098/33 Thermal rigid-rotor equilibrium in a uniform magnetic field Davidson and Krall, 1971 Trivelpiece, et al., 1975 Distribution function

Slide 9

HBEB099/33 Periodic Focusing Solenoid (weak focusing) Quadrupole (strong focusing) Single particle orbits v =60 o 0 sxssx 0 xxxx scosswAsx

Slide 10

HBEB0910/33 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. Approximate (small v ) R. C. Davidson, H. Qin, and P. J. Channell, Phys. Rev. Special Topics-Accel. Beams 2, 074401 (1999). Periodic Quadrupole 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 v ) R. C. Davidson, H. Qin, and P. J. Channell, Phys. Rev. Special Topics-Accel. Beams 2, 074401 (1999). Periodic Solenoidal Cold-fluid beam R. C. Davidson, Physics of nonneutral plasmas (Addison-Wesley, Reading, MA, 1990). 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). Uniform Other Beam EquilibriaThermal Beam Equilibria Equilibria Focusing Previous equilibrium theories

Slide 11

HBEB0911/33 Issues of previous theories 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 Self-similar density distribution 0 Constant-density contours are ellipses of the same aspect ratio

Slide 12

HBEB0912/33 Warm-fluid equilibrium theory* (Solenoidal focusing) Continuity equation Force balance equation Poissons equation Pressure tensor Ideal gas law is ignored in paraxial treatment *K. R. Samokhvalova, J. Zhou, and C. Chen, Phys. Plasmas 14, 103102 (2007)

Slide 13

HBEB0913/33 Warm-fluid equilibrium theory* (Solenodial focusing) Transverse beam velocity *K. R. Samokhvalova, J. Zhou, and C. Chen, Phys. Plasmas 14, 103102 (2007) Adiabatic equation of state RMS beam radius 0 2 n p V constsrsT brms 2 22 rsr eeV srV sr sr r brz

Slide 14

HBEB0914/33 Warm-fluid equilibrium theoretical results* (Solenoidal focusing) perveance focusing parameterrms beam radiusthermal rms emittance Poissons equation Beam rotation b self nq 4 2 Envelope equation Beam density *K. R. Samokhvalova, J. Zhou, and C. Chen, Phys. Plasmas 14, 103102 (2007) sTk s,rq sr Kr exp sr C s,rn Bb self brms th brms b 22 2 2 2 2 4 2 4 sr r ss bb cb 2 2 0 2 1 2 2 c s s b c z const cm srsTk bb brmsB th 22 2 2 2

Slide 15

HBEB0915/33 Kinetic equilibrium theory* (Solenoidal focsuing) Vlasov equation Single-particle Hamiltonian Paraxial approximation *J. Zhou, K. R. Samokhvalova, and C. Chen, Phys. Plasmas 15, 023102 (2008) ),,,( yx PPyx Coordinates Cartesian ~~ ),,,( ~~ yx PPyx Larmor Frame y x 2 ) ( s c Courant-Snyder transformation yx PPyx,,,

Slide 16

HBEB0916/33 Constants of motion and thermal distribution Angular momentum (exact): Scaled transverse Hamiltonian (approximate): Thermal distribution: J. Zhou, K. R. Samokhvalova, and C. Chen, Phys. Plasmas 15, 023102 (2008) const xy yPxPP const,,,, 2 sPPyxHswE yx 222 2 2222 2 4,, 22 1,,,,yxsw sr K syx qN K yxPP sw sPPyxH brms self b yxyx sw sw sr K sws ds swd brms z 322 2 1 2 PEexpCs,P,P,y,xf byxb constants are,, b C

Slide 17

HBEB0917/33 Beam envelope and density cold beam warm beam

Slide 18

HBEB0918/33 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. OShea, PRST-AB, 5, 064202 (2002)

Slide 19

HBEB0919/33 Comparison between theory and experiment for 5 keV, 6.5 mA electron beam* Experimental data z=6.4cm z=11.2cmz=17.2cm *S. Bernal, B. Quinn, M. Reiser, and P.G. OShea, 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)

Slide 20

HBEB0920/33 Chaotic phase space for a KV beam Qian, Davidson and Chen (1994) Pakter, Chen and Davidson (1999) Zhou, Chen, Qian (2003)

Slide 21

HBEB0921/33 Control of chaos in thermal beams (preliminary results) KV Beam Self-electric Field Map Thermal Beam Normalized Radius Normalized Momentum KV Beam Normalized Radius Normalized Momentum Wei & Chen, paper presented at DPP09

Slide 22

HBEB0922/33 Solenoidal LatticeQuadrupole Lattice Force-balance equation Equation of state (adiabatic process) constsrsT brms 2 Transverse flow velocity Beam density profile constsysxsT brms yb xb c sy sy yc sx sx xsyxeeV ,, Warm-fluid equilibrium theory (AG focusing) eeV sec sr sr r brb brms p c qnmn ext b self bbb BV VV 2

Slide 23

HBEB0923/33 Thermal beam equilibrium theoretical results (AG focusing) Beam density Poissons equation 4D thermal rms emittance perveance focusing parameter Envelope equations

Slide 24

HBEB0924/33 4 4 4 Dth KS K Beam equilibrium properties - Temperature effects Rms beam envelope increases with temperature. 4D rms emittance is conserved. Transverse beam temperature is constant across the cross section of the beam. const cm sysxsTk bb brms B Dth 22 2 4

Slide 25

HBEB0925/33 Beam equilibrium properties - Density profile Density profile on x- axisDensity profile on y- axis

Slide 26

HBEB0926/33 Beam equilibrium properties - Equipotential and density contours Equipotential contours are ellipses. Constant density contours are also ellipses.

Slide 27

HBEB0927/33 Elliptical symmetry but not self-similar The density is not self-similar Numerical proof of self-field averages % 100 1 brms y x b a

Slide 28

HBEB0928/33 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 UMER 6-quadrupole experiment* *S. Bernal, R. A. Kishek, M. Reiser, and I. Haber, Phys. Rev. Lett. 82, 4002 (1999) 10.48 13.43 17.13 26.83 35.28 42.43 49.88 57.98 66.08 73.98

Slide 29

HBEB0929/33 Comparison between theory and experiment Z=13.43cmZ=17.13cmZ=26.83cmZ=35.28cm

Slide 30

HBEB0930/33 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)

Slide 31

HBEB0931/33 Experimental opportunities 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. R. Bhatt, T.M. Bemis & C. Chen, IEEE Trans PS (2006) T.M. Bemis, R. Bhatt, C Chen & J..Zhou, APL (2007)

Slide 32

HBEB0932/33 Conclusions 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

Slide 33

HBEB0933/33 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, high- average-power electron sources and beams.