Thermal Beam Equilibria in Periodic Focusing Fields* C. Chen Massachusetts Institute of Technology...
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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.
Thermal Beam Equilibria in Periodic Focusing Fields* C. Chen Massachusetts Institute of Technology Presented at Workshop on The Physics and Applications
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
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.