Download Thermal Equilibrium Properties of Periodically

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.
Contributors
• 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
Space Charge 2013
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
Phase space for a KV beam
 preserve beam emittance
2.0
(a)
 prevent beam losses
 provide operational stability
 Control halo formation
(S/)1/2X'
 control chaotic particle motion
1.0
0.0
• Thermal equilibrium
 maximum entropy
 Maxwell-Boltzmann (“thermal”) distribution
-1.0
 most likely state of a laboratory beam
 smooth beam edge
-2.0
-2.0
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-1.0
0.0
x/a
1.0
2.0
Qian, Davidson and Chen (1994)
Pakter, Chen and Davidson (1999)
Zhou, Chen, Qian (2003)
Applications of high-brightness chargedparticle beams
 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)
 RF and Thermionic Photoinjectors
 Thermionic DC Injectors
 High Power Microwave Sources
Space Charge 2013
Adiabatic thermal beam theory*
(Periodic solenoidal focusing)
Angular momentum (exact):
Pq = xPy - yPx = const
(
)
Scaled transverse Hamiltonian  2
E w (s)H ^ x , y , Px , Py , s @ const
(approximate):
H ^ (x , y , Px , Py , s) =
(
)
1
K
K
2
2
2
2
2
2
2
self
+
+
+
+
f
+
+
[
]
)
P
P
x
y
x
,
y
,
s
w
s
x
y
(
)
(
x
y
2
2w2 (s)
2qN b
4rbrms
(s)
d 2 w (s)
K
1
+
k
=
s
w
s
w
s
(
)
(
)
(
)
z
2
ds 2
2rbrms
w 3 (s)
(s )
Thermal distribution:
f b (x , y , Px , Py , s) = C exp{- b [E - wb Pq ]}
b , C , wb are constants
* 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).
Space Charge 2013
Self-consistent equations*
K
4 th2  q f self (r , s )

 + 2
- 2
(
)
(
)

2
rbrms s 

b k B T^ s 

Beam density
 r 2
nb (r , s ) = 2
exp- 2
rbrms (s )  4 th
Poisson’s equation
 2f self = - 4p q n b
Beam rotation
w b rb20
1
W b (s ) = - W c (s )+ 2
2
rbrms (s )
Envelope equation
d 2 rbrms (s )
4 th2 + wb2 rb40 b b2 c 2
K
+ k z (s )rbrms (s ) =
2
3
2rbrms (s )
(s )
ds
rbrms
rms beam radius
rbrms (s ) = r
2 12
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C
focusing parameter
 W c (s )
(
)
kz s = 

 2b b c 
2
perveance
thermal rms emittance
2Nb q 2
K 3
 b mVb2
2
(s )
k B T^ (s )rbrms
 =
= const
2 2

b
2m b b c
2
th
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)
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Comparison between theory and experiment
for 5 keV, 6.5 mA electron beam*
Experimental
data
Normalized Density
Normalized Density
Experiment
Theory
1.0
0.5
0.0
-10
-5
0
x (mm)
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5
z=11.2cm
s=11.2 cm
1.5
10
Experiment
Theory
1.0
0.5
0.0
-10
-5
0
x (mm)
5
s=17.2 cm
z=17.2cm
1.5
Normalized Density
z=6.4cm
s=6.4 cm
1.5
10
Experiment
Theory
1.0
0.5
0.0
-10
-5
0
x (mm)
5
*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)
10
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).
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Focusing lattice and beam envelope
wb = 0
 0 = 80
SK / 4 th = 7.0
S k z (s ) = (2 / 3)  0 [1 + cos(2ps / S )]
1/ 2
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Poincare map in phase space
KV-like beam
Adiabatic thermal beam
Normalized momentum = (S / 4 th ) dr / ds
1/ 2
Normalized radius = r / 4 th S
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Poincare map in phase space
KV-like beam
Chaotic seas outside KV beam envelop.
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Adiabatic thermal beam
No chaotic motion in
adiabatic thermal Beam!
Parameters of 2D PIC simulation*
Vacuum Phase Advance
Scaled Normalized Beam Perveance
80 deg
7.0
Number of Particles
1,000,000
Number of Periods
20
Step Size
0.01
Mesh Size
0.02
Beam Pipe Radius
Number of Grid Points
5
250,000
*Barton, Field, Lang & Chen, IPAC 2012 paper WEPPR032;
Barton, Field, Lang & Chen, PRST-AB 15, 124201 (2012).
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Results of PIC simulation
Focusing parameter
rms envelope from PIC simulation
Same as the solution of the rms
envelope equation for the adiabatic
thermal beam.
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Results of PIC simulation (continued)
Horizontal Momentum distribution
Vertical momentum distribution
Both are Gaussian distributions.
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Results of 2D PIC simulation
Beam density distribution
(S / 4 th )1/ 2 D = 0.14
rbrms / D = 9.6
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Horizontal and vertical emittances
Proof of emittance conservation with
nonlinear space charge!
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)
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Innovative thermionic gun design*
Initial simulation
without magnetic field
Second iteration of simulation
with magnetic field
CATHODE
ANODE
ELECTRODE ELECTRODE
CATHODE
ANODE
ELECTRODE ELECTRODE
BEAM
TUNNEL
BEAM
TUNNEL
ANODE
APERTURE
ANODE
APERTURE
CATHODE
ELECTRON
BEAM
CATHODE
ELECTRON
BEAM
*C. Chen, T.M. Bemis, R.J. Bhatt, and J. Zhou, US. Patent No. 7,619,224 B2 (2009).
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OMNITRAK3D simulation for a cold beam
Particle distribution at z = 10 mm
Axial magnetic field profile
800
Bz (Gauss)
600
400
200
0
0
2
4
6
z (mm)
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8
10
1D adiabatic thermal C-L flow
1D adiabatic warm-fluid equations
mnV
V
f p
= -qn z
z z
nV
=0
z
Schematic of 1D C-L flow
f (0) = 0 = f (0)
f (d ) =  d
 2f / z 2 = -4pqn
 p
 3 =0
z  n 
p = k B nT
Emitter
T (0) = Tc
Chen, Pakter & Rizzato, IPAC proceedings, 2011.
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Collector
Theoretical predictions
Conservation laws:
J = qnV = const.
p / n3 = const.
1
3
2
mV + qf + k BT = const .
2
2
At the emitting surface:
Poisson equation:
Space Charge 2013
 3k BTc 
V (0 ) = 

 m 
1/ 2
n(0) =
J
 3k BTc 
q
 m 
1/ 2
 2f
- 4pJ
=
2
1/ 2 1/ 2
z
 - qf + 3k T  qf  qf - 6k T  
B c
B c
+ 
 

m
m
 

m
Examples of 1D adiabatic thermal C-L flow
Parameters:
J
= 2.6
J CL
k BTc
= 0 .1
- q d
Parameters:
J
= 1.15
J CL
k BTc
= 0.001
- q d
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First results of 1D self-consistent simulation
High-temperature example
• Model
 Charged sheets
• Parameters:
k BTc /( -q d ) = 1
J / J CL = 6.3
N P = 50,000
Chen, Pakter & Rizzato, IPAC proceedings, 2011.
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Results of refined 1D self-consistent simulation
k BTc /( -q d ) = 0.01
J / J CL = 1.4886
N P = 2,000,000
Space Charge 2013
R.V. Mok, T.R. Akylas & C. Chen
2D adiabatic thermal C-L flow theory and issues
mnV  V = -qnf -   
  (nV) = 0
 2f = -4pqn
 k B nT^

= 0

 0
0
k B nT^
0


0 

k B nT|| 
0
Need: Some generalized adiabatic equations of state
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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 emittancepreserving techniques
• Approach
 PIC simulation & Theory
Space Charge 2013
Work in progress by
R.V. Mok, T.R. Akylas & C. Chen
Conclusion
• 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
Space Charge 2013