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GWENAEL FUBIANI
L’OASIS GROUP, LBNL
6D Space charge estimates for dense electron bunches in vacuum
W.P. LEEMANS, E. ESAREY, B.A. SHADWICK, J. QIANG, G. DUGAN (Cornell)
Overview
- Motivation:
- space charge dominated evolution in vacuum of electron bunches produced in plasma accelerators
- ultra short, dense bunches with large energy spread
- Example: self modulated laser wakefield accelerator. Initial
bunch parameters at plasma exit:
- Q~1-10nC, transverse size ~5m, angular spread ~5-10 mrad, bunch length ~10m.
- Large energy spread: 0.1 to 100 MeV (Boltzmann momentum distribution)
Space charge expected to dominate bunch dynamics
Electrostatic potential energy per particle of a bunch of N electrons of radius R:
RNr
mcW e
10~2
For Q=5nC, R=5m, 2mcW
is about 1.8
implies a large space charge induced energy spread (MeV scale) , and a large angular spread (100’s of mrad scale) emittance growth
Semi-analytical 6D model of space charge forces for dense electron bunches with a large energy spread
1. A distribution of macro-particles in 6D phase space to model the beam.
2. Generate a set of ellipsoids at each time step: • Divide macro-particles into longitudinal momentum bins. • Each bin contains a small longitudinal momentum spread (e.g., 5%).
3. Space charge force in each momentum bin:• Macro-particle distribution is decomposed into ellipsoidal shells in position space• For each shell: an analytical expression for the electrostatic force is used.
4. To calculate the force on each macro-particle at each time step:• Iterate through each momentum bin• Calculate space charge force acting on that macro-particle in the bin rest
frame. • Transform this force into the lab frame. • Total force is the vector sum of Lorentz-transformed forces from all the momentum bins.
- Space charge forces are Lorentz transformed => has to be small.
- For large momentum spread: bin total bunch into momentum bins, each with a small spread. Each momentum bin is modeled as an ellipsoidal bunch; the ellipsoids from all bins are allowed to interact with each other via space charge forces.
- Each ellipsoid is also binned transversally using shells
- Use of adaptive bin size in momentum in order to compute ellipsoid of equal small momentum spread in the center of mass.
Calculation of space charge fields: divide macro-particles into longitudinal momentum bins, at each time step
Bunch description as a function of propagation distance
z
1 zu 11 zu 22 zu 33 zu
21izi u
Total force acting on one macro-particle
k skk
skkiz
k sk
sk
kix
skki
skki
AaaezzF
AaaexF
332
0
132
0
2)(
21
- Cylindrical symmetry is assumed
- A1 and A3 are the space charge coefficient of a shell
- ak and a3k the rms radii of the ellipsoid “k”- k relativistic factor- sk density of a shell- <z>k average position of the ellipsoid “k”
ak
a3k<zk><zk-1>
ak-1
a3k-1
Scheme of the binning of the electron bunch into shells
S. Chandrasekhar, Ellipsoidal Figures of Equilibrium (Dover Publications, 1969)
Fields from a single shell in rest frame
(1) (2) (3)
Distinction between the force in the interior, inside and outside a single shell
0 r
Adaptive bin size in momentum to compute ellipsoids of equal small momentum spread in the center of mass.
1 cmzu
Note: method close to SCHEFF routine in PARMELA but which can handle large energy spread e-beams
(1) (2)
(2)
(3)
(3)
zF
z
• Good agreement for <p> greater than 1 MeV is this parameter regime.• Difference in rms momentum ~ 2% (2%) for uz (ux) after 3 mm.• Accurate description for standard LINACs.• Fast calculation time: few minutes to propagate over 3 mm.
30%5,m5.2mrad2,m6
MeV5pC001
500
shell
'
Nz
xx
pQ
N
m
mm
z
p
Benchmark with electrostatic PIC code: Good agreement > few MeV
EPIC SASC
N > 50000 500
T ~ 6h (s.n*) ~ 5 min
*s.n=single node
• Difference in rms momentum are 9% (~5%) for uz (ux) after 0.3 mm.• Can be explained by formation of a head to tail density gradient along the propagation axis.• Corrections accounting for the longitudinal density gradient can be added.
30%5,m5.2mrad2,m6
MeV1pC001
500
shell
'
Nz
xx
pQ
N
m
mm
z
p
Benchmark with electrostatic PIC code: Moderate agreement < 1 MeV
Simulation of bunch with large energy spread (initial bunch distribution uniform)
mm30MeV2,:type
MeV1,MeV520,30
m10mrad2,m6
MeV61.0:nC1
50000
minmax
shellbin
'
.ctkTsmlwf
ppNN
zxx
pQ
N
zz
m
mm
z
p
• Highly non linear interaction between particles.• Space charge blowout occurs at a very early stage (50-100 m), after motion is dominantly ballistic.• Low energy tail formation.
Summary of the changes in the beam parameters as a function of propagation distance.
Spot size blows-up rapidly due to the large amount of low energy electrons
Note: no space charge x’(s)=x’0 and (s)=0
MeVuz 1
Short compression region induced by low energy electrons
The bunch density evolution is explored through space charge process
Initial distribution
Final distribution
After 0.3 mm of propagation:
• Beam has developed a low energy tail
• Total energy gain on the order of 1 MeV (self acceleration)
• Emittance growth
Note: approximately 8% of the electrons are going backwards
Low Energy (5 MeV) Colliding Pulse LWFA example: small energy spread & compact electron beam
mm330
%5,m5.2mrad2,m6
MeV25.5pC20
50000
shell
'
ctNz
xx
pQ
N
m
mm
z
p
Blowout is less significant than in the smlwfa regime due to lower charge density and higher average energy
After 3 mm: the 5 MeV low density electron bunch reaches the emittance dominated regime with 20 mrad of divergence and 18.5% of energy spread whereas the 40 MeV e-beam remains mainly unchanged.
Summary: Beam parameters versus propagation distance.
Colliding pulse produced e-beam.
Starting with a uniform density, nonlinear space charge forces generate non-uniformities within the electron bunch and emittance growth.
Electron bunch distribution at the final time step
Electron bunch density of energy after 3 mm of propagation
Conclusion
• A fast relativistic space charge code which can handle beam propagation in free space with arbitrarily large energy spread has been developed.
• We use a 6-D ellipsoidal model for the beam, and decompose the particle distribution into narrow longitudinal momentum bins (~ 5% spread) and into ellipsoidal shells.
• The space charge force is calculated for each shell from analytical expressions in the rest frame of each bin, then transformed to the lab frame and summed.
• The method was implemented to study the evolution of high density beams with arbitrary energy spreads such as self-modulated laser-wakefield generated bunches.
• A head to tail longitudinal density gradient has been observed due to relativistic effects. An analytical correction based on Taylor expansion can be added, bringing simulations to a more accurate description.