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.