1. 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)

2. 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 ~5  m, angular spread ~5-10 mrad, bunch length ~10  m. - Large energy spread: 0.1 to 100 MeV (Boltzmann momentum distribution)

3. Space charge expected to dominate bunch dynamics Electrostatic potential energy per particle of a bunch of N electrons of radius R: For Q=5nC, R=5  m, 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

4. 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.

5. - 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

6. Total force acting on one macro-particle - Cylindrical symmetry is assumed - A 1 and A 3 are the space charge coefficient of a shell - a k and a 3k the rms radii of the ellipsoid “k” -  k relativistic factor -  sk density of a shell - k average position of the ellipsoid “k” akak a 3k a k-1 a 3k-1 Scheme of the binning of the electron bunch into shells S. Chandrasekhar, Ellipsoidal Figures of Equilibrium (Dover Publications, 1969)

7. Fields from a single shell in rest frame (1)(2)(3) Distinction between the force in the interior, inside and outside a single shell 0r Adaptive bin size in momentum to compute ellipsoids of equal small momentum spread in the center of mass. Note: method close to SCHEFF routine in PARMELA but which can handle large energy spread e-beams (1)(2) (3)

8. Good agreement for 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. Benchmark with electrostatic PIC code: Good agreement > few MeV EPICSASC N> 50000 500 T~ 6h (s.n*)~ 5 min *s.n=single node

9. 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. Benchmark with electrostatic PIC code: Moderate agreement < 1 MeV

10. Simulation of bunch with large energy spread (initial bunch distribution uniform) 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.

11. 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 Short compression region induced by low energy electrons

12. 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

13. Low Energy (5 MeV) Colliding Pulse LWFA example: small energy spread & compact electron beam Blowout is less significant than in the smlwfa regime due to lower charge density and higher average energy

14. 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.

15. 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

16. 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.