Two-dimensional structures of electron bunches in relativistic plasma cavities

The spatial structure of an ultralow-emittance electron bunch in a plasma wakefield blowout regime is studied. The full Liénard-Wiechert potentials are considered for mutual interparticle interactions in the framework of the equilibrium slice model. This model uses the quasistatic theory which allows one to solve the Liénard-Wiechert potentials without knowledge of the electrons' history. The equilibrium structure we find is similar to already observed hexagonal lattices but shows topological defects. Scaling laws for interparticle distances are obtained from numerical simulations and analytical estimations.

Figure 1

Schematic depiction of (a) two alongside-propagating particles and (b) two particles propagating one behind the other in the same direction.

Figure 2

Schematic depiction of the random distribution on a circular disk inside the bubble.

Figure 3

Final distribution of a simulation with a momentum of $p=125$ MeV/$c$ and $n=1000$ electrons with ${\lambda }_{\text{pe}}=0.01$ cm.

Figure 4

Dependence of the mean particle distance for different momenta for $n=100$, 1000, and 4000 electrons with ${\lambda }_{\text{pe}}=0.01$ cm.

Figure 5

Dependence of the mean particle distance for different plasma wavelengths. The simulations were done at 125 MeV/$c$ for $n=100$, 1000, and 4000 electrons.

Figure 6

Dependence of the mean particle distance for a different number of electrons and a constant momentum of 125 MeV/$c$ with ${\lambda }_{\text{pe}}=0.01$ cm.

Figure 7

Topological defects in a lattice consisting of 500 electrons. Marked as red triangles are electrons with a topological charge of $-1$, and marked as blue squares are particles with a topological charge of $+1$.

Figure 8

Depiction of the number of electrons in dependence of the observed radius of the final distribution for different momenta. The dashed line depicts the reference curve of a constant surface density.