The filamented electron bunch of the bubble regime

Abstract

<jats:title>Abstract</jats:title><jats:p>We present a theory for describing the inner structure of the electron bunch in the bubble regime starting from a random distribution of electrons inside the bubble and subsequently minimizing the system's energy. Consequently, we find a filament-like structure in the direction of propagation that is surrounded by various shells consisting of further electrons. If we specify a two-dimensional (2D) initial structure, we observe a hexagonal structure for a high number of particles, corresponding to the close packing of spheres in two dimensions. The 2D structures are in agreement with the equilibrium slice model.</jats:p>

Finite-emittance Wigner crystals in the bubble regime

“…, N } and given plasma wavelength λ pe . This corresponds to finding a force balance between the external F ext resulting from the bubble potential (given via the quasi-static 3D bubble model for electron acceleration in homogeneous plasma of [23]) and the electron-electron interaction for every single electron F C (as shown in [20]):…”

“…In contrast to the previous publications [18,20,21], we now consider finite transverse emittance of the electrons. The normalized emittance can be defined as…”

“…For an ultra-low emittance of p ⊥ ≈ 10 −5 p (frame (a)), we see the previously observed central filament that is split into two halves. The splitting is due to the choice of longitudinal momentum p and plasma wavelength λ pe (which is discussed further in [20]). If we now increase the transverse momentum to p ⊥ ≈ 10 −4 p , we see that the structure begins to get corrugated and widened as we expected from our look into the structure of the electromagnetic fields.…”

“…For a large enough number of electrons different shells could already be seen in [20]. In this case, the outer shells would press the inner ones together, thus suppressing some effects of the finite emittance that would otherwise lead to breaking.…”

“…It was shown that smaller inter-particle distances compared to [17] are to be expected for the equilibrium structures. In [20,21] the 3D structure was examined using a Lorentz transformation which showed electron filaments in the direction of propagation and a formation of various shells around the central filament. These structures were analogous to those observed in typical ring accelerators, however with smaller inter-particle distances in the sub-nanometer regime [22].…”