The filamented electron bunch of the bubble regime

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.