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 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 two-dimensional structures are in agreement with the equilibrium slice model (ESM).

中文翻译：

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气泡区的丝状电子束

我们提出了一种理论，用于描述气泡区域中电子束的内部结构，从气泡内部电子的随机分布开始，然后使系统能量最小化。因此，我们在传播方向上发现了一种类似细丝的结构，它被由更多电子组成的各种壳包围。如果我们指定 2D 初始结构，我们会观察到大量粒子的六边形结构，对应于二维球体的紧密堆积。二维结构与平衡切片模型（ESM）一致。