Exact Relativistic Kinetic Theory of an Electron-Beam–Plasma System: Hierarchy of the Competing Modes in the System-Parameter Space

The stability analysis of an electron-beam–plasma system is of critical relevance in many areas of physics. Surprisingly, decades of extensive investigation have not yet resulted in a realistic unified picture of the multidimensional unstable spectrum within a fully relativistic and kinetic framework. All attempts made so far in this direction were indeed restricted to simplistic distribution functions and/or did not aim at a complete mapping of the beam-plasma parameter space. The present Letter comprehensively tackles this problem by implementing an exact linear model. Three kinds of modes compete in the linear phase, which can be classified according to the direction of their wave number with respect to the beam. We determine their respective domain of preponderance in a three-dimensional parameter space and support our results with multidimensional particle-in-cell simulations.

Figure 1

Normalized growth rate in the ($k_x$, $k_y$) plane for $n_b/n_p=1$, ${\gamma }_b=1.2$, $T_b=500\mathrm{keV}$ and $T_p=5\mathrm{keV}$. Isocontours are linearly spaced from 0.01 to 0.07.

Figure 2

Left: Hierarchy of the unstable modes in the ($n_b/n_p$, ${\gamma }_b$, $T_b$) parameter space for $T_p=5\mathrm{keV}$. The left surface delimits the two-stream-dominated domain (at low ${\gamma }_b$) and the oblique-mode-dominated domain, whereas the right surface delimits the filamentation-dominated domain (at high $n_b/n_p$) and the oblique-mode-dominated domain. Right: Plasma density profiles at the end of the linear phase as predicted by 2D PIC simulations run with three different sets of parameters: $n_b/n_p=0.1$, ${\gamma }_b=1.5$, and $T_b=500\mathrm{keV}$ (b); $n_b/n_p=1$, ${\gamma }_b=1.5$, and $T_b=100\mathrm{keV}$ (c); $n_b/n_p=1$, ${\gamma }_b=1.5$, and $T_b=2\mathrm{MeV}$ (d). In all cases, $T_p=5\mathrm{keV}$. In agreement with linear theory, the three resulting patterns evidence regimes dominated by two-stream, filamentation, and oblique modes, respectively.

Figure 3

Typical evolution of the fastest growing mode (arrow) for the oblique to two-stream (a) and oblique to filamentation (b) transitions, of which only the former is continuous.

Figure 4

3D PIC simulation initialized with $n_b/n_p=0.1$, ${\gamma }_b=3$, $T_b=50\mathrm{keV}$, and $T_p=5\mathrm{keV}$: isosurfaces of the beam and plasma density profiles at ${\omega }_{e}t=80$, 160, and 560. The system runs through three successive phases, each governed by a distinct instability class.