Results

Figures 89 and 90 show the electron phase-space distributions evaluated at equally spaced times for a two-stream instability calculation performed with $2\times 10^4$ electrons. It can be seen that the distribution initially takes the form of two uniform bands, corresponding to two counter-streaming electron beams. However, as time progresses, the bands spontaneously develop structure which grows in magnitude and eventually converts the phase-space distribution into a set of connected vortices. In this final state, the electrons are basically bouncing backwards and forwards in a quasi-periodic electric potential generated by non-uniformities in the electron density. In other words, the instability effectively destroys the two beams. For this reason, the two-stream instability is of major concern in particle accelerators, which often consist of counter-propagating charged particle beams.

Figure 90: The electron phase-space distribution evaluated at various times for a 1-dimensional two-stream instability calculation performed with $N=20000$, $J=1000$, $L=100$, $v_b = 3$, and $\delta t = 0.1$.