Breaking the Azimuthal Symmetry- Jumping off-axis or Staying Away from the Axis?

We performed experiments with electrons in a Malmberg-Penning trap, as well as 2D fluid simulations, and found conditions that lead to off-axis equilibrium states. We found that unstable initial distributions whose average distance from the axis is larger than about half the wall radius end up in off-axis states. These states consist of a small strong vortex and a large diffuse background, each having about half of the initial charge. We present a simple model showing that for small initial distributions, on-axis solutions maximize the background radius and thus the entropy. For extended initial distributions, no on-axis solutions are available and staying off-axis is necessary for the conservation of angular momentum. GOING OFF-AXIS Several theoretical studies discussed the possibility of off-axis equilibrium states of a cylindrically bounded guiding-center plasma [1-3]. Those solutions were obtained in the frame of statistical theories, based on the assumption of ergodicity, constrained by conservation laws. Those studies were not followed by experimental verification. We demonstrate that such off-axis solution can indeed be obtained, but under different conditions than those implied by previous studies. We also give a simple model, examining a limited phase space, that explains the behavior we observed, and emphasize the importance of the conservation of angular momentum as the reason for “going off-axis”. Here we show that sufficiently extended, unstable initial distributions of electrons in a Malmberg-Penning trap can end up in off-axis states. We corroborate our experiments with 2D fluid simulations showing the same behavior. The transition between initial distributions that lead to on-axis and off-axis solutions is quite sharp: the average distance of the electrons from the trap axis needs to be about half the wall radius, rw, for off-axis solutions to be obtained. The exact geometry of the initial distribution is not crucial: we obtained the same behavior whether we started with rings, bars (elongated rectangles) or tri-bars (three rectangles merging in the center of the trap). All the initial distributions had on-axis center-of-mass, and led to off-axis solutions when they were sufficiently extended. The electron experiments were performed with the photo-cathode machine [4]. With this machine, various initial distributions of electrons could be easily obtained. The time evolution is studied with repeated experiments, damping the electrons on a phosphor screen after varying hold times. Figure 1 shows an example for a typical time evolution, starting with a large ring of electrons. The ring breaks due to diocotron instability into discrete “vortices”. After a short period of mixing and merging these vortices end up in a single strong vortex, and an extended “background”. In the case of Fig. 1, this background reached the trap wall and has also a “hole” in it. The smaller coherent vortex, with density comparable to the initial one, is shifted off-axis. So is, although to a lesser degree, the center of mass of the total distribution of the electrons. All this happens on a short time scale (30 ms) over which it can be assumed that 2D fluid-like physics is sufficient to understand the phenomena. Over a much longer time scale (hundreds of ms) the center of mass slowly approaches the axis of the trap. Figure 2 shows an example for a time evolution, with a smaller ring of electrons as the initial distribution. Qualitatively, the evolution is similar to that of the larger distribution: the ring breaks into discrete “vortices”, the vortices merge and eventually end up in a single strong vortex with an extended “background”. Here, however, the smaller coherent vortex is on-axis. The background is more uniform than in Fig. 1 and does not reach the wall (for even smaller rings, we find backgrounds with smaller radius in the final state). FIGURE 1. Going off-axis: with a large ring of electrons as the initial condition. 0 20 40 60 80 100 100 200 300 400 500 600 100