Self-consistent time dependent two dimensional and three dimensional space charge distributions with linear force

Finding self-consistent distributions of beam particles interacting with each other via the space charge force is one of the challenges of accelerator physics. Exactly solvable models are used for simulation benchmarks, instability threshold calculations, etc. Since such distributions have been found only in one and two dimensions (Kapchinsky-Vladimirsky distribution), it is not possible to apply them to a general three dimensional motion. This paper shows how to construct new sets of self-consistent distributions, extending even to the three dimensional case.

Figure 1

Distribution with linear force. Arrows and circles show particle velocities and trajectories, respectively. The filled area shows the region of uniformly distributed particles.

Figure 2

Results of painting with linearly increasing emittance, as indicated by linearly decreasing emittance distributions, for (red distribution) equal tunes and same horizontal and vertical betatron phases, (green distribution) unequal tunes and 90° betatron phase separation, and (blue distribution) equal tunes and 90° betatron phase separation.