One-dimensional ordering of ultra-low-density ion beams in a storage ring

The two-particle model, first introduced by Hasse, is employed to predict the beam temperature at which a one-dimensional ordered state of ions will be established in a cooler storage ring. The proposed state does not have the ions (in the beam frame) at rest, but simply has them not passing each other; i.e., remaining in the same (ordered) sequence. The model is applicable to an ultra-low-density beam where collective Coulomb interactions are negligible. It is pointed out that the nature of the anomalous beam behavior observed in electron-cooling experiments at GSI (Darmstadt) and MSL (Stockholm) is approximately free from such parameters as the lattice design, ion species, beam density, and energy. On the basis of the model, which is put in Hamiltonian form, scaled, and numerically studied, a universal criterion of one-dimensional beam ordering at low line density is derived. Analytic work is employed to explain the numerical results and derives an approximate criterion.