Confinement and stability of crystalline beams in storage rings

We present a fully analytical approach to the study of tile confinement and stability of ''Crystalline Beams'' in storage rings, in terms of such fundamental accelerator concepts as tune shift and stopband. We consider a ''Crystalline Beam'' consisting of substrings, arranged symmetrically around the reference trajectory, and we examine the motion of a slightly perturbed test particle on one of them. Our approach quite naturally leads to the conclusion, that (a) storage rings need to be operated below the transition energy, and (b) the ''Crystalline Beam'' has the same periodicity as the storage ring. Each ''Crystalline Beam'' has an upper and lower limit of the spacing between the ions. The upper limit is determined by condition (b), and the lower limit is set by the stability of the test particle motion around the equilibrium