Static Criteria for the Existence of Coulomb Strings in Storage Rings

We derive four rigorous conditions for the stability of Coulomb strings in circular storage rings. These criteria are well met by the existing data from experiments in SIS, ESR, and CRYring but not by the NAP-M experiment. We calculate the potential of the joint transverse zigzag excitation and the longitudinal motion against each other of a string of charged particles as a function of their amplitudes and with the linear density as parameter. This potential exhibits a saddle point in amplitude space which, if overcome, destroys the order of the string. The conditions of stability are derived from the position and height of the saddle point which are fairly independent of the linear density. Our findings confirm the supposition that only the Coulomb interaction in the immediate vicinity of very close encounters of particles is important for the existence of strings.