Stability of 1D-ordered beams

Beam and storage ring parameters have to meet a number of stringent conditions before circulating three-dimensional crystalline beams can be obtained. Existing storage rings violate at least one of these conditions. The remarkable exception is the very low-energy ring PALLAS with 'radio frequency focusing', which apparently abides by the full set of conditions, and where 3D-ordering has indeed been realized. However 1D-ordering, - particles lining up in a 'string of non-overtaking links' - has been observed in several storage rings with electron cooling, even though the above conditions on the classical plasma parameter as well as on the lattice periodicity are largely violated. In this paper, we discuss conditions for 1D- ordering, including space-charge limits, instabilities and cooling requirements. We conclude that densities of some 10**4 ions/m may be possible; about 100 times more than those obtained in present experiments. Such dense strings are of practical interest for precision measurements of ion properties with single and with colliding beams.