The Criteria for the Existence of Coulomb Strings in Storage Rings

We derive four approximate conditions for the stability of Coulomb chains in circular storage rings. These criteria are well met by the existing data from experiments in SIS, ESR and Cryring. In the recent years Schottky measurements of the momentum spread δp/p at beams with very low density and extremely electron cooled heavy ions in the storage ring ESR [1] and in the synchroton SIS [2] of GSI have revealed that below a certain threshold in density intrabeam scattering ceases to act. Similar observations have been made in the Cryring [3] of Stockholm. The beam radius below threshold was determined to be smaller than some ten micrometers. The typical jump of the momentum spread has been explained as the ions form a linear string where the particles cannot pass each other any more [4]. Hereby classical Monte-Carlo trajectory calculations were performed with charged particles heading towards each other under constant focusing with the betatron frequency ωβ of the respective ring. As results, the probability of reflection (or transmission) drops (increases) sharply when going to larger interparticle distances. Although this model can explain the given experimental data it cannot yet predict the density and longitudinal, T||, and transverse, T⊥, thermal energies at which this effect shows up. The only hitherto known necessary relation for the existence -but not the stabilityof Coulomb strings, d > 1.4 aWS, (1) stems from the transition from strings to zigzags at λ < 0.709 , where λ = aWS/d is the linear density and aWS is the Wigner-Seitz radius aWS = 3q /2Mω β with q being the charge and M the mass of the ions.