Static Criteria for the Existence of Coulomb Strings in Storage Rings

Abstract

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.

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

“…To our knowledge, there are two separate theories developed for the 1D ordering effect [5,6,12]. As mentioned already, Hasse performed Monte Carlo simulations based on a "two-particle model," successfully reproducing the ESR outputs [5].…”

“…The model predicts the temperature below which ordering occurs, but not the density below which ordering can be achieved. A collective theory was constructed later by him, to predict the linear density and beam temperature at which the anomaly shows up [6]. Examining the excitation and stability of the collective "zigzag" mode, Hasse eventually proposed four criteria of 1D beam ordering: There is, however, the choice of "distance" in the evaluation of Coulomb energy, and different values of ⌫ (see below for two more) correspond to different choices for this distance.…”

“…[5] to give a simple and universal criterion of 1D ordering. The present model is valid for small linear density and has thus, to first order, nothing to do with Coulomb crystals and their stability where the collective space-charge interactions play a definitive role [6][7][8]. This assumption seems quite reasonable for the analysis of the present experiments, considering that the jump of the momentum spread takes place when the total number of stored ions reaches on the order of 1000 at the ESR, i.e., at an average interparticle distance of about 10 cm.…”

“…According to his theory, what had been established in these storage rings is a one-dimensional (1D) stringlike order; namely, individual ions still execute large transverse oscillations but no longer pass each other longitudinally. He then explored the collective motion of a Coulomb string to describe the fundamental mechanism behind the ESR and CRYRING observations [6].…”

Fermions with long and finite range interactions on a quantum ring

“…Quantum rings [20] and their generalisation to spheriums [21] are another example of idealised systems which can provide valuable insights into the physics of complex interacting many-body systems. A continuum quantum ring is a system of particle wave-functions constrained to be on a ring of radius R. Experimentally, such systems interacting via the Coulomb interaction can be realised, e.g., in circular storage rings with heavy ion beams [22,23], and can be studied with Coulomb string models [24]. One attractive aspect of quantum rings is that they can often be solved exactly, providing a benchmark to test various approximations to the quantum many-body problem.…”