Theory of longitudinal Schottky spectra of ordered ion beams in a storage ring

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Abstract
The longitudinal Schottky spectrum of an ultracold ion beam in a storage ring is calculated in the frame of harmonic oscillations of a one-dimensional (1D) Coulomb lattice. It is assumed that the extremely high cooling rate required to bring the beam into a one-dimensional ordered chain can be provided by electron or laser cooling. The anharmonic transversal oscillations with temperature much higher than the longitudinal one are taken into account within the self-consistent phonon approximation. The Schottky spectrum measured by the pickup system consists of bands located near the harmonics of the revolution frequency of the beam. The total intensity of each band characterizes the temperature distribution of the phonons in an ionic chain. The amplitude and the width of the peaks in the spectrum are a function of the strength of relaxation processes (cooling and heating) and the Coulomb correlations, respectively. It is suggested that a careful analysis of these spectra could be a signature of the presence of 1D ordering in the beam.