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During recent years, significant progress has been made in understanding the formation of crystalline beams in circular accelerators and storage rings. No crystalline beam exists in a constant gradient storage ring, but in an alternating gradient (AG) ring crystalline beams exist at all densities as long as the beam energy is smaller than the transition energy of the machine. The ground-state structure depends upon the machine lattice property, the beam energy, and the beam intensity (line density). At low density, the ground state is a one-dimensional (1D) chain with particles equally spaced along the azimuthal axis. As the density increases, the ground state first becomes a 2D zig zag in the plane of weaker transverse focusing, and then becomes 3D single- and multi-shell helices. The maximum spatial density of the crystal, on the other hand, is determined only by the machine lattice property and beam energy, independent of the beam intensity. Employing a previously developed formalism, the authors have performed ground-state and melting calculations of the expected crystalline beams in ion storage rings ASTRID and TSR

Crystalline beam properties as predicted for the storage rings ASTRID and TSR