Molecular dynamics is employed to study the nature and magnitude of beam cooling that is required in order to achieve a crystalline beam. Analysis is presented of a number of cooling systems now in use, or whose use is contemplated, none of which has been shown to be able to achieve the crystalline state. However, for an adequately strong cooling system that produces on average a constant angular velocity among the particles, a crystalline state can be achieved. In this paper, we present numerical values for a particular example under typical experimental conditions; these values appear to be achievable in practice. [S0031-9007(98)05644-0] PACS numbers: 52.60. + h, 29.20.Dh, 52.65. -y, 61.50. -fFor the last decade there has been interest in and experimental effort to achieve crystalline beams. These beams are sufficiently cold in the beam rest frame, so that the particles making up the beam "lock into" a position where the repelling Coulomb force just balances the external focusing force. Seen from the laboratory, the whole ordered structure circulates at great speed. The interest, besides intrinsically on this new state of matter, is primarily on the possibility of studying the physics of completely space-charge dominated beams, the possibility of studying Wigner crystal, and the possibility of obtaining high luminosity in colliders.The ground state of a crystalline beam was proposed by Dikanskiȋ and Pestrikov [1] based on an experimental anomaly observed on an electron-cooled proton beam at the storage ring NAP-M, and was first studied using the molecular dynamics (MD) method by Schiffer and coworkers [2]. At the same time, experimental efforts have succeeded in achieving very low beam temperatures, but not yet a crystalline state [3].In a long series of papers we explored the conditions for a crystalline beam ground state in a real storage ring [4][5][6][7]. Despite the large amount of work and the many publications in this field, there has not previously been a careful study of the nature and magnitude of the cooling force required to reach a crystalline state (see the papers in Ref.[6]). We have undertaken such a study and report on the results in this Letter.Particle motion can be described by a Hamiltonian [4,6] in the rest frame ͑x, y, z, t͒ of a circulating reference particle in which the orientation of the axes is rotating so that the axes are constantly aligned to the radial (x), vertical ( y), and tangential (z) direction. Consider in general a system of multispecies of particles under Coulomb interaction and external fields. Define a reference par-ticle with electric charge Z 0 e and atomic mass M 0 , and define for the ith species of particles with charge Z i e and mass M i , Z i ϵ Z i ͞Z 0 , and m i ϵ M i ͞M 0 . Measure dimensions in units of the characteristic distance j with j 3 r 0 r 2 ͞b 2 g 2 , time in units of r͞bgc, and energy in units of b 2 g 2 Z 2 0 e 2 ͞j, where r 0 Z 2 0 e 2 ͞M 0 c 2 is the classical radius, bc and gM 0 c 2 are the velocity and energy of the reference particle, and r is...
