We studied the structural, dynamical properties and melting of a quasi-one-dimensional system of charged particles, interacting through a screened Coulomb potential. The ground state energy was calculated and, depending on the density and the screening length, the system crystallizes in a number of chains. As a function of the density (or the confining potential), the ground state configurations and the structural transitions between them were analyzed both by analytical and Monte Carlo calculations. The system exhibits a rich phase diagram at zero temperature with continuous and discontinuous structural transitions. We calculated the normal modes of the Wigner crystal and the magneto-phonons when an external constant magnetic field B is applied. At finite temperature the melting of the system was studied via Monte Carlo simulations using the modif ied Lindemann criterion (MLC). The melting temperature as a function of the density was obtained for different screening parameters. Reentrant melting as a function of the density was found as well as evidence of directional dependent melting. The single chain regime exhibits anomalous melting temperatures according to the MLC and as a check we study the pair correlation function at different densities and different temperatures, which allowed us to formulate a different criterion. Possible connection with recent theoretical and experimental results are discussed and experiments are proposed.

Schweigert, Schweigert, and Peeters Reply: Rinn and Maass [1] claim that the Brownian dynamics (BD) simulation results of Schweigert et al. [2] are analyzed incorrectly. Furthermore, they claim that the definition for the intershell diffusion coefficient ͑D u ͒ used by Schweigert et al. [2] makes sense only when the particles remain in the same shell.The whole misunderstanding is based on the fact that Rinn et al. [1] believe that one needs to follow the trajectory of each individual particle in order to calculate D u . Within such an approach, one is in trouble when a particle jumps from one shell to another shell. In order to remove this switching of particles between shells they analyzed their data in two different ways: (1) ignoring shell jumps (open symbols in Fig. 2 of Ref. [1]); and (2) taking care of shell jumps (solid symbols in Fig. 2 of Ref. [1]).When Rinn et al. ignored shell jumps they found a very large "unrealistic" reduction of D u with decreasing G , 20. Notice that for G , 20 the diffusion coefficient D u attains values which are even smaller than in the G . 100 region, where the rigid crystal phase sets in. On physical grounds, this makes no sense.In their second approach, Rinn et al. calculated D u by "taking care of shell jumps." It is not clear what they mean with this and how they calculated D u . Did they remove the particles which performed a shell jump from their calculation of D u ? If so, it is not surprising that the results are different from those of Schweigert et al. [2]. As explained in Refs. [2,3], the radial fluctuations (and shell jumps) are essential for the stabilization of the intershell (or angular) diffusion in the reentrant region. The numerical results of Rinn et al. (solid symbols in Fig. 2 of Ref. [1]) saturate for G , 20 which is hard to understand physically.

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The plasma crystal formed by monodisperse particles trapped in the sheath of an rf discharge is known to show vertically aligned structures. Here, oscillations of the aligned particles are found below a threshold value of gas density as a precursor of the melting transition. Attractive forces due to the formation of a positive space-charge region below the upper particle are calculated from Monte Carlo simulations of ion trajectories in the sheath. The alignment as well as the oscillations of the plasma crystal are explained by a simple model based on the asymmetry of the forces. ͓S1063-651X͑96͒50307-7͔ PACS number͑s͒: 52.25. Vy, 52.35.Ϫg, 62.30.ϩd The formation of Wigner crystals in dusty plasmas has attracted much interest very recently. Dust particles immersed in a plasma interact by means of the Coulomb repulsion of the particle's charges acquired by electron and ion currents. Ikezi ͓1͔ theoretically predicted plasma conditions under which these particles should form regular lattices, the so-called plasma crystal. Experimentally these crystals were found by Chu and co-workers ͓2-4͔ in a magnetron rf discharge with trapped discharge-grown SiO 2 particles. Thomas et al. ͓5͔ and Melzer et al. ͓6,7͔ found plasma crystals in parallel plate rf discharges where dust particles intentionally added to the plasma are trapped in the sheath of the lower electrode, where mainly the upward-directed field force balances the gravitational force on the particles. The dust grains arrange in a flat crystal with a diameter of a few hundred interparticle distances and a thickness of up to 20 layers, with usual two-dimensional ͑2D͒ hexagonal order in the plane. In the vertical direction the particles are found to be aligned ͓4,7,8͔.In this paper we show that the alignment can be explained by nonreciprocal attractive forces on the particles due to ion streaming motion. These forces overcome the dust Coulomb repulsion. They are also responsible for the onset of particle oscillations about the aligned positions, which are compared with experimental findings.The measurements were performed in a parallel plate rf discharge at 13.56 MHz and a power input of 12 W with the lower electrode powered and the upper grounded. The discharge was operated in helium at pressures ranging from 30 to 150 Pa. Monodisperse spherical melamine/formaldehyde particles of 4.8 m and 9.4 m diameter were added to the plasma. The choice of different particle sizes provides a change of the gravitational force by a factor of about 8. The dust crystal is illuminated by a vertical or horizontal laser fan and is observed in scattered light with a video camera. The experimental setup has been decribed in detail in ͓7͔. The charge on the dust particles is determined from the resonance frequency in the potential well formed by the gravitational and electrical forces ͓6-8͔. The measured charge is Z Ϫ ϭ15 000 elementary charges corresponding to a surface potential of ϭϪ5 V for the 9.4-m particles (Z Ϫ ϭ3600,ϭϪ2.2 V for the 4.8-m particles͒. Figure 1 shows a side view of...