We derive dispersion relations for a system of identical particles confined in a two-dimensional annular harmonic well and which interact through a Yukawa potential, e.g., a dusty plasma ring. When the particles are in a single chain (i.e., a one-dim
ensional ring) we find a longitudinal acoustic mode and a transverse optical mode which show approximate agreement with the dispersion relation for a straight configuration for large radii of the ring. When the radius decreases, the dispersion relations modify: there appears an anticrossing of the modes near the crossing point resulting in a frequency gap between the lower and upper branches of the modified dispersion relations. For the double chain (i.e., a two-dimensional zigzag configuration) the dispersion relation has four branches: longitudinal acoustic and optical and transverse acoustic and optical.
We develop a stochastic model for the charge fluctuations on a microscopic dust particle resting on a surface exposed to plasma. We find in steady state that the fluctuations are normally distributed with a standard deviation that is proportional to
$CT_{e})^{1/2}$, where $C$ is the particle-surface capacitance and $T_{e}$ is the plasma electron temperature. The time for an initially uncharged ensemble of particles to reach the steady state distribution is directly proportional to $CT_{e}$.
One-dimensional and quasi-one-dimensional strongly-coupled dusty plasma rings have been created experimentally. Longitudinal (acoustic) and transverse (optical) dispersion relations for the 1-ring were measured and found to be in very good agreement
with the theory for an unbounded straight chain of particles interacting through a Yukawa (i.e., screened Coulomb or Debye-Huckel) potential. These rings provide a new system in which to study one-dimensional and quasi-one-dimensional physics.
We provide direct experimental evidence that the one-dimensional (1D) to two-dimensional (2D) zigzag transition in a Yukawa cluster exhibits power law behavior. Configurations of a six-particle dusty (complex) plasma confined in a biharmonic potentia
l well are characterized as the well anisotropy is reduced. When the anisotropy is large the particles are in a 1D straight line configuration. As the anisotropy is decreased the cluster undergoes a zigzag transition to a 2D configuration. The measured dependence of cluster width on anisotropy is well described by a power law. A second transition from the zigzag to an elliptical configuration is also observed. The results are in very good agreement with a model for particles interacting through a Yukawa potential.
We investigate the one- to two-dimensional zigzag transition in clusters consisting of a small number of particles interacting through a Yukawa (Debye) potential and confined in a two-dimensional biharmonic potential well. Dusty (complex) plasma clus
ters with $n le 19$ monodisperse particles are characterized experimentally for two different confining wells. The well anisotropy is accurately measured, and the Debye shielding parameter is determined from the longitudinal breathing frequency. Debye shielding is shown to be important. A model for this system is used to predict equilibrium particle configurations. The experiment and model exhibit excellent agreement. The critical value of $n$ for the zigzag transition is found to be less than that predicted for an unshielded Coulomb interaction. The zigzag transition is shown to behave as a continuous phase transition from a one-dimensional to a two-dimensional state, where the state variables are the number of particles, the well anisotropy and the Debye shielding parameter. A universal critical exponent for the zigzag transition is identified for transitions caused by varying the Debye shielding parameter.
