Zigzag transition of finite dust clusters

Experiments on the zigzag transition of finite dust clusters are presented. There, microspheres have been confined in an anisotropic trap in the sheath of a radio-frequency discharge plasma. Transitions of the clusters from a linear chain into a zigzag configuration are observed by increasing the particle number in the chain or by variation of the anisotropy of the confinement. The equilibrium configurations of the finite clusters, as well as the dynamical properties and the stability behavior near the transition have been investigated from a normal mode analysis. Furthermore, the longitudinal and transverse dispersion of the normal modes in the one-dimensional (1D) chain has been derived.

Figure 1

(a) Scheme of the experimental setup. (b) Cross section through the barrier: Illustration of the equipotential lines and cluster height above the confining barrier.

Figure 2

Snapshots of the dust cluster for $N=4$, 9, 10, and 18. The zigzag transition is observed for $N=10$ particles. The size of each image corresponds to $2.8\times 9.1{\mathrm{mm}}^2$.

Figure 3

Snapshots of the dust cluster with $N=13$ for $p=4.0$, 6.0, 6.8, and $7.5\mathrm{Pa}$. The zigzag configuration is observed for $p<7.5\mathrm{Pa}$. The size of each image corresponds to $2.8\times 9.1{\mathrm{mm}}^2$.

Figure 4

Critical anisotropy parameter $\beta$ as a function of particle number. The horizontal gray bars are the values of $\beta$ determined from the experiment with particle number variation for different $\kappa$ (compare Table ). The height of the bars indicate the rms errors of the experiment. The vertical gray bar indicates the region from $N=9$ to 10, where the zigzag transition occurs. The circles denote the $\beta$-values for $N=13$ of the experiment with gas pressure variation. The critical $\beta$ is taken at the highest pressure where the zigzag pattern still is observed, i.e., at $p=6.8\mathrm{Pa}$. The solid and dashed line show the critical $\beta$ from the models of Dubin and O’Neil [1] and Candido et al. [17], respectively.

Figure 5

Frequency of the lowest eigenmodes during pressure variation. The oscillation pattern of the corresponding eigenmodes is indicated in the insets. Below $7.5\mathrm{Pa}$ the cluster is in the zigzag state, above that pressure it forms a 1D chain.

Figure 6

Gray-scale plot of the spectral power density of the lowest perpendicular mode as a function of particle number in the cluster.

Figure 7

Gray-scale plot of the spectral power density as a function of wave vector $ka$ for the longitudinal (upper panel) and transverse (lower panel) modes. The solid line indicates the dispersion relation of the longitudinal and transverse wave in a linear chain according to Eq. (7) for $\kappa =0$. The high spectral activity at a frequency of $8\mathrm{Hz}$ in the transverse modes, as indicated by the arrow, is due to unwanted oscillations of the camera holder.