Instantaneous Normal Mode Analysis of Melting of Finite Dust Clusters

The experimental melting transition of finite two-dimensional dust clusters in a dusty plasma is analyzed using the method of instantaneous normal modes. In the experiment, dust clusters are heated in a thermodynamic equilibrium from a solid to a liquid state using a four-axis laser manipulation system. The fluid properties of the dust cluster, such as the diffusion constant, are measured from the instantaneous normal mode analysis. Thereby, the phase transition of these finite clusters is approached from the liquid phase. From the diffusion constants, unique melting temperatures have been assigned to dust clusters of various sizes that very well reflect their dynamical stability properties.

Figure 1

Density of states $\rho$ of the dust cluster with $N=19$ at three different temperatures achieved by laser heating. The different curves are vertically shifted by 0.5 for clarity. As usual, the unstable part ${\rho }_u(\omega )$ with the imaginary eigenfrequencies is plotted as ${\rho }_u(|\omega |)$ on the negative frequency axis. Corresponding trajectories of the cluster particles over 250 s are also shown.

Figure 2

Diffusion constant $D$ as a function of cluster temperature for both the $N=19$ (circles) and $N=20$ (squares) particle clusters calculated from INM. The symbol size approximately corresponds to the errors in $D$ (for fixed $c$). The dashed lines indicate a linear fit to the INM data to determine the melting temperature. The inset shows the ratio ${\rho }_u(|\omega |)/{\rho }_s(\omega )$ as a function of ${\omega }^2$ for $N=19$ together with the best exponential fit.

Figure 3

Melting temperature (circles) as a function of cluster size, i.e., particle number $N$. For comparison, the frequency of the lowest frequency mode is shown (squares, data from 25).