Resolving structural transitions in spherical dust clusters

Finite systems in confining potentials are known to undergo structural transitions similar to phase transitions. However, these systems are inhomogeneous, and their “melting” point may depend on the position in the trap and vary with the particle number. Focusing on three-dimensional Coulomb systems in a harmonic trap a rich physics is revealed: in addition to radial melting we demonstrate the existence of intrashell disordering and intershell angular melting. Our analysis takes advantage of a novel melting criterion that is based on the spatial two- and three-particle distribution functions and the associated reduced entropy which can be directly measured in complex plasma experiments.

Figure 1

Macrosopic 2D Yukawa OCP, $\kappa =1,N=2000$: (a) TCF in the crystal phase. (b) TCF in the strongly correlated liquid regime. The length scale is $a$. (c) Specific heat $c_V$ and reduced entropies of TCF and PDF [Eq. (13)]. Inset: Sketch of TCF parameters for a 2D system. In (a) and (b) the TCF is averaged over a finite $d_{\text{I}}$ range (see bars). In (c) no averaging is applied to $S^{\left(3\right)}$. In (a) and (b) the pair distances are in units of $a$.

Figure 2

Sketch of a spherical 3D cluster with two shells; particles on inner and outer shell are drawn in red and blue, respectively. Intrashell Voronoi diagrams (nearest neighbors of enclosed particles) are sketched in gray. In the C2P, the radii $r_{\text{I}}$ and $r_{\text{II}}$ as well as the pair angle $\vartheta$ are recorded (right). In the TCF three particles are chosen from the same shell (left), and two angular pair distances ${\vartheta }_{\text{I}},{\vartheta }_{\text{II}}$ and the bond angle $\phi$ are recorded.

Figure 3

Spherical 3D Coulomb cluster with $N=80$ particles on two shells. Left (right) column: $\Gamma =1000 \left(\Gamma =100\right)$. (a, d) C2P for a reference particle from the inner shell [selected by the $r_{\text{I}}$ integration (bar)]. (b, e) [(c, f)] TCF on the inner [outer] shell. The first particle pair is selected as nearest neighbors by the ${\vartheta }_{\text{I}}$ integration range (bar). The length scale is $r_0$.

Figure 4

(a) Occurrence of different states of the Coulomb cluster with $N=80$ particles. The states are divided into configurations with $N_1=19$ (solid red line, above the horizontal 0-line) particles or less (dashed) on the inner shell and those with $N=20$ (solid blue line, below the 0-line) or more (dashed). The fraction of the former (latter) configurations is plotted above (below) the zero line. The intrashell order is analyzed by the means of the Voronoi construction within the shell, which allows us to determine intrashell nearest neighbors. An intrashell configuration is classified by the number of bonds of each class and the eight most important configurations plotted as filled curves. The fraction of additional less important configurations with $N_1=19$ or 20 is represented by the shaded area. (b) Specific heat capacity and “capacity” $c_{\omega }^{\left(3\right)}$ from the TCF on the inner shell. The two intrashell disordering $\Gamma$ regions are indicated by vertical dashed lines.

Figure 5

(a) Reduced entropies for the cluster of Fig. 3 vs $\Gamma$ computed for the C2P (solid line), the radial density (dash-dot), and for the TCF on the inner (dashed) and outer shell (dotted). (b) Reduced specific heat (14) for the entropies from (a) compared to the specific heat. RM, radial melting; ID1 (ID2), intrashell disordering on inner (outer) shell.

Figure 6

Intershell angular melting (ISM) in the spherical 3D Coulomb cluster with $N=38$ particles. (a, b) C2P (the first particle radius is averaged over the inner shell) below (above) the melting temperature. The length scale is $r_0$. (c) Specific heat and reduced specific heat vs $\Gamma$. While radial melting (RM) is seen in $c_{\omega }$ and $c_{\omega }^{\left(2\right)}$, ISM is clearly visible only in $c_{\omega }^{\left(2\right)}$, based on the C2P.

Figure 7

Intrashell disordering transition (ID) in the outer shell of a spherical Coulomb cluster with $N=120$ particles around $\Gamma =50000$. (a) The particle number on the outer shell is constant for $\Gamma \gtrsim 400$ while transitions between different intrashell isomers take place at higher $\Gamma$. The intrashell Voronoi analysis allows for a classification of these isomers. (b) While the thermodynamic heat capacity $c_{\omega }$ captures only the radial melting (RM) at $\Gamma \approx 165$, the reduced heat capacity $c_{\omega }^{\left(3\right)}$ computed from the entropy of the TCF captures also the intrashell disordering.

Figure 8

Comparison of reduced heat capacities, $c_{\omega }^{\left(k\right)}$, and angular distance fluctuations (ADF), $\Delta \alpha$, Eq. (23), for the magic number Coulomb cluster with $N=38$ particles. (a) Intrashell disordering on the inner shell, measured by $c_{\omega }^{\left(3\right)1}$ from the TCF and by $\Delta {\alpha }_{11}$ for pairs of two particles within the inner shell. (b) The same for the outer shell. (c) Intershell angular disordering, measured by $c_{\omega }^{\left(2\right)}$ from the C2P and by $\Delta {\alpha }_{12}$, for pairs of particles belonging to different shells. While the (reduced) heat capacity $c_{\omega } \left(c_{\omega }^{\left(k\right)}\right)$ was obtained in an MC simulation with parallel tempering, the fluctuations $\Delta \alpha$ were obtained as averages over 40 Langevin molecular dynamics (LMD) simulation runs for each temperature. In panels (a) and (b) we also compare to data by Apolinario et al. in Fig. 5 of Ref. [35]. (The temperature was scaled by $\sqrt[3]{0.5}$ because of the different energy unit used in that reference.) Note the different temperature scale in panel (c).