Dimensional phase transition in small Yukawa clusters

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 clusters 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.

Figure 1

Schematic of the experimental setup. Nearly identical spherical dust particles are confined in a biharmonic potential well created in the rectangular depression between four conducting bars placed on the rf powered electrode. Experiments were performed for confinement geometries with bar separations $d=25.4\text{mm}$ and $d=14.0\text{mm}$.

Figure 2

[(a)–(g)] Measured particle positions for confining bar separation $d=25.4\text{mm}$. [(h)–(n)] Computed positions for ${\alpha }^2=9.24$ and $\kappa =3$. By matching $y_{rms}$ between the experiment and model, we find $r_0\approx 1.40\text{mm}$. Agreement between measured and computed configurations is excellent. For $n\le 5$ the configurations are linear, at $n=6$ a zigzag develops and for $n=19$ a fully elliptical cluster with a well-defined shell structure is seen. Both measured and computed figures have a 1:1 aspect ratio.

Figure 3

[(a)–(g)] Measured particle positions for a confining well with $d=14.0\text{mm}$. [(h)–(n)] Computed positions for ${\alpha }^2=30.7$ and $\kappa =4$. We estimate $r_0\approx 1.65\text{mm}$ by comparing $y_{rms}$ for the model and experiment. For $n\le 9$ the configurations are linear, at $n=10$ a zigzag develops. For $n=17$ the cluster remains in a zigzag configuration. Both measured and computed figures have a 1:1 aspect ratio.

Figure 4

Measured center-of-mass frequencies and longitudinal breathing frequencies vs particle number $n$ determined from thermally excited oscillations in (a) the $25.4\times 50.8{\text{mm}}^2$ confining well and (b) the $14.0\times 50.8{\text{mm}}^2$ well. Broken lines are average values, and the solid line in (b) is a linear fit to ${\omega }_{br}$.

Figure 5

Critical lattice parameter $a_c/r_{0T}$ for an unbounded straight chain [Eq. (11)] vs the transverse Debye shielding parameter ${\kappa }_T$. The data points are the experimentally measured values for the last straight configuration for $d=25.4\text{mm}$ (circle), and $d=14.0\text{mm}$ (diamond). The measured points lie close to, but above, the stability curve in the stable region.

Figure 6

Dependence of cluster width $y_{rms}$ on particle number $n$ comparing experiment and model. (a) Experimental data for bar separation $d=25.4\text{mm}$ and model solutions with ${\alpha }^2=9.24$ and $\kappa =3$ scaled using $r_0=1.65\text{mm}$. (b) Experimental data for bar separation $d=14.0\text{mm}$ and model solutions with ${\alpha }^2=30.7$ and $\kappa =4$ scaled using $r_0=1.40\text{mm}$. The dashed lines are power-law fits to the model points.

Figure 7

Cluster width $y_{rms}/r_0$ vs particle number $n$ for Debye shielding parameters $\kappa =0$, 1 and 4 with well anisotropy ${\alpha }^2=30.7$. The critical value of $n$ at which the zigzag transition occurs decreases as $\kappa$ increases. Solid lines are power-law curves fitted to the first five points following the zigzag transition.

Figure 8

Computed cluster width and length for anisotropy parameter ${\alpha }^2=9.2$ for $n=5$ and (inset) $n=6$ particles vs the Debye shielding parameter $\kappa$. Solid lines show the power-law fit to Eq. (13), while the dashed line has been added to guide the eyes. For $n=5$ particles there is a critical value ${\kappa }_c=4.22$ below which the cluster is one-dimensional and above which it is two-dimensional. For $n=6$, ${\kappa }_c=0.45$.

Figure 9

Computed cluster width and length for $n=5$ with shielding parameter $\kappa =3$ vs anisotropy parameter ${\alpha }^2$. A 2D-1D phase transition is with a critical value ${\alpha }_c^2=8.74$. The fitted power law (solid line) gives a critical exponent $\gamma =0.387$.