Mode couplings and resonance instabilities in dust clusters

The normal modes for three to seven particle two-dimensional (2D) dust clusters in a complex plasma are investigated using an $N$-body simulation. The ion wakefield downstream of each particle is shown to induce coupling between horizontal and vertical modes. The rules of mode coupling are investigated by classifying the mode eigenvectors employing the Bessel and trigonometric functions indexed by order integers ($m$, $n$). It is shown that coupling only occurs between two modes with the same $m$ and that horizontal modes having a higher shear contribution exhibit weaker coupling. Three types of resonances are shown to occur when two coupled modes have the same frequency. Discrete instabilities caused by both the first and third type of resonances are verified and instabilities caused by the third type of resonance are found to induce melting. The melting procedure is observed to go through a two-step process with the solid-liquid transition closely obeying the Lindemann criterion.

Figure 1

The point charge model for the ion wakefield employed for a typical experimental setup, where $Q$ is the dust particle charge, $q$ is the point charge, $l$ is the distance between the dust particle and the point charge, $Mg$ is gravity, and $E_x$ and $E_z$ are the horizontal and vertical electric fields, respectively.

Figure 2

Temporal evolution of the average kinetic energy for a five particle system without the ion wakefield with $E_z^{\prime }$ $=$ 1.705 × 10${}^5$ V/m${}^{-2}$.

Figure 3

Eigenvectors for the normal modes in a three particle cluster: (a) the horizontal rotation mode (mode 1), (b) the horizontal sloshing mode (modes 2 and 3), (c) the horizontal kink mode (modes 4 and 5), (d) the horizontal breathing mode (mode 6), (e) the vertical relative mode (modes 7 and 8), and (f) the vertical sloshing mode (mode 9).

Figure 4

Power spectra for a three particle cluster with $E_z^{\prime }$ $=$ 1 × 10${}^5$ V/m${}^{-2}$ (a) without and (b) with ion wakefield. The dots in (b) correspond to spectral frequencies without the ion wakefield. The color (grayscale) bar represents the power spectral density in units of (m/s)${}^2$/Hz.

Figure 5

The first type of resonance illustrated by (a) the power spectra and (b) $E_z^{\prime }$-$fr$ functions around the resonance points. The results shown are for a three particle cluster at the resonance between the vertical relative (blue triangles) and horizontal sloshing (red circles) modes. The decoupled shapes of the $E_z^{\prime }$-$fr$ functions are represented by the solid lines.

Figure 6

The second type of resonance illustrated by (a) the power spectra and (b) $E_z^{\prime }$-$fr$ functions around the resonance point for a three particle cluster at the resonance between the vertical relative (modes 7 and 8, blue triangles) and horizontal “kink” (modes 4 and 5, red circles) modes. The decoupled shapes of the $E_z^{\prime }$-$fr$ functions are represented by the solid lines.

Figure 7

The third type of resonance as illustrated by (a) the power spectra and (b) $E_z^{\prime }$-$fr$ functions around the resonance point for one of the resonances seen in a seven particle cluster. The blue triangles and red circles represent the coupled vertical and horizontal modes, respectively. The decoupled shapes of the $E_z^{\prime }$-$fr$ functions are represented by the solid lines.

Figure 8

$E_z^{\prime }$-$fr$ functions for (a) three, (b) four, and (c) five particle clusters and the average particle velocity as a function of $E_z^{\prime }$ for (d) three, (e) four, and (f) five particle clusters. In (a), (b), and (c), horizontal and vertical modes are represented by blue solid and red dashed lines, respectively. Intersection points corresponding to the first, second, and third types of resonance are marked by squares, triangles, and circles, respectively. In (d), (e), and (f), velocities in the horizontal and vertical directions are represented by blue solid and red dashed lines, respectively.

Figure 9

$E_z^{\prime }$-$fr$ functions for (a) six and (b) seven particle clusters and the average particle velocity as a function of $E_z^{\prime }$ for (c) six and (d) seven particle clusters. In (a-b), horizontal and vertical modes are represented by blue solid and red dashed lines, respectively. The intersection points corresponding to the first, second, and third types of resonance are marked by squares, triangles, and circles, respectively. In (c) and (d), velocities in the vertical and horizontal directions are represented by blue solid and red dashed lines, respectively.

Figure 10

Mode eigenvectors for a seven particle cluster, where the horizontal (vertical) modes are shown at top (bottom) and solid lines connect coupled modes.

Figure 11

The instability induced by the resonances $h$(2, 1)$v$(2, 1) and $h$(2, 2)$v$(2, 1) in a seven particle cluster, indicated by peaks in the vertical velocities. The maximum of the two peaks are aligned to show the difference in peak width.

Figure 12

(a) $u_{\mathrm{i},\mathrm{L}}$ and (b) $u_{i,}$${}_{\mathrm{rel}}$ as a function of the frame number (time) for the melting of a seven particle cluster induced by the $h\left(2,1\right)v(2,1)$ resonance instability.