Phonon spectra of two-dimensional liquid dusty plasmas on a one-dimensional periodic substrate

We investigate the phonon spectra of two-dimensional liquid dusty plasmas on a one-dimensional periodic substrate using numerical simulations. The propagation of the waves across the potential wells of the substrate is inhibited due to the confinement of the dust particles by the substrate minima. If the substrate wells are narrow or deep, one-dimensional chains of particles are formed in each minimum, and the longitudinal motion of an individual chain dominates the propagation of waves along the potential wells of the substrate. Increasing the width or decreasing the depth of the substrate minima allows the particles to buckle into a zigzag structure, and the resulting spectra develop two branches, one for sloshing motion and one for breathing motion. The repulsion between neighboring dust particles produces backward propagation of the sloshing wave for small wave numbers.

Figure 1

Snapshots of the dust particle positions (dots) along with an illustration of the width and depth of the 1D substrate potential. (a) $U_0=0.5E_0$ and $w=1.002b$. The particles form 1D chains at the bottom of each potential well of the substrate. (b) $U_0=0.5E_0$ and $w=2.004b$. For this value of $w$, the 1D chain of particles buckles into a zigzag shape. (c) $U_0=0.25E_0$ and $w=2.004b$. (d) $U_0=E_0$ and $w=2.004b$. As $U_0$ increases, the zigzag shape narrows. For each panel, the inset on the bottom left corner is the corresponding 2D distribution function [8] $g(x,y)$ of the total simulated area.

Figure 2

Longitudinal (a) and transverse (b) phonon spectra of our simulated 2D Yukawa liquid as functions of frequency $\omega /{\omega }_{pd}$ and wave number $kb$ in the absence of a substrate at $\mathrm{\Gamma }=200$, $\kappa =2.0$, and $\nu =0.027{\omega }_{pd}$.

Figure 3

Longitudinal and transverse phonon spectra ${\tilde{C}}_L({\mathbf{k}}_x,\omega )$ (a), ${\tilde{C}}_T({\mathbf{k}}_x,\omega )$ (b), ${\tilde{C}}_L({\mathbf{k}}_y,\omega )$ (c), and ${\tilde{C}}_T({\mathbf{k}}_y,\omega )$ (d) as functions of $\omega /{\omega }_{pd}$ and $kb$ for a periodic substrate with $U_0=0.5E_0$ and $w=1.002b$.

Figure 4

Longitudinal and transverse phonon spectra ${\tilde{C}}_L({\mathbf{k}}_x,\omega )$ (a), ${\tilde{C}}_T({\mathbf{k}}_x,\omega )$ (b), ${\tilde{C}}_L({\mathbf{k}}_y,\omega )$ (c), and ${\tilde{C}}_T({\mathbf{k}}_y,\omega )$ (d) as functions of $\omega /{\omega }_{pd}$ and $kb$ for a periodic substrate with $U_0=0.5E_0$ and $w=2.004b$.

Figure 5

Longitudinal and transverse phonon spectra ${\tilde{C}}_L({\mathbf{k}}_x,\omega )$ (a, c) and ${\tilde{C}}_T({\mathbf{k}}_x,\omega )$ (b, d) as functions of $\omega /{\omega }_{pd}$ and $kb$ for two different periodic substrates of $U_0=0.25E_0$ in (a, b) and $U_0=E_0$ in (c, d), where $w=2.004b$ is unchanged.

Figure 6

Longitudinal and transverse phonon spectra ${\tilde{C}}_L({\mathbf{k}}_y,\omega )$ (a, c) and ${\tilde{C}}_T({\mathbf{k}}_y,\omega )$ (b, d) as functions of $\omega /{\omega }_{pd}$ and $kb$ for two different periodic substrates of $U_0=0.25E_0$ in (a, b) and $U_0=E_0$ in (c, d), where $w=2.004b$ is unchanged.