Modes of three-dimensional dust crystals in dusty plasmas

Three-dimensional finite systems of charged dust particles confined to concentric spherical shells in a dusty plasma, so-called “Yukawa balls,” have been studied experimentally with respect to their dynamic properties. For that purpose, the three-dimensional thermal Brownian motion of the dust particles around their equilibrium positions was reconstructed with high accuracy. From that the dynamic properties in the form of mode patterns and frequencies are obtained from singular value decomposition and normal mode analysis. The dynamics of the Yukawa balls was found to be dominated by large-scale low-frequency shearlike modes.

Figure 1

(Color online) Scheme of the experimental setup.

Figure 2

(Color online) (a) Trajectories of the particles of the Yukawa ball with $N=31$ particles. The trajectories are shown over 1300 frames $\left(65\text{seconds}\right)$ in cylindrical coordinates. (b) Bond model of the dust cluster. The Yukawa ball forms two concentric shells with five particles on the inner and 26 particles on the outer shell.

Figure 3

Shear $\left({\Psi }_r^2\right)$ and compressional $\left({\Psi }_d^2\right)$ contribution of the 93 SVD modes of the $N=31$ Yukawa ball. For mode Nos. 5 and 51 the shear and compressional contribution are marked by an arrow. See text for details.

Figure 4

Relative signal energy of the 93 SVD modes of the $N=31$ Yukawa ball.

Figure 5

(a) 3D mode pattern of the intershell rotation mode (mode No. 5). Black arrow indicates the motion of the particles on the inner shell, gray that of the outer shell. (b) Projection of the 3D mode pattern on the $x\text{−}y$ plane. (c) Magnitude of shell rotation vectors in gray scale. The gray level of the figure background refers to zero rotation, hence lighter particles have counterclockwise rotation, darker particles rotate clockwise.

Figure 6

(a) 3D mode pattern of the most compressional mode (mode No. 51) and (b) the mode with highest signal energy (mode No. 1).

Figure 7

Spectral power density (gray scale) as a function of frequency for the modes of the Yukawa ball. The calculated spectral power density increases from black to white. The gray circles are a best fit of the calculated eigenfrequencies to the spectral power density.

Figure 8

Shear $\left({\Psi }_r^2\right)$ and compressional $\left({\Psi }_d^2\right)$ contribution of the normal modes of the $N=31$ Yukawa ball.

Figure 9

(a) 3D mode pattern of the intershell rotation mode (normal mode No. 7) and (b) the mode with highest frequency (mode No. 90).

Figure 10

Mode temperature and shear contribution ${\Psi }_r^2$ determined from the normal mode analysis and signal energy values $p_{\ell }$ determined from SVD as a function of the mode frequency.