Nonequilibrium finite dust clusters: Connecting normal modes and wakefields

The dynamic properties of finite three-dimensional dust clusters in a dusty plasma under the influence of an ion focus are studied by normal modes. The mode analysis has been extended to account for the ion focus using the point-charge model for the horizontal interaction of the dust particles. From that, an analytical model for a few-particle system is derived accounting for three distinct dynamical regimes at different focus strengths, namely, absolutely unstable and fully stable configurations as well as an unstable oscillatory regime. The techniques of normal mode analysis (NMA) and instantaneous normal modes (INM) extended by the ion focus have been applied to dust systems in the experiment and compared to the model. From this, the ion focus strength has been derived. The specific sensitivity of NMA and INM allows one to identify equilibrium configurations in this nonequilibrium environment for these finite clusters.

Figure 1

(a) Model of the dust-dust interaction. The ion focus is modeled as a point charge below the dust. Here the situation is illustrated for three vertically interacting particles. (b) and (c) Horizontal and vertical modes of the three-particle system. See text for details.

Figure 2

Particle trajectories over 300 s for a three-particle system at discharge powers of (a) 0.8 W, (b) 1.2 W, and (c) 1.6 W.

Figure 3

Spectra of the three-particle system ignoring the effect of an ion focus. Top panel: INM spectrum $\rho \left(\omega \right)$. By convention, purely imaginary mode frequencies obtained in the INM spectrum are plotted on the negative frequency axis, i.e., $i{\omega }_{\ell }\to -\left|{\omega }_{\ell }\right|$. The central panel shows the mode-resolved NMA spectrum (darker colors correspond to higher spectral density). The lower panel shows the PSD.

Figure 4

Spectra of the three-particle system including the effect of an ion focus. Top panel: INM spectrum $\rho \left(\omega \right)$. The real parts of the eigenvalues are given with solid lines. The imaginary part (conjugate pairs, dashed lines) of the mode with the lowest real part ($f\approx 4.5$ Hz) is also plotted. The central panel shows the mode-resolved spectrum (darker colors correspond to higher spectral density). The spectrum is split into two parts for the real part of the eigenvectors and the imaginary part. The lower panel shows the PSD where also the contributions of the real and imaginary parts are indicated.

Figure 5

Variation of the ion-focus strength $\xi$ for the three configurations shown in Fig. 2. The lower panels show the normal mode spectra of modes numbers 2 and 6. The upper panel shows the eigenvalues of the dynamical matrix $f_2$ and $f_6$ for these two modes as a function of the ion-focus strength $\xi$. For the horizontal modes real and imaginary parts of $f_2$ are indicated.

Figure 6

NMA spectrum of the five-particle system including the effect of an ion focus. The spectrum is split into two parts for the real part of the eigenvectors and the imaginary part. The particle trajectories of the system are shown in the inset. The white circles correspond to the INM eigenvalues.

Figure 7

NMA spectrum of the 30-particle system including the effect of an ion focus. The spectrum is split into two parts for the real part of the eigenvectors and the imaginary part. The particle trajectories of the system are shown in the inset. The white line corresponds to the INM eigenvalues (only shown for the real part, here, for clarity).