Mode couplings and resonance instabilities in finite dust chains

Employing a numerical simulation, the normal modes are investigated for finite, one-dimensional horizontal dust chains in complex plasma. Mode couplings induced by the ion flow within the sheath are identified in the mode spectra and the coupling rules are determined. Two types of resonance-induced instabilities are observed, one bidirectional and one unidirectional. Bidirectional instability is found to cause melting of the chain with the melting proceeding via a two-step process which obeys the Lindemann criterion. The relationship between the normal mode spectra observed in finite systems and the wave dispersion relations seen in larger systems was also examined using a dust chain model. For this case, the dispersion relation was obtained through multiplication of the mode spectra matrix by a transition matrix. The resulting dispersion relations exhibit both the general features observed in larger crystals as well as several characteristics unique to finite systems, such as discontinuities and strong energy-density fluctuations.

Figure 1

Eigenvectors for the $x$, $z$, and $y$ modes for a seven-particle chain.

Figure 2

Normal mode spectra for a three-particle chain with (a) $E_z^{\prime }$ = 6.0 $\times$ ${10}^5$ V/${\mathrm{m}}^{-2}$ and a seven-particle chain with (b) $E_z^{\prime }$ = 3.3 $\times$ ${10}^6$ V/${\mathrm{m}}^{-2}$.

Figure 3

(a), (b) Normal mode frequencies and (c), (d) average particle velocities as functions of $E_z^{\prime }$ for a (a), (c) three- and (b), (d) seven-particle chain. In (a) and (b), $x$ and $z$ modes are represented by blue solid and red dashed lines, respectively. In (c) and (d), velocities in the $x$ and $z$ directions are represented by blue solid and red dashed lines, respectively.

Figure 4

Normal mode spectra for a seven-particle chain with $E_z^{\prime }$ = 1.985 $\times$ ${10}^6$ V/${\mathrm{m}}^{-2}$.

Figure 5

Normal mode spectra for a three-particle chain with $E_z^{\prime }$ = 1.95 $\times$ ${10}^5$ V/${\mathrm{m}}^{-2}$.

Figure 6

$u_{\mathrm{rel}\left(\sigma \right)}$ as a function of time for the melting of a seven-particle chain induced by the $x$(4)$z$(5) resonance instability. $u_{\mathrm{rel}\left(\sigma \right)}$ in the $x$, $z$, and $y$ directions is represented by blue thick lines, red thin lines, and green dotted lines, respectively.

Figure 7

(a) Normal mode spectra for a 20-particle chain and the dispersion relations obtained from the spectra corresponding to the (b) $x$, (c) $y$, and (c) $z$ modes.

Figure 8

(a) The transition matrix for a 20-particle horizontal chain as defined by Eq. (4)) in the text. (b) The eigenvectors for the third, tenth, and 18th normal modes (left panel) and corresponding intensity over $k$ in the transition matrix (right panel).

Figure 9

The longitudinal dispersion relation obtained by inspection of the middle 20 particles within a 50-particle chain.

Figure 10

(a) Normal mode spectra for a 20-particle chain and the dispersion relations obtained from the spectra corresponding to the (b) $x$ and (c) $z$ modes upon the crossing of the $x$ and $z$ mode branches.