Crystal and fluid modes in three-dimensional finite dust clouds

When the number of dust particles becomes sufficiently large, it is convenient to treat the system as a continuum. In addition, when the cluster is not in a solid state, a fluid description might be more adequate.

Kählert and Bonitz [8, 37] treated the Yukawa cluster as a fluid droplet and derived the modes by solving the plasma fluid equations. In their approach, the total potential of the dust cloud with time-dependent density $n_{\rm d}( \skew3\vec{r}, t)$ is expanded into a set of radial and angular eigenfunctions. Multipole moments of the density can be defined in terms of these eigenfunctions as

$$\begin{equation} q_{lm}(t) = Q \sqrt{\frac{4\pi}{2l+1}} \int{n_{\rm d}( \skew3\vec{r}',t) \hat{i}_l(\kappa r') Y_{lm}^{*} (\theta,\phi)\mathrm{d} \skew3\vec{r}'}, \end{equation} \tag{ 2 }$$

with l and m being the angular mode numbers, $\hat {i}_l(\kappa r')$ the modified spherical Bessel function, and Y_lm(θ,ϕ) the spherical harmonics, respectively. In our analysis, weak screening was assumed by fixing the screening strength to κ = 0.6 but other values of κ did not reveal any qualitative differences [23, 24].

In order to derive the experimental fluid modes, the particle positions at each time step are taken to calculate the multipole moments q_lm, where the density n_d is treated as a series of δ-functions according to $n_{\mathrm { d}}( \skew3\vec {r},t) = \sum _{i}{\delta ( \skew3\vec {r} - \skew3\vec {r}_{i}(t))}$. For all fluid modes shown here, we calculated the power spectral density from 30 000 frames. The power spectral density q_lm(ω) of each fluid mode is taken as the Fourier transform of q_lm(t) via

$$\begin{equation} q_{lm}(\omega) = \frac{2}{TN} \left| \int_{0}^{T}{q_{lm}(t)\exp\left({-\mathrm{i}\omega t}\right) \mathrm{d}t}\right|^2 \end{equation} \tag{ 3 }$$

over a time interval T.

For the sake of a broader picture, the dynamics has also been analysed by normal modes [4, 27, 35, 62, 63]. Thereby, it is assumed that the particles only perform small oscillations around their local equilibrium positions. Thus, using the dynamical matrix, i.e. the second derivative of the total energy given in equation (1),

$$\begin{equation} H = \frac{\partial^2 E}{\partial r_{\alpha,i},\partial r_{\beta,j}}, \end{equation} \tag{ 4 }$$

the 3 × N eigenvalues and eigenvectors are calculated with α and β being the Cartesian coordinates x, y, z and i,j denoting the particle number. The eigenvalues define the frequencies of the different modes of the Yukawa cluster. The resulting 3D eigenvectors $\skew3\vec{e}_{i,l}$ describe the oscillation pattern of particle i in each normal mode l.

From the measured time series, the velocities $\skew3\vec{v}_{i} (t)$ of each particle are mapped onto the eigenmode pattern according to

$$\begin{equation} f_{l}(t) = \sum_{i = 1}^{N}{ \skew3\vec{v}_{i}(t) \skew3\vec{e}_{i,l}}. \end{equation} \tag{ 5 }$$

From this, the NMA spectral power density S_l(ω), i.e. the energy per frequency interval of each individual crystal mode l, follows from the Fourier transform of equation (5) similar to equation (3) above [27]. Contrary to the fluid mode technique, where the momentary particle positions inside the dust cloud are required, the NMA needs the full trajectories of each individual dust grain. For the NMA performed here, the full 3D trajectories of more than 1600 frames per experiment were used.

While both approaches are well applicable to dust clouds at high pressure, the analysis faces problems at low pressure since both methods are based on the isotropic Yukawa potential. In the latter case, the streaming ions make the potential anisotropic and lead to non-reciprocal inter-particle forces [52]. In principle, the ion focus can be taken into account by introducing an additional positive charge below each dust grain [46, 58, 64, 65] or by using the linear response formalism to calculate an improved interaction potential [50, 52]. However, the anisotropy of the clusters in the present experiment is not very pronounced, which is why we neglect these effects in our analysis. The comparison with the measurements is thus expected to yield useful information on the importance of wake effects in the theoretical description.

We have observed a spherical dust cluster and recorded the particle motion over 30 000 frames. From that, the particle positions have been determined. Then, the spectra of the fluid modes were calculated for all angular mode numbers up to the order l = m = 5 according to equations (2) and (3). Here, the q₀₀, q₁₀ and q₁₁ fluid modes are presented as representatives for the monopole and dipole modes (with q₀₀ being breathing mode like [8, 37], q₁₀ containing the sloshing mode in vertical direction and q₁₁ containing the sloshing mode in horizontal direction). As examples for fluid modes with a more complex structure, the spectra of the q_2m, m = 0,1,2 modes are depicted, corresponding to quadrupole modes.

Figure 3(a) shows the spectra of the fluid modes q₀₀, q₁₀ and q₁₁ and figure 3(b) the spectra of the fluid modes q₂₀, q₂₁ and q₂₂ for the unheated N = 60 cluster. Figures 3(c) and (d) show the same modes for the same cluster in the liquid regime at a laser heating power of P_L = 250 mW. In the absence of any external laser heating, figures 3(a) and (b), no fluid modes are detectable.

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For the spectra of the heated cluster shown in figures 3(c) and (d), mainly two fluid modes are excited. This can be seen from the elevated signal of the q₁₁ fluid mode at 2.5 Hz and the q₂₂ fluid mode at 3.0 Hz. The value of the peak positions for all fluid modes shown here and in the following were determined by fitting a Gaussian distribution to the peak. The peak width was typically found from the fit as σ = 1.4 Hz. The reason for the excitation of the horizontal q₁₁ and q₂₂ modes can be understood by the fact that the beams are pointing in the horizontal direction to the dust cloud. Therefore, the lasers transfer momentum to the cluster particles mainly in this direction.

Interestingly, the ratio of the two mode frequencies ω₂₂/ω₁₁ = f₂₂/f₁₁ = 3.0/2.5 Hz = 1.2, is in good agreement with the fluid theory which predicts two limiting cases: (a) for κ = 0 one gets $\omega _{ll} = \sqrt {3l/2l+1} \omega _0$, $\omega _{22}/\omega _{11} = \sqrt {3\cdot 2/(2\cdot 2+1)} = 1.095$ and for (b) κ → ∞ one finds $\omega _{ll} = \sqrt {l} \omega _0$, $\omega _{22}/\omega _{11} = \sqrt {2} \approx 1.41$ [8]. In particular, equating the theoretical ratio to 1.2 we obtain κ = 0.57 ≈ 0.6, which is in good agreement with earlier findings (due to the weak dependence of the frequency on the screening parameter (see figure 2 in [8]), the error is relatively large).

A possible cause not to find any peak in the q₀₀ spectra (which describes a monopole breathing oscillation) is the presence of neutral drag. As pointed out by Kählert and Bonitz low damping rates are necessary to detect fluid modes [37]. In our experiment, the neutral gas pressure was held constant at 6.4 Pa. From this, the Epstein friction coefficient can be calculated as ν = 8 s⁻¹ [66]. When we now assume that the peak of the q₁₁ mode is a sloshing oscillation at the trap frequency f₁₁ = 2.5 Hz ≈ ω₀/(2π) the normalized friction is ν/ω₀ ≈ 0.5. Even though the gas pressure is relatively low compared to most other Yukawa ball experiments [22, 23, 27], only a few broad peaks can be seen in the interesting frequency domain.

It is tempting to reduce the neutral gas pressure in order to decrease friction of the particles. This should lead to richer fluid mode spectra [37]. As discussed in section 2, lowering the neutral gas pressure leads to an increasing influence of the ion focus on the particles [36, 49]. Thus, we can study the influence of damping and the influence of anisotropic particle–particle interactions to the fluid modes at once. The spectra of the same fluid modes as above are shown in figure 4 for a cluster with N = 49 particles to compare directly with the results of the isotropic cluster.

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In the absence of any external laser heating, figures 4(a) and (b), indeed a few collective excitations appear in the elongated cluster. No peak occurs in spectra of the q₀₀ and q₁₀ fluid mode. An elevated signal can be found in the q₁₁ mode at about 3.5 Hz. The q₂₂ fluid mode, which represents a transverse quadrupole oscillation, has also a broad peak at the frequency domain of 4 Hz. Moreover, a sharp peak occurs in all quadrupole spectra and in nearly all spectra of the higher order modes at 7.8 Hz for this cluster indicating that this peak is not a pure fluid eigenmode.

Even though the friction is rather low no q₀₀ mode is detectable (for 4.8 Pa and assuming the trap frequency to be ω₀/(2π) ≈ 3.5 Hz one finds ν/ω₀ ≈ 0.27). The reason is in the elongation of the cluster along the z-axis and the particle-interaction, which is influenced by the ion focus. Such a cluster does not possess a uniform monopole mode.

High mode numbers generally correspond to more localized oscillations in the cluster. The observed sharp peak at 7.8 Hz in the spectra of the higher-order fluid modes for the N = 49 cluster thus belongs to an oscillation that does not involve large-scale motion. Moreover, the fact that quadrupole oscillations transverse to the ion flow q₂₂ are favoured over the longitudinal quadrupole oscillations seem to confirm that restoring forces are mainly in the direction perpendicular to the ion streaming direction even without laser excitation [48, 49, 60].

The origin of the transverse restoring forces can be attributed to the ion focus [46, 48, 49, 60]. The alignment can result in unstable oscillations—the Schweigert instability—since there are repulsive forces between the dust particles and attractive forces between ion cloud and the dust particles [47, 48, 52–59].

Additional evidence for the Schweigert instability in our experiment is given by the fact that the dust system is 'heated' by the instability [14, 31, 36, 46, 58, 64, 67–69]. For the dust cluster under investigation, the temperature (mean kinetic energy) of the cluster particles is found at $T_{\mathrm { kin}} = m \left \langle v^2\right \rangle /3 k_{\mathrm { B}} \approx 7370\,\mathrm {K}$, i.e. far above room temperature even in the unheated case [14, 67]. For comparison, the cluster in section 4.1 has T = 2930 K.

To underline the above argument, the motion of individual particles was analysed. Therefore, the four representative particles of the particle chain were chosen (marked in figure 2).

In figure 5(a) the transverse (to the ion stream) movement ρ(t) of the four particles within the cluster is shown over a time span of 2 s. The corresponding power spectra from the full trajectories (300 s) are depicted in figure 5(b).

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For the uppermost particle 1, the trajectory does not show any particular oscillations, consequently the power spectrum is nearly flat. In contrast, the lower particles 2–4 respond to the ion wakefield from particles placed above it. This can be seen from the trajectories correlated oscillatory motion and the corresponding peak in the spectra of the particles, respectively. Clearly, the frequency of the peak in the spectrum of the individual particle motion ρ(t) coincides with that of the q_2m fluid modes. Furthermore, the amplitude of the oscillatory motion increases with particle positions further down the ion stream. This supports that these oscillations are due to the Schweigert instability. A second peak in the spectrum for particles 3 and 4 at the first harmonic (≈ 16 Hz) hints at the nonlinear character of these oscillations.

The ion wakes induce local small-scale perturbations of the cloud potential, leading to signals in the fluid spectra for higher mode numbers, see figure 4. Even when the fluid mode technique starts from the assumption of an isotropic Yukawa model, equation (1), it allows for identification of such instabilities.

To display the influence of heating, the fluid mode spectra for the laser heated N = 49 cluster are shown in figures 4(c) and (d). Heating was applied in the same manner as for the spherical cluster. Here, the peak at roughly 3.2 Hz is even more pronounced in the q₁₁ mode. A slightly higher signal is also found in the spectra of the q₁₀ mode at 5 Hz.

Contrary to the unheated case, no sharp peaks occur at 7.8 Hz for higher angular mode numbers l = 2. Instead, several other modes are excited at 5.6 and 11.4 Hz in the q₂₀ mode, at 2.6 and 6 Hz in the q₂₁ mode and at roughly 3.6 Hz in the q₂₂ fluid mode.

The ratio of the two mode frequencies ω₂₂/ω₁₁ = 4/3.5 Hz = 1.14 for the unheated, and 3.6/3.2 Hz = 1.13 for the heated case, respectively, agree well with the ratio for the spherical cluster and literature [8]. However, one must take into account that the assumptions of the theory, i.e., isotropic confinement and interaction, are not fully fulfilled here.

According to the fluid theory of [8], the frequency for excitations in the q_lm fluid modes for the same l but different m should be independent of mode number m, since the underlying theory assumes isotropic confinement and isotropic particle–particle interaction. The fact that the frequencies of the q_1m and q_2m fluid modes change as m changes could either indicate that the confinement in the experiment is not perfectly isotropic or that the symmetry in the particle–particle interaction is broken due to the presence of an ion focus.

To analyse the fluid behaviour of finite dust clouds more closely, the evolution of the q₁₁ mode and the q₂₂ mode with temperature is investigated. Here, we restrict to the N = 49 cluster, since the fluid modes are more pronounced than for the spherical cluster, but the general results are qualitatively the same. The mode evolution is shown in figure 6 as a function of laser heating power. The power of the two beams was varied between 0 and 400 mW in steps of 50 mW thus covering a wide range of fluidity.

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For comparison, the kinetic temperature of the cluster particles as function of the applied laser power is shown in figure 6(c) together with the corresponding Coulomb coupling parameter. The Wigner–Seitz radius for the N = 49 cluster was estimated as b_WS = (3/4πn)^1/3 = 340 μm from the overall density. For the dust charge, the value of [14] Q = 6900e was used. The solid curve represents a parabolic fit of the temperature versus laser heating power. This parabolic dependence corresponds to a laser particle interaction via radiation pressure [38, 70]. It was shown previously [14] that the velocity distribution is not purely Maxwellian for the dust particles, but sufficient to assign a temperature to the 3D dust clusters.

In each spectrum of the q₁₁ fluid mode shown in figure 6(a), a peak is found at a frequency range of (3.3 ± 0.2) Hz. The position of the peak maximum does not change significantly with laser heating. This transverse sloshing mode was already discussed in the previous section. The peak height slightly increases as the laser power increases, and the peak gets broader. This is exactly what is expected for the fluid modes [37].

Figure 6(b) shows the evolution of the q₂₂ fluid mode. Here, a peak is found for all modes at a frequency interval of about (3.7 ± 0.2) Hz. As in figure 6(a), the peak height increases and broadens continuously. Similar behaviour was found for all fluid modes. The oscillation associated with the Schweigert instability at 7.8 Hz is seen for the unheated cluster at 0 mW. At higher heating this peak is destroyed, probably due to the frequent particle exchanges in the liquid regime.

One goal of this paper is to compare the fluid and crystal modes of finite dust clouds. Contrary to the fluid mode description, where the Yukawa ball is treated like a fluid droplet, the NMA describes the harmonic movement of individual dust grains in terms of crystal modes [4, 71]. To compare the crystal modes with the fluid mode results, a NMA is performed from the trajectories of the spherical and the aligned cluster for the unheated crystalline case (P_L = 0 mW) and the heated, liquid-like clusters (P_L = 250 mW). For both cases, the 3D positions of the cluster particles with respect to the centre of mass were assumed to be equilibrium positions and again weak screening κ = 0.6 was assumed to calculate the dynamical matrix H of equation (4).

The reason for choosing the NMA instead of the recently established instantaneous normal mode (INM) technique is that the NMA allows to investigate the mode resolved spectral properties, whereas the INM only enables to reveal information about the total distribution of the eigenfrequencies [32, 72].

Figure 7 shows the mode resolved NMA spectral power density for the unheated cluster and the heated cluster together with the total power density S(ω) (PSD), which follows from the summing up S_l(ω) over each individual mode $S(\omega ) = \sum _{l} {S_{l}(\omega )}$.

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For the crystalline cluster, figure 7(a), regions with higher power density can be found for each mode. These regions shift slightly to higher frequencies as the mode number increases. This is the expected trend and has been previously demonstrated for 2D and 3D clusters [27, 36]. The spectral width is much less than that of [27] due to the much lower pressure used in the present experiment. The corresponding PSD, figure 7(b), that is obtained from the integration over all modes, shows evenly distributed energy at each frequency in this representation. However, the interpretation of individual normal modes is not as straightforward as in the fluid mode description. For the heated cluster, see figure 7(c), the energy in the modes is not located in a narrow frequency band as for the solid cluster. Higher energy can be found in a much broader region up to 15 Hz for all modes. In the PSD, figure 7(d), most of the energy in the system is seen to be located in a broad frequency range with its maximum at about 2.6 Hz. Compared to the PSD for the solid cluster, figure 7(b), laser heating increases the values of the PSD about one order of magnitude since kinetic energy is deposited to the cluster particles. Frequent particle exchanges occurring in this liquid regime prevent the establishment of normal mode oscillations. Thus, the spectrum is much broader as for the cluster in the solid phase, so as the NMA is mainly suited for crystalline states.

The N = 49 cluster with aligned particle chains was investigated by means of NMA in the same manner as described above. For the solid cluster, figure 8(a), the features of the mode resolved NMA spectra of a solid weakly damped Yukawa cluster are recovered, too. Moreover, in this unheated case, a significant amount of energy is stored in a narrow frequency band around 7.8 Hz in most of all modes. Consequently, this feature is no eigenmode of the system [36]. The peak at 7.8 Hz becomes more distinct in the corresponding PSD, see figure 8(b). The contribution at 7.8 Hz is by far the most dominant input in the PSD (note that energy density in this narrow frequency band is about two magnitudes larger than for the unheated spherical cluster, figure 7). Thus, the Schweigert instability at 7.8 Hz can be identified within both, the crystal and the fluid mode approach.

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The mode resolved NMA spectra of the heated chain-like cluster, see figure 8(c), looks very similar to that of the spherical cluster with laser heating. In the PSD, figure 8(d), the maximum of the energy is now found at 3.3 Hz due to the different discharge parameters.

It is a remarkable result that even when both approaches assume isotropy and use different data as an input (fluid modes require the 3D density, i.e. the particle positions, and NMA uses the particles velocities), the dominant spectral properties of the system are well recovered:

In the fluid regime, the peak of the q₁₁ fluid mode coincides with the maximum of the PSD for both clusters. Moreover, in the unheated case of the elongated cluster, the sharp peak at 7.8 Hz due to the ion focus is covered by the NMA and the fluid mode approach as well. A difference in both methods lies however in the allocation of the total energy of the system onto the individual crystal or fluid modes.