⇦Chinese Physics Letters, 2017, Vol. 34, No. 7, Article code 075203 Effect of Particle Number Density on Wave Dispersion in a Two-Dimensional Yukawa System * Rang-Yue Zhang(张壤月)1, Yan-Hong Liu(刘艳红)2, Feng Huang(黄峰)1,3**, Zhao-Yang Chen(陈朝阳)4, Chun-Yan Li(李春燕)1 Affiliations 1College of Science, China Agricultural University, Beijing 100083 2School of Physics and Optoelectronic Engineering, Ludong University, Yantai 264025 3Key Laboratory of Agricultural Informationization Standardization (Beijing), Ministry of Agriculture, Beijing 100083 4Department of Physics, Beijing University of Chemical Technology, Beijing 100029 Received 27 February 2017 *Supported by the National Natural Science Foundation of China under Grant Nos 11675261 and 21403297, and the Scientific Research Foundation of Ludong University under Grant No LY2014010.
**Corresponding author. Email: huangfeng@cau.edu.cn Citation Text: Zhang R Y, Liu Y H, Huang F, Chen Z Y and Li C Y 2017 Chin. Phys. Lett. 34 075203 Abstract Effect of the particle number density on the dispersion properties of longitudinal and transverse lattice waves in a two-dimensional Yukawa charged-dust system is investigated using molecular dynamics simulation. The dispersion relations for the waves are obtained. It is found that the frequencies of both the longitudinal and transverse dust waves increase with the density and when the density is sufficiently high a cutoff region appears at the short wavelength. With the increase of the particle number density, the common frequency tends to increase, and the sound speed of the longitudinal wave also increases, but that of the transverse wave remains low. DOI:10.1088/0256-307X/34/7/075203 PACS:52.27.Gr, 52.27.Lw © 2017 Chinese Physics Society Article Text Dusty plasmas[1-7] contain a large number of charged dust particles and can be found in many environments, such as in space, the laboratory, and microelectronics fabrication. Since more electrons than ions inelastically collide with dust particles due to their very different thermal speeds, the dust grains are usually highly negatively charged. As a result, the electrostatic interaction energy of a dust system is much larger than its thermal energy, thus dusty plasmas are also useful as a model for investigating phenomena in strongly coupled systems.[6,7] Since the charged dust particles can strongly affect the properties of the background plasma, they have been intensively studied, especially the collective process such as linear and nonlinear waves and instabilities, formation of clusters and other self-organized structures.[6-15] The interaction between two dust particles in a plasma is usually modeled by the Yukawa (or screened Coulomb) potential $\varphi (r)$,[8] $$\begin{align} \varphi (r)=(Q^2/4\pi \varepsilon _0r)\exp(-r/\lambda _{\rm D}),~~ \tag {1} \end{align} $$ where $Q$ is the particle charge, $r$ is the interparticle distance, and $\lambda _{\rm D}$ is the Debye screening length (due to sheath formation around the dust). At or near thermodynamic equilibrium, the physical properties of a dust system can be characterized by two dimensionless parameters: the screening strength $\kappa =a/\lambda _{\rm D}$ and the coupling parameter ${\it \Gamma}=Q^2/4\pi \varepsilon _0 aT$, where $a$ is the mean interparticle distance, and $T$ is the system temperature in energy units. When ${\it \Gamma} \gg 1$, the dust system can be in a solid state and has an ordered structure.[9] There are two main types of dust waves.[10-14,16] One is the dust acoustic wave (DAW), which was theoretically predicted under the weak coupling approximation (i.e., ${\it \Gamma} < 1$), but can also exist in a more strongly coupled regime ${\it \Gamma} _{\rm c} \gg {\it \Gamma} \gg 1$, where ${\it \Gamma} _{\rm c}$ is the critical coupling parameter for crystallization.[10] Ruhunusiri et al.[11] obtained the susceptibilities for the ions, dusts and the dust charge fluctuation. They found that the inverse Landau damping and ion-neutral collisions can contribute equally to the DAW instability. Qian et al. included superthermal electrons and ions to analyze the DAW in self-gravitating dust plasmas.[12] In the strongly coupled regime, there can exist another wave, namely the dust lattice wave (DLW), which has also been widely studied theoretically and experimentally. For example, Misawa et al.[13] observed vertically polarized transverse DLWs in a 1D strongly coupled dust chain in a dc argon plasma at low gas pressure, and found that the transverse dust-wave propagates as a backward wave. Homman et al.[10] investigated the propagation of the DLWs in a 2D dust system in the sheath of an rf discharge. Abdikian et al.[14] explained the normal-mode features of dust lattice in a planar zigzag crystal chain and showed that propagation of the longitudinal and acoustic modes depends on the distance between the two chains, and the modes may not propagate if it is too short. Murillo[6] included a static local-field correction in the description of acoustic waves in a strongly coupled dust system. MD simulations are often invoked for investigating collective processes of dusts in plasmas. Winske et al.[16] found that the values of the coupling parameter and the number of simulation particles can cause the DAWs to have either a fluid- or lattice-like wave dispersion relation. Durniak et al.[17] used MD simulation to study the dynamics of charged dusts. Schmidt et al.[18] showed that before the transition to the Wigner lattice there can appear a transverse shear wave. Klumov et al.[19] investigated the formation and propagation of lattice waves in a three-dimensional complex plasma. Hartmann et al.[20] reported the dramatic transformation of the anisotropic phonon dispersion of the crystal lattice near the solid-liquid transition into the isotropic liquid dispersion. Hou et al.[21] measure a cutoff wavenumber in the transverse mode. Many of the MD simulation results have been found to be consistent with those from the theoretical models and experiments. However, the properties of the dust lattice waves still require further investigation. For example, the effect of the particle number density on the wave dispersion has not been studied in detail. In this Letter, we investigate the structure and wave characteristics of dust lattices. In particular, the dispersion relations for both the longitudinal and transverse DLWs are analyzed. In our simulation, a Yukawa potential[8,22] is used. We put 400 identical particles in a two-dimensional (2D) box of size $L\times L$, where $L=20a$ ($a$ is the interparticle distance) with periodic boundary conditions. Thus for given particle number density $n_{\rm d}$, the cutoff radius should be less than $L/2$, where $L=\sqrt {N/n_{\rm d}}$ with $N$ being the total particle number. In the MD simulation we shall use the NVT, or constant-temperature, assumption and use the Nosé–Hoover thermostat[23] to maintain the temperature at $T=0.0001$ (in the energy unit). The time step is $0.01\omega _{\rm pd}^{-1}$, where $\omega _{\rm pd} =\sqrt {Q^2/\varepsilon _0 Ma^3}$ is the plasma frequency, and for most runs we find that 6$\times$10$^{4}$ steps are adequate. In the following the length is normalized by the Debye length $\lambda _{\rm D}$, and the time is normalized by $t_0 =\sqrt {4\pi \varepsilon _0 M\lambda _{\rm D} ^3/Q^2}$. The dispersion relations of the transverse and longitudinal waves are then obtained by the current correlation functions together with the Fourier transformation. The current operator can be expressed as[18,24] $\overrightarrow j (q,t)=\sum\nolimits_{m=1}^N \overrightarrow v_m (t)e^{i\overrightarrow q \cdot \overrightarrow r_{m} (t)}$, where $\overrightarrow v_m (t)$ and $\overrightarrow r_{m} (t)$ are the velocity and the position of the particle $m$ at the time $t$, respectively, and ${\boldsymbol q}={\boldsymbol k}a$ is the normalized wave number. The transverse and longitudinal current correlation functions are then $$\begin{alignat}{1} \!\!\!\!\!\!C_{\rm t} (q,t)=\,&(2N)^{-1}\langle [\overrightarrow q \times \overrightarrow j _q (t)]\cdot [\overrightarrow q \times \overrightarrow j _{-q} (0)]\rangle,~~ \tag {2} \end{alignat} $$ $$\begin{alignat}{1} \!\!\!\!\!\!C_{\rm l} (q,t)=\,&N^{-1}\langle [\overrightarrow q \cdot \overrightarrow j _q (t)][\overrightarrow q \cdot \overrightarrow j _{-q} (0)]\rangle.~~ \tag {3} \end{alignat} $$ The correlation functions in the frequency space are then $$\begin{align} \widetilde{C}_{t,l} (q,\omega)=\int_0^\infty {e^{i\omega t}} C_{t,l} (q,t)dt.~~ \tag {4} \end{align} $$ The configuration for $T=0.0001$ and $n_{\rm d} =1.0$ is shown in Fig. 1. We can see that it is hexagonal[25] (each particle has six nearest neighbors, as marked by thin lines around the central grain in Fig. 1). As $n_{\rm d}$ increases, the whole system becomes more compact (the distance between the nearest-neighboring particles becomes shorter), and at the same time the lattice structure may also be modified due to the repulsive interaction force,[26] which can cause the change of the wave dispersion properties. In the following we investigate the latter in detail.
cpl-34-7-075203-fig1.png
Fig. 1. The lattice structure for $n_{\rm d}=1.0$.
Before presenting the dispersion relation, we first consider the method for choosing the values of the wave vector $q$. In the wave dispersion curves $\omega$ versus $q$ for different $n_{\rm d}$ shown in Fig. 3, we have used the interparticle distance $a$ of $n_{\rm d} =1.0$ to normalize $q$, so that $0\leqslant q < 2\pi$ for $n_{\rm d} =1.0$. For 400 particles in the square box, within a period there are twenty $q$ values each in the parallel and perpendicular directions. The Brillouin zones for different $n_{\rm d}$ are different. That is, the twenty $q$ values in the three $\omega$ versus $q$ curves are staggered under three $n_{\rm d}$ values. Without affecting the behavior of the $\omega$ versus $q$ curves, for a rough comparison one can use the same twenty $q$ values of the three curves. Thus for the same $q$ value, the system with small (large) $n_{\rm d}$ will have a small (large) Brillouin zone. This approach for choosing the $q$ values is valid especially for systems containing a large number of particles (i.e., $q$ has almost continuous values). Accordingly, in the following we shall use the same twenty $q$ values for different $n_{\rm d}$. The transverse and longitudinal waves can propagate in the parallel ($\theta =0$, where $\theta$ is the angle between the wave vector and the primitive translation vector shown in Fig. 1) as well as perpendicular ($\theta =\pi /2$) directions.[27,28] Figure 2 shows the $C_{l,t} (q,\omega)$ spectra of the transverse and longitudinal waves along the parallel and perpendicular directions. As is seen, the $C_{l,t} (q,\omega)$ spectra for $n_{\rm d} =0.5$ (a$_{1}$)–(d$_{1}$) and $n_{\rm d} =1.0$ (a$_{2}$)–(d$_{2}$) in the four branches nearly have a single peak. However, when the particle number density increases to $n_{\rm d} =1.8$ (a$_{3}$)–(d$_{3}$), small new peaks appear. It means that as $n_{\rm d}$ is increased, the lattice structure is modified. In addition, the fluctuation of the amplitude can be seen on the current correlation functions in Fig. 2. During performing our MD simulation, the nonlinear effect has been included. Similar phenomena on the amplitude perturbation are also investigated in the nonlinear waves such as dust ion-acoustic (DIA) solitons.[29,30] The temporal evolution on the perturbation of the amplitude shows that it is damped monotonically with time for the weakly dissipative hybrid DIA nonlinear wave.[29] When in the presence of electromagnetic radiation (positively charged dust), the dissipative processes during the nonlinear wave propagation are depressed, that is, the nonlinear waves are damped more slowly than those in the case of negatively charged dust.[30]
cpl-34-7-075203-fig2.png
Fig. 2. The wave spectra of the longitudinal ($C_{\rm l}(q,\omega)$) and transverse ($C_{\rm t} (q,\omega)$) current correlation functions for parallel ($\theta =0$) and perpendicular ($\theta =\pi /2 $) propagations: (a$_{1}$)–(a$_{3}$) $C_{\rm l} (q,\omega)$, $\theta =0$; (b$_{1}$)–(b$_{3}$) $C_{\rm t} (q,\omega)$, $\theta =0$; (c$_{1}$)–(c$_{3}$) $C_{\rm l} (q,\omega)$, $\theta =\pi /2$; (d$_{1}$)–(d$_{3}$) $C_{\rm t} (q,\omega)$, $\theta =\pi /2$. Here (a$_{1}$)–(d$_{1}$), (a$_{2}$)–(d$_{2}$) and (a$_{3}$)–(d$_{3}$) are for the particle number densities 0.5, 1.0, and 1.8, respectively. The wave number $q$ for each wave branch varies from 0.314 to 5.966.
cpl-34-7-075203-fig3.png
Fig. 3. The dispersion relation of longitudinal (l) and the transverse (t) waves in parallel ($\theta =0$) and perpendicular ($\theta =\pi /2$) directions for the number densities 0.5 ($\square $), 1.0 ($\times$), and 1.8 ($\vartriangle $).
The dispersion relations $\omega$ versus $q$ of the longitudinal and transverse waves can be obtained from Fig. 2 and are plotted in Fig. 3. The interparticle distance $a$ for $n_{\rm d} =1.0$ is used in the normalizations of $q$ and $\omega _{\rm pd}$. As mentioned, for $n_{\rm d} =1.0$, $q$ varies from 0 to $2\pi$. For $n_{\rm d} =0.5$, the Brillouin zone is smaller in size and $q$ varies from 0 to 4.44. Similarly, for $n_{\rm d} =1.8$, $q$ varies from 0 to 8.42. In contrast to the longitudinal waves, anisotropy (i.e., $\theta$-dependence) is significant for the transverse waves.[31] For $n_{\rm d} =1.0$, the dispersion curves $\omega ({\rm l},\pi /2)$ versus $q$ and $\omega ({\rm l},0)$ versus $q$ have a maximum at about $q=3.14$ and they are symmetrical with respect to this maximum. This can be attributed to the formation of standing waves at the boundary of the first Brillouin zone.[8] On the right side of the maximum, $\omega$ decreases for short wavelength and the dispersion relation becomes negative, i.e., the directions of the phase velocity and the group velocity are opposite.[8,16] For $n_{\rm d} =0.5$, the curves increase again (after the decrease), indicating that the first Brillouin zone for $n_{\rm d} =0.5$ is smaller than that for $n_{\rm d} =1.0$. The maximum of the dispersion curve for this case is at $q\approx 2.22$. For $n_{\rm d} =1.8$, the two dispersion curves start to flatten after the maximum, showing that when $n_{\rm d}$ is sufficiently high, the group velocity becomes zero at short wavelengths. Such dispersion properties are similar to the investigation on other particle numbers.[16] It gives three cases with particle numbers of 128, 64, and 32, i.e., one particle per cell, one particle for every two cells, and one particle every four cells. It is in agreement with our result of the dispersion with changing particle number density. In addition, the increase of particle number density will cause the interparticle distance to be shorter, and accordingly decrease the screening strength $\kappa =a/\lambda _{\rm D}$. The smaller $\kappa$ will cause the higher $\omega$, which is also in agreement with the result in Refs. [8,20]. From Fig. 3 it can be seen that at a specific $q$ value, the wave frequency increases with the particle number density (within the range considered). Due to the fact that waves excited in the Yukawa systems are collective motions of the constituent particles and can be characterized by space-time correlation functions,[22] during the oscillation, distances between the grains change and the grains' interaction potential changes as well. From this point of view, the increase of particle number density (within the range considered) can make the particles' interaction stronger and the collective effect becomes remarkable, which can be reflected in the wave frequency. The similar result is shown in the investigation on dust lattice waves, that is, the frequency of the lattice oscillation increases considerably for large grain charges,[31] which means the stronger particles' interaction.
cpl-34-7-075203-fig4.png
Fig. 4. (a)–(f) Dispersion relation of the longitudinal (l) and the transverse (t) waves in the parallel ($\theta =0$) direction for $0.5\leqslant n_{\rm d} \leqslant 1.8$.
We also find that the longitudinal and transverse waves intersect at a common frequency $\omega _{\rm int}$, corresponding to the Einstein frequency.[32] Figure 4 shows the parallel transverse and longitudinal waves for different $n_{\rm d}$. In Fig. 4(a) there are two intersects, caused by the shorter Brillouin range for small $n_{\rm d}$ due to our normalization. For $n_{\rm d} =0.5$, when $q$ varies in (0, 4.44), there is only one intersect. We see that the common frequency (the first intersect for $n_{\rm d} =0.5$) moves towards large $q$ (or short wavelength) as $n_{\rm d}$ increases. Figure 5 is for $\omega _{\rm int}$ versus $n_{\rm d}$. It can be seen that $\omega _{\rm int}$ increases with $n_{\rm d}$ for $0.5\leqslant n_{\rm d} \leqslant 1.8$, or the common frequency is very low when $n_{\rm d}$ is very small. In the investigation, the common frequency decreases with increasing the screening strength $\kappa$.[8] This may be due to the fact that with increasing the screening strength the interaction potential becomes less, thus the wave frequency decreases. In our MD simulation, a change of $n_{\rm d}$ leads to the change of the interparticle potential: smaller $n_{\rm d}$ means a larger interparticle distance and thus weaker interaction potential and lower common frequency, which is similar to the result in Ref. [8].
cpl-34-7-075203-fig5.png
Fig. 5. Variation of the common wave frequency $\omega _{\rm int}$ with the particle number density, obtained from the dispersion relation in Figs. 4(a)–4(f).
cpl-34-7-075203-fig6.png
Fig. 6. The sound speeds of the transverse and longitudinal waves with different $n_{\rm d}$.
In the limit $q\to 0$, the sound speeds of the waves tend to their acoustic limit.[33] Figure 6 shows the sound speeds of the parallel and perpendicular transverse and longitudinal waves as a function of the particle number density. It can be seen that the sound speeds of longitudinal waves are larger than those of the transverse waves. Liu et al.[8] showed that for a perfect hexagonal lattice, the theoretical sound speeds of two longitudinal (or transverse) waves will be approximately the same. Figure 6 also shows that the sound speeds of longitudinal waves increase with the particle number density, but those of the transverse waves tend to remain at low values. In conclusion, the dust lattice structure and dispersion properties of both longitudinal and transverse dust waves in a 2D Yukawa dust system have been investigated. It is found that modification of the lattice structure due to the increase of particle number density can lead to the change of the wave properties, and when the density is high, a cutoff region appears in the dispersion relations at the short wavelength. It is also found that the dispersion curves of the longitudinal and transverse waves can intersect, and the intersection, or common frequency tends to increase with the particle number density. The sound speeds of longitudinal waves also increase with the particle number density, but those of the transverse waves remain at low values. The present investigation should be helpful for a better understanding of the relationship between wave dispersion and the density, and we can roughly judge the change of density from the dispersion relations. References Contribution of the Dust Grains to the Damping of the Electromagnetic Waves Propagating in PlasmaCharge of Dust on Surfaces in PlasmaEffects of the Charge-Dipole Interaction on the Coagulation of Fractal AggregatesManipulating Dust Charge Using Ultraviolet Light in a Complex PlasmaExperimental studies of charged dust particlesLongitudinal collective modes of strongly coupled dusty plasmas at finite frequencies and wavevectorsCoulomb solid of small particles in plasmasWave dispersion relations in two-dimensional Yukawa systemsDispersion relations of longitudinal and transverse waves in two-dimensional screened Coulomb crystalsLaser-excited dust lattice waves in plasma crystalsDispersion relations for the dust-acoustic wave under experimental conditionsDust-acoustic waves in self-gravitating dusty plasmas with Lorentzian electrons and ionsExperimental Observation of Vertically Polarized Transverse Dust-Lattice Wave Propagating in a One-Dimensional Strongly Coupled Dust ChainLattice modes in a zigzag crystal of dust particlesSelf-Organization of Charged Particulates in the Presence of External ForceNumerical simulation of dust-acoustic wavesMolecular-Dynamics Simulations of Dynamic Phenomena in Complex PlasmasLongitudinal and transversal collective modes in strongly correlated plasmasCrystallization waves in a dusty plasmaCollective Modes in 2-D Yukawa Solids and LiquidsWave spectra of two-dimensional dusty plasma solids and liquidsMolecular dynamics evaluation of self-diffusion in Yukawa systemsCanonical dynamics: Equilibrium phase-space distributionsWave Dispersion Relations in Yukawa FluidsExperimental determination of the charge on dust particles forming Coulomb latticesMD Simulation of Charged Dust Particles With Dipole MomentsDust lattice wave dispersion relations in two-dimensional hexagonal crystals including the effect of dust charge polarizationWaves in two-dimensional hexagonal crystalEvolution of weakly dissipative hybrid dust ion-acoustic solitons in complex plasmasWeakly dissipative dust-ion-acoustic solitons in complex plasmas and the effect of electromagnetic radiationLinear and nonlinear dust lattice waves in plasma crystalsCollective Modes in Strongly Correlated Yukawa Liquids: Waves in Dusty PlasmasLow-frequency modes in two-dimensional Debye-Yukawa plasma crystals
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