Plasma Crystal in (3 + 1) Dimensions

A quasi-two-dimensional plasma crystal in (3 + 1) dimensions has been experimentally observed for the first time; i.e., three spatial coordinates of each microparticles of the crystal for a long time have been determined. As a result, fine details of the structural evolution of such a system upon its melting and crystallization have been identified.

INTRODUCTION

Interest of various research groups has been focused on the plasma crystal since its discovery [1–4]. On the one hand, this is due to interest in processes occurring in the plasma containing microparticles (such a plasma is called a complex or dusty plasma) because this plasma is abundant in nature. Interstellar clouds, gaseous dust clouds, planetary rings, atmospheres of comets, and dusty structures in upper layers of atmospheres and ionospheres of planets (e.g., noctilucent clouds in the Earth) are to a certain extent complex plasmas [5, 6]. On the other hand, systems of microparticles (in particular, plasma crystals) studied in laboratories, usually being strongly coupled systems, are of great interest in the context of condensed matter physics because microparticles in experiments with such a complex plasma are easily identified by optical methods. As a result, one can follow the behavior of each individual microparticle and thereby study processes of melting, crystallization, development of various instabilities, excitation of collective modes in a system, etc., at the most detailed (“atomic”) level [7, 8].

The complex plasma in laboratories is conventionally obtained by adding microparticles to a gas-discharge plasma of inert gases at a low pressure. The recombination of electrons and ions of the plasma on the surface of microparticles results in their fast charging; as a result, a microparticle can acquire a large typical charge of Z_d ~ 10³–10⁴e, where e is the elementary charge. The large charge of microparticles often leads to the crystallization of an ensemble of microparticles with the formation of a plasma crystal. A two-dimensional or quasi-two-dimensional plasma crystal is of particular interest in the context of the physics of two-dimensional melting (e.g., [9, 10]) and the properties of a two-dimensional liquid [11]. Such an almost planar system is formed upon the injection of microparticles into a gas-discharge plasma in a near-electrode region, where the gravitational force acting on particles is balanced by the electric field (e.g., [8]). Modern high-resolution video cameras allow one to determine the trajectories of individual particles. The trajectories of particles in such quasi-two-dimensional systems only in the horizontal (x, y) plane (plane of the monolayer of microparticles) were experimentally studied to date, whereas their motion in the transverse direction (in the vertical z direction) was ignored or only the integral optical characteristics of vertical displacements of the ensemble of microparticles were analyzed. We emphasize that the knowledge of vertical displacements of microparticles in such systems is crucially important for understanding their evolution (in particular, upon their melting and crystallization). In this work, a quasi-two-dimensional plasma crystal in (3 + 1) dimensions is experimentally observed for the first time; i.e., three spatial coordinates x, y, and z of each microparticles of the crystal for a long time t are determined. A very important feature of this experiment is that the parameters of a discharge remain unchanged during the observation of microparticles. Below, we briefly describe the experimental setup and discuss the first results of observations.

EXPERIMENT

The main part of the experimental setup is the modified new-generation plasma chamber Zyflex [12], where a capacitively coupled RF discharge in argon at a frequency of 13.56 MHz is created. This chamber was designed to study the dynamic and structural properties of two- and three-dimensional dusty structures in the gas-discharge plasma. To study quasi-two-dimensional systems, we used a solid electrode, which makes it possible to obtain more homogeneous dusty structures; the upper electrode was removed, and the upper part of the chamber was made transparent for video observation of the dusty structure. Figure 1 shows the view of the setup, the positions of high-resolution video camera, and a typical two-dimensional dusty plasma crystal observed in this setup.

Fig. 1.

(Color online) (а) View of the new-generation gas-discharge chamber Zyflex and an almost planar (in the (x, y) plane) two-dimensional plasma crystal formed after the injection of monodisperse polymer microparticles, which is shown in the inset in a magnified form. A disk shape of the plasma crystal is determined by the axial symmetry of the nearly parabolic confinement potential $U(r) \propto$ r² [13, 14]. (b) Optical system that makes it possible to determine three coordinates of microparticles at each time. This system includes four synchronized high-resolution video cameras record from different directions microparticles illuminated by laser radiation.

The characteristic RF voltage applied to the electrode ${{V}_{{pp}}}$ is 60–200 V, and the gas pressure can be varied in the range of 0.1–250 Pa. Microparticles (usually monodisperse polymer particles with a size of several microns) are injected into the glowing discharge by means of an electromagnetic dispenser; the number of microparticles injected into the discharge is controlled by the time of application of the voltage to an electromagnet and the number of injections. The electrode 114 mm in diameter allows one to create dusty plasma structures containing up to several thousand particles (with a characteristic distance between them of about 100 μm). A 2-mm-high metallic ring located at the edge of the electrode induces an additional horizontal electrostatic potential confining the motion of microparticles in the central region of the discharge. As shown in [13, 14], the confinement potential $U(r)$ in the center of the discharge chamber in the plane of the monolayer of microparticles is close to parabolic $U(r) \propto {{r}^{2}}$, and its axial symmetry in this plane, in particular, explains the disk shape of the two-dimensional plasma crystal shown in Fig. 1.

The video diagnostics and optical tomography of microparticles were performed with four calibrated synchronized high-speed (180 fps) high-resolution ($2048 \times 2048$ pixel) video cameras, which recorded the evolution of microparticles through transparent walls of the Zyflex chamber from different directions. A scheme implementing such a tomography is presented in Fig. 1b. Microparticles were illuminated by a 3-W laser whose beam was swept to a horizontal sheet with a controlled thickness in the range of 1.5–3.5 mm in the field of view of the video cameras. The studied optically thin system of microparticles makes it possible to accurately determine the spatial positions of particles at any time.

To determine the position of the image of a particle in a frame with a subpixel accuracy, the known method of identification of particles in experiments with the complex plasma [15, 16] based on the separation of a group (cluster) of pixels corresponding to each individual microparticle is applied. In this case, the coordinates of the microparticle are calculated as weight-averaged coordinates of these pixels with their brightness used as their weight.

The three coordinates of a microparticle can be calculated knowing the two-dimensional coordinates of its images from two calibrated video cameras (stereopair). To calibrate the cameras, we recorded a special checkerboard calibration pattern. The processing of the data thus obtained makes it possible to determine the distortion of lenses, the internal parameters of the cameras, and their relative positions [17, 18]. The three coordinates were calculated by the triangulation method [19] using data from two cameras 1 and 3 (see Fig. 1b), which were located at the largest angle to each other.

For the triangulation of the coordinates of the microparticle, it is necessary to find its images from cameras 1 and 3. According to the computer stereovision theory, images of a certain point in space lie on the so-called epipolar lines ${{{\mathbf{l}}}^{1}}$ and ${{{\mathbf{l}}}^{2}}$ for the first and second calibrated video cameras, respectively [19, 20]. These epipolar lines can be determined using the fundamental 3 × 3 matrix F such that ${{{\mathbf{l}}}^{1}} = {{F}^{{\text{T}}}}{{{\mathbf{x}}}_{2}}$ and ${{{\mathbf{l}}}^{2}} = F{{{\mathbf{x}}}_{1}}$, where ${{{\mathbf{x}}}_{1}}$ and ${{{\mathbf{x}}}_{2}}$ are the coordinates of the images of a point from the first and second cameras in the homogeneous representation (${\mathbf{x}} = (x,y{{,1)}^{{\text{T}}}}$). Figure 2 illustrates the search for correspondences between images of microparticles from cameras 1 and 3. The circle in the image from camera 1 marks a microparticle whose image from camera 3 should be found. To this end, we plot the epipolar line ${\mathbf{l}}_{{{\text{c}}1}}^{3}$ and find the images of microparticles located within the allowable deviation from this line. If these images are more than one, it is necessary to use information from the remaining cameras. Images located near the epipolar lines ${\mathbf{l}}_{{{\text{c}}1}}^{2}$ and ${\mathbf{l}}_{{{\text{c}}1}}^{4}$ are sought from cameras 2 and 4, and the epipolar lines ${\mathbf{l}}_{{{\text{c}}2}}^{3}$ and ${\mathbf{l}}_{{{\text{c}}4}}^{3}$ corresponding to these images are plotted. The desired image from camera 3 lies at the intersection of three epipolar lines. This optical tomography of the plasma crystal allows one to determine with a high accuracy the coordinates of microparticles at each time instant, i.e., to describe the system in 3 + 1 dimensions.

Fig. 2.

Images of a part of the structure of dust particles obtained at the same time by four cameras. The circle in camera 1 image marks the analyzed particle. Circles in other images mark points sufficiently close to the epipolar lines ${\mathbf{l}}_{{{\text{c}}1}}^{2}$, ${\mathbf{l}}_{{{\text{c}}1}}^{3}$, and ${\mathbf{l}}_{{{\text{c}}1}}^{4}$. Lines ${\mathbf{l}}_{{{\text{c}}2}}^{3}$ and ${\mathbf{l}}_{{{\text{c}}4}}^{3}$ certainly indicate the desired image from camera 3.

EXPERIMENTAL RESULTS

Some results of the observation of the evolution of the two-dimensional plasma crystal are presented below. It is important that the experiment was performed in argon at the unchanged parameters of the RF discharge (pressure p ≈ 0.7 Pa and V_pp ≈ 65 V). Microparticles with sizes of (7.01 ± 0.08) μm were injected into the glowing discharge; the formed plasma crystal included approximately 3 × 10³ particles. It is known that such microparticles in quasi-two-dimensional systems hardly affect the discharge plasma. In an observation time of about 1 min, we detected the stages of crystal, its heating, subsequent melting, and recrystallization of the melt. This is the first observation of phase transitions in experiments with quasi-two-dimensional systems at constant parameters of the discharge. Melting under such conditions means that the system of microparticles is unstable with respect to their vertical vibrations. The mechanism of such an instability will be discussed in more detail elsewhere. We now only note that the possible mechanism of melting of this system can be the mode-coupling instability; as a result of the development of this instability, the energy of the vertical vibrations of microparticles is efficiently transferred to horizontal modes [24].

Figure 3 presents combined trajectories of some microparticles in the plasma crystal until the stages of its melting and subsequent recrystallization, which corresponds to the evolution of the system for approximately 1 min or 10⁴ videoframes. To distinguish the motion of microparticles in the vertical confinement potential along the z axis, a fixed shift is added to the z coordinate of each microparticle in each next frame; i.e., $z({{t}_{k}}) = z({{t}_{k}}) + {{\delta }_{z}}(k - 1)$ for the kth frame, where ${{\delta }_{z}} = 0.5$ mm. The colors of the resulting combined trajectories correspond to velocities of microparticles in the horizontal (x, y) plane according to the color scale on the right. The system is melted at a certain time when transverse vibrations of particles result in the heating of the system in the plane of the crystal. Further, the system is recrystallized to the plasma crystal according to these trajectories and the general form of the system. Most microparticles at the melt stage leave the laser sheet region and their coordinates cannot be determined. This stage corresponds to the gap in trajectories at z ≈ 600 cm. This stage follows after a strong increase in the transverse velocity ${{v}_{z}}$ indicated by strong oscillations on trajectories of microparticles at z ≈ 500–600 cm. In addition, before the melting of the system, the plasma crystal is split in the center with the formation of two layers with a square lattice. The structure of the plasma crystal at this time is shown in Fig. 4a. This splitting is due to the radial inhomogeneity of the crystal (the density of particles is maximal in the center and decreases monotonically toward the periphery [14, 21, 22, 25]). In this case, the condition of the transition of the monolayer to a bilayer system occurs in the center of the system, where the density of particles is higher, as a result of the development of structural instability (the so-called $1\Delta \to 2\square$ transition) [23, 26]. In this case, the monolayer of charged microparticles with the triangular lattice is split into two layers with square lattices shifted with respect to each other. In our case, the periphery of the system holds the triangular symmetry and remains the monolayer. This is a specificity of the system of charged particles in the horizontal parabolic confinement. Figure 4b shows the results of simulation of the three-dimensional Yukawa system in the horizontal confinement with the parameters close to experimental ones. As in the experiment, the simulation demonstrates splitting in the center of the system into two layers with shifted square lattices. As in the experiment, splitting into two layers with shifted square lattices is observed in the center of the system, whereas the monolayer with the triangular lattice remains in the periphery of the system. The insets of Fig. 4 show the side views of the considered systems clearly demonstrating the splitting effect in the center of the plasma crystal.

Fig. 3.

(Color online) Trajectories of some microparticles of the plasma crystal until its melting and subsequent crystallization, which corresponds to the evolution of the system for approximately 1 min or 10⁴ videoframes. To distinguish the motion of microparticles in the vertical confinement potential along the z axis, a fixed shift is added to the z coordinate of each microparticle in each next frame; i.e., $z({{t}_{k}}) = z({{t}_{k}}) + {{\delta }_{z}}(k - 1)$ for the kth frame, where ${{\delta }_{z}} = 0.5$ mm. The colors of the resulting combined trajectories correspond to velocities of microparticles in the horizontal (x, y) plane according to the color scale on the right. This allows one to determine the time of melting of the plasma crystal when vertical (along the z axis) vibrations of particles result in the heating of the system in the plane of the monolayer. Further, the system is recrystallized as seen in trajectories. Most microparticles at the melt stage leave the laser sheet region and their coordinates cannot be determined. This stage corresponds to the gap in trajectories at z ≈ 600 cm. This stage follows after a strong increase in the vertical velocity ${{v}_{z}}$, indicated by strong oscillations on trajectories of microparticles at z ≈ 500–600 cm.

Fig. 4.

(Color online) (a) Fragment of the plasma crystal obtained in the experiment at the stage before melting when the system of microparticles in the center of the crystal is separated because of the radial inhomogeneity of the crystal (density of particles is maximal in the center and decreases monotonically toward the periphery [14, 21, 22]); in this case, the condition of the transition of the monolayer to a bilayer system occurs because of the development of an instability (the so-called buckling instability or the $1\Delta \to 2\square$ transition, i.e., when the monolayer of charged microparticles with a triangular lattice is split into two layers with a square lattice [23]). (b) Results of simulation of the three-dimensional Yukawa system in the horizontal confinement potential close to parabolic U(r) ~ (r – r_c)², where r_c is the center of the system, with parameters close to experimental values. As in the experiment, splitting into two layers with a shifted square lattice is observed in the center of the system, whereas the monolayer with the triangular lattice remains in the periphery of the system. The color of particles corresponds to the vertical z coordinate according to the legend in panel (a). The insets show the side views of the considered systems clearly demonstrating the splitting effect.

The square lattice in the quasi-two-dimensional system of microparticles was recently observed in [27] (the $1\Delta \to 2\square$ transition was observed in the center of the initially planar crystal with the triangular lattice upon the weakening of the vertical confinement). The authors of [27] ignored the properties of the horizontal confinement and explained this effect by an additional interaction between particles caused by ion focusing on microparticles [8]. As shown above, this is unnecessary: a conventional Coulomb system (or a Yukawa system) in the parabolic confinement has the indicated properties, and the shifted square lattice observed in the experiment evidences a weak effect of ion focusing on the structure of the crystal (otherwise, after the splitting of the crystal into two layers, these two layers would be located one above the other because of the vertical pairing of particles).

To summarize, the evolution of the quasi-two-dimensional plasma crystal in (3 + 1) dimensions (three spatial coordinates + time) until its melting and recrystallization at constant parameters of the RF discharge has been experimentally demonstrated for the first time using our optical tomography system. This opens a new stage of investigation of the laboratory complex plasma. It has been shown that the melting of the crystal occurs after its splitting in the center of the system into two layers with shifted square crystal lattices; i.e., the structural instability of the crystal and the $1\Delta \to 2\square$ transition are observed. This splitting is caused by the horizontal parabolic confinement, which is responsible for the inhomogeneity of the crystal in the radial (from its center) direction. In this case, since the density of microparticles is maximal in the center of the crystal, the condition for the development of this instability occurs in the center, whereas the peripheral part of the crystal holds the planar structure with the triangular lattice.