Dispersion relation of square lattice waves in a two-dimensional binary complex plasma
DDispersion relation of square lattice waves in a two-dimensional binarycomplex plasma
Z.-C. Fu, A. Zampetaki,
2, 3
H. Huang, and C.-R. Du
1, 4, a) College of Science, Donghua University, 201620 Shanghai, People’s Republic of China Max-Planck-Institut für Extraterrestrische Physik, 85741 Garching, Germany Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, 40225 Düsseldorf,Germany Member of Magnetic Confinement Fusion Research Centre, Ministry of Education, 201620 Shanghai,People’s Republic of China (Dated: 31 August 2020)
Binary complex plasmas consist of microparticles of two different species and can form two-dimensional square latticesunder certain conditions. The dispersion relations of the square lattice waves are derived for the longitudinal andtransverse in-plane modes, assuming that the out-of-plane mode is suppressed by the strong vertical confinement. Theresults are compared with the spectra obtained in Langevin dynamics simulations. Furthermore, we investigate thedependence of the dispersion relation on the charge ratio and mass ratio of the two particle species.PACS numbers: 52.27.LwKeywords: Complex plasmaComplex plasmas consist of a mixture of weakly ionizedgases and microparticles. The latter acquire a charge dueto the flow of the surrounding ions and electrons which arenegative, owing to the higher thermal velocity of electrons .Considering the plasma screening effect, the interaction be-tween the microparticles can be described via the Yukawapotential . In the laboratory, the charged particles are usu-ally suspended in the sheath above the lower electrode ofa radio-frequency (rf) discharge, where the gravity force isbalanced by the electric force. In strongly coupled complexplasmas, monodisperse microparticles can be vertically con-fined to a single layer and form a hexagonal lattice, knownas plasma crystal . Due to the stretched time scales andlow damping, two-dimensional (2D) complex plasma crystalsprovide an unique opportunity to study generic processes insolids and liquids at the kinetic level . With external manip-ulations by electric fields or laser beams, various phenomenasuch as melting and recrystallization , microstructure undershear , Mach cone excitations , and entropy production have been investigated both experimentally and theoretically.One of the most defining properties of plasma crystals isthe dispersion relation of the microparticles’ collective oscil-lations, in the form of lattice waves. This has been derivedanalytically and measured directly using video microscopyin the case of monodisperse complex plasmas . Remark-ably, due to the strong ion flow in the sheath, the interactionsbetween microparticles are altered by the so-called wake ef-fect, resulting in the coupling of the horizontal and verticalmodes . This eventually triggers a mode-coupling insta-bility (MCI) and causes the crystal to melt .A binary complex plasma consists of microparticles oftwo different species. With an appropriate selection oftheir mass and size, these particles can form, in the labora-tory, a bilayer or a quasi-two-dimensional (q2D) crystalline a) Electronic mail: [email protected] suspension . The phonon spectra for these structureshave been measured experimentally and studied by a quasi-localized charge approximation approach and molecular dy-namics simulations . A mode coupling between the hori-zontal modes in the two layers, mediated by the plasma wakes,has been proposed .Meanwhile, taking advantage of the plasma etching effect,the two particle species can be suspended at the same heightfor a certain amount of time . Under these conditions, bi-nary complex plasmas have been found to form square lat-tices with a quadruple symmetry as the one presented inFig. 1. A strong vertical confinement can efficiently suppressthe vertical motions and thus the expected MCI . A natu-ral question that then arises is how the horizontal wave modesare modified in such 2D square lattices compared to the well-studied case of hexagonal lattices in monodisperse 2D com-plex plasma crystals. FIG. 1. (a) The sketch of a square lattice with particle spacing a in a2D binary complex plasma consisting of the particle species A and B .(b) The reciprocal lattice in the k -space, where the basis vectors are b , = π a − ( / , ± / ) . Due to the square symmetry, we considerthe wave vectors k at 0 ◦ and 45 ◦ in the first Brillouin zone (the shadedarea surrounded by a dashed line). In this article, we derive the dynamical matrix of a 2D bi-nary complex plasma crystal in order to study the dispersion a r X i v : . [ phy s i c s . p l a s m - ph ] A ug relation of the formed square lattice and to investigate its de-pendence on the involved particle charges and masses. In ourapproach we assume a very strong confinement, resulting inthe particles being confined on a 2D plane, and allowing us toneglect the effect of the plasma wakes.The interaction between the charged microparticles is as-sumed to be of a Yukawa form. Thus, each particle i of species p = A or B and a charge Q p , is a source of a Yukawa potential ϕ p , i ( r ) = Q p (cid:12)(cid:12) r − r p , i (cid:12)(cid:12) exp (cid:0) − (cid:12)(cid:12) r − r p , i (cid:12)(cid:12) / λ D (cid:1) (1)with a screening length λ D at its 2D position r p , i = x p , i (cid:98) x + y p , i (cid:98) y . Consequently the equations of motion for the i -th par-ticle of species A and the i -th particle of species B , in the 2D plane of confinement, read d r A , i dt + ν d r A , i dt = − Q A M A ∇ A , i (cid:32) ∑ j (cid:54) = i ϕ A , j ( r A , i ) + ∑ j ϕ B , j ( r A , i ) (cid:33) , d r B , i dt + ν d r B , i dt = − Q B M B ∇ B , i (cid:32) ∑ j ϕ A , j ( r B , i ) + ∑ j (cid:54) = i ϕ B , j ( r B , i ) (cid:33) , (2)where ν is the frictional drag coefficient and M A , B and Q A , B are the mass and the charge of the particle species A and B ,respectively. Note also that the symbols ∇ A , i , ∇ B , i denote thegradients with respect to the vectors r A , i and r B , i , accordingly.Our scope here is to investigate the vibrational properties ofthe 2D binary complex plasma crystal around its square latticeequilibrium, shown in Fig. 1. In this crystalline configuration,the microparticles of the species p = A or B are located at thepositions R p = X p (cid:98) x + Y p (cid:98) y with X A = ( m + n ) a , Y A = ( m − n ) a , X B = − X A and Y B = Y A where m , n are arbitrary integersand a denotes the square lattice constant (Fig. 1). Linearizingaround this equilibrium and introducing the plane wave ansatz d p = d ( ) p exp [ − i ω t + i ( k x X p + k y Y p )] (3)for the displacement d p of the particles of species p from theirequilibrium positions R p , we arrive, in view of the square lat-tice symmetry, at the 4 × D = Q A M A F sA + Λ Q F oB G sA + Λ Q G oB − Λ Q F lB − Λ Q G lB G sA + Λ Q G oB F sA + Λ Q F oB − Λ Q G lB − Λ Q F lB − Λ M Λ Q F lB − Λ M Λ Q G lB Λ M Λ Q F sA + Λ M Λ Q F oB Λ M Λ Q G sA + Λ M Λ Q G oB − Λ M Λ Q G lB − Λ M Λ Q F lB Λ M Λ Q G sA + Λ M Λ Q G oB Λ M Λ Q F sA + Λ M Λ Q F oB . (4)In this expression we have denoted the mass ratio as Λ M = M A / M B and the charge ratio as Λ Q = Q A / Q B . The elements F op , F lp , and F sp with p = A or B are given by the followingsums of the effective spring constant F ( X , Y ) over the latticepositions ( X p , Y p ) , excluding the central position: F op = ∑ X p , Y p F ( X p , Y p ) , F lp = ∑ X p , Y p F ( X p , Y p ) cos ( k x X p + k y Y p ) , F sp = F op − F lp . (5)Such summations also apply to the rest of the matrix elements F o , l , sp and G o , l , sp . The corresponding effective spring constants read F ( X , Y ) = R − e − R / λ D [ X ( + R / λ D + R / λ D ) − R ( + R / λ D )] , F ( X , Y ) = F ( Y , X ) , G ( X , Y ) =( XY / R ) e − R / λ D ( + R / λ D + R / λ D ) (6)with R = √ X + Y .The eigenvalues Ω j of the dynamical matrix D are con-nected with the eigenfrequencies ω j of the crystal through therelation Ω j = ω j ( ω j + i ν ) . For simplicity, we approximatein the following theoretical results the ω j with the respectivevalues of Ω j , under the assumption that the damping in oursystem is very small ( ν (cid:28) ω j ) . Since D is a 4 × Ω j ( k x , k y ) , two of which aretransverse and two longitudinal.An example of these spectra for Λ M = Λ Q =
8, atthe two characteristic wave vector angles for the square lat-tice, θ = ◦ and θ = ◦ , is shown in Fig. 2 (a),(b) andFig. 2 (c),(d), respectively. Each of the longitudinal [Fig. 2(a),(c)] and the transverse modes [Fig. 2 (b),(d)], possessestwo branches, due to the two-particle unit cell of the binarysquare lattice. From these the lower one corresponds to theacoustic branch, with the two particles of the unit cell oscil-lating in phase, and the higher one to the optical branch, withthe two particles oscillating out of phase. FIG. 2. The phonon spectra obtained from the Langevin dynam-ics simulation (color map) and the theoretically calculated dispersionrelations, i.e. the eigenvalues Ω j of the dynamical matrix D (whitecurves). Upper panels show (a) the longitudinal and (b) the trans-verse modes at the wave vector angle θ = ◦ . Lower panels show (c)the longitudinal and (d) the transverse modes at θ = ◦ . In order to corroborate our theoretical calculation, weperform a 2D Langevin dynamics simulation with periodicboundary conditions using LAMMPS in the NVT ensemble .In our simulation, the two species of particles are modelledas negative point-like charges and are arranged in a squarelattice in a single layer, as illustrated in Fig. 1. Note thatwe do not apply any external excitation to trigger the forma-tion of lattice waves. Instead, we measure the phonon spectrafrom the collective thermal motions of the particles by ap-plying a Langevin thermal bath of temperature T =
500 Kwith a damping rate ν = . − . We also set the inter-particle distance to a = . λ D = . N = M A = M B = × − kg, whereas their charges are set to Q A = e and Q B = e , respectively. Further detailsof the simulation can be found in the references .The wave spectra V k , ω of the simulated square lattices inthe binary complex plasma are computed using the 2D Fouriertransform V k , ω = / ST (cid:90) S (cid:90) T v ( r , t ) exp [ − i ( k · r + ω t )] dsdt , (7) FIG. 3. The dependence of the dispersion relation of the binarysquare crystal on the charge ratio Λ Q =
6, 7, 8 and 9. Upper pan-els show (a) the longitudinal and (b) the transverse modes at θ = ◦ .Lower panels show (c) the longitudinal and (d) the transverse modesat θ = ◦ . The mass ratio used is Λ M = where S and T are the linear size of the area in the simula-tion and the period over which particle motion is summed,respectively . As shown in Fig. 2, both the acoustic and theoptical branches of the wave spectra can be clearly identifiedin our simulation results. They are also in a very good agree-ment with our theoretical results using the dynamical matrixapproach, indicating that the finite temperature and the damp-ing have a marginal influence on the vibrational properties ofthe system, as long as the square lattice structure persists.In order to study the dependence of the dispersion relationon the disparity of two particle species, we fix the mass andthe charge of the particle species B and vary in our theoreticalcalculations the mass ratio Λ M and the charge ratio Λ Q . Thedispersion relation for both the longitudinal and the transversemodes are shown in Fig. 3 for Λ M = Λ Q and in Fig. 4 for Λ Q = Λ M .As we observe in Fig. 3, the eigenfrequencies of the longi-tudinal and transverse optical branches increase significantlyas the charge ratio Λ Q increases, at both θ = ◦ and θ = ◦ .In contrast, the eigenfrequencies of the acoustic branches in-crease only moderately. Particularly, the frequencies of thetransverse acoustic branch at θ = ◦ [Fig. 3 (d)] seem to behardly affected by the variation of Λ Q .Regarding the dependence of the dispersion relation of thebinary complex plasma crystal on the mass ratio Λ M (Fig. 4),the frequencies of all the optical branches decrease as Λ M in-creases. However, for the acoustic branch, the dispersion rela-tion does not show any dependence on the mass ratio, exceptfor the transverse acoustic branch at θ = ◦ . For this branch,as depicted in Fig. 4 (d), the frequencies decrease with Λ M atthe same extent as they do for the optical branch. FIG. 4. The dependence of the dispersion relation of the binarysquare crystal on the mass ratios Λ M =
1, 1 .
5, 2 and 2 .
5. Upper pan-els show (a) the longitudinal and (b) the transverse modes at θ = ◦ .Lower panels show (c) the longitudinal and (d) the transverse modesat θ = ◦ , respectively. The charge ratio used is Λ Q = To summarize, we have studied the dispersion relationsof square lattice waves in a 2D binary complex plasma us-ing the dynamical matrix approach. Both acoustic and opti-cal branches are observed for the longitudinal and transversemodes of the system, as expected for a binary system. Theresults are found to be in a good agreement with Langevindynamic simulations, allowing us to verify our calculations.Furthermore, we have investigated the dependence of the dis-persion relation on the charge ratio and the mass ratio of thetwo particle species. The results provide a comprehensive un-derstanding of the phonon spectra in square lattices of 2D bi-nary complex plasmas.For the study of the square lattice observed in quasi-2Dcomplex plasmas , the effect of the ion wakes in theplasma sheath must be explicitly considered, since it alters theparticles’ interactions, rendering them non-reciprocal . Inthe simplest approach the wakes can be modeled as point-likepositive charges located directly beneath the microparticles and the dispersion relation of the lattice waves can be calcu-lated through the dynamical matrix, following a similar pro-cedure as the one employed in this article. Our results cantherefore serve as a benchmark for the more involved calcu-lations for quasi-2D complex plasmas with non-reciprocal in-teractions, which will be addressed in future works. ACKNOWLEDGMENTS
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