PHYSICS OF PLASMAS 14, 103708 共2007兲
Theoretical study of the wave dispersion relation for a two-dimensional strongly coupled Yukawa system in a magnetic field Ke Jiang, Yuan-Hong Song, and You-Nian Wanga兲 State Key Lab of Materials Modification by Beams, School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116023, China
共Received 22 June 2007; accepted 28 August 2007; published online 30 October 2007兲 A theoretical model is presented to investigate the wave dispersion relation of a two-dimensional 共2D兲 strongly coupled Yukawa system, taking into account a constant magnetic field pointing perpendicular to the 2D Yukawa system, within the framework of the quasilocalized charge approximation. Numerical results represent the dependence of the dispersion relation on the magnetic field strength, the coupling parameter, and the screening parameter. Both the high-frequency and low-frequency branches are shown as a result of the coupling of the longitudinal and transverse modes due to the Lorenz effect. The results obtained from the theoretical analysis agreed well with the molecular-dynamics simulation. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2789999兴 I. INTRODUCTION
Complex 共dusty兲 plasma is an ionized gas in which submicrometer–to–micrometer-sized particles, usually called dust grains, are embedded.1–5 In this plasma, microparticles with the charge Q interact with each other via the screened Coulomb 共or Yukawa兲 potential 共r兲 = 关Q2 / r兴exp共−r / D兲, where r is the interparticle distance, e is the electron charge, and D is the Debye screening length. The variety of complex plasmas can be classified according to two parameters. One is the screening parameter, = a / D, where a = 共d0兲−1/2 is the average interparticle distance with d0 being the equilibrium density of the 2D dust layer. The other parameter is the Coulomb coupling ⌫, defined as the ratio of the interparticle interaction energy to the particle thermal kinetic energy, ⌫ = Q2 / 共aTd兲, where Td is the dust layer temperature 共in units of energy兲. When ⌫ Ⰷ 1, the dust system is said to be strongly coupled and, consequently, the particles tend to arrange themselves into solid-like or crystalline structures, as shown in the microphotographs reported in Refs. 2–5. Ever since strongly coupled dusty plasmas were first created in 1994, the wave phenomena in these structures have been extensively studied, both theoretically6–9 and experimentally.10,11 Two wave modes have been identified in 2D dust plasmas: A longitudinal 共compressional兲 wave in which particles are displaced parallel to the direction of the wave vector k, and a transverse 共shear兲 wave in which particles are displaced perpendicular to k. A comprehensive theory of both modes of waves in Yukawa crystals has been developed by Wang et al.12 and subsequently verified in considerable detail in the experiments conducted by Nunomura et al.13 More recently, the influence of an external magnetic field on the wave dispersion of 2D plasma crystals was considered by Uchida et al.14 They first reported two new wave modes named “upper-hybrid and lower-hybrid dust lattice wave,” a兲
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which are high-frequency and low-frequency branches of waves that are coupled by the well known longitudinal and transverse modes. Note that, in their model, they consider the particles to distribute with perfect hexagonal lattice structure and waves to propagate parallel to a fundamental lattice vector. In the present work, the quasilocalized charge approximation 共QLCA兲 method, which was proposed by Kalman and Golden and has successfully described the collective excitations in 2D and 3D Yukawa systems,15–18 is employed to further study the wave dispersion relation of a 2D strongly coupled complex plasma in a magnetic field. In addition, damping effects due to the collisions of dust particles with neutrals are taken into account by means of a phenomenological factor.19 Our model is a way to study the dispersion relation of waves in magnetized dusty plasma, and special attention has been paid to the dependence of the dispersion relation on the magnetic field strength, coupling parameter ⌫, and screening parameter . Additionally, the random-phase approximation 共RPA兲 and the molecular-dynamics 共MD兲 simulation results were also reported for comparison with our QLCA results. Compared with the model in Ref. 14, it should be emphasized that the dusty plasma structure and the wave propagation with respect to the crystal axes are not necessary in our model. The paper is organized as follows. In Sec. II, a general expression is derived on the basis of QLCA for the dielectric tensor in a 2D dust layer in an external magnetic field. In Sec. III, the expressions of the dispersion relation under four different conditions are presented. Numerical results for these quantities are discussed in Sec. III for different plasma parameters. Finally, a short summary is presented in Sec. IV. II. BASIC THEORY
Consider a dust layer in the plane z = 0 of a Cartesian coordinate system with R = 兵x , y , z其, which is immersed in a large volume of plasma with density n0. Bulk conditions are reached for such distances from the dust layer that 兩z兩 Ⰷ D,
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© 2007 American Institute of Physics
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Phys. Plasmas 14, 103708 共2007兲
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and we assume that the plasma is quasineutral there, namely, ne⬁ = ni⬁ = n0. The dust layer consists of Nd particles at the equilibrium positions r j = 兵x j , y j其, with j = 1 , 2 , . . . , Nd. Let d共r , t兲 and ud共r , t兲 be, respectively, the density and the velocity field 共having only the x and y components兲 of the dust layer at the position r = 兵x , y其 and at time t. The equilibrium d 具 共r 兲典, where density of the layer is given by d0 = 兺Nj=1 j 共r j兲 = ␦共r − r j兲 is the single-particle density and 具…典 is an ensemble average over the dust structure realizations in equilibrium. The particles interact with each other through a screened Coulomb potential 共rij兲, and j共t兲 is the displacement amplitude of the jth dust particle from its equilibrium position r j. In the following, we use the QLCA to determine the single-particle positions j共t兲. The QLCA is based on the physical assumption that the dust particles are trapped in a local potential where they undergo small oscillations around their equilibrium sites r j, such that the oscillation amplitudes 兩 j共t兲兩 remain much smaller than the interparticle distance a. Following the procedure of Refs. 15–18, we can describe the microscopic motion of a single particle by
冋
− 2k共兲 = − D共k兲 + +
册
d0共k兲 kk :k共兲 + i␥k共兲 md
关k共兲 ⫻ B兴 d0 , 1/2 Fext共k, 兲 + ic B0 共mdNd兲 共1兲
where k = 兵kx , ky其 and md is the dust particle mass, while the collective coordinates k共兲 are defined via the 2D Fourier transform
j共t兲 =
1
2冑mdNd
兺k
冕
˜ − DT共k兲兴2 − 关DL共k兲 − DT共k兲 + 20共k兲兴关 ˜ − DT共k兲兴 关 − 22c = 0, ˜ = 共 + i␥兲 and where
c =
QB cmd
20共k兲 = with
pd =
冉
2pd共kD兲2
1 共k兲 D共k兲 = 兺 d0md qq关g共兩q − k兩兲 − g共q兲兴, V2D q
共6兲
冑1 + 共kD兲2 , 2Q2d0 m d D
冊
1/2
共7兲
being the dusty plasma frequency. Of the various damping mechanisms that may result in unstable waves in the complex plasmas, we only discuss the neutral gas friction in this article, which is certainly the major damping mechanism in experiments. And it is known that the wave propagation may only exist with relatively weak damping. Thus, we can set 2 ⬇ r2 + 2iri by assuming that the imaginary part of the dispersion relation 兩i兩 is much smaller than the real part. Accordingly, when 兩2i 兩 ⬇ 0, we can obtain the real part of the dispersion relations 共k兲, which is divided into two branches: High-frequency h共k兲 and low-frequency l共k兲, from Eq. 共4兲, as
2h共k兲 = 21 关DL共k兲 + DT共k兲 + 20共k兲 + 2c 兴
共2兲
In Eq. 共1兲, Fext共k , 兲 represents the external force, and D共k兲 is a 2D QLCA dynamical matrix, as a function of the static pair correlation function of the dust layer, g共r兲, or its Fourier transform g共k兲, given by15–18
共5兲
is the dust cyclotron frequency caused by the external magnetic fields. And also, 0共k兲 is the frequency of the longitudinal-acoustic wave in the RPA description,
+ dk共兲exp共ik · r j − it兲.
共4兲
1 2
冑关DL共k兲 − DT共k兲 + 20共k兲 + 2c 兴2 + 42c DT共k兲, 共8兲
2l 共k兲 = 21 关DL共k兲 + DT共k兲 + 20共k兲 + 2c 兴 −
1 2
冑关DL共k兲 − DT共k兲 + 20共k兲 + 2c 兴2 + 42c DT共k兲. 共9兲
The imaginary part of the two branches is 共3兲
−2 with 共k兲 = 2Q2 / 冑k2 + D being the 2D Fourier transform of the potential 共r兲, and V2D the area of the 2D dust layer. The second term on the right-hand side of Eq. 共1兲 comes from the friction force due to the collisions of dust particles with neutral atoms/molecules in the plasma, where the factor ␥ is the Epstein drag coefficient.19,20 The last term represents the Lorentz force acting on dust particles. B0 = B0zˆ is the strength of the external magnetic field. The longitudinal and transverse projections of the QLCA dynamical matrix, DL共k兲 and DT共k兲, were defined in our previous articles.21 The 2D dispersion relation of a dusty plasma in a magnetic field can be obtained from the motion equations in the plane z = 0,
i共k兲 = −
␥ 2
冋
⫻ 1±
2c
冑关DL共k兲 − DT共k兲 + 20共k兲 + 2c 兴2 + 42c DT共k兲
册
.
共10兲 These two wave modes are the results of the coupling of longitudinal and transverse modes due to the Lorenz force. We then consider the situation in which both the magnetic field and the damping effect are weak, i.e., c → 0 and ␥ = 0. Two well known independent longitudinal and transverse modes can be derived, respectively, from Eq. 共4兲 as follows:
L共k兲 = 冑DL共k兲 + 20共k兲,
共11兲
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Theoretical study of the wave dispersion relation…
T共k兲 = 冑DT共k兲.
Phys. Plasmas 14, 103708 共2007兲
共12兲
In the small k region, both the longitudinal and transverse waves exhibit an acoustic dispersion relation. These results were described previously by Kalman and Rosenberg in Ref. 15. Furthermore, if the correlation effects between the dust particles are neglected, namely DL共k兲 = DT共k兲 = 0, Eqs. 共8兲 and 共9兲 could be reduced to the well-known result, predicted by the RPA theory as follows:
2 = 2c + 20共k兲.
共13兲
It is clear that the mode given in Eq. 共13兲 is in the nature of a hybrid cyclotron-plasmon mode in 2D, resulting from superposing cyclotron oscillation due to magnetic field upon a 2D dust acoustic wave. Consequently, the dispersion relation turns into a cutoff at c in the limit of long wavelengths k → 0, which can be easily seen in our figures.
III. RESULTS AND DISCUSSIONS
The parameters used in our numerical computation are, respectively, the coupling parameter ⌫, the screening parameter , and the parameter  = c / pd, which is introduced here for convenience and represents the magnetic field strength B0. All these parameters are in agreement with recent experiments in dusty plasmas.22–25 The numerical results generated from Eqs. 共8兲, 共9兲, and 共13兲 are shown below to illustrate, respectively, the dispersion relation in QLCA 关Eqs. 共8兲 and 共9兲兴 and RPA descriptions 关Eq. 共13兲兴. Note that we set ␥ = 0 in all our simulation results due to our main attention focused on the influence of the external magnetic field. Here we plot all graphs up to ka = 10 due to the fact that most of our analysis is based on the theory of QLCA, which is essentially a linear-dielectric-response theory. By linear, we mean that the perturbations or fluctuations of the system quantities such as particle density are small compared with the equilibrium ones. Considering the fact that we are studying an equilibrium system, where there is no external perturbation applied, the linear approximation should hold well since our analysis is not based on the kinetic theory of plasma, in which one needs to reply closely on the longwavelength approximation to get the analytical expression of the dispersion relation. Also there is no explicit longwavelength approximation in our analysis. Therefore, our conclusion should be still valid for short wavelength 共large k兲 at least at the hierarchy of QLCA. There should be a minimum wavelength 共or maximum wave number兲 for QLCA to be valid, due to the neglect of the Landau damping in QLCA.26 However, in strongly coupled systems, this minimum wavelength is expected to be very small.26 Even though it is very hard to determine the exact value of the minimum wavelength analytically, MD simulation27 and especially some experimental observations 共see, for example, Refs. 10, 11, and 13兲 have validated it to be well above ka = 10. Thus, we plot all graphs up to ka = 10.
FIG. 1. 共Color online兲 Dependence of the dispersion relation on the magnetic field strength, which is represented by the parameter  = c / pd: 共a兲  = 0.3, 共b兲  = 0.5, and 共c兲  = 1.0. Here, the coupling parameter ⌫ = 1000 and the screening parameter = 1 are kept fixed. The dispersion relation is normalized by pd for , and a for k, where h and l label, respectively, the high-frequency and low-frequency branches. The dashed lines and solid lines are, respectively, for the results from the RPA and QLCA descriptions. The symbols show the results from MD simulation.
Figure 1 shows the influence of the external magnetic field on the wave dispersion relation for  = 0.3, 0.5, and 1.0, with ⌫ = 1000 and = 1 kept fixed. It is clear that the magnetic field strength exerts substantial influence on the high-
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Jiang, Song, and Wang
FIG. 2. Dependence of the dispersion relation on the coupling parameter ⌫: 共a兲 ⌫ = 10, 共b兲 ⌫ = 100, and 共c兲 ⌫ = 1000. Here, the screening parameter = 1 and the parameter  = c / pd = 0.5 are kept fixed. 共Others are as in Fig. 1.兲
frequency branch, but not much on the low-frequency branch. It has been also found that the RPA description is only correct in the long-wavelength regime. Furthermore, the results obtained from the theoretical analysis agreed well with the MD simulation contributed by Lu-Jing Hou in Fig. 1共a兲. We next analyze the dependence of the wave dispersion relation on the coupling parameter ⌫ of a 2D dust layer. Figure 2 shows the results for the dispersion relation normalized by pd for different coupling constants: ⌫ = 10, 100, and 1000, with the parameter  = c / pd = 0.5 and the screening parameter = 1 kept fixed. For comparison, we also show the corresponding results in Fig. 2 produced by the RPA description. In Fig. 2, one can see that both of the two branches show little difference for different ⌫ except some small oscillations, within the framework of QLCA. It can also be seen that the RPA description can only work in the long-wavelength regime. In order to illustrate the screening effects, we next vary the screening parameter for = 1, 2, and 3, with ⌫ = 1000
Phys. Plasmas 14, 103708 共2007兲
FIG. 3. Dependence of the dispersion relation on the screening parameter : 共a兲 = 1, 共b兲 = 2, and 共c兲 = 3. Here, the coupling parameter ⌫ = 1000 and the parameter  = c / pd = 0.5 are kept fixed. 共Others are as in Fig. 1.兲
and  = c / pd = 0.5 kept fixed. It is shown in Fig. 3 that the amplitude of the dispersion relation becomes small with increasing. Based on the definition of 共 = a / D兲, one can conclude that with lower amplitudes can be achieved by increasing the average interparticle distance a. Note that for ⬎ 1, the system is still strongly coupled. Usually we treat ⌫ and separately, but sometimes ⌫* = ⌫共1 + + 2兲exp共−兲 is used as the coupling parameter, which is called the “effective coupling parameter” or the “modified coupling parameter.”28,29 It is introduced to describe the structural properties of the Yukawa system, such as whether the system is crystallized or strongly coupled 共⌫* Ⰷ 1兲. For ⌫ = 1000 and = 2 , 3 in Fig. 3, ⌫* = 947.1 and 647.4, respectively, which means the system is still strongly coupled. IV. CONCLUDING REMARKS
In summary, we have performed a theoretical description of a 2D strongly coupled dusty plasma, with a constant magnetic field pointing perpendicular to the 2D dust layer, which
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takes into account the correlation between the dust particles and the damping effects due to the dust collisions with neutral plasma particles. The expressions of the dispersion relation are derived by using the QLCA method. It has been found that both high-frequency and low-frequency branches exist as a result of the coupling of the longitudinal and transverse modes due to the Lorenz force acting on the dust particles. Special attention in our analysis is paid to the dependencies of the dispersion relation on magnetic field strength, the coupling parameter ⌫, and the screening parameter . In comparison with our QLCA results, the random-phase approximation results and the MD simulation are also presented. The details of our MD simulation method will be shown in a future paper. ACKNOWLEDGMENTS
We would like to thank L. J. Hou for his MD simulation results and helpful discussion. K.J. thanks V. N. Tsytovich, M. Horanyi, and S. Khrapak for fruitful discussions. This work is supported by the Research Fund for the Doctoral Program of Higher Education of China, Grant No. 20050141001 共Y.N.W兲. H. Ikezi, Phys. Fluids 29, 1764 共1986兲. Y. Hayashi and K. Tachibana, Jpn. J. Appl. Phys., Part 2 33, L804 共1994兲. 3 J. H. Chu and L. I, Phys. Rev. Lett. 72, 4009 共1994兲. 4 H. Thomas, G. E. Morfill, V. Demmel, J. Goree, B. Feuerbacher, and D. Mohlmann, Phys. Rev. Lett. 73, 652 共1994兲. 5 A. Melzer, T. Trottenberg, and A. Piel, Phys. Lett. A 191, 301 共1994兲. 6 F. M. Peeters and X. Wu, Phys. Rev. A 35, 3109 共1987兲. 1 2
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