PIC/MC simulation of dusty RF discharges Yu. I. Chutov*, W. J. Goedheer + *Faculty of Radio Physics, Taras Shevchenko Kiev University, Volodymyrs’ka Str. 64, 252017 Kiev, Ukraine [email protected]+ FOM-Institute for Plasma Physics "Rijnhuizen", P.O. Box 1207, 3430 BE Nieuwegein, The Netherlands [email protected]Radio frequency (RF) discharges are often used in plasma chemical reactors due to their high efficiency to create uniform plasmas in large volumes. Charged sheaths separate RF plasmas from walls and electrodes controlling the charged particle fluxes from the plasma and its interaction with the walls and electrodes. Dust particles can appear in RF plasmas as the product of the plasma-wall interaction and their subsequent penetration into the plasma or be created due to coagulation of various components in chemically active plasmas. The dust particles can essentially influence the properties of RF plasmas due to the continuous selective collection of background electrons and ions that can cause an essential change of an electron energy distribution function for fast electrons and therefore cause a change of the electron non-elastic process rates. Therefore a kinetic treatment of RF discharges with dust particles is very actual. In this work a 1D PIC/MC method is developed for computer simulations of the dusty RF discharge using the method for discharges without dust particles. A RF discharge in argon with dust particles distributed uniformly in the interelectrode gap is simulated at parameters providing a possibility to consider the discharge as a physical model of processing plasmas. The spatial distribution of various discharge parameters, including the charge on the dust particles, has been computed at various densities of dust particles and the electric current in the external discharge circuit. Obtained results show that the dusty RF discharge has a non-uniform quasi-neutral central part with a low electric field and non-stationary sheaths with a strong electric field separating the electrodes from the central part. The dust particles essentially influence the spatial distribution of the discharge parameters, in particular an increase of a dust particle density causes an expansion of sheaths. The electric current in the external electric circuit changes at a large enough density of dust particles. The dust particle charge changes non-monotonously across the interelectrode gap and has a maximum at the sheath edge, due to the spatial distribution of plasma parameters and a peculiarity of the electron energy distribution function in the quasi-neutral central part of the RF discharge. The electron energy distribution function is shown to be the same for fast electrons in the quasi-neutral central part due to their free motion in this part. Therefore the electron flux to a negatively charged dust particle is equivalent in the central part of a RF discharge, unlike the ion flux which depends on the ion density. The difference between these fluxes causes a non-monotonous change of the dust particle charge.
Transcript
PIC/MC simulation of dusty RF discharges
Yu. I. Chutov*, W. J. Goedheer+
*Faculty of Radio Physics, Taras Shevchenko Kiev University,Volodymyrs’ka Str. 64, 252017 Kiev, Ukraine
[email protected]+FOM-Institute for Plasma Physics "Rijnhuizen", P.O. Box 1207,
Radio frequency (RF) discharges are often used in plasma chemical reactors due to their high efficiencyto create uniform plasmas in large volumes. Charged sheaths separate RF plasmas from walls andelectrodes controlling the charged particle fluxes from the plasma and its interaction with the walls andelectrodes. Dust particles can appear in RF plasmas as the product of the plasma-wall interaction and theirsubsequent penetration into the plasma or be created due to coagulation of various components inchemically active plasmas. The dust particles can essentially influence the properties of RF plasmas dueto the continuous selective collection of background electrons and ions that can cause an essential changeof an electron energy distribution function for fast electrons and therefore cause a change of the electronnon-elastic process rates. Therefore a kinetic treatment of RF discharges with dust particles is very actual.
In this work a 1D PIC/MC method is developed for computer simulations of the dusty RF discharge usingthe method for discharges without dust particles. A RF discharge in argon with dust particles distributeduniformly in the interelectrode gap is simulated at parameters providing a possibility to consider thedischarge as a physical model of processing plasmas. The spatial distribution of various dischargeparameters, including the charge on the dust particles, has been computed at various densities of dustparticles and the electric current in the external discharge circuit.
Obtained results show that the dusty RF discharge has a non-uniform quasi-neutral central part with a lowelectric field and non-stationary sheaths with a strong electric field separating the electrodes from thecentral part. The dust particles essentially influence the spatial distribution of the discharge parameters, inparticular an increase of a dust particle density causes an expansion of sheaths. The electric current in theexternal electric circuit changes at a large enough density of dust particles.
The dust particle charge changes non-monotonously across the interelectrode gap and has a maximum atthe sheath edge, due to the spatial distribution of plasma parameters and a peculiarity of the electronenergy distribution function in the quasi-neutral central part of the RF discharge. The electron energydistribution function is shown to be the same for fast electrons in the quasi-neutral central part due to theirfree motion in this part. Therefore the electron flux to a negatively charged dust particle is equivalent inthe central part of a RF discharge, unlike the ion flux which depends on the ion density. The differencebetween these fluxes causes a non-monotonous change of the dust particle charge.
Magnetic bubbles and non-uniform plasma production in magnetic field
Experiments on crystallization in a dusty plasma indicate that below a critical neutral pressure, Pcrit, thedust component in a dusty plasma behaves as a weakly coupled fluid1 with dust temperature much higherthan the temperatures of the neutrals, ions, or electrons. As the neutral pressure is increased such that P >Pcrit the dust grains are seen to crystallize forming a coulomb lattice. �The phase transition from the solidto fluid state (melting) has been addressed,2 but an important outstanding issue is the physical process thatgoverns the phase transition from the fluid state to the solid state, which is seen in both experiments andsimulations. Our analysis, in conjunction with numerical simulations3, indicates that a two-streaminstability between the ions and dust is responsible for dust heating for P < Pcrit and thereby preventing thedust component from crystallizing. As the neutral pressure is increased the ion-neutral and dust-neutralcollision frequencies increase. For P > Pcrit the collision frequencies are sufficiently large to stabilize thetwo-stream instability. This removes the heat source, which enables the conditions for strong coupling inthe dust component to be achieved. Consequently, the dust component can now condense into a coulombcrystal. We will analyze the relevant streaming instabilities, quantify Pcrit in terms of the backgroundplasma parameters, and discuss their application vis-à-vis our simulations3 and experiments.
_________________________________*Supported by Office of Naval Research and NASA
1. H. Thomas and G.E. Morfill, Nature (London) 379, 806, 1996.2. V.A. Schweigert et al., Phys. Rev. E 54, 4155, 1996: Phys. Rev. Lett., 80, 5345, 1998.3. G. Joyce, M. Lampe, and G. Ganguli, IEEE Trans. Plasma science, April 2001 (in press).
Simulation of Phase Transitions in a Streaming Dusty Plasma*
G. Joyce, G. Ganguli and M. LampeNaval Research Laboratory, Washington DC 20375-5346
Dust grains, under certain conditions, can form various configurations in the vicinity ofthe plasma sheath. In this region of the plasma, ions stream toward the boundary with velocitiesnear or above the ion sound speed, cs. It has been observed that at pressures below some criticalpressure, Pcrit, the dust grains acquire a large random kinetic energy and behave as a fluid. AbovePcrit, the dust grains become cold and form a strongly coupled crystalline state1. We have used theNRL DSD simulation code2 to generate these states. We have also used the code to study aninfinite, homogeneous dusty plasma in the presence of streaming and have compared these resultswith our theoretical analysis3. We have shown that the origin of the grain heating and thetransition from the fluid to crystalline phases is due to the dust-ion two-stream instability. Belowthe stability boundary the dust grains are strongly heated by the resulting electrostatic fields andexhibit fluid behavior. The instability is quenched by a combination of dust-neutral and ion-neutral collisions which are dependent on the neutral gas pressure. At pressures above the criticalpressure, the grains are cooled and become crystalline.
Fig. 1 Dust grains form a gas at P=530mT Fig. 2 Dust grains form a crystal at P=550mT
The dust is streaming in the z-direction. The left picture in both figures shows the dust grains in the planeparallel to the streaming direction. The right picture in both figures shows a superposition of the dustgrains in two planes perpendicular to the streaming direction.
*Supported by Office of Naval Research and NASA.1. A. Melzer, A. Homann, and A. Piel, Phys. Rev. E 53 2757 (1996)2. G. Joyce, M. Lampe, and G. Ganguli, IEEE Trans. Plasma Science, April (2001) (in press)3. G. Ganguli, G. Joyce, and M. Lampe, “Role of Streaming Instabilities in Dusty Plasma
Phase Transition” , (2001) (this workshop)
Particle dynamics within a well-characterized potential well
G. A. Hebner, M. E. Riley, D. Johnson, Pauline Ho, and R. J. BussSandia National Laboratories
Single and multiple particle dynamics within a well-characterized plasma potential have been measuredusing a number of experimental techniques. Experiments were performed in a modified GEC rf referencechamber. For this work, the lower electrode contains a spherical radius depression that forms a potentialwell. For a range of plasma conditions, the plasma sheath approximately conforms to the depression toform a spherical well. The trajectories of single particles as they fall to the bottom of the well areobtained using frame digitization and analysis techniques. The measured trajectories as functions ofpressure and radius of curvature are in good agreement with theoretical predictions.
Multiple particle arrangements show a single layer, hexagonal close packed structure. These structuresare formed without the traditional particle confinement rings. The nearest neighbor spacing as functionsof radial position, number of particles and radius of curvature were determined using image analysis. Theshape of the radial compression within our potential well provides information about the details of theparticle interaction dynamics and particle sheaths. In general, measurements of the radial compressionare in good agreement with model predictions. Additional experimental measurements focused oncharacterization of the plasma potential well will be presented.
This work was performed at Sandia National Laboratories and supported by the Division of MaterialSciences, Office of Science, U. S. Department of Energy and Sandia National Laboratories (DE-AC04-94AL85000). Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed MartinCompany, for the United States Department of Energy.
Plasma dispersion function for a kappa-Maxwellian distribution:steps towards a wakefield study
M.A. Hellberg* and R.L. Mace***School of Pure & Applied Physics, University of Natal, Durban, South Africa
([email protected]) **Discipline of Physics, School of Physical Sciences, University of Durban-Westville, South Africa
One of the curiosities of dusty plasmas is the attraction between adjacent negatively-charged dustparticles, that may lead to strings of dust grains, and hence crystals. A possible explanation of thisphenomenon lies in the electrostatic wakefield formed by ions streaming past the first dust particle,leading to an attractive potential behind it. To find the potential, one needs to consider the dielectricfunction ε for waves in a plasma, based on a kinetic model1,2. Various distribution functions have beenused previously. We consider this problem afresh, and introduce the kappa-Maxwellian distribution.
In the experiments, the electron distribution may be isotropic, but not necessarily Maxwellian. It may beof power-law form, and is well represented by a generalized Lorentzian (or kappa) distribution - theparameter kappa enables one to represent different power-laws. It has been shown3 that the plasmadispersion function, Zκ, for an isotropic kappa distribution is proportional to a Gaussian hypergeometricfunction for arbitrary real kappa. As the ions are streaming through a sheath region, it is reasonable toassume for them a power-law form along the flow direction. However, one may expect equilibration, andhence isotropy, in the perpendicular plane. We thus model the ions by a 1-d kappa distribution along theflow, with, in the perpendicular plane, a 2-d Maxwellian distribution, having temperature T⊥ � T|| :fκM (v||,v⊥) = π-3/2 θ||
-1 θ⊥-2 [Γ(κ+1)/{κ3/2Γ(κ-1/2)}] [1+{v ||
2/(κθ||2)}] -κ exp{-(v⊥/ θ⊥)2},
where the effective thermal speeds satisfy θ||2 = {(2κ -3)/κ}{KT ||/m} and θ⊥
2 = 2KT⊥/m.
As a first step towards studying the wakefield problem, we require a plasma dispersion function for akappa-Maxwellian distribution. Following our earlier work3, we have now shown that4
ZκM(ζ) = i (κ - 1/2) κ-3/2 2 F 1[1, 2κ; κ + 1; (1/2) (1 - {ζ/ i κ1/2})].
Hence we have4 evaluated a number of special cases, and found power-law and asymptotic expansions,and the derivative of Z�0.
Interestingly, for plasmas with kappa-Maxwellian distributions, the general dispersion relation for one-dimensional electrostatic waves becomes 1 - Σα (ωpα
2/k2θα2)Z'κM (ω/kθα) = 0.
It will be noted that ZκM thus fits the one-dimensional dispersion relation naturally, without any addedfactors involving κ, unlike the isotropic three-dimensional plasma dispersion function, Zκ
3.As the salient feature in finding ZκM is the one-dimensional kappa distribution function, it is applicable toany distribution for which the perpendicular velocity part is separable and of the form g (v⊥
2).
The generalized plasma dispersion function ZκM should also find application to wave studies inmagnetized plasmas, for instance to planned dust plasma crystal experiments in a magnetic field and tospace plasmas. The magnetic field defines a preferred direction in space, and the perpendicularMaxwellian distribution facilitates the solution of the perpendicular velocity integrals, which are virtuallyintractable for the isotropic kappa distribution.
1. S. Benkadda, V.N. Tsytovich & S.V. Vladimirov, Phys. Rev. E, 60, 4708 (1999).2. S.V. Vladimirov and M. Nambu, Phys. Rev. E, 52, 2172 (1995).3. R.L. Mace & M.A. Hellberg, Phys. Plasmas, 2, 2098 (1995).4. M.A. Hellberg & R.L. Mace, To be submitted to Phys. Plasmas (2001).
Removal of dust particles resulting in a sudden change of the external magnetic field
Deyong Liu, Xiaogang Wang, Dezhen Wang, and Yue LiuState Key Lab of Materials Modification by Beams,
Dalian University of Technology, Dalian 116024, China
Removing dust particles to create a clean plasma source is an important issue of plasmaprocessing [1]. The motion of dust particles due to a sudden change of an external magnetic fieldis investigated in this presentation. As the vertical magnetic field is turned on /off suddenly, acurl of horizontal electrical field is generated and an BE × drift of electrons/ions is built up. Dueto the existence of dust particles, the BE × drift is non-neutral and a charge separation can beformed to force the dust particle to move.
In the strongly magnetized case, if the variation of the magnetic field is slow enough tomagnetize both electrons and ions, the charge separation will force the dust particles to move inthe same BE × direction as ions and electrons. In the weakly magnetized case however, onlyelectrons are magnetized, the charge separation may lead to an ambipolar diffusion process [2].The dust particle then moves to the direction opposite to the BE × drift. In the first case, we usea “modified” MHD treatment to “add up” electrons and ions to set up a “simplified two-fluids”model. In the second case, the “multi-fluids” (for electrons, ions, and dust particles) treatmentmust be applied. Numerical results and theoretical analysis will be presented.
This work was supported by the National Natural Science Foundation of China, Grant No.19875006.
[1]. H. Fujiyama, Y. Maemura, and M. Ohtsu, Jpn. J. Appl. Phys. 38 (1999) 4550[2]. Y. Maemura, S. –C. Yang, and H. Fujiyama, Surf. Coat. Tech. 98 (1998) 1351
([SHULPHQWV RQ 'XVW ,RQ�$FRXVWLF 6ROLWDU\ :DYHV
<RVKLKDUX 1DNDPXUD DQG $� 6DUPD
7KH ,QVWLWXWH RI 6SDFH DQG $VWURQDXWLFDO 6FLHQFH� 6DJDPLKDUD� .DQDJDZD ��������
Compressional pulses were launched in a 2D plasma crystal consisting of 8.69 µm polymermicrospheres suspended in a low pressure Ar rf plasma. The pulses were excited by the radiation pressureof a modulated Ar laser beam incident on the particles. Particle positions and velocities were determinedfrom direct imaging using video microscopy. The pulse propagation speed was measured, similar to thegroup velocity of compressional waves launched with sinusoidal excitation.
As expected for nonlinear waves1, we found that a pulse propagates through the lattice at a speedthat increases with pulse amplitude, for Mach numbers M > 0.07 (Fig.1). This nonlinear behavior wasobserved only for bigger values of screening parameter κ, κ ≥ 1.26 for the current experimentalconditions. As another indication of nonlinear wave propagation, we found that the particle velocity is notproportional to the relative variation in the particle number density (Fig.2). Nonlinearity appears fornumber density increase bigger than 10 %. Note that the effect is bigger in the excitation region, wherethe amplitudes are largest. The slope of linear part of dependencies in Fig.2 corresponds to sound speed ofcompressional waves.
Fig.2. Pulse amplitude (particle speed) vs.pulse amplitude (relative variation ofparticle number density δn/n). Both aredecreasing as the pulse propagates throughthe plasma crystal. Deviation from lineardependence is another indication ofnonlinearity1.
Fig.1. Dependence of pulse propagationspeed on pulse amplitude, both normalizedby the pulse propagation speed C0 for thelowest laser power. Data for differentvalues of κ are shown by different colors.Fit line according to theory of nonlinearsound waves in 3D liquid1. Thisdependence is an indication ofnonlinearity.
0.9
1
1.1
1.2
1.3
0 0.02 0.04 0.06 0.08 0.1Peak particle speed / C
Pul
se p
ropa
gatio
n s
peed
/ C
0
0
0
0.5
1
1.5
2
0 0.05 0.1 0.15 0.2
Par
ticle
spe
ed (
mm
/s)
Number density dn/n
inside excitation region
outside excitation region
Wave dispersion relations in a 2D plasma crystal
S. Nunomura, J. Goree, S. Hu, X. Wang, and A. BhattacharjeeDepartment of Physics and Astronomy, The University of Iowa, Iowa City, Iowa 52242
Dispersion relations of longitudinal and transverse waves were investigated in a two-dimensional (2D)plasma crystal. We excited both types of wave modes by applying the radiation pressure of a laser to aplasma crystal. The laser power was modulated sinusoidally. To investigate the detailed properties ofwave propagation phenomena, dispersion relations for both types of modes were measured at variousexperimental conditions, i.e., at different values of the shielding parameter, damping rate and wavepropagation direction. The results agreed well with the theory recently developed by Wang et al1.Comparing to the theory, we discovered that an unknown wave damping mechanism exists, in addition toEpstein drag, and it is prominent in a low gas pressure regime.
Experiments were performed using a parallel-plateradio-frequency discharge. Polymer microsphereswere shaken into the plasma, where they werelevitated by the electric field in the sheath above thelower electrode. The particles arranged in a singlehorizontal layer, with a hexagonal lattice. They wereimaged using a video camera, which recorded theparticle motion. Waves were launched by irradiatingthe crystal with a modulated laser. The motion of theparticles in response to the modulated laser beam wasanalyzed to measure the propagation speed anddispersion of the waves. The orientation of the lasersheet determined whether we excited a longitudinal ortransverse wave.
An example of both dispersion relations is shown inFig. 1. A curvature in the dispersion relation, i.e.,"dispersion," was observed above ω/ω0 ∼ 1 for thelongitudinal mode. In contrast, the transverse modeexhibited a linear relationship of ω vs kr, i.e.,"dispersionless," characteristic over a wide range ofkra/π (0 < kra/π < 1) 2. Looking at the slope of adispersion relation at kr = 0, which defines the soundspeed, we notice that the sound speed Cl of the longitudinal mode is faster than Ct of transverse modes. Inour experiments, the sound speed is a few cm/s for longitudinal mode and several mm/s for transversemode, respectively. So, the ratio of Cl / Ct is approximately 5 for κ ~ 1. The wave frequency andwavelength are typically a few Hz and several mm, respectively, for both modes.
[1] X. Wang, A. Bhattacharjee, and S. Hu, Phys. Rev. Lett. 86, 2569 (2001).[2] S. Nunomura, D. Samsonov, and J. Goree, Phys. Rev. Lett. 84, 5141 (2000).
0
1
2
0 1 2 3 4
0
10
20
30ω/
ω 0
ω (s
-1)
k (mm-1)
longitudinal mode
experiment
theory k
rk
i
θ=0º
sound speed
0
1
2
0
10
20
30
0 0.2 0.4 0.6 0.8 1
ω/ω 0
ω (s
-1)
transverse mode
ka/π
Fig. 1. Dispersion relations in a 2D palsma crystal.Waves propagete parallel to lattice alignment.Fig. 1. Disperson relations in a 2D plasma crystal.Waves propagate parallel to lattice alignment.
Nonlinear Oscillations of Dust Particles in the Sheath of an RF-Discharge
A. Piel, C. Zafiu, A. MelzerInstitute for Experimental and Applied Physics, Christian-Albrechts-University
In dusty plasmas, large particles of about 10µm diameter can only be suspended against gravity by thestrong electric fields found in the sheath region of the discharge. These particles are known to carry somethousand elementary charges. Linear oscillations in the vertical direction have been used before todetermine the charge of the particles from their natural resonance1. A similar technique was used recentlyto derive the particle charge from damped oscillations in the sheath2. Nonlinear vertical oscillations ofparticles in the sheath were recently used to explore the sheath structure3.
In this contribution, we demonstrate that the shift of thenonlinear resonance of the particles in the sheath is amanifestation of the position dependence of the particlecharge. In Fig. 1a, the nonlinear resonance curve shows ashift towards lower frequencies and hysteresis. The measuredamplitudes are compared with a model that takes a positiondependent charge of the particle into account and solves thenonlinear equation of motion numerically. The effective‘potential energy‘ curve in Fig. 1b, which is obtained fromthe above analysis, reflects the weakening of the electric fieldforce by a reduced charge at positive excursions from theequilibrium position, while the softening at negativeexcursions can be attributed to the approach of thequasineutral plasma region. The dashed curve represents thelocal harmonic oscillator potential at the equilibrium position.
The model is based on a third-order polynomialrepresentation for the position dependent particle charge anda linear (or parabolic) representation for the sheath electricfield. Detailed investigations with different particle sizesshow that the model of Ref. 3, which assumes a constantparticle charge but a highly nonlinear electric field in the
sheath, is not capable to describe our experiments. On the contrary, the nearly linear increase of theelectric field acording to the model of Liebermann4 together with a strongly nonlinear position dependentcharge gives a suitable description of the observed nonlinear resonance as well as of the up-downasymmetry of the resonance amplitudes for the different particle sizes.
1A. Melzer, T. Trottenberg, A. Piel, Phys. Lett. A191, 301 (1994); Th. Trottenberg, A. Melzer, A. Piel,Plasma Sources Sci. Technol. 4, 450 (1995); H. Schollmeyer, A. Melzer, A. Homann, A. Piel, Phys.Plasmas 6, 2693 (1999)2E.B. Tomme, B.M. Annaratone, J.E. Allen, Plasma Sources Sci. Technol. 9, 87 (2000)3A.V. Ivlev, R. Sütterlin, V. Steinberg, M. Zuzic, G. Morfill, Phys. Rev. Lett. 85, 4060 (2000)4M. A. Liebermann, IEEE Trans. Plasma Sci. PS-16, 638 (1988)
Vertical transverse wave modes in a 2D complex (dusty) plasma.
D. Samsonov, A. Ivlev, R.A. Quinn, G.E. Morfill
Max-Planck-Institute for Extraterrestrial PhysicsGiessenbachstrasse 1, Garching 84750, GERMANY
Vertical transverse waves were studied in a 2D strongly coupled dusty plasma. Theexperiments were performed in an rf parallel plate discharge [1]. A monolayer dustlattice was formed from monodisperse plastic microspheres and levitated in the electrodesheath. The vertical waves were launched in the lattice. The wave structures weredirectly imaged with a high speed video camera and analyzed with particle trackingsoftware. It was found that the waves have an inverse (optical) dispersion.
The theory describing the experiment is also discussed here. It is based on a set ofequations of motion written for a linear string [2] taking into account ion wakes belowthe particles suspended in the sheath.
[1] D. Samsonov, J. Goree, Z. W. Ma, A. Bhattacharjee, H. M. Thomas and G. E. Morfill,Phys. Rev. Lett., 83(18) pp. 3649-3652 (1999).
[2] A.V. Ivlev and G. Morfill, Anisotropic dust lattice modes, Phys. Rev. E63, 016409(2001)
Crystallization of a mono-layer plasma crystal
R. A. Quinn, D. Samsonov, D. Goldbeck, M. Zuzic, G. E. MorfillMax Planck Institute
One of the primary advantages of complex plasmas is their use as a model system for solid-liquid-gas phase transitions. Traditionally, phase transitions in plasma crystals have been studiedby melting crystalline states, usually by lowering the neutral gas pressure. At lower pressures,instabilities act to heat the particles, thus melting the crystal. This technique has thedisadvantage, however, of significantly altering the plasma environment at the same time, whichcan have an uncontrolled effect on the particle interaction potential.
In the present experiment, a single layer (2D) plasma crystal is melted to the gaseous phase whilekeeping the plasma power and neutral gas pressure constant. Afterwards, the particles graduallyrecrystallize as the neutral gas cools them down. Here, the early stages of the recrystallizationprocess are studied. Initial empirical results for the time evolution of the particle temperature andstructural properties of the system during recrystallization will be presented.
Crystallization of a 3D plasma crystal
D. Goldbeck, R. A. Quinn, M. Zuzic, G. E. MorfillMax Planck Institute
Complex plasmas is their are often used as model systems for solid-liquid-gas phase transitions.Usually, these transitions have been studied by melting the plasma crystals by lowering theneutral gas pressure. The lower pressure enables instabilities to overcome frictional damping,thus heating the particles and melting the crystal. However, lowering the pressure alsosignificantly alters the plasma environment which in turn effects the particle interaction potential.
Here, the initial stages of the crystallization of a 3D complex plasma are analyzed, in a non-changing plasma environment (neutral gas pressure and plasma power constant.). A 3D imagingmethod has been developed in order to analyze the dynamics of crystallization in all three degreesof freedom simultaneously. The technique provides both the 3D particle positions and 3Dvelocities in the viewed volume. The first empirical results for the time evolution of the particletemperature and microdynamics of the system will be presented.
Solitons and double layers in multiply-charged dusty plasmas
P.H. Sakanaka, I. Spassovska, F. Al-Yousef b, and W. Sahyouni b
Finite amplitude electrostatic solitons and double layers theory in a multi-component
unmagnetized dusty plasma is presented 1 , by supposing that the constituents of dusty plasmas
are the electrons, positive ions, and an admixture of negatively and positively charged dust
grains which are simultaneously present. It is shown that stationary solutions of the multi-fluid
dusty plasma equations and Poisson's equation can be expressed in terms of the energy integral
of a classical particle with a modified Sagdeev potential. The latter is analyzed both analytically
and numerically to demonstrate the existence of rarefactive and compressive electric potentials
which travel faster than the effective dust-acoustic velocity. Compressive dust-acoustic solitons
exist only when there is a significant fraction of positively charged dust grains. Furthermore, the
four fluid dusty plasma system, with both the negatively and positively charged dust grains, also
provides the possibility of double layers. Conditions under which solitons and double layers arise
are given, and their profiles are displayed graphically. The results of investigation should be
helpful in identifying the salient features of nonlinear structures in low-temperature space and
laboratory dusty plasmas in which positively and negatively charged dust grains coexist. In
particular, we have applied the theory in the laboratory plasma reported by Oohara et al.2 , and
we can predict that a double layer might be possible to be launched, in their experiment.
1 P.H.Sakanaka and P.K. Shukla, Physica Scripta, T84, 181-183 (2000).
2 W. Oohara, N. Tomioka, T. Hirata, R. Hatakeyama, and N. Sato,
Proceedings of the 2000 International Congress on Plasma Physics,
Quebec, October, 2000.
Nonlinear evolution of dust waves driven by cross-field electron currents
W. A. Scales and G.S. ChaeBradley Department of Electrical and Computer Engineering
Virginia Tech, Blacksburg VA 24061-0111
There has been considerable application of cross-field electron current two-steam instabilities tointerpreting irregularities in the earth's ionosphere. In particular, the Farley-Buneman instabilityhas been investigated extensively with theoretical and nonlinear numerical simulation modelsdue to its role in producing irregularities in the earth's E-region. Recently [1], an analog of theFarley-Buneman instability in a dusty plasma has been proposed. Due to the observation of dustin the earth's ionosphere, this instability, which involves the production of dust acoustic waves,will most likely have important applications to meteor trails and noctilucent clouds. This Hallcurrent instability is driven by a static background electric field which ultimately produceselectron ExB drifts in a magnetized dusty plasma. This work considers the important aspects ofthe nonlinear evolution of this low-frequency Hall current instability. A two-dimensionalnumerical simulation model in the plane perpendicular to the background magnetic field hasbeen developed. The electrons and ions are treated as a two-component fluid while the dust istreated with the Particle-In-Cell PIC technique. The simulation is driven by a static backgroundelectric field which produces an ExB drift on the electrons. The simulation model shows lineargrowth of dust acoustic type waves during the early time evolution. The results are alsoconsistent with earlier theoretical predictions [1] of the characteristics of these waves. Thenonlinear evolution is considered for a wide variety of dust and background plasma parameters.Principal parameters considered include the electron-neutral collision frequency, the dust-neutralcollision frequency, and the strength of the driving background electric field. The nonlinearevolution shows dust heating, nonlinear mode-coupling effects, and the development ofsecondary waves that ultimately result in the saturation of the instability. Comparisons of thenonlinear evolution of the low-frequency Hall current instability to past theoretical and nonlinearnumerical simulation work on the Farley-Buneman instability will be discussed. Possibleapplications to ionospheric irregularities will also be described.
[1] Rosenberg, M. and P.K. Shukla, Low Frequency Hall current instability in a dusty plasma,Journal of Geophysical Research, in press, 2000.
A kinetic model for low-frequency modes in dusty self-gravitating plasmas
V. V. YaroshenkoInstitute of Radio Astronomy of the National Academy of Science of Ukraine,
Chervonopraporna 4, Kharkov, Ukraine 310002.
G. Jacobs and F. VerheestSterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281, B--9000 Gent, Belgium.
Many laboratory and space plasmas contain a substantial amount of dust particles, the presence of whichleads to a wealth of new and exotic phenomena. The massive dust grains can amass the lighter plasmaspecies and become in this process highly charged yet their charge-to-mass ratio remains very small.Because of their very heavy nature, it is necessary to take the self-gravitational interactions of the dustgrains into account because they are not negligible, compared to the electric forces.
The inclusion of self-gravitational effects in dusty plasmas changes the dispersion relations and entails thepossibility of a gravitational collapse in space plasmas. In order to trigger a gravitational instability, theplasma dimensions have to exceed a critical lengthscale, the well known Jeans length.
We studied the stability of low-frequency modes in self-gravitating dusty plasmas within the kineticdescription [1]. Comparison of these results with the stability of the analogous hydrodynamic modelshows that the stability regions of both models differ only slightly, but on the other hand the growth ratesare quite different. In the kinetic model, all stable dust Langmuir and dust-acoustic waves are damped dueto the collisionless Landau damping. Quantitatively this mechanism is influenced by self-gravitation andstrongly dependent on the plasma parameters in general.
For one of the more popular low-frequency modes, the dust-acoustic mode, we also included thepossibility of a continuous size (mass)-distribution of the dust grains [2,3,4], thus rendering a morerealistic picture since in many applications dust grains are omnifarious in shape and size. The size-distribution is modelled as a decreasing power law in a range of particle sizes, which is a goodapproximation according to observational data.
Self-gravitation and the distribution of particle sizes are obviously intertwined concepts and this isreflected in the results as the shape of the size-distribution modifies the coupling between plasma andgravitational waves substantially. In agreement with earlier research, the existence of a dust-distributioninstability and a new stable mode due to the size spectrum of the dust particles is recovered.
[1] V V Yaroshenko, G Jacobs and F Verheest, Kinetic approach to low-frequency waves in dusty self-gravitating plasmas, Phys. Rev E (accepted, 2001)[2] F Melandsø, T K Aslaksen and O Havnes, A kinetic model for dust acoustic waves applied toplanetary rings, J. Geophys. Res., 98, 13315-13323 (1993)[3] A Brattli, O Havnes and F Melandsø, The effect of a dust-size distribution on dust acoustic waves, J.Plasma Phys., 58, 691-704 (1997)[4] V V Yaroshenko, G Jacobs and F Verheest, Dust-acoustic modes in self-gravitating plasmas with dustsize distributions, Phys. Rev E (submitted, 2001)
Saturn Ring and Spoke Simulation Experiment in Fine Particle Plasmas
Toshiaki YOKOTA and Ayumi ANDOUFaculty of Science, Department of Physics
Ehime University,Bunkyo-cho 3, Matuyama, Ehime 790-8577 Japan
We are experimenting of the Saturn ring and spoke by using two component fineparticle plasmas generated by a boat method. Two component plasmas which werecomposed of positively charged particles and negatively charged particles were generatedby UV irradiation of fine aluminum particles.
The ring form of fine particle plasmas was able to create by a rotation of smallinsulator sphere in which a small permanent magnet was inserted. The Experimentalparameters for ring formation coincides almost with the simulated values. We areexperimenting its motion and charge of trapped particles. Nd-YAG pulse laser light wasshot to the fine particle plasma ring to simulate the spoke and to investigate the origin ofspoke. We were able to create a gaseous plasma hole in the fine particle ring, and we areexperimenting of detail of its motion.
Shear Pulse Sustained in the Two-Dimensional Dust Lattice
S. ZhdanovMoscow State Engineering & Physics Institute (Technical University),
The theory describing propagation of pulsed shear wave in a dust lattice layer is proposed. The predictions ofdeveloped theory are in a good agreement with the recent experimental observations1.
The plasma crystal as a collection of strongly coupled micron-sized charged particles dispersed into plasma is ofgreat interest because it gives the simplest example of classical particle collection with high-ordered structure. Beinginserted into the plasma, dust particles collect free plasma charges and acquire mainly the same electric charge. Due
to the repulsion, the cloud of dusts with the same sign of charges on the particles could not be in equilibrium withoutaction of additional confining external field. This is why the structure of plasma crystal is defined not only by theinterparticle forces but also by the symmetry of this external field. Well known, if the external force confining the
dusts against repulsion is preferentially stronger in definite direction, the dust cloud collapses and forms thin quasitwo-dimensional monolayer. This “lattice layer” of strongly coupled charged particles can sustain longitudinalwaves and transversal waves. In the case of in plane particle motion, the latter are referred to as shear waves.
Recently, shear waves have been successfully observed in experiments1.The report is devoted to the theoretical study of pulsed shear wave propagating across a dust lattice layer.
Experimentally observed pulsed waves have finite, but relatively small amplitude. Therefore, it can be proposed that
these waves could be described well in the frames of linear theory. We follow just the same way in thisinvestigation. Proposed theory allows:
i. to explain the wave-form of observed shear pulse, including initial pulse formation, decay of initial pulse
on two oppositely propagating sub-pulses, and formation of restoring motion (fist five plots in the Fig.shown below);
ii. to estimate the magnitude of external force exciting a wave;
iii. to determine accurately the value of the damping rate.In concern with the last statement, following moment should be noted. Usually, to demonstrate a damping, the
decaying plots for the wave amplitude are
considered. But it is not the direct way toestimate quantitatively the value of dampingrate: As it is well known, the damping rate
coincides just with the decaying rate of the waveenergy, not with the wave amplitude. Becausethe wave energy is proportional to squared wave
amplitude, the decaying rate of the amplitudehas to be less than the damping rate. In theproposed theory, it has been proved that
damping rate can be successfully estimated byusing decaying plot for the integral of
longitudinal particle velocity distribution over transversal coordinate, ∫= dytyVI x ),( (shown in the last plot of
the Fig.) Here, estimated damping rate is about 2.51 s-1, and this value exceeds the Epstein drag coefficient, ~1.22s-1, estimated in1. It is of great interest, because we obtain a direct way to estimate the damping caused by plasma.
1 S. Nunomura, D. Samsonov, and J. Goree, Phys. Rev. Lett. 84, 5141 (2000).
