a survey of dusty plasma physics.pdf

PHYSICS OF PLASMAS VOLUME 8, NUMBER 5 MAY 2001

A survey of dusty plasma physics *P. K. Shukla†

Institut fur Theoretische Physik IV, Fakulta¨t fur Physik und Astronomie, Ruhr-Universita¨t Bochum,D-44780 Bochum, Germany and Department of Plasma Physics, Umea˚ University, S-90187 Umea˚, Sweden

~Received 16 October 2000; accepted 7 November 2000!

Two omnipresent ingredients of the Universe are plasmas and charged dust. The interplay betweenthese two has opened up a new and fascinating research area, that of dusty plasmas, which areubiquitous in different parts of our solar system, namely planetary rings, circumsolar dust rings, theinterplanetary medium, cometary comae and tails, as well as in interstellar molecular clouds, etc.Dusty plasmas also occur in noctilucent clouds in the arctic troposphere and mesosphere,cloud-to-ground lightening in thunderstorms containing smoke-contaminated air over the UnitedStates, in the flame of a humble candle, as well as in microelectronic processing devices, inlow-temperature laboratory discharges, and in tokamaks. Dusty plasma physics has appeared as oneof the most rapidly growing fields of science, besides the field of the Bose–Einstein condensate, asdemonstrated by the number of published papers in scientific journals and conference proceedings.In fact, it is a truly interdisciplinary science because it has many potential applications inastrophysics~viz. in understanding the formation of dust clusters and structures, instabilities ofinterstellar molecular clouds and star formation, decoupling of magnetic fields from plasmas, etc.!as well as in the planetary magnetospheres of our solar system@viz. Saturn~particularly, the physicsof spokes and braids in the B and F rings!, Jupiter, Uranus, Neptune, and Mars# and in stronglycoupled laboratory dusty plasmas. Since a dusty plasma system involves the charging and dynamicsof massive charged dust grains, it can be characterized as a complex plasma system providing newphysics insights. In this paper, the basic physics of dusty plasmas as well as numerous collectiveprocesses are discussed. The focus will be on theoretical and experimental observations of chargingprocesses, waves and instabilities, associated forces, the dynamics of rotating and elongated dustgrains, and some nonlinear structures~such as dust ion-acoustic shocks, Mach cones, dust voids,vortices, etc!. The latter are typical in astrophysical settings and in several laboratory experiments.It appears that collective processes in a complex dusty plasma would have excellent futureperspectives in the twenty-first century, because they have not only potential applications ininterplanetary space environments, or in understanding the physics of our universe, but also inadvancing our scientific knowledge in multidisciplinary areas of science. ©2001 AmericanInstitute of Physics.@DOI: 10.1063/1.1343087#

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I. INTRODUCTION

The interplay between plasmas and charged dust grhas opened up a new and fascinating research area, thadusty~or complex! plasma. A dusty~or complex! plasma is anormal electron-ion plasma with an additional charged coponent of small micron-sized particulates. This extra comnent, which increases the complexity of the system eventher, is responsible for the name ‘‘complex plasma.’’ Dus~complex! plasmas are ubiquitous in different parts of ocosmic environment,1–5 namely, in planetary rings, in circumsolar and the Phobos dust rings, in the interplanemedium, in cometary comae and tails, and in interstellar mlecular clouds. In fact, the dark bands of dust, which bloparts of the Orion, Lagoon, Coalsack, Horsehead, and Enebulae, indicate that dust must have been abundant innebulae that coalesced to form the Sun, planets, and o

*Paper HI1 1, Bull. Am. Phys. Soc.45, 156 ~2000!.†Invited speaker.

1791070-664X/2001/8(5)/1791/13/$18.00

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stars. On the other hand, during the Voyager 1 and 2 flyof the outer planets and the ICE flyby of comet GiaobinZinner, it has been demonstrated that the plasma wavestrument can detect small dust particles strikingspacecraft.6 Complex dusty plasmas also occur in the flamof a humble candle, in the zodiacal light, in cloud-to-groulightening in thunderstorms containing smoke-contaminaair over the United States, in volcanic eruptions, and in blightening. A recent investigation7 suggests that ball lightening is caused by oxidation of nanoparticle networks fronormal lightening strikes on soil.

Furthermore, meteoritic dust is thought to be presenthe Earth’s mesosphere at altitudes of;80–95 km. It hasbeen conjectured that in the cold summer mesopauseparticles can form around meteoritic dust particles, withicy dust particles possibly influencing the charge balancethe region.8,9 On the other hand, the presence of charged dparticles in the polar summer mesopause has been invokeexplain aspects of the very strong polar summer radar echreferred to as polar mesosphere summer echoes~PMSE!,

1 © 2001 American Institute of Physics

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1792 Phys. Plasmas, Vol. 8, No. 5, May 2001 P. K. Shukla

which occur at altitudes of 80–93 km. Recently, the preseof charged dust in the mesosphere has been detected brect rocket probe measurements, and both negativelypositively charged dust grains have been reported.10,11 Therole of charged dust in mesospheric electric fields is recnized by Zadorozhny.12 The formation of an artificial dustyplasma in the ionosphere was also revealed duringSpacelab 2 mission when the space shuttle orbital manesystem engines were fired.13

Since dust particles are a main element of interest insolar system and in the interstellar medium, there are a nber of future missions~viz. the European Space AgencROSETTA mission for detecting dust on comet 46Wirtanen in 2012, the Cassini spacecraft mission arrivingSaturn in 2004 for exploring in detail the possible dust sidust charge, dust dynamics, as well as other collectivecesses involving charged dust grains! that will provide indetail the properties and global dynamics of charged dgrains. It is anticipated that future rocket campaigns in noern Scandinavia will provide more information regarding tmesospheric dust, while experiments on the InternatioSpace Station will determine the dusty plasma behaviorder microgravity conditions.

Lately, the physics of dusty~complex! plasmas has appeared as one of the most rapidly growing fields of scienbesides the field of Bose–Einstein condensates, as demstrated by the number of published papers in scientific jonals and conference proceedings. It has a tremendous imin astrophysics and low-temperature laboratory discharincluding processing plasmas in the semiconductor indusWhile in the latter one wants to clean up charged dusts whare anathema to microchips, charged dust grains aredeliberately created in low-temperature radio frequencyglow discharges to understand the basic physical proceassociated with the presence of those grains. In laboradischarges, one is able to study the growth of dust graunder gas densities and temperatures typical of the nefrom which the solar system was formed. The particulalook like tiny cauliflowers14 pressed together in irregulastrings—a growth pattern that offers clues to the ratewhich dust particles in interstellar space turned intoclumps of matter, which are large enough to assembleplanets due to gravity. Irregular structures of charged dparticulates also appear in tokamaks.15

In fact, major boosts to dusty plasma research came athe discovery of the dust acoustic wave~DAW!,16 the dustion-acoustic wave~DIAW !,17 the dusty plasma crysta~DPC!,18–20and the dust lattice wave~DLW!.21 We note thatthe idea of the dust acoustic wave~DAW! was put forwardby the present author in the Capri Meeting on Dusty Plasmin July 1989, while Ikezi22 theoretically predicted the Coulomb crystallization of charged dust grains in a strongcoupled dusty plasma system when the ratio betweenCoulomb interaction and the dust thermal energies exce170. A number of laboratory experiments have spectaculverified the theoretical predictions of the DAWs,23–29 theDIAWs,30,31 and the DLWs.32–34

Dusty ~complex! plasmas are fully or partially ionizedlow-temperature gases comprising neutral gas molecu


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electrons, ions, and extremely massive charged submiand micron sized dust grains. The latter, which are a billtimes heavier than the ions, acquire several thousandelectron charges. The dust grain charging occurs duevariety of physical processes35–43 including the collection ofthe background plasma electrons and ions by dust grainsphotoelectron emission, secondary electron emission,thermionic emission, etc. Dust grains can be charged bnegatively and positively. The grains act as a source of etrons when they are charged positively due to the irradiatof ultraviolet ~UV! radiation. Both the positive and negativdust grains can coexist in laboratory and space plasmaappears that the dust grain charging is a new physical proin a dusty plasma, which marks a distinction betweenlatter and the usual multicomponent electron-ion plasma ctaining two ion species.

In this paper, we present the basic physics of dusty pmas and describe the progress that has been made in theof collective processes in dusty plasmas during the lastcade. Specifically, we shall discuss the properties of a duplasma, illustrate various charging processes, and focuswaves, instabilities, and coherent nonlinear structures.manuscript is organized in the following fashion. In Sec.we briefly describe the general properties of a dusty plasIn Sec. III, we discuss various charging processes. SecIV contains dusty plasma waves in an unmagnetized cThe excitation of dusty plasma waves are considered in SV. Section VI points out the importance of collective effecwith regard to the generation of a wakefield in dusty plamas. In Sec. VII, we discuss theories for dust ion-acoushocks and holes, and present experimental observationthose nonlinear structures. Section VIII presents the wspectra in a magnetized dusty plasma. Finally, Sec. IX ctains a summary of our investigation.

II. PROPERTIES OF DUSTY PLASMAS

The constituents of dusty plasmas are neutral gas mecules, electrons, ions, and massive~compared to the ions!charged dust grains. There are three characteristic lescales for such a combined dust and plasma mixture. Thare the dust grain radiusR, the dusty plasma Debye radiulD , and an average intergrain distanced. The latter is relatedto the dust number densitynd by ndd3;1. The dusty plasmaDebye radiuslD is given by44

1

lD2

51

lDe2

11

lDi2

, ~1!

where lDe5(Te/4pne0e2)1/2@lDi5(Ti /4pni0e2)1/2# is theelectron ~ion! Debye radius, whereTe (Ti) is the electron~ion! temperature,ne0 (ni0) is the unperturbed electron~ion!number density, ande is the magnitude of the electrocharge. WhenTe;Ti and ne0;ni0, we havelDe;lDi ,while for Te@Ti and ni0.ne0 we havelD;lDi@lDe . Industy plasmas, we typically haveR!lD . One can treat thedust from a particle dynamics point of view whenR!lD

,d, and in that case we have a plasma containing isolascreened dust grains, or a dust-ion plasma. On the ohand, collective effects of charged dust grains become


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1793Phys. Plasmas, Vol. 8, No. 5, May 2001 A survey of dusty plasma physics

portant whenR!d,lD . Here, charged dust particulatewhich are essential ingredients of the total plasma mixtucan be treated as massive point particles similar to multcharged negative~or positive! ions in a multispecies plasmaThe dusty plasma quasineutrality condition for negativcharged dust grains is

ni05ne01Zd0nd0 , ~2!

wherenj 0 is the unperturbed number density of the partispeciesj ( j equalse for the electrons,i for singly chargedions, andd for the dust grains! and Zd0 is the number ofunperturbed charges residing on the dust grain surface. Wmost of the electrons from the ambient plasma are attacto the dust grain surface, we may haveZd0nd0@ne0. How-ever, it should be noted that the depletion of the electrcannot be complete~e.g., Whipple et al.43!, because theminimum value of the ratio between the electron andnumber densities turns out to be (me /mi)

1/2 when the grainsurface potential approaches zero, whereme (mi) is the elec-tron ~ion! mass. Here, the dusty plasma may be regarapproximately as a two component plasma composednegatively charged dust grains and ions; the latter shielddust grains. Such a situation is common in the Saturn rias well as in low-temperature laboratory discharges. Onother hand, in thermal or UV irradiated dusty plasmas,grains emit electrons and they are charged positively.shielding of positive grains comes from the electrons, anequilibrium we have45 ne0'Zd0nd0, since the ion numbedensity is completely depleted.

For a spherical geometry, the solution of the linearizPoisson equation

¹2fd21

ld2fd50 , ~3!

where fd is the Debye–Hu¨ckel ~or screened Coulomb! orYukawa potential

fd~r !5fg~r 0!r 0

rexpS 2

r 2r 0

ldD , ~4!

fg(r 0) is the floating potential of the dust particle at thlocationr 0, and the effective dusty plasma Debye radius~in-cluding the dust charge fluctuation effects23,46! is

ld5lD

~11 f dn2 /n1!1/2. ~5!

Here, we have definedf d54pnd0lD2 R. Furthermore,

n15~R/A2p!@~vpi /lDi !1~vpe /lDe!exp~efs /Te!#

is the dust charge relaxation frequency arising from the dgrain surface potential (fs) changes, and

n25~R/A2p!@~vpi /lDi !~12efs /Ti !

1~vpe /lDe!exp~efs /Te!#

is a frequency associated with changes in the orbit limimotion ~OLM! currents due to the presence of the oscillat


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potential, andvpi (vpe) is the ion ~electron! plasma fre-quency. The term f dn2 /n1 arises due to dust chargfluctuations.46–53

The dust grains in a dusty plasma could be either weaor strongly correlated depending on the strength of the Clomb coupling parameter

G5Q2

dTdexp~2k!, ~6!

whereQ5Zd0e, Zd0 is the number of unperturbed chargresiding on the dust grain surface,Td is the dust temperatureandk5d/ld . A dusty plasma can be considered as weacoupled as long asG<1. However, whenG@1 andk<1,charged dust microspheres strongly interact with each otand we have the possibility of forming Coulomb lattices instrongly coupled dusty plasma. Strongly coupled dusty pmas are created in low-temperature dusty plasma dischafor studying the formation and dynamics of dusty plasmcrystals. They are also found in laser implosion experimeas well as in colloidal systems.

There have been arguments that a dusty plasma is slar to a multi-ion plasma. However, this assertion has torefuted, because a dusty plasma is significantly differfrom a multi-ion plasma in that the presence of masscharged dust grains produce new collective phenomenacompletely different time and length scales. An examplethe DAW16 in which the dust mass provides the inertiwhile the restoring force comes from the pressures ofinertialess electrons and ions. In laboratory dusty plasdischarges,24–29 the DAW frequency is typically 10–20 Hzand video images of the DAW wavefronts are possible.24,28

Also, the dust charge fluctuation dynamics47–53 and dust–dust interactions54–57give rise to new effects. The dust maand shape distributions58 as well as their rotation59,60and theplasma boundary61 also introduce new effects in dusty plamas. Furthermore, there is a dust lattice wave21,62–64whosecounterpart exists only in solids.65 Besides, dusty plasmasupport a great variety of nonlinear structures including dacoustic66 and dust ion-acoustic31,67,68 shock waves as welas Mach cones69 and vortical structures.70–73 Finally, in astrongly coupled dusty plasma we have the possibilitynew attractive forces~viz. the wakefield,74,75 the dipolarinteraction59,76–78 etc.! as well as phase transitiophenomena79,80 in dusty plasma crystals. In contrast to solstate crystals, the latter have many unusual properties.ion charges in solid state crystals are of order unity, whiledusty plasma crystals charges on dust grains are huge~hun-dreds of thousand of the electron charge! and associated interaction energies~typically 900 eV! are many orders ofmagnitude larger than in solid state crystals. Also the lattspacing in dusty plasma crystals is of the order of a mmcontrast to the 0.1 nm scale in solid state crystals. Therefdusty plasma crystals have ordered plasma inhomogenedue to the shielding. It turns out that knowledge of baplasma physics, probe theory, statistical mechanics, asas solid state physics and condensed matter physics isquired for understanding numerous collective processedusty plasmas.


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III. CHARGING OF DUST GRAINS

Dust particles are charged due to a variety of procesincluding the bombardment of the dust grain surfacebackground plasma electrons and ions, photoelectron esion by UV radiation, ion sputtering, secondary electron pduction, etc. In low-temperature laboratory plasmas, dparticles are mainly negatively charged when any plaselectrons hitting the surface of the dust grains are attacheit and simply lost from the background plasma. Here,charging process depends on the charging cross sectwhich are determined by the impact parameter of the partthat approaches the grain to distances smaller than theticle size. Thus, the charging cross sections for electronsions are given by, respectively,

se~qd ,v !5pR2S 112eqd

Rmev2D , ~7!

and

s i~qd ,v !5pR2S 122eqd

Rmiv2D , ~8!

for v2.2euqdu/Rme[v*2 , whereas forv2,v

*2 we have

se(qd ,v)50; herev5uvu andqd is the dust charge. Clearlythe electrons must have a minimum speedv* in order toarrive at the grain surface. The charging equation is tgiven by

~] t1vd•¹!qd5I e1I i[I d~qd! , ~9!

where

I d~qd!5 (s5e,i

qsE vss~qd ,v ! f s~v!d3v ~10!

is the plasma current through the dust particle surface,vd isthe dust velocity,qe52e,qi5e, and f s(v) is the velocitydistribution of the particle speciess.

If the dusty plasma is close to equilibrium, then the dtribution function f s can be approximated by a Maxwelliadistribution (f s0) with the drift velocity v0 between theplasma and dust. We have

f s05ns0S 1

2pv ts2 D 3/2

expS 21

2v ts2 ~v2v0!2D , ~11!

where ns0 is the unperturbed number density andv ts

5(Ts /ms)1/2 is the thermal speed of the particle species.

Assuming that the streaming velocities of electrons aions are much smaller than their respective thermal velties, we have the following expressions for the equilibriuelectron and ion currents,35,51 respectively,

I e052pR2eS 8Te

pmeD 1/2

ne0 expS eqd0

RTeD , ~12!

and

I i05pR2eS 8Ti

pmiD 1/2

ni0S 12eqd0

RTiD . ~13!


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On the other hand, if the ion streaming velocityv0 is muchlarger than the ion thermal velocity, then the approximexpression for the ion current is51

I i0'pR2eni0v0S 122efg

miv02 D . ~14!

For an arbitrary value ofv0, we have to use a somewhamore complicated expression51 for I i0, namely,

I i05pR2eS 8Ti

pmiD 1/2

ni0FF1~u0!2F2~u0!eqd0

RTiG , ~15!

where F1(u05v0 /A2v t i)5(Ap/4u0)(112u02)erf(u0)

10.5exp(2u02) andF2(u0)5(Ap/2u0)erf(u0) are written in

terms of the error function erf(u0)5(2/Ap)*0u0exp(2j2)dj.

It is easy to show by Taylor expansion aroundv050 thatboth the functionsF1 and F2 approach unity asu0 ap-proaches zero.

For the equilibrium, we can setI e01I i050 and obtain,on using Eqs.~12! and ~13!, the expression

v te exp~efg /Te!5ni0

ne0v t i S 12

efg

TiD , ~16!

which determines the surface potentialfg of isolated dustgrains. The electrons are initially collected by a dust gradue to their higher thermal velocity relative to the ions. Sinthe grain is electrically floating, it charges to a negativesurface potential,fg,0, in order to repel further electroncollection and enhance ion collection. A sphere in a thermized hydrogen plasma floats tofg522.51T/e, where wehave assumed thatTe5Ti5T andni0'ne0.

The grain mean charge,qd , is related to its surface potential,fg,0, by the grain capacitance,C, which for spheri-cal isolated grains is simplyR, and thusqd5Rfg . Thismodel for the grain charge applies to the case wheregrains are sufficiently far apart~in comparison with the De-bye lengthlD of the dusty plasma!. On the other hand, whenthe spacing between the grains is comparable to or lesslD , the dust grains are closely packed. Here, the differeF5fg2f0 between the surface potentialfg and the plasmapotentialf0 has a smaller magnitude than in the case wd@lD , and consequently the average charge on a dust gqd5FR is smaller than for isolated grains.1 For this case, wereplace fg in Eq. ~16! by F and use ni0 /ne05(12qdnd0 /ene0) to obtain the variation ofF against the am-bient plasma number density for a fixed dust number dennd0. Barkanet al.30 have described the results of a laboratoexperiment on the charging of dust grains in a fully ionizesteady state, magnetized plasma column. By varying thetio d/lD , they experimentally demonstrated the predictreduction42,43 in the grain charge for the case of ‘‘closepacked’’ grains (d/lD,1).

Next, we discuss conditions under which the dust graare charged positively. In an inert gas with dispersed dgrains, the latter are charged positively36,48–50either by UV-induced photoemission from the dust, or by thermionic emsion from radiatively heated dust grains. Both of themechanisms would require dust grains of low work functiW. Examples of low work function material include variou


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metals that typically haveW,5 eV ~e.g. Ag, Cu, Al, Ca, andCs carbides with W;2.18– 3.50 eV, borides withW;2.54– 2.92 eV, and oxides of metals with work functioranging fromW51 eV to W54 eV.

We focus on the creation of a dust-electron plasma~elec-trons and positively charged dust! when the grains arecharged positively due to the photoelectric emission of etrons in the presence of a flux of UV photons with enerW,\n,I , where \ is the Planck constant,n is the fre-quency, andI is the ionization potential of an inert gas. Forunidirectional photon flux, the electron photoemission crent for fg[fp.0 is35,36

I ph5epR2QabsYJ exp~2efp /Tpe! , ~17!

whereQabs is the efficiency of absorption for the UV radiation (Qabs;1 for 2pR/l.1, wherel is the wavelength ofthe radiation!, J is the UV photon flux,Y is the yield of thephotoelectrons, andTpe is their average energy. The expnential factor in Eq.~17! takes into account the fact that thphotoelectrons have sufficient energy to overcome the potial barrier of positively charged grains with the surface ptential fp . When the ratio between the electron mean fpath ~in the gas! to the dust radius is large, the orbit-limitecurrent to a positive dust probe is35,36

I pe52~8p!1/2ene0v teR2S 11

efp

TeD . ~18!

By setting I ph1I pe50, we then obtain the equilibriumchargeZd

Zd~11bZd!5Ap

8

QabsYJ

nd0v teexp~2bZdTe /Tpe! , ~19!

whereb5e2/RTe and we have usedQ5Zde5Cfp[Rfp

andne05Zdnd05Rfpnd0 /e.Next we suppose that the dust grains are positiv

charged due to thermionic emission induced by laser hea~i.e., infrared or visible! or by thermal radiative heating. ThRichardson–Dushman form for the thermionic emission crent is35

I th54pR2A0Tg2 expS 2

~W1efp!

TgD , ~20!

whereA054peme /\3 and Tg is the grain temperature. Busing I th1I pe50, we obtain36

y~11y!exp~ty!5A2pb

ev tend0A0Tg

3 expS 2W

TgD , ~21!

where y5efp /Te and t5Te /Tg . The laser intensity re-quired to heat the grains to the temperatureTg can be esti-mated by balancing the heating rate due to a unidirectiophoton flux and the blackbody radiative loss rate.

Recent experiments on the Russian space stationhave generated a dusty plasma with positively charged grunder solar radiation in the microgravity environment.39 Fur-thermore, Sickafooseet al.40 have conducted a laboratory eperiment to demonstrate that photoelectric emission~UV il-lumination! causes an isolated spherical grain in vacuum


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attain a positive charge. The mean measured charge is ptive with a value of 5.3(61.6)3104 e for zinc grains,5.0(61.0)3104 e for copper grains, and 4.1(61.0)3104 efor graphite grains.

IV. WAVES IN DUSTY PLASMAS

The addition of charged particulates in an electron-plasma is found to modify or even dominate wave propation, wave instability, wave scattering, etc. The modificatiooccur owing to the inhomogeneity associated with the rdom distribution of charged particulates and the departfrom the conventional quasineutrality condition in aelectron-ion plasma due to the presence of chargedgrains. On the other hand, in multicomponent dusty plasmthere appear new wave modes that are associated withdust grain dynamics. Here, we discuss the physics of theacoustic, dust ion acoustic, and dust lattice waves whhave been experimentally observed.

During the ‘‘Capri Dusty Plasma Physics Meeting’’ iJuly 1989, the author proposed consideration of the wphenomena taking into account the dust grain dynamicsBoltzmann electron and ion distributions. In such a situatiwe can have extremely low phase velocityvp5v/k ~in com-parison with the electron and ion thermal velocities!, low-frequency (v; tens of Hz, which is much smaller than thdust plasma frequency in low-temperature dusty plasmacharges! DAWs16 in which the restoring force comes fromthe electron and ion pressures while the dust mass provthe inertia to maintain the DAWs. The wavelengths (2p/k)of the latter are;0.6 cm, and they appear on a very lontime scale so that their wavefronts can be seen with naeyes.24

The spectra of dusty plasma waves are obtained by Frier analyzing the Vlasov, Poisson, and Maxwell equatiosupplemented by the dust charging equation. In a duplasma, the properties of the electrostatic waves are demined from

e~v,k!f50 , ~22!

wheref is the wave potential,

e~v,k!511xe1x i1xd1xqe1xqi ~23!

is the dielectric constant,xe , x i , xd are the electron, ionand dust susceptibilities, respectively, andxqs is the linearsusceptibility associated with the dust charge fluctuationnamics. In a dusty plasma the electrons and ions are wecoupled, and forv/k!v te ,v t i , we have

xe'1

k2lDe2

, ~24!

and

x i'1

k2lDi2

, ~25!

which correspond to Boltzmann distributed electrons aions; i.e., the corresponding electron and ion number denperturbations arene0ef/Te and2ni0ef/Ti , respectively.


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When the dust grains are weakly coupled, we obtainv/k@v td , the dust susceptibility

xd'2vpd

2

@v~v1 ind!23k2v td2 #

, ~26!

wherev td5(Td /md)1/2 is the dust thermal speed,nd is theeffective dust collision frequency, and vpd

5(4pZd02 e2nd0 /md)1/2 is the dust plasma frequency.

For negatively charged dust grains, we have51,52

xqe1xqi5f dn2

k2lD2 ~n12 iv!

. ~27!

For nd ,kv td!uvu!n1, we obtain the dust acoustic wave frquency

v5kldvpd

~11k2ld2!1/2

, ~28!

which for f dn2 /n1!1 andk2lD2 !1 reduces to16

v5kCD . ~29!

Here, CD5Zd(nd0 /ni0)1/2(Ti /md)1/2(11ne0Ti /ni0Te)1/2 is

the dust acoustic velocity.Using the expression~29!, the wave phase velocityvp

5v/k can be estimated81 if one knows the plasma and duparameters. In the experiment of Chuet al.,23 one hasR'1mm, nd0md;1 g/cm3 ~and thusmd'4310212 g!, Te

52.6 eV, Ti50.26 eV, andTd51022Te . The charge oneach dust grain isZd;8100. The average intergrain distanis of the order of 300mm, from whichnd0;3.73104 cm23.The plasma density is roughlyne0;53109 cm23. For thisset of parameters, Eq.~29! depictsvp5v/k'7 cm/s, whichis to be compared with thevp5 f w3lw512 Hz30.5 cm56 cm/s observed in the experiment. Furthermore, inplasma with ne0Ti!ni0Te , we have CD

5Zd(Ti /md)1/2(nd0 /ni0)1/2. For dust grains of;5 mm size,md'10212 kg, Zd'43104, andnd0 /ni0'1028, we obtainCD;8 cm/s, which is in good agreement with thobservations24 which report vp;9 cm/s. Hence, forlw

'0.6 cm we obtain the DAW frequencyf w;15 Hz.The spatial attenuation rate of the DAWs in a collision

dusty plasma is determined from26

k251

ld2

v~v1 ind!

@vpd2 2v~v1 ind!#

, ~30!

wherek5kr1 ik i , kr(ki) is the real~imaginary! part of thewave number, andv is real.

Next, we consider the dust ion-acoustic waves~DIAWs!which occur in the frequency domainkv td ,kv t i ,vpd!uvu!kv te ,vpi , where vpi is the ion plasma frequency. Othese time scales, the electrons follow the Boltzmann disbution, while the ions~dust grains! are inertial~immobile!.Thus, xe is given by Eq.~24!, while the ion susceptibilityreads31

x i'2vpi

2

v~v1 in i !23k2v t i2

, ~31!


r

a

l

i-

where n i is the effective ion-~dust/neutral! collision fre-quency.

Neglecting the dust charge fluctuation effect, we obtthe frequency of the DIAWs17 for uvu.n i

v5klDevpi

~11k2lDe2 !1/2

, ~32!

which for k2lDe2 !1 reduces to

v5kCS , ~33!

whereCS5(ni0 /ne0)1/2(Te /mi)1/2 is the dust-modified ion-

acoustic speed. We note in Eq.~33! that the DIA wave phasevelocity, v/k, increases with the relative dust concentratiobecause in dusty plasmas, we haveni0.ne0.

Barkan et al.30 performed an experiment in a dusplasma device to investigate the propagation and dampindust ion-acoustic waves. They found that the phase veloof the latter increases in accordance with Eq.~33!. As aconsequence of the phase velocity increase, the ion Landamping rate is significantly reduced. The observed DIAfrequencies were in the range 3–5 kHz~depending on thevalue ofZdnd0 /ni0).

In a plasma with positively charged dust grains and eltrons, the shielding of the dust grains comes from the etrons. Here, the frequency of low-phase velocity~in compari-son with the electron thermal velocity! DAWs is45 v5klDevpd /(11k2lDe

2 )1/2. Inclusion of the dust charge fluctuation somewhat modifies the shielding length.

Collective modes in a strongly coupled dusty plasmwere theoretically investigated by several authors.54–57

Rosenberg and Kalman55 investigated the effect of strondust coupling on the DAWs by supposing that charged dgrains interact with each other via a screened Coulo~Yukawa! potential

fD~r !5Q2

rexp~2r /lD! , ~34!

with the exponential factor taking into account the screenof the dust charge by the plasma electrons and ions whichweakly correlated. The quasilocalized charge~QLC! ap-proximation was then used to derive the following dispersrelation for DAWs:55

v5vpdS k2d2

k2d21k21D~k,G!D 1/2

, ~35!

wherek5d/lD is a measure of dust charge screening byplasma, and the term arising from strong coupling is givapproximately byD(k→0)' f sk

2d2 with f s'2(4/45)@0.910.05k2# whenk<1 andGd5Zd

2e2/dTd@1. In the regimekd!k ~i.e., kld!1), Eq. ~35! gives v'kCD(11 f sk

2)1/2.The latter shows that the effect of strong dust coupling isreduce the DAW phase velocity, sincef s,0. The decreaseof the phase speed ask increases may be related to an icrease in the compressibility of the dust fluid as the rangethe intergrain potential decreases. In the regimekd@k ~i.e.,klD@1), we obtain from Eq. ~35! v'vpd(11 f sk

2k2lD2 )1/2, which shows that the effective dust plasm


dula

-eheth

al

dcyf

he

p-is

it

taal2

vet

i

lyb

relrvsalined

y

s,

-the

en-

thea.hetwoion


frequency is reduced due to a decrease of the effectivecharge with stronger screening. The DAW dispersion retion of Murillo56 in the strong coupling limit is v5kCD /(11k2lD

2 )1/2(11k2d2/16)1/2, which does not de-pend explicitly onG.

On the other hand, Kaw and Sen57 employed a generalized hydrodynamic~GH! description, which incorporates thnonlocal viscoelasticity with memory effects arising from tstrong correlation among dust particles, and obtaineddust susceptibility

xd52vpd

2

v22gdmdk2v td2 1 ivk2h* /~12 ivtm!

, ~36!

wheregd is the adiabatic index,md511U(G) is the com-pressibility,U(G)5Ec /nd0Td is the so-called excess internenergy, Ec is the correlation energy,h* 5@(4hs/3)1jb#/mdnd0 and tm5@(4hs/3)1jb#/@nd0Td(12gdmd)14U(G)#, andhs ,jb are the coefficients for the shear anbulk viscosity, respectively. For longitudinal low-frequenwaves (v!kv te ,kv t i), the linear dielectric responses oweakly coupled electrons and ions~which obey the Boltz-mann law! are given by Eqs.~24! and ~25!. Thus, from 11xe1x i1xd50, we obtain foruvu,tm!1

v25k2S gdmdv td2 1

vpd2 lD

2

11k2lD2 D 2 ivh* k2 , ~37!

which depicts the damping of the modified DAWs. On tother hand, foruvutm@1, one obtains

v25k2CD

2

11k2lD2 F11gdmd

ld2

lD2 ~11k2lD

2 !S 114

15U~G! D G ,

~38!

whereld5v td /vpd is the dust Debye radius.Kaw and Sen57 have also shown that their GH descri

tion also admits a low-frequency ‘‘shear’’ mode whicheither a damped wavev'2 ih* k2 for uvu!tm

21 , or apropagating wavev'k(gdEc /nd0md)1/2 for uvu@tm

21 . Thelatter is analogous to elastic wave propagation in solids wthe correlation energyEc playing the role of the elasticmodulus. Transverse waves have been experimenobserved82 in a two-dimensional screened Coulomb crystThey were excited by applying a chopped laser beam to adusty plasma. Measurements of the dispersion relation rean acoustic, i.e., nondispersive, character over the enrange of measured wave numbers, 0.2,kd/p,0.7. Recentworks83 have further focused on wave dispersion relationsstrongly coupled plasmas.

Next, we discuss the properties of DLWs in the strongcoupled plasma crystal in which the particulates interactmeans of their mutual~shielded! Coulomb repulsion. Undethese conditions wave type motions of the dust grains rtive to each other are possible, and they have been obsein several experiments.32–34 In linear particle arrangementas well as in monolayer plasma crystals these waves hbeen identified as dust lattice waves under strong coupconditions. The dispersion relation of the DLW is obtainfrom the equation of motion for a dust particle


st-

e

h

lly.Dal

ire

n

y

a-ed

veg

]2xn

]t21nd

]xn

]t5

Fc

md, ~39!

where nd is the dust-neutral collision frequenc('2A2ngR2cg with the Epstein drag law;ng is the neutralgas density,R!lm5 mean free path of the gas moleculeandcg is the thermal velocity of the gas molecule! and21

Fc5Zd

2e2

d3@~212k1k2!exp~2k!#~xn2122xn1xn11! ,

~40!

describes the force acting onnth particle due to its interaction between neighboring particles in the presence ofDebye–Hu¨ckel ~or Yukawa!# interaction potentialfD(x)5(Q/uxu)exp(2uxu/lD).

Following the standard approach65 for longitudinalwaves on an infinite linear chain, we obtain from Eq.~39! thedispersion relation for the DLWs as

v21 ivnd5vDL2 , ~41!

where

vDL2 5

vpd2

pnd0d3~212k1k2!exp~2k!sin2~kd/2! , ~42!

is the squared dust lattice frequency.21 Dust lattice waves indust plasma crystals have been observed experimentally33,34

in rf discharges. In the latterf DL5vDL/2p is typically lessthan 50 Hz, which is larger thannd;4 Hz. The knowledgeof the dust lattice frequency is useful in deducing the screing of the particles in the rf sheath.

V. INSTABILITIES IN DUSTY PLASMAS

In the past, Bharuthramet al.84 and Rosenberg85 dis-cussed the possibility of dusty plasma wave excitation inpresence of equilibrium ion drifts in a uniform dusty plasmSpecifically, they theoretically predicted the excitation of tdust acoustic and dust ion-acoustic waves due to thestream and kinetic instabilities. The appropriate dispersrelation for the DAWs in the kinetic regime is85,86

111

k2lD2

1 i1

k2lDi2 S p

2 D 1/2v2kui0

kv t i2

vpd2

v250 , ~43!

whereui05eE0 /min i is the streaming ion drift velocity inthe presence of a constant electric fieldE0. Equation~43!admits an oscillatory instability of the DAW whenui0

.uv r1 iv i u/k, where v r5kCD /(11k2lD2 )1/2 and the

growth rate is

v i'S p

8 D 1/2 v r3

vpd2

1

k2lDi2

ui0

v t i. ~44!

For ni0 /ne0@1 andTe@Ti , the expression Eq.~44! for thegrowth rate (;v r) of the ion streaming driven DAWs(v r /2p;15 Hz, v r /k;9 cm/s, and 2p/k;0.6 cm! is con-sistent with observations24,29 which reportsu05eE0 /min in

;23105 cm/s forE0;1 V/cm.


u

t

th

e

sbls

tie

i-

d

f

tevOe

e,

uai

-

edmre-

sed

qs.re

lyti-ed

ller

ma

the

stsmain

s.tic

erain

ls,


On the other hand, the dispersion relation in the ion-dtwo-stream regime is87

12vpi

2

As~v2kui0!2 S 12 in i

v2kui0D2

vpd2

Asv2

50 , ~45!

where As511(klDe)22. In the absence of ion-dus

collisions, Eq. ~45! for kui0@v gives v r;v i

;(vpivpd2 )1/3/AAs, with a maximum growth rate atkui0

;vpi /AAs. Furthermore, in a collisional dusty plasma wiv!kui0;vpi /AAs,n i , Eq. ~45! has the approximatesolution87

v'vpd

11 i

A2S vpi

n iD 1/2 1

As3/4

, ~46!

which admits a dissipative instability.However, in a nonuniform dusty plasma sheath, ther

dust charge gradient appears.88 Shukla89 has shown that freeenergy stored in the latter can be coupled to the DAWConsequently, the dusty plasma sheath becomes unsta88

In order to understand the physics of an unstable duplasma sheath, we first consider its equilibrium properwhich are governed by nd0u05constant, u0u081 (Q0 /md) f081gx50, u0Q085I e01I i0[I 0 , and e(ni0

2ne0)1Q0nd050, whereu0 is the component of the equlibrium dust fluid velocity along thex axis, u085]u0 /]x,Q0(5fg /R) is the unperturbed dust charge,fg is the un-perturbed grain potential,f085]f0 /]x[2E0x is the unper-turbed sheath electric field,mdgx is thex component of thegravity force, andmd is the dust mass. The unperturbeOLM currents are I e052pR2ene(f0)3(8/p)1/2v te exp(eQ0 /RTe) and I i05pR2eni(f0)3(8/p)1/2v t i(12eQ0 /RTi). It follows that in the absence othe equilibrium dust fluid velocity, we haveQ0E0x5mdgx

and I e01I i050. The latter determines the equilibriumcharge on the dust grain surface, while the former dictathat the balance between the sheath electric and graforces is responsible for the levitation of the dust grains.the other hand, in the presence of a uniform dust flow, thappears a dust charge gradientQ085I 0 /u0, which can be ex-pressed as Q0852pR2e(8/pu0

2)1/2@Nev te exp(eQ0 /RTe)2Nivti(12eQ0 /RTi)#, where Ne5ne0 exp(ef0 /Te) and Ni

5ni0 exp(2ef0 /Ti).In order to study the instability of our equilibrium stat

as described above, we let the number densitynd5nd01n1,the dust fluid velocity ud5u0x1u1, the potential f5f0(x)1f1, and the dust chargeqd5Q0(x)1Q1, wherend0 and u0 are uniform, andn1 , u1 ,f1, and Q1 are smallperturbations of their equilibrium values. The relevant eqtions for the perturbed quantities associated with the DAWa nonuniform dusty plasma are89

dtn11“~nd0u1!50, ~47!

dtu11~Q0 /md!“f11 x~f08/md!Q150, ~48!

“

2f15kD2 f124p~Q0n11nd0Q1!, ~49!

and


st

a

.e.tys

sitynre

-n

~dt1n1!Q11Q08u1x52n2Rf1 , ~50!

wheredt5] t1u0]x andkD2 (5lD

22) is the square of the Debye wave number. We notice that Eqs.~48! and~50! containthe terms (f08/md)Q1 andQ08u1x , which are associated withperturbed electrostatic forces involving the unperturbsheath electric field and the convection of the equilibriudust charge gradient, respectively. These two forces aresponsible for the novel absolute instabilities, as discusbelow.

The local dispersion relation can be obtained from E~47! to ~50! by supposing that the perturbed quantities aproportional to exp(iky2ivt). For k2/kD

2 !1 we have89

12vD

2

v21S 11

VD2

v2 D f dn2

n12 i ~v1V!50, ~51!

wherevD5kvpd /kD[kCD is the dust acoustic frequency,16

VD2 5Q0kQf08/md , kQ5Q08/Q0 , f d54pnd0R/kD

2 , and V5VD

2 /v. Equation~51! is a cubic polynomial inv, whichcan be analyzed numerically. However, some useful anacal results, which exhibit novel instabilities, can be derivfrom Eq. ~51! in several limiting cases.

We consider that the wave frequency is much smathan the dust charge relaxation frequencyn1, which is typi-cally the case in low temperature laboratory dusty plasdischarges. Here, Eq.~51! takes the form

~v22vD2 !~n12 iVD

2 /v!1~v21VD2 ! f dn250. ~52!

Several useful results follow from Eq.~52!. First, for VD

50 ~which ensures that there are no dc electric field anddust charge gradient!, we have from Eq.~52! the modifiedDAW frequency v5vD /(11 f dn2 /n1)[V0. Second, forv'vD1 ig, whereg,vD ,uVDu, Eq. ~52! gives the growthrate

g5~vD

2 1VD2 !VD

2 f dn2

2~n12vD

2 1VD4 !

, ~53!

of a novel DAW instability forVD2 .0. The latter is fulfilled

if E0xQ08,0. Clearly, the dc electric field and the ducharge gradient must oppose each other for the dusty plato become unstable. As an illustration, we mention thatlaboratory experiments,90 we typically havene0;ni0'108

cm23, Te;10Ti'1 eV, R;1210mm, lD;102– 103mm.Accordingly, for f dn2 /n1;1, n1;103 s21, vD;60 s21,andVD510 s21 the growth time, deduced from Eq.~52!, isa fraction of a second. This is consistent with observation88

Finally, we discuss the instability of the dust acouswaves in a dusty plasma which contains nonspherical~elon-gated! rotating dust grains.60 Here, the dipole moments of thdust grains are nonzero. The motion of a charged dust gin the electromagnetic fields (E52¹f2c21] tA and B5¹3A, wheref andA are the scalar and vector potentiarespectively, andc is the speed of light in vacuum! is de-scribed by the Lagrangian


e

nd

.of

of

hec

n

ria

nccio

ahtynae

ese

ynn-

foreshichorrainn-

iveteredthed.the

a

din ater

resstiche

stynon-

or-ob-

t offor--gh athe

,


L5mdv2

21

1

2I abVaVb1

q

cv•A~r ,t !2qf1m•B

1Fd11

2 (i

Dqi~Dr i•¹!G•S E1v

c3BD , ~54!

wheremd5( iDmi andq5( iDqi are the total mass and thcharge of the grain, I ab5( iDmi@(Dr i)

2dab

2(Dr i)a(Dr i)b# is the tensor of the moment of inertia,d5( iDqiDr i is the dipole moment of the elongated grain, am5(1/2c)( iDqi(Dr3Ui)(Ui5V3Dr i is the rotational ve-locity of the grain! is the magnetic moment of the grainFurthermore,Dmi and Dqi are the mass and the chargethe ith part of the grain, respectively,r i5r1Dr i andui5v1V3Dr i are its coordinate and the velocity (v andr are thevelocity and position of the center of mass,Dr i is the coor-dinate of thei th part of the grain relative to the centermass, andV is the angular velocity of the dust grain!. Wenote that Eq.~54! has been derived by assuming that tscale of inhomogeneity of the electromagnetic field is mularger than the grain sizeR, as well as by using the relatiodtd5V3d.

Mahmoodiet al.60 have derived the kinetic equation fothe dust grain distribution in the presence of the LagrangEq. ~54!. Subsequently, the perturbed dust distribution fution is obtained for the case where the wave phase veloexceeds the thermal velocity of the dust particles. By choing the unperturbed distribution function of the form

f d05nd0~2pmdTd!23/2~2pITd!21/2

3expF2p2

2mdTd2

~pw2pw0!2

2ITdG , ~55!

whereTd is the dust temperature,pw05IV0, andV0 is thepreferred angular frequency of the rotating dust grains. Mmoodi et al.60 derived the dielectric tensor for the dusplasma following the standard method. For the longitudiwaves (v!kc) the modified dispersion relation for thDAW reads60

111

k2lD2

2vpd

2

v22

k'2

k2 F V r2

~v2V0!21

V r2

~v1V0!2G50, ~56!

where V r5(4pd2nd0/4I )1/2. From Eq.~56! it follows thatthe dust grain rotation gives a contribution only for wavwith k'

2 Þ0. ForV050 that contribution is expressed in thchange of the dust acoustic frequency

v5vDAS 11k'

2

k2

V r2

Vpd2 D 1/2

, ~57!

where vDA5klDvpd /(11k2lD2 )1/2. However, in the pres-

ence of the dust grain rotation, Eq.~56! admits complexsolutions for any rotation frequencyV0, satisfying the con-dition

V0,vDAF11S k'2

k2

V r2

vpd2 D 1/3G 3/2

. ~58!


h

n-tys-

-

l

Letting v5V01 ig in Eq. ~56!, whereg!V0, we obtain thegrowth rate forvDA(11k'

2 V r2/k2vpd

2 )1/2'V0

g531/222 4/3S k'2

k2

V r2

vpd2 D 1/3

V0 . ~59!

Equation~59! exhibits that the growth rate of the instabilitis proportional toV r

2/3V0. Thus, energy of the dust rotatiocan flow into dust acoustic oscillations, driving them at nothermal level.

VI. WAKEFIELD

Besides the Debye–Hu¨ckel and far-field potentials,75

there also appears an oscillatory wake potential74,75 in adusty plasma. The wakefield, which may be responsiblethe attraction of charged dust grains of like polarity, arisdue to the resonance interaction of a test dust charge wmoves with a velocity close to the modified ion acousticdust acoustic speeds. The physics of the charged dust gattraction is similar to the electron attraction in supercoductors in which Cooper pairs are formed due to collectinteraction involving phonons. In dusty plasmas, the latare replaced by the DIAWs and DAWs. Negatively chargdust grains feel an attractive force in the negative part ofoscillatory potential,74,75where the positive ions are focuseThe wake potential of a test charge in the presence ofDAW in an unmagnetized plasma is75

fw~r50,j t ,t !5qt

j tcos~j t /L ! , ~60!

where j t5uz2v ttu, qt is the charge of the test particle,L5lD@(v t2V0)22CD

2 #1/2/CD is the lattice spacing,v t is thetest particle velocity,V0 is the equilibrium ion streamingvelocity, r and z are the radial and axial coordinates incylindrical geometry. Foruv t2V0u;30 cm/s,lD;300 mm,andCD;6 cm/s, we find thatL;1 mm, which is in agree-ment with observations.23 The concept of the wakefield ansubsequent attraction of negatively charged dust grainslinear chain has been verified both by compusimulations90,91 and in dusty plasma experiments.92,93

VII. NONLINEAR STRUCTURES

In this section, we discuss possible nonlinear structuin dusty plasmas. Specifically, we focus on dust ion-acou~DIA ! shocks and DIA holes. The latter are due to ttrapped~vortex-like! ion distribution.

Dust ion-acoustic shocks in an unmagnetized duplasma may arise when there is a balance between thelinearity ~associated with the harmonic generation! and thekinematic viscosity introduced by the dust-ion drag. The fmation of shock structures in dusty plasmas has beenserved by Nakamuraet al.31 and Luo et al.67 Luo et al.67

have presented an experimental investigation of the effecnegatively charged dust grains on the ion acoustic shockmation in a Q machine. Luoet al.67 observed that ion acoustic compressional pulses steepened as they traveled throudusty plasma if the percentage of the negative charges inplasma on the dust grains was>75%. On the other hand


seerrokknn

ncsitthb

tyth

rs-

vi

.-

reeryu

-

rl

ob

ed

ted

ckstyam-s,ob-

estheons

ngustybed-

x-ntthe

gf aniesfre-ncy

Onsuald inthe

df.


Nakamuraet al.31 found that in the linear regime, the phavelocity of the DIA waves increases and the wave suffheavy damping when the dust number density in an election plasma is considerably increased. Furthermore, Namura et al31 found that an oscillatory ion-acoustic shocwave in a usual argon plasma transforms into a monotoshock front when it travels through the dusty plasma columThe formation of the shock structure is due to a balabetween the harmonic nonlinearity and a kinematic viscothat is produced by ion-dust collisions. The dynamics ofshock structure associated with DIA waves is modelledthe Kortweg–de Vries~K–dV!-Burgers equation68

~2l]t1n02h0]j2!F1g0F]jF1]j

3F50, ~61!

wherel5(d13s)1/2, d5ni0 /ne0 ,s5Ti /Te , n0;n id /vpi ,h0;md /vpilDe

2 , g5(3d21112s)/d, F5ef/Te , withn id being the ion-dust collision frequency,vpi is the ionplasma frequency, andmd is the kinematic viscosity of thedusty plasma. We note that Eq.~61! is derived from thehydrodynamic~comprising the Boltzmann electron densidistribution, the continuity and momentum equations forions! and Poisson equations, by employing the standardductive perturbation method and the stretched variablej5e1/2(x2lt) and t5e3/2t, wheree is a smallness parameter. Equation~61! without then0 term admits an oscillatoryshock structure. On the other hand, when the kinematiccosity overwhelms the dispersion, Eq.~61! assumes the formof a Burgers equation forn050. The Burgers equation, vizEq. ~61! without then0F and ]j

3F terms, admits a monotonic shock profile, as observed by Nakamuraet al.31

Nakamuraet al.31 also observed ion holes, which adescribed in terms of the nonlinear DIA involving thtrapped~vortex-like! ion distribution. The latter might appeaas a nonlinear saturated state of the two-stream instabilita dusty plasma. In the presence of trapped ions, the ion nber density perturbation for small amplitudes~viz. c!1) is68

ni'ni0@12aw2b~2w!3/21O~w2!#, ~62!

where a512A2MW(M /A2),b5(4/3Ap)(12a2M2)exp(2M2/2) for a<0, and the Dawson integral is denoted by W(y)5exp(2y2)*0

ydwexp(w2). Furthermore, w5ef/Ti , M is the Mach number~the speed of the nonlineastructure/the ion thermal speedv t i), and the electric potentiais assumed to be negative, restricted by2c<w<0, wherecplays the role of the normalized~by Ti /e) amplitude. Wenote thata is a parameter which determines the numbertrapped ions. A plateau in the resonant region is givena50, and a,0 corresponds to a vortex-like excavattrapped ion distribution.

Substituting ne'ne0(12F1F2/2) and Eq. ~62! intoPoisson’s equation, we readily obtain68

]r2C2FC1bC3/250, ~63!

whereC52w,r5x/lDi , and F5a1dTi /Te . A possiblesolution of Eq.~63! is a potential hole

w5225F2

16b2sech4~AFx/4lDi !. ~64!


sn-a-

ic.eyey

ee-

s-

inm-

fy

The Mach number and the maximum amplitude are relaby c1/254b/5F.

The above discussions of the formation of DIA showaves and DIA holes are limited to a weakly coupled duplasma. However, a recent laboratory experiment by Ssonovet al.69 has shown the formation of Mach cone shockor V-shaped disturbances created by superdust acousticjects, in two-dimensional Coulomb crystals. Mach conwere double, first compressive then rarefactive, due tostrongly coupled crystalline state. There are also indicatiof probe-induced particle circulation~Law et al.70! as well asvortex formation71–73 in strongly coupled plasma systems.

VIII. COLLECTIVE PROCESSES IN A DUSTYMAGNETOPLASMA

The foregoing investigations have dealt with chargiand numerous collective processes in an unmagnetized dplasma. However, laboratory and space plasmas are emded in an external magnetic fieldzB0, where z is the unitvector along thez direction andB0 is the strength of themagnetic field. While the charging of dust grains in an eternal magnetic field is not fully explored at the presestage, there have been several attempts to examineplasma wave spectra,94–105 wakefields,106,107 and nonenve-lope solitons108 in a magnetized dusty plasma, by ignorinthe dust charge fluctuation dynamics. The presence oexternal magnetic field can significantly affect the velocitof the plasma species. For example, when the wavequency is much smaller than the electron gyrofrequevce5eB0 /mec, we can have modified lower-hybrid~LH!waves (vci ,vcd!uvu!vce)

94 and electrostatic ion-cyclotron ~EIC! waves (v;vci!vce ,kzv te)

97 whose fre-quencies are, respectively,

v5vpi

~11de!1/2S 11

mine0kz2

meni0k'2 D 1/2

, ~65!

and

v5vciS 11ni0

ne0

k'2 cs

2

vci2 D 1/2

, ~66!

where de5vpe2 k'

2 /vce2 k2[ak'

2 /kz2 , and kz and k' are the

parallel ~to the external magnetic fieldB0z direction! andperpendicular components of the wave vectork(5 zkz

1k'), cs5(Te /mi)1/2 is the ion acoustic speed,vci

5eB0 /mic, andvcd5ZdeB0 /mdc. Sincene0,ni0 in dustyplasmas, the LH wave frequency is somewhat reduced.the other hand, we see that the phase velocity of the uEIC waves is increased when a dust component is addean electron-ion plasma. The LH waves can be excited inpresence of streaming~with the velocityVd0, which is pro-duced by the equilibrium electric fieldE0) dust grains. For(kz /k')2,(me /mi)ni0 /ne0 the maximum growth rate is102

gmax'(A3/24/3)vLH(Zd2nd0mi /ni0md)1/3, where vLH

5vpi /A11a'kyVd0. We note that the growth rate is valifor (vpd /vpi)

2/3.vci /vLH . Using the parameters of Re13, namely Vd0;4 km/s, B0;0.35 G, vpi /vci;500,vpe /vce;2.5, we find thatvLH;6273104 s21. The con-


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dition for the maximum growth rate then givesky;15– 18m21, which is approximately twice the wave number for tArecibo 430 MHz radar. Since the radar scatters with wanumbersks52k0, this LH instability, whose growth time istypically 0.25 ms, is relevant for the observations reportedRef. 13. Furthermore, we note that streaming electronsdrive EIC waves, which have been observed in a duplasma laboratory experiment96 that contains an externamagnetic field (B0<0.4 T!. The critical electron drift forexcitation of the current driven EIC instability isVcr

5v t i(6112ne0Ti /ni0Te), which is lower than the case witno dust component. The measured perpendicular waveleturned out to be;2.5 cm.

There are several ultralow-frequency (!vci) electro-static waves in a uniform dusty magnetoplasma. First,frequency (vcd!v!vci ,vcekz /k' ,kzv te) of the DAW in amagnetized plasma is100

v5klDe

~11k2lDe2 1k'

2 rs2!1/2S vpi

2 kz2

k21vpd

2 D 1/2

, ~67!

wherers5lDevpi /vci . Equation~68! shows that forkz /k@vpd /vpi , we have

v5ksCS

~11k'2 rs

21k2lDe2 !1/2

. ~68!

On the other hand, in the opposite limit, viz.kz /k!vpd /vpi , Eq. ~68! gives100

v'klDevpd

~11k'2 rs

21k2lDe2 !1/2

. ~69!

Second, forvpevpikzk'@vpdvcik2, we have the modified

convective cells

v'S ne0mi

ni0meD 1/2 kz

k'

vci!vci , ~70!

which propagate almost perpendicular toz.In a nonuniform dusty magnetoplasma, theE3B0 cur-

rent remains finite. This leads to the appearance of alow-frequency~in comparison with the ion gyrofrequencvci) electrostatic wave98

v5kykd

k'2

vci , ~71!

which is flute-like. Here,kd5ni021](Zdnd0)/]x. On the

other hand, the frequency of ES drift waves is95

v52CS2kyk i /vci(11bs), where k i5] ln ni0 /]x and bs

5k'2 CS

2/vci2 . Shukla and Varma98 and Shuklaet al.95 have

discussed the properties of vortices associated with modconvective cells@given by Eq.~72!# and ES drift waves.

The above discussions of wave motions in a magnetidusty plasma assume immobile dust grains. The inclusiothe dust grain dynamics will give rise to dust-Alfve´n,94 dustcyclotron,51 and dust whistler104 waves. The frequency(vcd!v!vci) of the latter isv5kz(kz

21k'2 )1/2c2vcd /vpd

2 .One should also comment on the possibility of perfor

ing laboratory experiments in dusty plasmas where the ef


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nanty

th

e

w

d

dof

-ct

of the magnetic field may be important on the dust motioMerlino et al.24 have suggested that using relatively smdust grains (;0.01mm!, it may be possible to investigatprocesses involving magnetized dust~e.g., for B050.5 T,Td50.025 eV, the mass density of the grain material;23103 kg/m3, we have the dust gyroradius'1 cm and thedust gyrofrequencyf cd5vcd/2p'10 Hz; hence video imaging of dust cyclotron waves should be possible as well!.

In a magnetized dusty plasma the Coloumbian and oslatory wake potentials are significantly modified. In the folowing, we briefly discuss the formation of wakefields in thpresence of EIC waves in a uniform dusty magnetoplasHere, the wake potential forj05uz2utu@lDe ,rs5Cs /vci

is107

fw'qt

j0

lDe2

rs2

cos~j0 /L0! , ~72!

whereL052u/3vci is the effective attraction length, andu isthe z component of the test charge velocity. Equation~68!also holds in the presence of magnetic field alignedstreaming~with the velocityVi0), in which caseu is replacedby Vi0 in j0 andL0. We see that the magnetic field significantly affects the amplitude of the wake potential as wellthe attraction length. Finally, we mention that the theorywake potentials in a nonuniform dusty magnetoplasmapresented by Salimullah and Shukla.111 They found that boththe Coulombian and oscillatory wake potentials are signcantly affected by the presence of a dc magnetic field anddensity inhomogeneity in a dusty plasma.

IX. DISCUSSION AND CONCLUSIONS

In this paper, we have described various collective pcesses in dusty plasmas. Starting from the occurrencedusty plasmas, we have described the properties of dplasmas as well as discussed charging of dust grains andwaves, new instabilities, and some coherent nonlinear sttures that are observed in an unmagnetized dusty plasThe focus was on the dust acoustic, dust ion acoustic,dust lattice waves. The low-frequency dust acoustic waare excited due to the streaming and kinetic instabilities iuniform plasma, while there appears a new class of instaity in a nonuniform dusty plasma sheath which contains aelectric field and an equilibrium dust charge gradient. Tlatter is maintained by the equilibrium electron and ion crents that reach the dust grain surface, as well as by a fiequilibrium dust flow that is driven by the dc electric field.is found that when the forces associated with the dc elecfield and the dust charge gradient oppose each other, a dplasma is subjected to an absolute instability. This instabihas been observed by Nunomuraet al.88 near the sheathboundary in a dusty plasma. Physically, instabilities aribecause the dc sheath electric field does work on thegrains to create dust charge fluctuations which cannot kin phase with the potential of electrostatic disturbances inonuniform dusty plasma with a dust charge gradient. Thfree energy stored in the latter is coupled to unstable elecstatic waves when the dc electric field~in association withthe dust charge fluctuation! produces a charge imbalance


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the dusty plasma. Furthermore, we have discussed thepersion properties of a dusty plasma containing elongaand rotating dust grains. It is found that the rotational eneof the dust grains can be coupled to plasma oscillations wthe wave frequency is close to the rotational angularquency of the dust grain. The instability of the longitudinwaves occurs only in the case when the wave vector liethe plane of the dust grains. Also pointed out is the imptance of collective effects with regard to the generationwakefields in dusty plasmas. The concept of wakefieldsassociated charged dust attraction in a linear chain hasverified by computer simulations and in laboratory expements. Finally, we have illuminated the physical propertof DIA shocks and DIA holes that were observed in a labratory dusty plasma device by Nakamuraet al.31 Attemptshave been made to correlate several theoretical predictwith laboratory observations and computer simulations idusty plasma. Due to the limitation of pages here, weable to discuss only a selected class of wave phenomenaweakly coupled dusty magnetoplasma. We have neithercussed ionization nor ion-drag related instabilities109 whichseem to be observed experimentally.110 Several laboratoryexperiments112–114have demonstrated the effects of poloidion flows associated withE3B0 drift on dust particle behavior in magnetized plasmas. However, we do hope thatpaper shall stimulate further studies of collective proces@linear waves~viz. DLWs, shear waves, etc.! and their insta-bilities as well as numerous nonlinear structures~viz. soli-tons, shocks, voids, and vortices!# in a strongly coupleddusty magnetoplasma. In closing, we stress that the resulthe present investigation should also be useful in understing the salient features of low-frequency fluctuations andsociated nonlinear structures~viz. shock waves! in planetaryrings, in interstellar dust-molecular clouds, in supernovaplosions, in cosmic particle acceleration, and in cometplasmas where massive charged particulates are commo

ACKNOWLEDGMENTS

The author dedicates this paper to D. Asoka Mendisthe occasion of his 65th birthday, as he regards Asokagreat scholar who has significantly contributed to thevancement of the knowledge in cosmic dusty plasmas.also thanks Horst Fichtner, A. A. Mamun, and Lennart Stflo for reading the manuscript and for valuable advices. Fthermore, the benefit of useful collaboration with Tito Medonca, Greg Morfill, David Resendes, Marlene Rosenbeand Davy Tskhakaya is gratefully acknowledged.

This work was partially supported by the Max-PlanInstitut fur Extraterrestrische Physik at Garching~Germany!and the Deutsche Forschungsgemeinschaft~Bonn, Germany!through the Sonderforschungsbereich 191 as well as byNATO Project entitled ‘‘Studies of Collective Processes in Dusty Plasmas’’ through Grant NSA~PST.CGL974733!5066, and by the European Uniothrough the Human Potential Research and Training Nworks for carrying out the project entitled ‘‘Complex Plamas: The Science of Laboratory Colloidal Plasmas and Msospheric Charged Dust Aerosols’’ through Contract N


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HPRN-CT-2000-00140. The author also acknowledgessupport of the International Space Science Institute~ISSI!,Bern ~Switzerland! for the international team ‘‘Dust PlasmInteraction in Space.’’

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