M. Salimullah et al- Ultra-Low-Frequency Electrostatic Modes in a Magnetized Dusty Plasma

8/3/2019 M. Salimullah et al- Ultra-Low-Frequency Electrostatic Modes in a Magnetized Dusty Plasma

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http://www.ictp.trieste.it/~pub_offIC/97 /193

United Nations Educational Scientific and Cultural OrganizationandInternational Atomic Energy Agency

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

ULTRA-LOW-FREQUENCY ELECTROSTATIC MODESIN A MAGNETIZED DUSTY PLASMA

M. Salimullah1Department of Physics, Jahangirnagar University,Savar, Dhaka-1342, Bangladesh

andInternational Centre for Theoretical Physics, Trieste, Italy,M.R. Amin

Department of Physics, Jahangirnagar University,Savar, Dhaka-1342, Bangladesh,Max Planck Insti tute for Extraterrestrial Physics,

85740 Garching, GermanyandInternational Centre for Theoretical Physics, Trieste, Italy,M. Salahuddin

Department of Physics, Jahangirnagar University,Savar, Dhaka-1342, Bangladeshand

A. Roy ChowdhuryDepartment of Physics, Jadavpur University, Calcutta-700 032, India

andInternational Centre for Theoretical Physics, Trieste, Italy.

MIRAMARE - TR IESTENovember 1997

Regular Associate of the ICTP.


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A B S T R A C TA study on the extremely low-frequency possible electrostatic modes in a finite tem-

perature magnetized dusty plasma taking the charged dust grains as the third componenthas been carried out using the appropriate Vlasov-kinetic theory for the dynamics of theelectrons, ions and the dust particles. It is found tha t the inequalities of charge andnumber density of plasma species, and the finite-Larmor-radius thermal kinetic effectsof the mobile charged dust grains, introduce the existence of very low-frequency electro-stat ic eigenmodes in the three-component homogeneous magnetized dusty plasma. Therelevance of the present investigation to space and astrophysical situations as well aslaboratory experiments for dust Coulomb crystallization has been pointed out.


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A dusty plasma is a three-component plasma with electrons, ions, and a dispersedphase of very massive charged grains of solid matter. Dusty plasmas are usually encoun-tered in the space and astrophysical situations 1" 3 , fusion devices4, industry and mod-ern technologies5, 6 as well as in the dust coagulation and macroscopic Coulomb crystalformat ions7"13 . Th e size of the heavy grains is typically in the m icrometer ran ge. Th edust grains are massive (md/m p ~ 10 6 10 12) and can be charged up to Zd ~ 10 3 10 5by various charging processes, such as, electron and ion plasma currents, photoelectriceffects, second ary imp act ionizatio ns, transie nt even ts, etc. Th e presence of this verymassive, charged, low-density grains in the plasma introduces new time and space scalesin the plasm a behavior leading to new waves and instabilities. This has been the sub-ject of many recent investigations. The process of charging of grains and g rain chargefluctuations are also interesting and have been investigated recently 1"3 .

In most of the earlier studies on collective effects in dusty plasmas, fluid descriptionof plasmas was considered 14 "18 . Since the mass of a dust grain can be many orders higherthan that of an ion, the finite-Larmor-radius (FLR) effects must become significant in asmall but finite temperature magnetized dusty plasma. In this situation, it is anticipatedth at the presence of externa l m agnetic field with th e inclusion of FL R th erm al kinetic effectcan also open a number of additional or new modes which usually cannot be found bythe cold-fluid magnetohydrodynamics (MHD) theory. Kinetic theory including the finiteLarmor radius effects was employed earlier 19 for the case of transverse propagation andthe presence of the dust-lower-hybrid mode was shown to exist in a hot magnetized dustyplasma. M.R. Amin 20 attempted a study of specific low-frequency electrostatic modes forpropagation in almost perpendicular and almost parallel to the magnetic field startingfrom an ap pro xim ate expression for the dielectric function [Eq.(2) of his pa pe r2 0]. To thebest knowledge of the authors, there has not been any systematic study on the variouspossible low-frequency electrostatic modes at different frequency regimes including theFL R effects in a hot m agnetized dusty p lasma . In this Letter, we investigate the very low-frequency electrostatic modes whose frequency ranges are in between the dust cyclotron


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and ion cyclotron frequencies as well as below the dust cyclotron frequency in a uniformlymagnetized dusty plasma by considering dust dynamics with Vlasov- kinetic theory. Inpart icular, we investigate the ultra- low- frequency electrostatic dust modes propagatingobliquely to the external magnetic field by including the FLR therm al kinetic effects.

We consider a fully ionized plasma consisting of positive ions, electrons and negativelycharged dust grains embedded in a uniform ambient magnetic field. Although the size,mass, and the charge of the dust grains vary from one grain to another, we assumet h a t the dust grains all have uniform size and a constant negative charge, and t h a t theyne i th e r break up nor coalesce. When the grain size is much smaller t h a n the wavelengthof perturbations and the interparticle distance, t h e n the dust grains may be treated asnegatively charged poin t masses (like negative ions). In th e equilibrium, t he plasmais quasineutral and the conservation of the particle number density must always hold.Thus, for negatively charged dust grains, the quasineutrality con dition demands t h a tn e0 + Zdnd0 = Zini0, where Zi an d Zd refer to the charge numbers of ions and dustparticles; nj0 is the un per turbed particle number density. We assume con stant chargeon the dust particles and thereby neglect any damping of the wave modes t h a t mayarise because of grain charge fluctuations. Our approximation can be justified for thosedusty plasmas where the charging frequency of the grains is very small compared to thewave- frequency under con siderat ion .21 However, in a dusty plasma, the dust electroniccharge, mass, and radius, etc. may have distribut ions, i.e., these may be time- dependentquantities and should be treated as dynam ical variables. The charge fluctuation of grainsalone leads to a dissipation of th e low- frequency m odes. 2 1"2 7

Let us now consider any electrostatic mode ( ,k) propagating obliquely to an externalmagnetic field Bs || z. We assume t h a t the wavevector k of the mode lies on the xzplane in a three- dimensional Cartesian coordinate system. The dispersion relat ion of thiselectrostatic mode is obtained from 28

e(u,k) = 1+ ]T = 0, (1) = e ,i ,d


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where = ^paf ^ 2~ 2a2v2 \ \ V ln(ba)ew(- ba) ( 2 )

is t h e p l a s m a s u s c e p t i b i l i t y fo r s p e c i e s ; = e,i,d r e p r e se n t s p l a s m a s p e c i e s , p =(4 Z 2e2n 0/ m )1/ 2, b = k"j_v" / 2uj2a; I n(b ) is th e nth- ord er modified Bessel functionwith it s argument b and Z is the plasma dispersion function with it s argument [( n c )/ k \\ va]; v = (2 T / m ) 1 / 2 and c = Z eBs/ m c, are respectively, the thermal ve-locity and cyclotron frequency of particles of species ; m and T are the mass an dt e m p e r a t u r e of a particle, e is the magnitude of electron charge and c is the speed of lightin a vacuum. We now consider Eq.(1) explicitly for the dispersion relations of variouspossible low- frequency elect rostatic modes in the following different conditions :

I. Dust- lower- hybrid mode ( cd


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T h e n , using Eq.(2), the electron an d ion plasma susceptibilities are given by

Xi = ^ ~ ^ ( j = e ' *} ( 4 )

an d for the unmagnetized dust dynamics, the dust susceptibility is given by2

d = 2 . (5)UJ

Using Eqs.(4) and (5), the dispersion relation, Eq.(1) for the mode under considerationreduces to

U DL H 1 __ _ ^ i J _ ! ^ ! ^ (- , , niomek\ Zd 2( 6 )

dndo me V neo m

where DL H2 = cd ci(Zdndo/nio)/(1 + neome/ niomi), (7)

may be referred to as the dust- lower- hybrid frequency.19 Equation (6) is the dispersionre lat ion for the mode in the frequency range cd < < ci and with the strong FLRt h e r m a l kinetic effects for the dust particles. We observe t hat for k\\ = 0 the mode becomesa static vibration and the frequency becomes directly proport ional to ^JujCiUJcd. Therefore,t h i s mode may be called a dust- lower- hybrid m o d e . We recover the same dust- lower- hybridfrequency, Eq.(3) of Ref.19 for transverse propagation {k\\ ~ 0) in a cold dusty plasma.F or exact transverse propagation with cd < 1 > kVi/u)Ci, thelow-frequency mode is given by Eq.(4) of Ref.19

With the inclusion of the dust charge fluctuations and following Jana et al.24, Ma-hanta et al.27 have studied the low-frequency electrostatic lower-hybrid-like mode havingk v , ci, cd


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I I . Ultra- low- frequency dust- modes (


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These modes may cause enhanced low frequency electrostatic noise from the dust-plasma environments. In the process of dust coagulation and crystallization, the resonantin te ract ion of dust grains and these modes may provide a new mechanism of dust a t t r a c -t ion causing the formation of structures.

III. Dust- cyclotron mode ( ~ cd)Like the electron- or ion- cyclotron wave, there can exist a dust- cyclotron mode in a uni-formly magnet ized dusty plasma und er th e following conditions :

UJ ~ UJcd] UJ,\ UJ UJcd \^$> k\\Vd'ik \ \ v e, k\\Vi > (adiabatic electrons and ions); and

k\Ve,ij^ce,c% 1.Using these conditions in Eq.(2), we obtain the electron, ion and dust susceptibilities

as

X

T h u s ,given by

for

e -

d ^

i i 2 ^

p eft, Ue

pdK Vd

>k2v2

k2vi6 ) ( 6 ) exp (- &,)]. (14)

J)

U J - ujcd

k2vd2, the dust- cyclotron mode for the long parallel wavelength is

1(bd) exp (- bd)neoT d/ Z2ndoT e - Io(bd) exp ( - 6 d ) , ' ( 1 5 )

where bd = k2v2d/2uj2cd.Assuming that the dust charging frequency is small compared to the dust cyclotron

frequency, the Trom s0- damping21~ 27 due to the dust charge fluctuation can be neglected.T h e detailed studies on Troms-damping of this dust cyclotron mode is beyond the scopeof the present paper. As the dust cyclotron frequency is much smaller than the cyclotronfrequencies of electrons and ions, the Landau damping of this mode on electrons and ionswould be negligible.30 Moreover, since ~ cd and ^ $> k\\Vd, the collisionless Landaudamping of the mode (oc exp (u)2/ k2vd)) to the dust particles is also negligible, conse-quently, t he dust- cyclotron mode may be a natural mode of the magnetized dusty plasma.


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IV. Dust- Bernstein mode ( ~ n cd)In this case, we use the following conditions to obtain the various susceptibilities :

U J ~ n cd; > k \ \ ve>i ; , | n cd | > k \ \ vd.

U n d e r these conditions only th e dust m otion cont ribute to the ultra- low- frequencysusceptibilities, and thus the electrostat ic dust- Bernstein mode can, similar to electron/ ionBernstein modes, 28 be given by

UJ = n cd [1 + I n( b d) exp ( - b d) ] . (16)In this case, the inertia of the dust particles with strong FLR effects ( k v d ^ $> cd) is in-volved and the dynamics of th e electrons and ions con tribute insignificantly for this short-perpend icular wavelength dust- Bernstein mode. Th is ultra- low- frequency dust- Bernsteinmode may give rise to a new resonant frequency causing low- frequency electrosta tic noiseemerging perpendicular to the magnetic field.

We have studied th e dispersion relation s of the possible low- frequency electrosta ticmodes in a hot magnetized dusty plasma using the Vlasov- kinetic theory. Th e plasmahas been considered as multispecies with electron s, ions, and heavily charged massive dustgrains as the plasma components. It has been found t h a t the inequalities of charges andnumber densities of th e different plasma species an d the FLR th erm al kinetic effects of th emassive and heavily charged dust grains introduce th e existence of th e electrosta tic eigen-modes [Eqs. (6), (9), (13), (15) and (16) ] in the three- componen t, uniform, finite tem-p e r a t u r e , and magnetized dusty plasma. We have considered different frequency rangesof th e low- frequency eigenmode ( ,k ). First, we consider the case when cd


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hybrid frequency ~ ^Jujciujcd- We note here t h a t for exact transverse propagation thismode was predicted in th e earlier investigation .19

We have found two types of ultra- low- frequency modes in the frequency ran ge 1, we have found a low- frequencydust- mode whose dispersion relation is given by Eq.(15). This mode may be called anelectrostatic dust- cyclotron mode. In the comparatively h igher frequency range and shortperpendicular wavelength, ~ n cd, > k \\va, a dispersion relation, Eq.(16) for thedust- Bernstein mode in the high- density regime, pd ^> k2vd2 is obtained.

In th e present investigation, th e dust grains were taken as a third- componen t of th ehomogeneous dusty plasma having con stan t charge and mass. For simplicity, we haveassumed t h a t th e charge on th e dust grains is not affected by the waves. The variation oft h e dust charges can lead to additional damping 2 1"2 7 apart from the collisionless Landaudam ping of th e low- frequency electrosta tic modes studied here.

The low- frequency modes stud ied here can have man y application s in space and astro-physics as well as in laboratory experiments for Coulomb- dust crystallization and dust-coagulation in hot magnetized dusty plasmas9"1 1 '31 '32 '33. Various collective effects includ-ing dust Coulomb crystallization and the parametric mode coupling interactions throughthese ultra- low- frequency modes in magnetized dusty plasmas will be an important fieldof research in the future and the work in these lines is in progress.Acknowledgments:This work was done within the framework of th e Associateship Scheme of th e In ternat ion alC e n t r e for Theoretical Physics, Trieste. Italy. The authors would like to acknowledge thehospitality at I C T P where a part of the work was completed. Financial support from theSwedish In tern ationa l Development Cooperation Agency is acknowledged. M.R.A. wouldalso like to acknowledge th e support of th e Alexander von Humboldt F o u n d a t i o n . Theauthors are grateful t o Professor B. Dasgupta for fruitful discussions.


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M. Salimullah et al- Ultra-Low-Frequency Electrostatic Modes in a Magnetized Dusty Plasma