research article quantum treatment of kinetic alfvén waves...

Research ArticleQuantum Treatment of Kinetic Alfveacuten Waves Instability ina Dusty Plasma Magnetized Ions

N Rubab1 and G Jaffer2

1Department of Space Science Institute of Space Technology Islamabad Pakistan2Department of Space Science University of the Punjab Lahore Pakistan

Correspondence should be addressed to N Rubab drnrubabgmailcom

Received 19 July 2016 Accepted 15 November 2016

Academic Editor Anna Cimmino

Copyright copy 2016 N Rubab and G Jaffer This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited The publication of this article was funded by SCOAP3

Kinetic Alfven wave instability is examined rigorously in a uniform nondegenerate quantum dusty plasma A linear dispersionrelation of kinetic Alfven wave in inertial regime is derived by incorporating Bohm potential in the linearized Vlasov model It isfound that the quantum correction119862119876 appears due to the insertion of Bohmpotential inVlasovmodel and causes the suppression inthe Alfven wave frequency and the growth rates of instability A number of analytical expressions for various modes of propagationare derived It is also found that the system parameters that is streaming velocity dust charge number density and quantumcorrection significantly influence the dispersion relation and the growth rate of instability

1 Introduction

Theoretical and laboratory studies on current-driven electro-magnetic instabilities have been a popular subject in plasmaresearch [1 2] Electromagnetic and electrostatic plasmainstabilities driven by the parallel or cross-field currentsgive rise to narrow banded spontaneous emissions and havereceived a lot of attention in the past few decades It has beenspeculated that low frequency electromagnetic waves that isAlfven waves are the most important electromagnetic waveswhich take part in various dynamical processes occurring inthe interior of the sun [3ndash5] Recently kinetic Alfven waves(KAWs) have been modified via quantum effects associatedwith electrons for example Hussain et al [6] made anattempt to study spin and nonrelativistic quantum effects onKAWs within the frame work of kinetic theory of Alfvenwaves and their analysis suggested that the spin quantumeffects suppress the Alfven wave frequency in a hot andmagnetized plasma

Usually for quantum effects to be significant in plasmathe temperature to density ratio is considered to be quitesmall The quantum effects are expected to be significant onvery small scales where debye length 120582119863 = radic11989611987941205871198991198902 and

gyroradius 120588119895 should be smaller or of the order of the deBroglie wavelength 120582119863119861 = ℏradic2120587119898119895119896119879 At such small scalesthe quantum kinetic theory is better suited for the propertreatment of these microphenomena To study various aspectof quantum plasma numerous efforts have beenmade whoseapplications are ranges from plasmonic device applications todense astrophysical plasmas [7ndash16]

Recently various researches have been performed tostudy the collective effects in a quantum plasma In theseinvestigations the researchers have focused on the lineardispersion effects of several low frequency modes and insta-bilities as a benchmark problem in quantum plasma It hasbeen treated by means of Bohm diffraction or quantumstatistical effects that include the Fermi pressure [17] Variousfluid plasma models have been developed by introducingthe quantum effects through quantum corrections This canbe done either through the quantum force produced bydensity fluctuations originating from the Bohm potential orthrough the spin of particles which includes the magneticdipole force andmagnetization energy At large scales plasmais treated as an electrically conducting fluid and can bedescribed bymagnetohydrodynamics while the situation hasto be interpreted within the frame work of kinetic theory

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2016 Article ID 6793572 6 pageshttpdxdoiorg10115520166793572

2 Advances in High Energy Physics

when microscopic scales are involved The proper treatmentof such phenomena requires the development of kinetictheory based on quantum effects [18] The presence of dustparticles is known to significantly alter the electrostatic andelectromagnetic modes [19ndash23] This is due to the fact thattheir presence gives rise to some novel modes associated withdust particle dynamics and finite divergence of the cross-field plasma current density More recently Mahdavi andAzadboni investigated the nondegenerate quantum effectson Weibel instability by considering its application in theabsorption layer of fuel pallet In their study the thermalenergy was taken larger than the potential and Fermi energyby taking the quantum effects negligibly small and showedthat nondegenerate quantum effects and density gradientstend to stabilize the Weibel instability [24]

In this paper we have made an effort to examine therole of quantum diffraction associated with electrons in adense hot and magnetized plasma and its effect on lowfrequency modes We discuss the characteristics of shortwavelength dust kinetic Alfven waves by emphasizing thequantum diffraction effects arising through Bohm potentialof electrons which is obviously dominant over the ions andthe dust species due to large mass difference The dispersionrelation of quantum DKAW (QDKAW) is derived by incor-porating Bohm quantum potential into the linearized Vlasovequation Also the coupling of electrostatic quantum dustacousticmode (QDAW)with kineticAlfvenwave (KAW) andthe theoretical aspect of KAW instability in an inertial regimeis discussed

Themanuscript is organized as follows basic assumptionsleading to the general dispersion relation are presented inSections 2 and 3 while results and conclusions are discussedin Section 4

2 Basic Assumptions

We consider an electromagnetic kinetic Alfven wave in a col-lisionless electron-ion quantum dusty plasma The stronglymagnetized ions are considered to be Maxwellian and abeam of nondegenerate electrons drifting across an externalmagnetic field (B0 ) with constant ion drift velocity1198810 that is 1198810 perp B0 while the dust component is coldand unmagnetized The plasma beta 120573119894 is assumed to bevery small that is 120573119894 ≪ 1 where 120573119894 = 4120587119899119894011987911990511986120 Anelectromagnetic wave with a wave vector k lies in 119909119911 planek = 119896perp + 119896 making angle 120579 with 119909-plane We haveadopted two potential representationswhich are used in a lowbeta plasma to express the electromagnetic perturbations todescribe the electric field E that is 119864perp = minusnablaperp120593 and 119864 =minusnabla120595 where 120593 = 120595

The linearized Poisson equation is

minus (1198962perp120593 + 1198962120595) = 4120587119890 [1198991198901 minus 1198991198941 minus 11988511988901198991198891] (1)

where 1198991198901 1198991198941 and 1198991198891 are electron ion and dust numberdensities respectively In (1) 1198851198890 = minus1198761198890119890 (with 1198851198890 asthe number of electron charge on a grain) is the equilibriumcharge on an average dust grain and 119890 is the electron chargeFor electromagnetic waves we may ignore the dust charge

fluctuation effects that is 1198851198891 = 0 [25 26] as it causesdamping (non-Landau type) which is relevant for electro-static waves Now on combining Amperersquos and Faradayrsquos lawwe get [27]

11988821198962perp119896 (120593 minus 120595) = 4120587120596 [1198691198941 + 1198691198891] (2)

where 1198691198951 represent the field aligned current densities forplasma species Since the electrons are streaming perpendic-ular to the field direction therefore 1198691198901 = 0The linearizedVlasov equation for quantum plasma after incorporatingBohm quantum potential in the direction of field will obtainthe following

1205971198911198901120597119905 + V1205971198911198901120597119911 + [ 119902119890119898119890119864119911 +

ℏ2411989821198901198990nablanabla21198991198901]1205971198911198900120597V = 0 (3)

The perturbed distribution function of streaming electronsis given by the aid of Vlasov equation and the zeroth orderdistribution function 1198911198900 which is the usual distributionfunction and is assumed to obey a Maxwellian or a FermiDirac distribution

1198911198900 = 119860119890 exp(V2 + (Vperp minus 1198810)2V2119905119890

) (4)

where

119860119890 = 1198991198900 ( 1198981198902120587119879119890)32 (5)

Since the ions are hot and magnetized with finite Larmorradius effects therefore we can solve Vlasov equation interms of guiding center coordinates by ignoring quantumeffects and obtain the perturbed distribution for any electro-magnetic wave [28 29]

1198911198941 = (1198991198940119890119879119894 )sum119897 sum119899119896V120595 + 119899Ω119888119894120593120596 minus 119899Ω119888119894 minus 119896V exp (119894 (119899 minus 119897) 120579)

sdot 119869119899 (119896perpVperpΩ119888119894 ) 119869119897 (119896perpVperpΩ119888119894 )1198911198940

(6)

where1198911198940 is the equilibriumMaxwellian distribution functionfor electrons Ω119888119894 is the electron cyclotron frequency and 119869119899is the Bessel function of first kind of order 1198993 Basic Theory and Dispersion Relation

Theperturbednumber density for electrons including nonde-generate quantum effects has been recast by the aid of Vlasovequation and is determined by the relation 1198991198901 = int1198911198901119889k as

1198991198901 = 21198901198991198900120593119898119890V2119905119890 (1 + 120578119885 (120578)1 + 119862119876 (1 + 120578119885 (120578))) (7)

where 119862119876 = ℏ2119896341198982119890V2119905119890119896 which shows the quantumcorrection in the number density of electrons and 119885(120578)is the plasma dispersion function for cross-field streaming

Advances in High Energy Physics 3

nondegenerate electrons [30] with argument 120578 = (120596 minus119896perp1198810)119896perpV119905119890By using (6) the number density of hot and magnetized

ions is

1198991198941 = minus11989911989401198901198981198941119896V2119905119894

sdot sum119899

[119896V119905119894120595 (1 + 120585119894119899119885 (120585119894119899)) + 119899Ω119888119894120593119885 (120585119894119899)] 119868119899 (119887119894) 119890minus119887119894 (8)

where 119868119899 is the modified Bessel function with argument 119887119894 =1198962perpV21199051198942Ω2119888119894 and 119885(120585119894119899) is the usual dispersion function for aMaxwellian plasma with 120585119894119899 = (120596 minus 119899Ω119888119894)119896V119905119894

Theperturbed number density of cold and unmagnetizeddust grains by using hydrodynamic model is as follows

1198991198891 = 119899119889011987611988901198981198891205962 (1198962perp120593 + 1198962120595) (9)

Since the nondegenerate electrons are streaming alongthe 119909-direction therefore the longitudinal component ofcurrent density perturbation is taken to be zero that is 1198691198901 =0 The parallel components of current density for ions anddust species are

1198691198941 = minus11989911989401198902119879119894119896sum119899 [(1 + 120585119894119899119885 (120585119894119899)) (119896V119905119894120585119894119899120595 + 119899Ω119888119894120593)]sdot 119868119899 (119887119894) 119890minus119887119894

1198691198891 = 119899119889011987621198890119898119889120596 119896120595(10)

Using the explicit expressions of 1198991198901 1198991198941 and 1198991198891 in (1) and1198691198941 1198691198901 and 1198691198891 in (2) allows obtaining the following systemof equations

119860120593 + 119861120595 = 0119862120593 + 119863120595 = 0 (11)

where

119860 = 1198962perp + 21205962119901119890

V2119905119890( 1 + 120578119885 (120578)1 + 119862119876 (1 + 120578119885 (120578)))

+ 119899120596119888119894119896 119885 (120585119894119899) 119868119899 (119887119894) 119890minus119887119894 + 1198962perp12059621199011198891205962

119861 = 1198962 + 21205962119901119894

V2119905119894(1 + 120585119894119899119885 (120585119894119899)) 119868119899119890minus119887119894 minus 1198962 120596

21199011198891205962

119862 = 11988821198961198962perp + 21205962119901119894

V2119905119894

119899120596Ω119888119894119896 (1 + 120585119894119899119885 (120585119894119899)) 119868119899119890minus119887119894 119863 = 119896 (1198962perp1198882 + 1205962119901119889) minus 120596 11205822119863119894sum119899

V1199051198942 1205851198941198991198851015840 (120585119894119899) 119868119899119890minus119887119894

(12)

And by eliminating 120593 and 120595 from the homogeneousequation (11) the dispersion relation reduces to

119860119863 minus 119861119862 = 0 (13)

which after substituting the values of 119860 119861 119862 and 119863 from(12) into (13) and some simple algebra is given by

1 + 212059621199011198941198962V2119905119894sum119899

[(1 + 12059621199011198891198962perp1198882)119899Ω119888119894119896V119905119894119885 (120585119894119899) 119868119899119890minus119887119894

+ (1 + 120585119894119899119885 (120585119894119899)) 119868119899119890minus119887119894] + 212059621199011198901198962V2119905119890[(1 minus 12059621199011198891198962perp1198882)

sdot ( 1 + 120578119885 (120578)1 + 119862119876 (1 + 120578119885 (120578)))] + (1 +12059621199011198891198962perp1198882)

1198962perp1198962

(1minus 12059621199011198891205962 ) = 0

(14)

and represents the general dispersion relation of KAWstreaming instabilities in a quantumdusty plasma In the limit12059621199011198891198962perp1198882 ≪ 1 we get the dispersion relation of modified twostream instabilities in a dusty plasma [31] and in a dust-freeplasma the instability results are as in classical plasma [32]

When the free energy associated with streaming isincreased that is 1198810 ≫ V119905119890 the maximum growth ratesalso continue to increase and the dispersion equation for allplasma components becomes nonresonant (120585119894119899119890 gt 1) Theunstable mode in a nonresonant regime is typically knownas fluid instability In the cold plasma limit

119860 = 1198962perp1198911198941205962 (1205962 minus 1205962119889119897ℎ minus12059621205962119901119890119891119894(120596 minus 119896perp1198810)2 minus 1198621198761198962perpV2119905119890)

119861 = 1198962perp119891119894 minus 1198962 12059621199011198891205962

119862 = 11988821198961198962perp119863 = minus119896 (1198962perp1198882 + 1205962119901119889)

(15)

where 1205962119889119897ℎ = (1205962119901119889Ω2119888119894)1205962119901119894 and 119891119894 = 1205962119901119894Ω2119888119894Thus from the dispersion relation (13) we obtain a

biquadratic equation for 120596 in terms of various plasmaparameters

1198861205964 + 1198871205963 + 1198881205962 + 119889120596 + 119890 = 0 (16)where

119886 = 120572 + 1198962perp1198882119887 = minus21198963perp11988101198882 minus 2119896perp1198810120572 minus 211989621198812119860119894119896perp1198810119888 = (minus1205962119889119897ℎ minus 11989621198812119860119894119862119876 minus 119862119876 minus 1198962perp11988120 minus 120596

2119901119890119891119894 )120572

minus 119896211988121198601198941205962119901119889 minus 1198621198761198962perp1198882 minus 1198964perp119888211988120 119889 = 2119896perp11988101205962119889119897ℎ120572 + 211989621198812119860119894119896perp11988101205962119901119889119890 = (119862119876 minus 1198962perp11988120 ) 1205962119889119897ℎ120572 minus 11989621198962perp119881011988121198601198941205962119901119889

(17)

where 120572 = 1198962perp1198882 + 1205962119901119889


In the absence of streaming electrons the solution of (16)is given by

1205962 = 1205962119889119897ℎ + 2119862119876 + 11989621198812119860119894(1 + 1198962perp1205822119889) (18)

which is quantum corrected low frequency shear Alfven wavein a dusty plasma where 119881119860119894 = 1198610radic41205871198991198940119898119894 is the Alfvenspeed and 120582119889 = 119888120596119901119889 is the dust skin depth In the limit119862119876 = 0 we obtain the dispersion relation of KAW in inertialregime that is 1205962 = 1205962119889119897ℎ + 11989621198812119860119894(1 + 1198962perp1205822119889) When 119862119876 = 01198962perp1205822119889 ≪ 1 we get the dust-modified dispersion relation ofshear Alfven waves that is 1205962 = 1205962119889119897ℎ + 11989621198812119860119894 which is anatural mode of any dusty magnetoplasma and 1205962119889119897ℎ providesconstraint for the electromagnetic wave propagation Ouranalysis also indicates that cross-field streaming effects arenot coupled with perpendicular wavenumber but appear asan additional term in the dispersion relation The reasonmay be the absence of 119869119890 due to streaming in cross-fielddirection and the Bohm potential induced Vlasovmodel It istherefore tempting to introduce quantum correction whichmay significantly modify the classical modes and relatedinstabilities

In order to observe the effect of dust grains on the wavein a low beta plasma the limiting case of 1198962perp1205822119889 ≫ 1 can beobtained from (18) in the form

1205962 = 1205962119889119897ℎ + Ω119894Ω119889 (11989911988901198851198891198991198940 ) 11989621198962perp (19)

The second term on RHS can be recognized as theconvective cell frequency which strongly depends on dustparameters It can be recognized that dust inertia may playa major role in the wave dynamics while the stationarydust excludes this mode The heavy dust grains can be animportant factor in diminishing the effect of wave frequencyand unstable regions of propagation

4 Results

In the presented work we have investigated the cross-fieldstreaming instability and dispersion propertied of KAW dueto nondegenerate electrons in a quantum dusty magneto-plasma We have derived the KAW instability growth ratesusing various parameters close to dense astrophysical plasmathat is 119899119890 = 1026 119899119894 = 1001times1026119885119889 = 103 1198991198890 = 03times1023and 1198610 = 1013 G [33] The influence of quantum Bohmpotential is found to act against the KAW instability whileclassical effects reinforce the unstable regions as depictedin Figure 1 The cross-field nondegenerate electron beamstreaming with velocity 1198810 helps in the growth of unstableregions and a further increase inhibits the stabilization whichmaintains the wave amplitude as illustrated in Figure 2It has also been observed that quantum parameter 119862119876 isinclined to suppress the instability In Figure 3 we canobserve that the growth rates are enhanced with the increasein number density followed by increase in cross-filed ionstreaming velocity Physically when the dust concentration

0068

0069

007

0071

0072

0073

02 04 06 080

20

40

60

80

100

CQ

kperp

Figure 1 Contour plots between perp and 119862119876 for 1198810 = 10 119885119889 = 103and 119899119889 = 10minus8119899119894

0 0 00

1

2

3

4

5

6

7

02 04 06 080

10

20

30

40

50

V0

kperp


is increased it also enhances the depletion of electrons due toattachment to the dust grain surface which in turn increasesthe streaming velocity and ultimately is responsible for theexcitation of an electromagnetic waveThe role of dust chargeis also found to enhance the wave activity which can be seenin Figure 4 Our results indicate that KAWs in inertial regimeare less affected by the Bohm potential effects and hence 119862119876appears to suppress the Alfven wave frequency in a hot andmagnetized nondegenerate quantumplasmawhich is evidentfrom Figure 5


04

0608

1

12

14

02 04 06 08

20000

40000

60000

80000

nd

kperp

Figure 3 Plots between and 119899119889 for 1198810 = 10 119862119876 = 10 and 119885119889 =103

04

0608

1

12

14

16

18

02 04 06 08

20000

40000

60000

80000

Zd

kperp

Figure 4 Effect of 119885119889 on the dispersion characteristics

To summarize we have investigated the effects of quan-tum contribution on the growth and dispersion of lowfrequency kinetic Alfven wave in inertial regime by usingVlasov-Maxwell equations We have neglected the quantumeffects of ions as they are heavier than electrons It is foundthat quantum effects suppress the instability in a hot andmagnetized plasma The density fluctuations are not accom-panied by pure electromagnetic nonstreaming dense plasmatherefore applying quantum corrections makes sense toelectromagneticwaveswhen density fluctuations are involvedespecially in the inertial turbulent range We therefore have

02 04 06 08

305

310

315

320

325

kperp

CQ = 0 (Maxwellian)

CQ = 0

Figure 5 Effects of 119862119876 on the real part of the dispersion relation

recast the perturbed nondegenerate electron density whichhas a strong dependence on quantum correction parameter119862119876 Our analysis is based on linear approximation and webelieve that we have contributed to the analysis of KAWIin a nondegenerate quantum plasma and there is a largeset of nonlinear investigations which should be addressedThe present analysis could be applied to dense astrophysicalstreaming and ICFplasmaswhere quantumdiffraction effectsare nonnegligible and significant

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S P GaryTheory of Space PlasmaMicroinstabilities CambridgeUniversity Press 1993

[2] A L Brinca F J Romeiras and L Gomberoff ldquoOn wavegeneration by perpendicular currentsrdquo Journal of GeophysicalResearch Space Physics vol 108 no 1 article 1038 2003

[3] L Chen ldquoAlfven waves a journey between space and fusionplasmasrdquo Plasma Physics and Controlled Fusion vol 50 no 12Article ID 124001 2008

[4] N F Cramer The Physics of Alfven Wave Wiley-VCH BerlinGermany 2001

[5] WGekelman S Vincena B VanCompernolle et al ldquoThemanyfaces of shear Alfven wavesardquo Physics of Plasmas vol 18 no 5Article ID 55501 2011

[6] AHussain Z Iqbal G Brodin andGMurtaza ldquoOn the kineticAlfven waves in nonrelativistic spin quantum plasmasrdquo PhysicsLetters A vol 377 no 34ndash36 pp 2131ndash2135 2013

[7] F Haas G Manfredi and M Feix ldquoMultistream model forquantum plasmasrdquo Physical Review EmdashStatistical Physics Plas-mas Fluids and Related Interdisciplinary Topics vol 62 no 2pp 2763ndash2772 2000

[8] F Haas L G Garcia J Goedert and G Manfredi ldquoQuantumion-acoustic wavesrdquo Physics of Plasmas vol 10 no 10 pp 3858ndash3866 2003

[9] F Haas ldquoA magnetohydrodynamic model for quantum plas-masrdquo Physics of Plasmas vol 12 no 6 Article ID 062117 2005


[10] M Marklund and G Brodin ldquoDynamics of spin-12 quantumplasmasrdquo Physical Review Letters vol 98 no 2 Article ID025001 2007

[11] M Marklund B Eliasson and P K Shukla ldquoMagnetosonicsolitons in a fermionic quantum plasmardquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 76 no 6Article ID 067401 2007

[12] G Manfredi ldquoHow to model quantum plasmasrdquo Fields InstituteCommunications vol 46 pp 263ndash287 2005

[13] G Brodin and M Marklund ldquoSpin solitons in magnetized pairplasmasrdquo Physics of Plasmas vol 14 no 11 Article ID 1121072007

[14] G Brodin M Marklund B Eliasson and P K ShuklaldquoQuantum-electrodynamical photon splitting in magnetizednonlinear pair plasmasrdquo Physical Review Letters vol 98 no 12Article ID 125001 2007

[15] P K Shukla ldquoA new dust mode in quantum plasmasrdquo PhysicsLetters A vol 352 no 3 pp 242ndash243 2006

[16] P K Shukla and L Stenflo ldquoShear Alfven modes in ultra-coldquantummagnetoplasmasrdquoNew Journal of Physics vol 8 no 7article 111 2006

[17] L G Garcia F Haas L P L de Oliveira and J GoedertldquoModified Zakharov equations for plasmas with a quantumcorrectionrdquo Physics of Plasmas vol 12 no 1 Article ID 0123022005

[18] S Kumar and J Y Lu ldquoQuantum treatment of kinetic Alfvenwaverdquo Astrophysics and Space Science vol 341 no 2 pp 597ndash599 2012

[19] M Rosenberg and P K Shukla ldquoIon-dust two-stream instabilityin a collisional magnetized dusty plasmardquo Journal of PlasmaPhysics vol 70 no 3 pp 317ndash322 2004

[20] S Ali and P K Shukla ldquoStreaming instability in quantum dustyplasmasrdquo European Physical Journal D vol 41 no 2 pp 319ndash324 2007

[21] P K Shukla and A A Mamun Introduction to Dusty PlasmaPhysics Institute of Physics Bristol UK 2002

[22] C R Choi and D-Y Lee ldquoSolitary Alfven waves in a dustyplasmardquo Physics of Plasmas vol 14 no 5 Article ID 0523042007

[23] C Rim Choi C-M Ryu N C Lee and D-Y Lee ldquoIon acousticsolitary waves in a dusty plasma obliquely propagating to anexternalmagnetic fieldrdquo Physics of Plasmas vol 12 no 2 ArticleID 22304 2005

[24] M Mahdavi and F Khodadadi Azadboni ldquoThe quantumeffects role on Weibel instability growth rate in dense plasmardquoAdvances in High Energy Physics vol 2015 Article ID 746212 6pages 2015

[25] F Melandsoslash T K Aslaksen and O Havnes ldquoA new dampingeffect for the dust-acoustic waverdquo Planetary and Space Sciencevol 41 no 4 pp 321ndash325 1993

[26] M Salimullah M K Islam A K Banerjee and M NambuldquoKinetic Alfven wave instability in a dusty plasmardquo Physics ofPlasmas vol 8 no 7 pp 3510ndash3512 2001

[27] A Hasegawa and L Chen ldquoKinetic processes in plasma heatingby resonant mode conversion of Alfven waverdquo Physics of Fluidsvol 19 no 12 pp 1924ndash1934 1976

[28] K Zubia N Rubab H A ShahM Salimullah and GMurtazaldquoKinetic Alfven waves in a homogeneous dusty magnetoplasmawith dust charge fluctuation effectsrdquo Physics of Plasmas vol 14no 3 Article ID 032105 2007

[29] C S Liu and V K Tripathi ldquoParametric instabilities in amagnetized plasmardquo Physics Reports vol 130 no 3 pp 143ndash2161986

[30] D Summers and R M Thorne ldquoThe modified plasma disper-sion functionrdquo Physics of Fluids B vol 3 no 8 pp 1835ndash18471991

[31] M Rosenberg and N A Krall ldquoModified two-stream instabil-ities in dusty space plasmasrdquo Planetary and Space Science vol43 no 5 pp 619ndash624 1995

[32] A A Galeev and R N Sudan Basic Plasma Physics North-Holland Publishing Company Amsterdam The Netherlands1983

[33] S A Khan A Mushtaq and W Masood ldquoDust ion-acousticwaves in magnetized quantum dusty plasmas with polarityeffectrdquo Physics of Plasmas vol 15 no 1 Article ID 013701 2008

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when microscopic scales are involved The proper treatmentof such phenomena requires the development of kinetictheory based on quantum effects [18] The presence of dustparticles is known to significantly alter the electrostatic andelectromagnetic modes [19ndash23] This is due to the fact thattheir presence gives rise to some novel modes associated withdust particle dynamics and finite divergence of the cross-field plasma current density More recently Mahdavi andAzadboni investigated the nondegenerate quantum effectson Weibel instability by considering its application in theabsorption layer of fuel pallet In their study the thermalenergy was taken larger than the potential and Fermi energyby taking the quantum effects negligibly small and showedthat nondegenerate quantum effects and density gradientstend to stabilize the Weibel instability [24]

In this paper we have made an effort to examine therole of quantum diffraction associated with electrons in adense hot and magnetized plasma and its effect on lowfrequency modes We discuss the characteristics of shortwavelength dust kinetic Alfven waves by emphasizing thequantum diffraction effects arising through Bohm potentialof electrons which is obviously dominant over the ions andthe dust species due to large mass difference The dispersionrelation of quantum DKAW (QDKAW) is derived by incor-porating Bohm quantum potential into the linearized Vlasovequation Also the coupling of electrostatic quantum dustacousticmode (QDAW)with kineticAlfvenwave (KAW) andthe theoretical aspect of KAW instability in an inertial regimeis discussed

Themanuscript is organized as follows basic assumptionsleading to the general dispersion relation are presented inSections 2 and 3 while results and conclusions are discussedin Section 4

2 Basic Assumptions

We consider an electromagnetic kinetic Alfven wave in a col-lisionless electron-ion quantum dusty plasma The stronglymagnetized ions are considered to be Maxwellian and abeam of nondegenerate electrons drifting across an externalmagnetic field (B0 ) with constant ion drift velocity1198810 that is 1198810 perp B0 while the dust component is coldand unmagnetized The plasma beta 120573119894 is assumed to bevery small that is 120573119894 ≪ 1 where 120573119894 = 4120587119899119894011987911990511986120 Anelectromagnetic wave with a wave vector k lies in 119909119911 planek = 119896perp + 119896 making angle 120579 with 119909-plane We haveadopted two potential representationswhich are used in a lowbeta plasma to express the electromagnetic perturbations todescribe the electric field E that is 119864perp = minusnablaperp120593 and 119864 =minusnabla120595 where 120593 = 120595

The linearized Poisson equation is

minus (1198962perp120593 + 1198962120595) = 4120587119890 [1198991198901 minus 1198991198941 minus 11988511988901198991198891] (1)

where 1198991198901 1198991198941 and 1198991198891 are electron ion and dust numberdensities respectively In (1) 1198851198890 = minus1198761198890119890 (with 1198851198890 asthe number of electron charge on a grain) is the equilibriumcharge on an average dust grain and 119890 is the electron chargeFor electromagnetic waves we may ignore the dust charge

fluctuation effects that is 1198851198891 = 0 [25 26] as it causesdamping (non-Landau type) which is relevant for electro-static waves Now on combining Amperersquos and Faradayrsquos lawwe get [27]

11988821198962perp119896 (120593 minus 120595) = 4120587120596 [1198691198941 + 1198691198891] (2)

where 1198691198951 represent the field aligned current densities forplasma species Since the electrons are streaming perpendic-ular to the field direction therefore 1198691198901 = 0The linearizedVlasov equation for quantum plasma after incorporatingBohm quantum potential in the direction of field will obtainthe following

1205971198911198901120597119905 + V1205971198911198901120597119911 + [ 119902119890119898119890119864119911 +

ℏ2411989821198901198990nablanabla21198991198901]1205971198911198900120597V = 0 (3)

The perturbed distribution function of streaming electronsis given by the aid of Vlasov equation and the zeroth orderdistribution function 1198911198900 which is the usual distributionfunction and is assumed to obey a Maxwellian or a FermiDirac distribution

1198911198900 = 119860119890 exp(V2 + (Vperp minus 1198810)2V2119905119890

) (4)

where

119860119890 = 1198991198900 ( 1198981198902120587119879119890)32 (5)

Since the ions are hot and magnetized with finite Larmorradius effects therefore we can solve Vlasov equation interms of guiding center coordinates by ignoring quantumeffects and obtain the perturbed distribution for any electro-magnetic wave [28 29]

1198911198941 = (1198991198940119890119879119894 )sum119897 sum119899119896V120595 + 119899Ω119888119894120593120596 minus 119899Ω119888119894 minus 119896V exp (119894 (119899 minus 119897) 120579)

sdot 119869119899 (119896perpVperpΩ119888119894 ) 119869119897 (119896perpVperpΩ119888119894 )1198911198940

(6)

where1198911198940 is the equilibriumMaxwellian distribution functionfor electrons Ω119888119894 is the electron cyclotron frequency and 119869119899is the Bessel function of first kind of order 1198993 Basic Theory and Dispersion Relation

Theperturbednumber density for electrons including nonde-generate quantum effects has been recast by the aid of Vlasovequation and is determined by the relation 1198991198901 = int1198911198901119889k as

1198991198901 = 21198901198991198900120593119898119890V2119905119890 (1 + 120578119885 (120578)1 + 119862119876 (1 + 120578119885 (120578))) (7)

where 119862119876 = ℏ2119896341198982119890V2119905119890119896 which shows the quantumcorrection in the number density of electrons and 119885(120578)is the plasma dispersion function for cross-field streaming



ions is

1198991198941 = minus11989911989401198901198981198941119896V2119905119894

sdot sum119899

[119896V119905119894120595 (1 + 120585119894119899119885 (120585119894119899)) + 119899Ω119888119894120593119885 (120585119894119899)] 119868119899 (119887119894) 119890minus119887119894 (8)



1198991198891 = 119899119889011987611988901198981198891205962 (1198962perp120593 + 1198962120595) (9)



1198691198891 = 119899119889011987621198890119898119889120596 119896120595(10)


119860120593 + 119861120595 = 0119862120593 + 119863120595 = 0 (11)

where

119860 = 1198962perp + 21205962119901119890

V2119905119890( 1 + 120578119885 (120578)1 + 119862119876 (1 + 120578119885 (120578)))

+ 119899120596119888119894119896 119885 (120585119894119899) 119868119899 (119887119894) 119890minus119887119894 + 1198962perp12059621199011198891205962

119861 = 1198962 + 21205962119901119894

V2119905119894(1 + 120585119894119899119885 (120585119894119899)) 119868119899119890minus119887119894 minus 1198962 120596

21199011198891205962

119862 = 11988821198961198962perp + 21205962119901119894

V2119905119894


V1199051198942 1205851198941198991198851015840 (120585119894119899) 119868119899119890minus119887119894

(12)


119860119863 minus 119861119862 = 0 (13)


1 + 212059621199011198941198962V2119905119894sum119899

[(1 + 12059621199011198891198962perp1198882)119899Ω119888119894119896V119905119894119885 (120585119894119899) 119868119899119890minus119887119894


sdot ( 1 + 120578119885 (120578)1 + 119862119876 (1 + 120578119885 (120578)))] + (1 +12059621199011198891198962perp1198882)

1198962perp1198962

(1minus 12059621199011198891205962 ) = 0

(14)




119861 = 1198962perp119891119894 minus 1198962 12059621199011198891205962

119862 = 11988821198961198962perp119863 = minus119896 (1198962perp1198882 + 1205962119901119889)

(15)



1198861205964 + 1198871205963 + 1198881205962 + 119889120596 + 119890 = 0 (16)where


2119901119890119891119894 )120572


(17)

where 120572 = 1198962perp1198882 + 1205962119901119889



1205962 = 1205962119889119897ℎ + 2119862119876 + 11989621198812119860119894(1 + 1198962perp1205822119889) (18)



1205962 = 1205962119889119897ℎ + Ω119894Ω119889 (11989911988901198851198891198991198940 ) 11989621198962perp (19)


4 Results


0068

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CQ

kperp


0 0 00

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V0

kperp




04

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nd

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20000

40000

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kperp



02 04 06 08

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kperp

CQ = 0 (Maxwellian)

CQ = 0



Competing Interests


References








































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





ions is

1198991198941 = minus11989911989401198901198981198941119896V2119905119894

sdot sum119899

[119896V119905119894120595 (1 + 120585119894119899119885 (120585119894119899)) + 119899Ω119888119894120593119885 (120585119894119899)] 119868119899 (119887119894) 119890minus119887119894 (8)



1198991198891 = 119899119889011987611988901198981198891205962 (1198962perp120593 + 1198962120595) (9)



1198691198891 = 119899119889011987621198890119898119889120596 119896120595(10)


119860120593 + 119861120595 = 0119862120593 + 119863120595 = 0 (11)

where

119860 = 1198962perp + 21205962119901119890

V2119905119890( 1 + 120578119885 (120578)1 + 119862119876 (1 + 120578119885 (120578)))

+ 119899120596119888119894119896 119885 (120585119894119899) 119868119899 (119887119894) 119890minus119887119894 + 1198962perp12059621199011198891205962

119861 = 1198962 + 21205962119901119894

V2119905119894(1 + 120585119894119899119885 (120585119894119899)) 119868119899119890minus119887119894 minus 1198962 120596

21199011198891205962

119862 = 11988821198961198962perp + 21205962119901119894

V2119905119894


V1199051198942 1205851198941198991198851015840 (120585119894119899) 119868119899119890minus119887119894

(12)


119860119863 minus 119861119862 = 0 (13)


1 + 212059621199011198941198962V2119905119894sum119899

[(1 + 12059621199011198891198962perp1198882)119899Ω119888119894119896V119905119894119885 (120585119894119899) 119868119899119890minus119887119894


sdot ( 1 + 120578119885 (120578)1 + 119862119876 (1 + 120578119885 (120578)))] + (1 +12059621199011198891198962perp1198882)

1198962perp1198962

(1minus 12059621199011198891205962 ) = 0

(14)




119861 = 1198962perp119891119894 minus 1198962 12059621199011198891205962

119862 = 11988821198961198962perp119863 = minus119896 (1198962perp1198882 + 1205962119901119889)

(15)



1198861205964 + 1198871205963 + 1198881205962 + 119889120596 + 119890 = 0 (16)where


2119901119890119891119894 )120572


(17)

where 120572 = 1198962perp1198882 + 1205962119901119889



1205962 = 1205962119889119897ℎ + 2119862119876 + 11989621198812119860119894(1 + 1198962perp1205822119889) (18)



1205962 = 1205962119889119897ℎ + Ω119894Ω119889 (11989911988901198851198891198991198940 ) 11989621198962perp (19)


4 Results


0068

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CQ

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0 0 00

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02 04 06 08

20000

40000

60000

80000

Zd

kperp



02 04 06 08

305

310

315

320

325

kperp

CQ = 0 (Maxwellian)

CQ = 0



Competing Interests


References








































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





1205962 = 1205962119889119897ℎ + 2119862119876 + 11989621198812119860119894(1 + 1198962perp1205822119889) (18)



1205962 = 1205962119889119897ℎ + Ω119894Ω119889 (11989911988901198851198891198991198940 ) 11989621198962perp (19)


4 Results


0068

0069

007

0071

0072

0073

02 04 06 080

20

40

60

80

100

CQ

kperp


0 0 00

1

2

3

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5

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7

02 04 06 080

10

20

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50

V0

kperp




04

0608

1

12

14

02 04 06 08

20000

40000

60000

80000

nd

kperp


04

0608

1

12

14

16

18

02 04 06 08

20000

40000

60000

80000

Zd

kperp



02 04 06 08

305

310

315

320

325

kperp

CQ = 0 (Maxwellian)

CQ = 0



Competing Interests


References








































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




04

0608

1

12

14

02 04 06 08

20000

40000

60000

80000

nd

kperp


04

0608

1

12

14

16

18

02 04 06 08

20000

40000

60000

80000

Zd

kperp



02 04 06 08

305

310

315

320

325

kperp

CQ = 0 (Maxwellian)

CQ = 0



Competing Interests


References








































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics

































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



research article quantum treatment of kinetic alfvén waves...

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