Excitation of lower hybrid waves by a spiraling ion beam in a magnetizeddusty plasma cylinderSuresh C. Sharma and Ritu Walia Citation: Phys. Plasmas 15, 093703 (2008); doi: 10.1063/1.2983139 View online: http://dx.doi.org/10.1063/1.2983139 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v15/i9 Published by the American Institute of Physics. Additional information on Phys. PlasmasJournal Homepage: http://pop.aip.org/ Journal Information: http://pop.aip.org/about/about_the_journal Top downloads: http://pop.aip.org/features/most_downloaded Information for Authors: http://pop.aip.org/authors

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Excitation of lower hybrid waves by a spiraling ion beamin a magnetized dusty plasma cylinder

Suresh C. Sharma and Ritu WaliaDepartment of Physics, Maharaja Agrasen Institute of Technology, PSP Area Plot No.-1, Sector-22,Rohini, Delhi-110086, India

�Received 21 July 2008; accepted 25 August 2008; published online 30 September 2008�

A spiraling ion beam propagating through a magnetized dusty plasma cylinder drives electrostaticlower hybrid waves to instability via cyclotron interaction. Numerical calculations of the growth rateand unstable mode frequencies have been carried out for the Princeton Q-1 device using theexperimental dusty plasma parameters �e.g., Barkan et al., Planet. Space Sci. 43, 905 �1995��. It isfound that as the density ratio ��=nio /neo, where ni0 is the ion plasma density and ne0 is the electrondensity� of negatively charged dust grains to electrons increases, the unstable mode frequency of thelower hybrid waves increases. In addition, the growth rate of the instability also increases with thedensity ratio �. In other words, the presence of negatively charged dust grains can further destabilizethe lower hybrid wave instability. The growth rate has the largest value for the modes where Jl�pnro�is maximum �here pn=xn /r0, where pn is the perpendicular wave number in cm−1, r0 is the plasmaradius, and xn are the zeros of the Bessel function J1�x�� i.e., whose eigenfunctions peak at thelocation of the beam. The growth rate scales as one third power of the beam current. © 2008American Institute of Physics. �DOI: 10.1063/1.2983139�

I. INTRODUCTION

A perpendicular ion beam driven lower hybrid mode hasbeen observed1 with unmagnetized beam and target ions in anonisothermal rf discharge plasma �Te�Ti�, unlike a fusionplasma, and the observed instability was nonresonant. Seileret al.2 reported experimental results on the excitation oflower hybrid instability by a spiraling ion beam in a lineardevice �viz., Princeton Q-1 device�. In this case, the fre-quency measurement shows that the instability occurs at justabove the cyclotron harmonics ��=n�ci+��, probably as acoupling of the beam cyclotron mode with the lower hybridmode supported by the plasma.

Recently, there has been growing interest in studyingelectrostatic waves in multicomponent plasmas. The pres-ence of negatively charged dust grains can significantly in-fluence the collective properties of plasma in which they aresuspended. Fluctuations of the dust grain charge are found tobe a source of wave damping or growth.3–25 Barkan et al.9

reported experimental results on the current driven electro-static ion cyclotron �EIC� instability in a dusty plasma,where they found that the presence of negatively chargeddust grains enhanced the growth rate of the instability. Theeffect of charged dust on the collisionless EIC instability wasinvestigated by Chow et al.10,11 with the Vlasov theory,where the critical electron drift velocity was evaluated in thepresence of either positively or negatively charged dustgrains. In the case of negatively charged dust, they found thatthe critical electron drift velocity decreased as the ratio ofpositive ion density to electron density increased, showingthat the mode was more easily destabilized in plasma con-taining negatively charged dust grains.

D’Angelo12 investigated dispersion relations for low-frequency electrostatic waves �two electrostatic ion cyclotronand two ion acoustic modes� in magnetized dusty plasma. In

the presence of negatively charged dust grains, he found thatthe mode frequencies increased as the density ratio of nega-tively charged dust grains to positive ions increased. Merlinoet al.13 presented theoretical and experimental results on lowfrequency electrostatic waves in a plasma containing nega-tively charged dust grains. The presence of negativelycharged dust grains modifies the properties of current drivenEIC instability through the quasineutrality condition eventhough the dust grains do not participate in the wave dynam-ics. Song et al.14 studied the current driven EIC instability innegative ion plasmas in a Q machine and found that thecritical electron drift velocity decreased with the ratio ofnegative ion density to the positive ion density. Later,Sharma and Srivastava15 developed a model of ion-beamdriven EIC instability in a cylindrical plasma with negativeions. They compared their theoretical results with the experi-mental observations by Song et al.14 and found good agree-ments. More recently, Sharma and Ajay16 developed a modelof ion beam driven ion-acoustic waves in a plasma cylinderwith negative ions. In this case, Sharma and Ajay found thatthe phase velocity of the sound wave in the presence of posi-tive and negative ions increase with the relative density ofnegative ions.

Chow and Rosenberg17 studied current driven EIC insta-bility in a negative ion plasma and found that as the densityratio of heavy ions to the light ions increased, the modefrequency increased, whereas the critical electron drift veloc-ity decreased. They also showed that the wavelength for themaximum growth rate shifted to a larger value as the densityratio increased.

Experiments by Suszcynsky et al.18 on the current drivenEIC instability in a plasma with two positive ion speciesindicated that the critical electron drift velocity increasedwith the density ratio of heavy ions to light ions.

PHYSICS OF PLASMAS 15, 093703 �2008�

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Ma and Yu19 have developed a self-consistent theory ofion acoustic waves and Langmuir wave instability in unmag-netized dusty plasmas. Vladimirov et al.20 have studied ionacoustic waves in dust-contaminated plasmas. Charging andtrapping of macroparticles in near electrode regions of fluo-rocarbon plasmas with negative ions has been studied byOstrikov et al.21 Later, a self-consistent theory of linear ionacoustic waves in complex laboratory plasmas containingdust grains and negative ions has been presented byVladimirov et al.22

More recently, Rosenberg et al.23 have studied ion-duststreaming instability in a plasma containing dust grains withlarge thermal speeds using the kinetic treatment.

Sharma and Tripathi26 have developed a model of gyrat-ing ion beam driven lower hybrid waves in a magnetizedplasma cylinder without charged dust grains. In the presenceof energetic gyrating ion beam lower hybrid modes aredriven to unstable via cyclotron harmonic interaction.

In this paper, we study the excitation of lower hybridwaves by a spiraling ion beam in a magnetized dusty plasmacylinder. A spiraling ion beam propagating through a magne-tized dusty plasma cylinder drives electrostatic lower hybridwaves to instability via cyclotron interaction. In Sec. II wecarry out the instability analysis. The plasma and beam re-sponses are obtained using fluid treatment. We incorporate amodel of dust charge fluctuations by following Whipple etal.3 and Jana et al.4 We obtain the growth rate of the insta-bility using first-order perturbation theory. Results are givenin Sec. III. Conclusions are given in Sec. IV.

II. INSTABILITY ANALYSIS

We consider a cylindrical dusty plasma column of radiusa1 with equilibrium electron, ion, and dust particle densitiesbeing given as ne0, ni0, and nd0 immersed in a static magneticfield Bs in the z direction. The charge, mass, and temperatureof the three species are denoted by �−e, me, Te�, �e, mi, Ti�,and �−Qd0, md, Td�, respectively. A spiraling ion beam with

velocity v� =v�0�+vz0

z, mass mb, and radius r0 propagatesthrough the dusty plasma cylinder �cf. Fig. 1�. The densityprofile of the beam is given by nb0= �N0 /2�r0���r−r0�,where r0=v�0 /�cb, �cb is the beam cyclotron frequency andN0 is the number of beam ions per unit axial length. In equi-librium, there is overall charge neutrality, i.e.,

eni0 + enb0 = ene0 + Qd0nd0.

Let us assume a perturbation of the electrostatic potential isgiven by

�1 = �0�r�e−i��t−l�−kzz�. �1�

The response of plasma electrons to the perturbation is gov-erned by the equation of motion

me� �v�

�t+ �v� . ��v�� = − eE� −

e

cv�xB� s. �2�

On linearization, Eq. �2� yields the perturbed velocity

v�1� =e

me����1 � �ce + i����1

�2 − �ce2 � , �3�

vz1 = −ekz�1

me�, �4�

where �ce�=eBs /mec� is the electron cyclotron frequency andsubscript 1 refers to perturbed quantities. Substituting theperturbed velocities given by Eqs. �3� and �4� in the massconservation equation, we obtain the perturbed electron den-sity as

ne1 =ne0e

me� ��

2 �1

�2 − �ce2 −

kz2�1

�2 � . �5�

The ion response can be taken to be unmagnetized and isgiven as

ni1 = −ni0e

mi�2 ���

2 �1 − kz2�1� . �6�

Following the treatment of Sharma and Tripathi,27 the ionbeam density perturbation is given by

nb1 =N0

��1 − l�cb�2

e��r − r0�mb2�r0

� l2

r2 + kz2��1, �7�

where �1=�−kzvb0.Similarly, the dust density perturbation is given by

nd1= −

nd0Qd0k�2 �1

md�2 . �8�

Here dust is treated as unmagnetized because ��pi��cd

with �cd=Qd0Bs /mdc being the dust gyrofrequency. Now ap-plying probe theory to a dust grain, the charge on a dustgrain Qd is known to be balanced with the plasma currentson the grain surface3,4 as

−dQd

dt= Ie + Ii. �9�

Electron and ion currents on the grain surface are given by

Ie = − �a2e� 8Te

�me�1/2

ne exp� e��g − V�Te

�and

Spiraling ion beam + plasma with dust grains

BS || Z^>—

Z^

^r

FIG. 1. Schematic of spiraling ion beam and plasma cylinder with nega-tively charged dust grains.

093703-2 S. C. Sharma and R. Walia Phys. Plasmas 15, 093703 �2008�

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Ii = �a2e� 8Ti

�mi�1/2

ni�1 −e��g − V�

Ti� ,

where a is the dust grain sphere radius, ne and ni are electronand ion densities in the absence of dust grains, ��g−V� is thedifference between the grain surface potential and plasmapotential. In equilibrium, the electron and ion grain currentsare equal, i.e., Ie0= Ii0, where Ie0 and Ii0 denote the equi-librium electron and ion currents on the grain surface. Forexample, we may write

Ie0 = �a2e� 8Te

�me�1/2

ne0 exp� e�g0

Te� .

Then the charge fluctuation is governed by the equation

dQd1

dt+ �Qd1 = − Ie0�ni1

ni0−

ne1

ne0� , �10�

where

� =Ie0eCg

� 1

Te+

1

Ti − e�g0� �11�

and Qd1=Qd−Qd0 is the perturbed dust grain charge, Cg

= �a�1+a /De�� is the capacitance of the dust grain,28 De is

the electron Debye length. Substituting d /dt=−i� in Eq.�10�, we obtain the dust charge fluctuation,

Qd1 =Ie0

i�� + i���ni1

ni0−

ne1

ne0� . �12�

Substituting the value of ne1 and ni1 from Eqs. �5� and �6� inEq. �12�, we obtain

Qd1 =Ie0

i�� + i���−e

mi�2 ���

2 �1 − kz2�1�

−e

me� ��

2 �1

�2 − �ce2 −

kz2�1

�2 �� . �13�

Using Eqs. �5�–�8� and �13� in the Poisson’s equation,

�2�1 = 4��ne1e − ni1e − nb1e + nd0Qd1 + Qd0nd1� .

We obtain a second order differential equation in �1, whichcan be rewritten as

�2�1

�r2 +1

r

��1

�r+ �p2 −

l2

r2��1

= −2N0

Lmb��1 − l�cb�2

��r − r0�r0

� l2

r2 + kz2��1, �14�

where

p2 =

− kz2 +

�pe2 kz

2

�2 +�pi

2 kz2

�2 +i

�� + i���pi

2

�2

ne0kz2

ni0+

i

�� + i���pe

2 kz2

�2

1 −�pe

2

�2 − �ce2 −

�pi2

�2 −i

�� + i���pi

2

�2

ne0

ni0−

i

�� + i���pe

2

��2 − �ce2 �

, �15�

L = 1 −�pe

2

�2 − �ce2 −

�pi2

�2 −i

�� + i���pi

2

�2

ne0

ni0

−i

�� + i���pe

2

��2 − �ce2 �

, �16�

�pe= �4�ne0e2 /me�1/2 and �pi= �4�ni0e2 /mi�1/2 are the elec-tron and ion plasma frequencies and

=Ie0

e�nd0

ne0� . �17�

In addition, we have ignored �pd2 �=4�nd0Qd0

2 /md� as�pi

2 /�2��pd2 /�2.

In the absence of the beam, the solution of Eq. �14� isgiven by �1=AJl�pnr�, pn= p. At r=a1, �1 must vanish,hence Jl�pna1�=0, i.e., pn=xn /a1�n=1,2 , . . . �, xn are the ze-ros of the Bessel function Jl�x�. In the presence of the beam,the solution wave function �1 can be expressed in a series oforthogonal sets of wave function,

�1 = �m

AmJl�pmr� . �18�

Substituting the value of �1 from Eq. �18� in Eq. �14�, mul-tiplying both sides of Eq. �14� by rJl�pnr� and integratingover r from 0 to a1, retaining only the dominant mode �m=n�, we obtain

p2 − pn2 = −

�pb2

L��1 − l�cb�2

� l2

r02 + kz

2�Jl2�pnr0�

Jl+12 �pna1�

, �19�

where �pb2 =4N0e2 /a1

2mb.Substituting the value of p2 from Eq. �15�, Eq. �19� can

be rewritten as

��2 − �2���1 − l�cb�2 =�2�pb

2

k2

� l2

r02 + kz

2�Jl2�pnr0�

Jl+12 �pna1�

, �20�

where

093703-3 Excitation of lower hybrid waves… Phys. Plasmas 15, 093703 �2008�

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� = ��pi2 kz

2

k2 + �pe2 kz

2

k2 +i

�� + i���pi

2 ne0

ni0

kz2

k2

+i

�� + i���pe

2 kz2

k2�1/2

,

k2 = pn2 + kz

2. �21�

In the limit of vanishing beam density, Eq. �20� yields tworoots,

� = kzvbo + l�cb,

�22�=� .

giving the beam cyclotron mode and the lower hybrid modein presence of dust, respectively. When nbo�0, we expand �as

� = � + � ,

=kzvbo + l�cb + � ,

where � is the small frequency mismatch due to the finiteright-hand side of Eq. �20�. Then Eq. �20� gives the growthrate

� = Im� = ���pb2 � l2

r02 + kz

2�2�pn

2 + kz2�

Jl2�pnr0�

Jl+12 �pna1�

1/3

�3

2. �23�

The coupling parameter �cf. Eq. �17�� can be rewrittenafter using the charge neutrality condition and the value ofequilibrium electron current as

= 0.397�1 −1

��� a

vte��pi

2 �mi

me� , �24�

where ��=nio /neo�.The charging rate �cf. Eq. �11�� can also be rewritten in

the limit, namely, Te=Ti, i.e., electron temperature is equal tothe ion temperature, for example, Q-Machine plasmas,

� = 0.79a��pi

Di��1

���mi

me

Ti

Te�1/2

. �25�

If ��=nio /neo�=1, i.e., →0 we recover the expressions forthe growth rate �cf. Eq. �12�� and dispersion relation �cf. Eq.�10�� of Sharma and Tripathi26 after putting vb0=0, i.e., spi-raling ion beam is standing in the beginning and hence �= l�cb �cyclotron harmonic interaction�.

III. RESULTS AND DISCUSSIONS

In the calculations we have used dusty plasma param-eters for the experiment of Barkan et al.9 Using Eq. �21�, wehave plotted in Fig. 2, the dispersion curves of lower hybridwaves for the following parameters: ion plasma density ni0

=107 cm−3, electron plasma density ne0=107–106 cm−3. Theion and electron temperatures are assumed to be Te=Ti

=0.2 eV, the static magnetic field Bs=5�103 G �cf. Ref. 2�,plasma radius a1=2 cm, mi /me�7.16�104 �potassium�, theaverage dust grain size a=1 m, mode number n=3, i.e., thethird zero of the Bessel function. We vary � from 1 to 10.

The applicability of the expressions for electron and iongrain currents �cf. Eq. �9�� flowing into the grain surface is asfollows: The orbit motion limited �OML� theory was initiallyderived for Maxwellian plasma particles �i.e., for electronsand ions�. It has been studied in detail in connection withelectrostatic Langmuir probes which are usually stationary orquasistationary. The probe theory for magnetized plasma hasnot been understood so far.29,30 Thus the applicability of theOML theory to dynamical or wave systems in magnetizedplasma is limited. Because both the fluctuating fields and theexternal magnetic fields can affect the plasma particle distri-bution and orbits, hence both of these are important in thederivation of the plasma currents �electron and ion currents�entering into the probe. But a complete OML theory of dustcharging in a magnetized plasma is still unknown.29,30

Here we have discussed only the application of the OMLtheory to magnetized plasma in which lower hybrid wavespropagate using a standard model in which dust particles sizeand shape effects have been neglected. Such a model is rea-sonably valid in the limit a�De�, where a is the dustgrain size, De is the electron Debye length, and is thewavelength of the fluctuations provided that the spread inQd /md �dust charge to mass ratio� for the dust particles in theequilibrium plasma may be neglected.

Moreover, Jana et al.4 have given the limits of the OMLtheory for magnetized dusty plasma. According to Jana et al.

TABLE I. Unstable wave frequencies � �rad/s� and axial wave numberskz�cm−1� for different values of �.

� kz �cm−1� � �rad /s��106

1 0.297 10.646

2 0.429 10.876

4 0.633 11.297

6 0.814 11.685

8 0.963 12.144

10 1.137 12.50

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Kz (cm-1)

10δ =

1δ =

8δ =4δ =

2δ =

6δ =

710×

()

./s

rad

ω

FIG. 2. Dispersion curves of lower hybrid waves and a beam mode for thefollowing parameters: ni0=107 cm−3, ne0=107–106 cm−3, Te=Ti=0.2 eV,Bs=5�103 G, a1=2 cm, mi /me�7.16�104, a=1 m, n=3, and Eb

=100 eV.

093703-4 S. C. Sharma and R. Walia Phys. Plasmas 15, 093703 �2008�

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the charging equation �cf. Eq. �9�� can be valid if a��L

�where �L= �vte /�ce� is the electron Larmor radius, a is thedust grain size, vte is the electron thermal velocity, and �ce isthe electron cyclotron frequency�. We have chosen param-eters in such a way that the above mentioned condition issatisfied.

We have also plotted the beam mode for potassium beamenergy Eb=100 eV �cf. Ref. 2�. The frequencies and the cor-responding wave numbers of the unstable wave are obtainedby the point of intersections between the beam mode andplasma mode and are given as Table I. From Table I, we cansay that the unstable wave frequencies of the lower hybridwaves in presence of dust grains increases with the relativedensity of negatively charged dust grains �. In addition, Bar-kan et al.,9 and Chow and Rosenberg10 have also found thatthe wave frequency was about 10%–20% larger than the ion-cyclotron frequency in the presence of dust grains. FromTable I, it can also be seen that the axial wave vectorkz�cm−1� increases with increasing �. This result is similar tothe theoretical result of Chow and Rosenberg,10 where thewave is more unstable because the axial wave vectorkz�cm−1� increases with increasing �.

Using Eq. �23�, we have plotted in Fig. 3 the growth rate��s−1� of the lower hybrid waves as a function of the relativedensity of negatively charged dust grains for the same pa-rameters used for plotting Fig. 2 plus unstable wave frequen-cies and wave numbers of the lower hybrid waves �cf. TableI� in addition to beam density nb0=4�106 cm−3 �cf. Ref. 2�and beam radius r0=1.5 cm. From Fig. 3, it can be seen thatthe growth rate of the unstable mode increases with �.

The growth rate of the unstable mode increases withbeam density and scales as the one-third power of the beamdensity �cf. Eq. �23��.

IV. CONCLUSION

In conclusion, we may say that the electrostatic lowerhybrid waves are driven to instability in a magnetized dustyplasma cylinder via cyclotron interaction. Our growth rateresults are qualitatively similar to the experimental observa-tions of Barkan et al.9 and theoretical predictions of Chowand Rosenberg.10 The frequency of the unstable mode in-creases with the relative density of negatively charged dustgrains �cf. Table I�. In other words, the presence of nega-tively charged dust grains can further destabilize the lowerhybrid wave instability. The growth rate has the largest valuefor the modes where Jl�pnro� is maximum, i.e., whose eigen-functions peak at the location of the beam.

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6.7

6.8

6.9

7

7.1

7.2

1 2 3 4 5 6 7 8 9 10

( )0 0/i en nδ =

510×(

)./

rads

γ

FIG. 3. Growth rate ��s−1� of the unstable mode as a function of the relativedensity of negatively charged dust grains for the same parameters as in Fig.2 and for beam density nb0=4�106 cm−3.

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