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Current-Driven Ion-Cyclotron Waves in Presence of a Transverse
DC Electric Fields in Magnetized Plasma with Charge Fluctuation
Suresh C. Sharma* and Satoshi Hamaguchi
Science and Technology Center for Atoms, Molecules, and Ions Control,
Graduate School of Engineering, Osaka University, 2-1 Yamada-oka, Suita,
Osaka 565-0871, Japan
*Currently on Leave from the Physics Department, GPMCE (G.G.S. Indraprastha
University,
Delhi), INDIA
ABSTRACT
The current-driven electrostatic ion-cyclotron (EIC) instability is studied in
presence of a transverse dc electric fields in a collisional magnetized dusty plasma.
We derive the appropriate charging equation self consistently by using the Vlasov
equation.
1. INTRODUCTION
Fluctuations of the dust grain charge are found to be a source of wave
damping or growth1-2 . Current-driven EIC waves in presence of a transverse dc
electric field in magnetized plasma without dust has been studied experimentally3
and theoretically4 earlier.
2. INSTABILITY ANALYSIS
Dusty plasma densities ne0 , ni0 , and nd0 , static magnetic field B is in
the z-direction, charge, mass and temperature for three species are (-e,m,Te),
(e,mi ,Ti), and (-Qd0 ,md ,Td), collisional frequency (= υe for electrons and =0 for
ions), dc electric fields E0 is in the x-direction, drift of magnitude vE=cE0/B is in
the y-direction. In addition drift vde along the magnetic field direction.
Basic Equations
m
v
d 
dt
 q E  ce v  B 
 ( p )
n
1
 m e v .
(1)
n .(n v )  0
 
t
(2)
f
1
 v.f  [ Fext  q ( E  v  B)].f  0
t
m
(3)
.E  4
(4)
where α=e, i
Charging equation5
dQd
1  Q   I
0e
d1
dt
ne1
ne0
 Ii1,
(5)
By solving Eqs. (1)-(5), we obtain
єr(ω,k) + i єi(ω,k) =0,
(6)
where
 p2
 pi 2
 r ( , k )  1  k 2v 2  ( 2
z
te
  p2 e1
( 2  2 ) k z 4vte4
[3 
ci
2
2)
k2
kz 2
 ( 2
 2 )
 pi 2

 ( 2  2 ) k 2v 2
z te
 pi 2
(22 ci2 )

4 a2nd 0

kz
4e( g 0 ) k2
]k 2 ,
mi vti 2
z
(7)
2
 i ( , k ) 
 p2 e1
kz 4vte4
 p2

 ( 2  2 ) k 2v 2
z te
  p2 e1
 ( 2  2 ) k 4v 4
z te
4a2nd 0
4e( g 0 ) k2
 p2

 ( 2  2 ) ( 2  2 ) kz [3  m v 2 ] k 2 ,
z
ci
i ti
2
(8)
We write ω=ωr + i (    << ωr ). The real and imaginary part of the frequencies
are given by
r  k y vE  [
2 ni 0
ci  k cs ne 0
2
2
k2

2
 ( 2  2 ) 4a nd 0 kz
4e(  g  0) 1/ 2
] ,
mi vti2
cs 2 nei 00 [3 
n
(9)
and
   12
ne 0 mi
ni 0 m
r (1
[{1 
k yv E
)
k z vde (1 R )
r
k 2
r
[(r  k y v E )2 ci 2 ]2 k z 2
2
z

1

} k e4v r 4 (1  (  ) )  ( 
2
te
ne 0 mi  e

( 2  2 ) ni 0 m k z 4vte4
2
2
)
( 2 r  ci )
2
r (1
 [( k v
r
2
k yv E
y E)
r
2
)
ci 2 ]2
(10)
3
]
,
R=kyvE/kzvde. If ni0/ne0 =1 and nd0→0, i.e., β→0, we recover the expression for the
growth rate and the dispersion relation [cf. Eqs. (23) and (24)] for the ion-cyclotron
waves in presence of a transverse dc electric fields of Sharma et al.4
3. RESULTS
1. Dusty plasma parameters6
potassium ion plasma density ni0 ≈ 109–1010 cm-3, ne0 ≈ 109 cm-3, relative
density of negatively charged dust grains δ(=ni0/ne0)= 0 to 8.0, Te≈ Ti ≈ 0.2 eV,
B =0.1x103 G, ωp =1.78x109 rad/s, ωci =0.23x105 rad/s (potassium), nd0 ~5x104
cm-3, a≈1x10-4 cm, vE =3.3x104 cm/sec, mi /m (potassium) ~1x105 , e(g -0)/mi
vti 2 = -1.71, ky =0.1 cm-1 , kz =1.0 cm-1 ( i.e., k  2  k z 2 ). We have plotted in Fig.
1 the normalized real frequency ( ωr / ωci ) of the Current driven EIC waves as a
function of δ. The normalized wave frequency (ωr /ωci ) increases about 10 %
when δ changes from 1 to 4.0. Barkan et al.6 have found that the wave frequency
was some 10- 20% larger than the ion-cyclotron frequency in presence of dust.
Hence our unstable wave frequency results, qualitatively and quantitatively, are
similar to the experimental observations of Barkan et al. We have plotted in Fig.
2 the normalized growth rate (γ/ ωci ) of the current driven EIC wave instability
as a function of relative density of negatively charged dust δ for the same
parameters as Fig. 1 and for e ~1x108 sec-1 . The normalized growth rate (γ/
ωci ) decreases monotonically with the relative density of negatively charged
dust δ.
1.6
1.55
1.5
 r / ci
1.45
1.4
1.35
1.3
1.25
1.2
1.15
0
1
2
3
4
 ( ni 0 / ne0 )
4
5
6
7
8
Fig. 1 The normalized real frequency ( r / ci ) of the current driven ioncyclotron waves as a function of the relative density of negatively
charged dust grains δ.
0.5
0.45
0.4
 / ci
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
8
 ( ni 0 / ne0 )
Fig. 2 The normalized growth rate ( γ/ωci ) of the current driven ioncyclotron wave instability as a function of the relative density of
negatively charged dust δ for the same parameters as Fig. 1 and for
e ~1x108 sec-1.
References
1.
M. R. Jana, A. Sen, and P. K. Kaw, Phys. Rev. E 48, 3930 (1993).
S. C. Sharma amd M. Sugawa, Phys. Plasmas, 6, 444 (1999).
3.
M.E. Koepke and W.E. Amatucci, IEEE Trans. Plasma Sci. 20, 631 (1992).
4.
S. C. Sharma , M.Sugawa and V.K.Jain, Phys. Plasmas 7, 457 (2000).
5.
M.Tribeche and T.H. Zerguini, Phys. Plasmas, 8, 394 (2001).
6.
A. Barkan, N. D’Angelo, and R. L. Merlino, Planet. Space Sci. 43, 905
(1995).
2.
5
6