Solitary waves and double layers in a dusty electronegative plasma
A. A. Mamun,
*
P. K. Shukla, and B. Eliasson
†
Institut für Theoretische Physik IV, Ruhr-Universität Bochum, D-44780 Bochum, Germany
Received 5 August 2009; revised manuscript received 13 September 2009; published 27 October 2009
A dusty electronegative plasma containing Boltzmann electrons, Boltzmann negative ions, cold mobile
positive ions, and negatively charged stationary dust has been considered. The basic features of arbitrary
amplitude solitary waves SWs and double layers DLs, which have been found to exist in such a dusty
electronegative plasma, have been investigated by the pseudopotential method. The small amplitude limit has
also been considered in order to study the small amplitude SWs and DLs analytically. It has been shown that
under certain conditions, DLs do not exist, which is in good agreement with the experimental observations of
Ghim and Hershkowitz Y. Ghim Kim and N. Hershkowitz, Appl. Phys. Lett. 94, 151503 2009.
DOI: 10.1103/PhysRevE.80.046406 PACS numbers: 52.27.Lw, 52.35.Sb, 52.35.Mw
Recently, electronegative plasmas 1–4plasmas with a
significant amount of negative ions whose contribution can-
not be neglected in any way have attracted a great deal of
attention not only because of their potential applications in
microelectronic and photoelectronic industries 5 but also
because of their occurrence in both laboratory devices and
space environments 1–10. The electronegative plasmas,
which are observed in both laboratory devices and space, are
not pure in general. They are contaminated in most cases
by solid impurities dust, which are not practically neutral
but are charged 11 by absorbing electronegative plasma
electrons and positive as well as negative ions 12–20.
Therefore, in general, electronegative plasmas are, in fact,
dirty or dusty electronegative plasma 12–18,20. On the
other hand, it has been predicted by a number of authors
21–23 that negative ions in such electronegative plasmas
are in Boltzmann equilibrium. This prediction has been con-
clusively verified by a recent laboratory experiment of Ghim
and Hershkowitz 24. Motivated by this recent laboratory
experiment 24, we consider a dusty more general elec-
tronegative plasma containing Boltzmann electrons and Bolt-
zmann negative ions, cold mobile positive ions, and nega-
tively charged stationary dust, and examine the possibility
for the formation of ion-acoustic in the absence of dust and
dust-ion-acoustic 25,26in the presence of dust solitary
waves SWs and double layers DLs in a dusty electrone-
gative plasma DENP.
We consider a one-dimensional, collisionless, unmagne-
tized DENP composed of Boltzmann electrons, Boltzmann
negative ions, cold mobile positive ions, and negatively
charged stationary dust. Thus, at equilibrium we have
n
i0
= n
e0
+ n
n0
+ z
d
n
d0
, where n
i0
, n
e0
, n
n0
, and n
d0
are, respec-
tively, positive ion, electron, negative ion, and dust number
density at equilibrium, and z
d
is the number of electrons
residing onto the surface of a stationary dust. We are inter-
ested in examining the nonlinear propagation of a low
phase speed in comparison with electron and negative-ion
thermal speeds, long wavelength in comparison with
Dm
= k
B
T
e
/ 4n
i0
e
2
1/2
with T
e
being the electron tempera-
ture, k
B
being the Boltzmann constant, and e being the mag-
nitude of the electron charge perturbation mode on the time
scale of the ion-acoustic IA waves. The time scale of the IA
waves is much faster than the dust plasma period so that dust
can be assumed stationary. The nonlinear dynamics of the
low-frequency electrostatic perturbation mode in such a
DENP is described by
n
i
t
+
x
n
i
u
i
=0, 1
u
i
t
+ u
i
u
i
x
=-
x
, 2
2
x
2
=
e
exp +
n
exp - n
i
+
d
, 3
where n
i
is the positive-ion number density normalized by its
equilibrium value n
i0
, u
i
is the positive-ion fluid speed nor-
malized by the ion-acoustic speed C
i
= k
B
T
e
/ m
i
1/2
, is the
electrostatic wave potential normalized by k
B
T
e
/ e, x is the
space variable normalized by
Dm
, and t is the time variable
normalized by the ion plasma period
pi
-1
= m
i
/ 4n
i0
e
2
1/2
,
e
=1 / 1+ + ,
n
= / 1+ + ,
d
= / 1+ + .
= n
n0
/ n
e0
, = z
d
n
d0
/ n
e0
, = T
e
/ T
n
, T
n
is the negative-ion
temperature, and m
i
is the ion mass. The linear dispersion
relation for the low phase speed in comparison with the
electron and negative-ion thermal speed and long wave-
length in comparison with
Dm
modified IA waves 24,25
is V
p
= u
i0
+
e
+
n
-1/2
, where V
p
= / kC
i
is the normal-
ized phase speed of the perturbation mode under consider-
ation, u
i0
is the ion drift speed 27 normalized by C
i
, is
the wave frequency, and k is the propagation constant.
To derive an energy integral 28,29 from Eqs. 1–3, we
first make all the dependent variables depend only on a
single variable = x - U
0
t, where U
0
is the nonlinear wave
speed normalized by C
i
. We note that U
0
is not the Mach
number, since it is normalized by C
i
. If U
0
would be normal-
ized by the phase speed V
p
of the IA waves, it would then
be called the Mach number M. So the relation between U
0
and M is M = U
0
/ V
p
. Now, using the steady-state condition
*
Permanent address: Department of Physics, Jahangirnagar Uni-
versity, Savar, Dhaka-1342, Bangladesh.
†
Also at Department of Physics, Umeå University, SE-901 87
Umeå, Sweden.
PHYSICAL REVIEW E 80, 046406 2009
1539-3755/2009/804/0464066 ©2009 The American Physical Society 046406-1