
Research Article


Effect of Nonthermal Electron on Dustacoustic Shock Waves in Dusty Plasma


Louis E. Akpabio
and
Akpan N. Ikot


ABSTRACT

A dusty plasma system containing nonthermal electron distributions, Boltzmann distributed ions and mobile charge fluctuating positive dust has been considered. The nonlinear propagation of the dustacoustic (DA) waves in such dusty plasma has been investigated by employing the reductive perturbation method. The effect of nonthermal electrons on the height and thickness of DA shock waves are also studied. It has been found that the thickness of the Dust acoustic shock composition decreases as the nonthermal parameter increases, while the amplitude of the shock composition thickness varies with the charge fluctuating dust.





Received:
February 09, 2012; Accepted: June 13, 2012;
Published: July 19, 2012 

INTRODUCTION
Dust and plasmas exist together in the universe and they make dusty plasmas.
Dusty plasmas are found in cometary tails, asteroid zones, planetary ring, interstellar
media, lower part of Earth’s ionosphere and magnetosphere, etc. (Goertz,
1989; Mende’s and Rosenberg, 1994; Shukla,
2001; Horanyi and Mendis, 1986; Verheest,
2001; Horanyi, 1996). Noticeable applications of
dusty plasmas are also found in laboratory devices (Barkan
et al., 1995; Merlino et al., 1998;
Homann et al., 1997). There has been a rapidly
growing interest in the field of dusty plasmas because of its great variety
of new phenomena associated with wave and instabilities (Verheest,
1992; Pieper and Goree, 1996; Bliokh
and Yaroshenko, 1985). The existing plasma wave spectra are not only modified
by the presence of charged dust grains in a plasma (DeAngels
et al., 1988; Shukla and Stenflo, 1992),
it also brings about new novel eigen modes such as Dust Acoustic (DA) wave (Rao
et al., 1990; Barkan et al., 1996),
Dust Ion Acoustic (DIA) waves (Shukla and Silin, 1992;
Barkan et al., 1996) etc.
From the first theoretical study on the ultra low frequency DA waves in dusty
plasma by Rao et al. (1990) and motivated by
the experimental observations of these waves (Barkan et
al., 1996; Pieper and Goree, 1996), numerous
investigations have been carried out to study the different aspects of the physics
of dusty plasmas during the past few years. However, most of the investigations
were mainly on dusty plasmas with negatively charged dust grains (Amin
et al., 1998; Popel and Yu, 1995; Ma
and Liu, 1997), in this regard, nonlinear solutions and double layers in
dusty plasmas have been investigated by several authors (Bharuthram
and Shukla, 1992; Mamun et al., 1996). However,
in the space plasmas environments, some plasma systems are found with positively
charged dust grains (Mendis and Horanyi, 1991; Chew
et al., 1993; Haunes et al., 1996;
Horanyi et al., 1993). Such dust grains with
net positive charge are due to processes such as irradiation by Ultraviolet
(UV) light; thermionic emission produced by radiative heating as well as secondary
emission of electrons from the surface of the dust grains (Verheest,
1992; Shukla and Mamun, 2002).
Recently, Paul et al. (2009), investigated
the nonlinear propagation of DA waves accounting for the charge fluctuating
positive dust and Boltzmanndistributed electrons and ions. For this purpose,
they derived the Burgers equation, by employing the reductive perturbation method
(Washimi and Taniuti, 1996). They showed that, the dust
charge fluctuation is a source of dissipation and is responsible for the formation
of collisionless DA shock waves in such dusty plasma. Since in a real dusty
plasmas; the electron behaviour can be powerfully modified by the nonlinear
potential of the localized DA composition by generating a population of fast
vigorous electrons, the present paper is mainly to determine how the electron
nonthermality effect can be expected to modify the result of Paul
et al. (2009). This simplification involves a little increase in
algebraic intricacy of the pertinent formulas. This notwithstanding, the basic
principles do not change.
BASIC EQUATION We consider unmagnetized collisionless dusty plasma consisting of nonthermal electrons, Boltzmanndistributed ions and charge fluctuating positively charged mobile dust. We assume for simplicity that all the grains have the same charge, equal to q_{d} = +z_{d}e, with z_{d} representing the charge state of the dust component. Hence, charge neutrality at equilibrium is given by n_{e0} = n_{i0}+z_{d0} n_{d0}, where n_{e0}(n_{i0}) is the equilibrium electron (ion) number density, n_{d0} is the dust density at equilibrium, z_{d0} represent equilibrium charge state of the dust component. All the dust grain is assumed to be spheres of radius r_{d}. The basic equations for onedimensional DA waves for such a dusty plasma is given as:
where, φ is the electrostatic potential, n_{d}, n_{e},
n_{i} are respectively, the number density for the plasma species for
dust, electrons and ions, u_{d} is the dust fluid speed. The nonthermal
electron distribution is given as (Carins et al.,
1995):
Where: and the Boltzmann distributed ion as:
where, α_{1} is a parameter determining the number of nonthermal
electrons present in our plasma model, k_{B} is the Boltzmann constant
and T_{e} (T_{i}) is the electron (ion) temperature. Neglecting
all other charging processes, we assume that the dust is charged by photoemission
current (I^{+}_{p}), thermionic emission current (I^{+}_{t})
and electron absorption current (I¯_{e}) only. The charge state
z_{d} of the dust component is not constant but varies according to
the following equation (Paul et al., 2009; Shukla
and Mamun, 2002):
Where:
where, h is the Planck’s constant, T_{ph} is the photon temperature,
w_{e} is the work function, J is the UV photon flux, Y is the yield
of photons. The typical values of w_{e}, J and Y are given respectively
as 8.2 eV, 5.0x10^{4} photons /cm^{2}/s and 0.1. For convenience,
we express the set of Eq. 1 to 7 in normalized
form by introducing the following normalized variables: N_{d} = n_{d}/n_{do},
u_{d}u_{d}/C_{d}, φ = eφ/k_{B}T_{e},
Z_{d} = z_{d}/z_{do}, X = x/λ_{Dd}, T =
tw_{pd}, λ_{Dd} = (k_{B}T_{e}/4πz^{2}_{do})^{1/2},
C_{d} ≡ (z_{do}k_{B}T_{e}/m_{d})^{1/2}
and w_{pd} = (4πz^{2}_{do}n_{do}e^{2}/m_{d})^{1/2}
to obtain the following equations:
where, σ is T_{e}/T_{i}; μ_{i} is n_{io}/z_{do}n_{do};
μ_{i} is n_{io}/z_{do}n_{do}; μ is
π r^{2}_{d}/z_{do}w_{pd}; P is JY; Q is
2e^{we/kBTe}(2πm_{e}k_{B}T_{e})/h^{2
3/2}(8k_{B}T_{e}/πm_{e})^{1/2}, R
is n_{e0}(8k_{B}T_{e}/πm_{e})^{1/2},
α is z_{do}e^{2}/r_{d}k_{B}T_{ph};
β is z_{do}e^{2}/r_{d}k_{B}T_{e}.
NONLINEAR DUST ACOUSTIC SHOCK WAVES
To study the dynamics of nonlinear dust acoustic shock waves in the presence
of nonthermal electrons, Boltzmann distributed ions and charge fluctuating
positive dust grains; we employ the reductive perturbation technique (Washimi
and Taniuti, 1996). We introduce the stretched coordinates (Das
et al., 1997) ξ = ε(XV_{0}T) and τ = ε^{2}T,
where ∈ is a small parameter and V_{0} is the DA shock waves velocity
normalized by C_{d}. The variables N_{d}, U_{d}, Z_{d}
and φ are expanded as:
Now, substituting these expansions in to Eq. 1114
and collecting the terms of different powers of∈, in the lowest order,
we obtain:
Where:
Where:
The next order in ε,O(ε^{2}) yields a system of equations that leads to Burgers equation as follows:
Making using of Eq. 1623 we eliminate
N_{d2}, U_{d2}, Z_{d2} and φ_{2} to obtain
the following equation:
where, the nonlinear coefficient A and the dissipation coefficient B are given by:
The Burgers equation which describes the nonlinear propagation of the DA shock waves in the dusty plasma under consideration is given as Eq. 24. It can be observed that, the right hand side of Eq. 24 which represent the dissipative term is due to the presence of non thermal parameter (β_{n}), the ratio of electron and ion temperature (σ) and the charge fluctuating positive dust (μ_{i}). RESULTS AND DISCUSSION
Our expression for Z_{d1} as Eq. 18 agrees with
what is obtained by Paul et al. (2009) when nonthermal
parameter β_{n} is set to zero and σ set to 1. We strongly
feel that the last term in the denominator for Eq. 18 should
be [+μR(1+β)] as against what is obtained by Paul
et al. (2009) as [μR (1+β)]. Likewise, the last term for
f as (μRβ) reported by Paul et al. (2009)
should be (+μRβ) as in our report for Γ. Equation
19 which gives the linear desperation relation for DA waves is greatly altered
by the presence of the electron nonthermal parameter, ratio of electron and
ion temperature, as well as the positive dust charge fluctuation. For stationary
shock wave solution of Eq. 24, we set ζ = ξU_{0}τ’
and τ’ = τ to obtain the equation:
The latter equation can be integrated, using the conduction that φ is
bounded as wζ→ or by the application of Tanh method (Malfliet,
1992, 2004; Malfliet and Hereman,
1996a, b) to yield:
Where:
and:
Equation 29 represents a monotonic shocklike solution with
the shock speed, the shock height and the shock thickness given by U_{0},
φ_{0} and Δ_{sh}, respectively. It is obvious from
Eq. 29 that, the presence of electron non thermal parameter
significantly modifies the shock wave amplitude and its width.
To see the influence of the nonthermal parameter on the DA shock waves, we
chose σ as 1.5 and vary β_{n}. The following parameters; U_{0}
= 0.1 m sec^{1}, P = 5.00x10^{17} m^{2 }sec^{1},
Q = 1.07x10^{31} m^{2} sec^{1}, R = 2.48x10^{13}
m^{2} sec^{1}, V_{0} = 0.8, β = 1.2x10^{9}
c^{2} kg^{1} m^{3} sec^{2}, μ = 2.5x10^{12}
m^{2} sec^{1} corresponding to the mesosphere event has been
chosen from Paul et al. (2009). Figure
1 shows that as the β_{n} increases; the positive shock width
decreases, while the amplitude of the positive shock width varies as μ_{i}
increases.

Fig. 1: 
Variation of positive shock thickness (Δ_{sh})
potential profile with nonthermal electron parameter (β_{n})
for different values of μ_{i} 

Fig. 2: 
Variation of amplitude φ_{0} of the positive
shock wave with (β_{n}) for different values of μ_{i} 

Fig. 3: 
Variation of amplitude φ_{0} of the negative
shock wave with β_{n} for different values of μ_{i} 
The variation of amplitude (φ_{0}) of the positive shock height waves are presented in Fig. 2. From this Fig. 2, it can be observed that, the amplitude of the positive shock height decrease with increase in β_{n} up to 0.5 and after this point it increases. Also, we can observe that, when μ_{i }the amplitude of the shock height wave, is smaller than what is obtained for μ_{i} = 0.8 due to the fact that the dust charge fluctuation is a sources of dissipation DA waves.
Figure 3 demonstrate the variation of negative amplitude
of the shock height with β_{n}. It shows that the shock height
increases with β_{n} up to 1.75 after which, there is no variation
of the shock height with β_{n} for μ_{i} = 0.8. When
μ_{i} = 1.2, the amplitude of the negative shock height suddenly
increases with β_{n} from 1.78 and then drops to exactly the same
amplitude on the negative axis. After which the negative amplitude of the shock
height increase with β_{n} to infinity. Since space plasma are
more realistically modeled by making use of nonthermal velocity distributions
as discussed by Maharaj et al. (2006), we have
seen that the nonthermal electron distribution function significantly modifies
the result obtained by Paul et al. (2009).
CONCLUSION
We have extended the recent work of Paul et al.
(2009) to see under what conditions the electron nonthermality effect can
be expected to modify the results of their analysis. We have shown here that,
the basic feature; of the nonlinear DA waves are modified by both presence
of the nonthermal electron and the charge fluctuating dust in dusty plasmas.
Present results are summarized as follows:
• 
The width of the DA shock structures decreases as the non
thermal parameter increases, while the amplitude of the shock structures
width varies as μ_{i} increases. This is due to the fact that
dust charge fluctuation is a source of dissipation and lead to the development
of DA shock waves in the dust plasma 
• 
It is also shown that, the positive amplitude of the shock height decreases
with increase in β_{n} up to a point and then increases. While,
when the charge fluctuation (μ_{i}) is increased to 1.2, the
negative shock height exhibits the occurrence of kink as in Fig.
3. The findings in this paper are important in understanding nonlinear
DA wave phenomena in space plasmas 

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