Indian Journal of Pure & Applied Physics
Vol. 51, July 2013, pp. 488-493
Effect of non-thermal electrons and warm negative ions on ion-acoustic solitary
waves in multi-component drifting plasma
Basudev Ghosh1*, Sreyasi Banerjee2 & Sailendra Nath Paul1
1
2
Department of Physics, Jadavpur University, Kolkata 700 032, India
Department of Electronics, Vidyasagar College (Day), Kolkata, India
*E-mail:[email protected]
Received 1 June 2012; revised 12 March 2013; accepted 2 May 2013
Using the Sagdeev’s pseudopotential approach, effects of non-thermal electrons and warm negative ions on the
conditions for existence and structure of first and second order ion-acoustic solitary waves have been investigated in a multicomponent drifting plasma. It is shown that there exists a critical concentration of negative ions which decides the existence
and nature of the ion-acoustic solitary waves. It is found that the non-thermal electrons, the concentration of negative ions
and the temperature of negative ions have significant contributions towards the excitation and structure of the ion-acoustic
solitary waves. The plasma under consideration can support the formation of compressive, rarefactive as well as W-type
solitons with certain restricted values of plasma parameters. The results are important in the context of ionospheric and
magnetospheric plasmas.
Keywords: Non-thermal electrons, Negative ions, Ion-acoustic solitary waves, Sagdeev potential
1 Introduction
In recent years, the formation of solitary wave
structures has been a topic of great interest due to its
relevance in cosmic applications1, ion heating in
linear turbulent heating devices2 and confinement of
plasma in tandem mirror devices3. Space plasmas are
of multispecies type and offer a rich source for
studying solitary waves. There has been considerable
interest in the study of ion-acoustic solitary waves in
multispecies plasmas including the effects of negative
ions4. Negative ions in plasma can be found in the
D-region of ionosphere5, plasma processing reactors6,
and neutral beam sources7. Negative ions are formed
due to electron attachment to neutral particles when
an electronegative gas is introduced into electrical gas
discharge. It has been found that even a small amount
of negative ions may have significant effect on the
formation of non-linear ion-acoustic wave structures8.
In realistic situation, ions have finite temperature and
it has been found that ion temperature can
significantly modify the characteristics of nonlinear
ion-acoustic structures9. It has been found that the
electron and ion distributions play a crucial role in
characterizing the physics of the nonlinear wave
structures. It can significantly influence the conditions
required for the formation of solitons and doublelayers. Moreover, it is also known that electron and
ion distributions can be significantly modified in the
presence of large amplitude waves. The presence of
non-Maxwellian electrons in plasma gives rise to
many interesting characteristics in non-linear
propagation of waves including the excitation of ionacoustic solitons in plasma.
In recent years, non-linear wave structures have
been studied by using different non-Maxwellian
distributions such as non-thermal electrons10, nonisothermal electrons11,12 and q-non-extensive electron
velocity distributions13. The presence of nonisothermal electrons has been shown to significantly
change the reflection properties of ion-acoustic
solitary waves in magnetized inhomogeneous
plasmas14-19. For electrostatic wave propagation in
plasma, it has been shown that non-Maxwellian
distribution presents a better fit to the experimental
data while standard Maxwellian distribution only
provides a crude description20. The solitary structure
with density depression in the magnetosphere as
observed by the Freja21 and Viking22 satellites has
been explained by Cairns et al.23 by assuming nonthermal electron distribution. Such electron
distribution with enhanced population of energetic
electrons has been observed in the magnetosphere24.
Non-thermal distribution is a common feature of
auroral zone25.This type of distribution is also
common to many space and laboratory plasmas in
which wave damping produces electron tail26. Drift
GHOSH et al.: EFFECT OF NON-THERMAL ELECTRONS ON ION-ACOUSTIC SOLITARY WAVES
motion of ions plays an important role on the
formation of ion-acoustic solitons in negative ion
plasma27.
Thus, it is interesting and more realistic in various
physical situations to investigate ion-acoustic solitary
waves in a multi-component drifting plasma by
considering simultaneous presence of non-thermal
electrons and warm negative ions. The excitation and
characteristics of the ion-acoustic solitary waves in a
multi-component drifting plasma consisting of nonthermal electrons and warm positive as well as
negative ions, have been studied in the present paper.
2 Basic Equations
We consider a collisionless unmagnetized plasma
consisting of warm positive and negative ions and
non-thermal electrons. The ions are assumed to have
constant streaming motion in equilibrium state. The
dynamics of ion-acoustic waves in such a plasma are
described by the following set of normalized basic
equations:
489
K BTe / mi , the densities by the equilibrium ion
density no, all the length by the Debye-length
K BTe / 4π no e 2 , the time variable is normalized to
the ion plasma period ω p−1 = (mi / 4π n0 e2 )1/ 2 and the
potential φ by K BTe / e where K B is the Boltzmann
constant.
3 Sagdeev Potential and Soliton Solution
To study time-independent solitary structure, we
make all the dependent variables depend only on a
single variable
η = x − Vt
… (6)
where V is the Mach number with respect to the ionacoustic speed. We also use the steady state condition
and impose the boundary conditions:
ni → nio , n j → n jo , ui → uio , u j → u jo , pi → pio ,
p j → p jo , φ → 0, d φ / dη → 0
at x →∝
∂ns ∂
+ ( ns us ) = 0
∂t ∂x
… (1)
∂us
∂u
σ ∂ps Z s ∂φ
+ us s + s
=
∂t
∂x Qs ns ∂x Qs ∂x
… (2)
nio = 1 + Z j n jo
∂ps
∂p
∂u
+ us s + 3 ps s = 0
∂t
∂x
∂x
… (3)
Using the transformation given in Eq. (6) and
boundary conditions given in Eqs (1-5,7) and
following standard procedure,we derive :
∂ 2φ
= ne + ¦ Z s ns
∂x 2
s
… (4)
d 2φ
= (1 − βφ + βφ 2 ) eφ
2
dη
The charge neutrality condition of the plasma is:
As electrons are assumed to be non-thermally
distributed, the electron density ne is given by23 :
ne = (1 − βφ + βφ 2 ) eφ
… (7)
… (5)
+¦
s
… (8)
2
ª ­
½ º
« °§ V − u + 3σ s pso · + 2Z sφ ° »
¸
¾
so
3
1 « ®¨
¨
Qs nso ¸¹
Qs ° »
¿ »
Z s nso2 Qs2 « ¯°©
«
»
2
12σ s pso « ­
3σ s pso · 2Z sφ ½° »
°§
« − ®¨¨ V − uso −
¸ +
¾»
nso ¸¹
Qs ° »
« °¯©
¿¼
¬
… (9)
where β = 4 p / (1 + 3 p) , β measures the deviation
from thermalized state and p determines the number
of non-thermal electrons present in the plasma.
In the Eqs (1-5), the subscript s stands for i and j
which represent positive and negative ions,
respectively. The parameters ns , us , ps ,σ s are,
respectively the concentration, velocity, pressure and
temperature of positive and negative ions. The
concentration of non-thermal electrons is represented
by ne, φ denotes the electrostatic potential, Qi=1 for
positive ions and Qj = m j / mi , Zi = −1 and Zj= 1. The
The qualitative nature of the solution of Eq. (9) can
be most easily seen by introducing the Sagdeev
potential. Eq. (9) can be written in the form:
velocities are normalized by ion-acoustic speed
where the Sagdeev potential ψ (φ ) is given by:
d 2φ
∂ψ
=−
= −ψ ′(φ )
2
dη
∂φ
… (10)
INDIAN J PURE & APPL PHYS, VOL 51, JULY 2013
490
ψ (φ ) = (1 + 3β ) − eφ + β (φ − 1) eφ − β (φ 2 − 2φ + 2 ) eφ
−¦
s
3
2
s
3
2
so
ª ­§
« °®¨ V − uso + 3σ s pso
Qs nso
3 12σ s pso « °¨©
¬¯
Q n
· 2Z sφ ½°
+
¾
¸¸
Qs °
¹
¿
2
3
2
3
­°§
3σ s pso
− ®¨ V − uso −
¨
Qs nso
°¯©
·
¸¸
¹
§
3σ s pso
− ¨ V − uso +
¨
Qs nso
©
3
2
2
2 Z sφ ½°
+
¾
Qs °
¿
· §
3σ s pso
¸¸ + ¨¨ V − uso −
Qs nso
¹ ©
º
»
»
¼
… (11)
·
¸¸
¹
3
Considering the small amplitude theory ( φ < 1 ), one
may expand the Sagdeev potential up to fourth order
in φ . It yields:
d 2φ
= S1φ − S2φ 2 + S3φ 3 + ⋅⋅⋅⋅⋅⋅
2
dη
… (12)
and
1
1
1
ψ (φ ) = − S1φ 2 + S2φ 3 − S3φ 4 − ....
2
3
4
where
Z s 2 nso
S1 = (1 − β ) − ¦
3σ p
2
s
Qs (V − uso ) − s so
nso
ª
Z n
1«
S 2 = − «1 − ¦
3
2
s
«¬
2Qs 2 3σ s pso
3
3 2
s
so
§
3σ s pso
− ¨ V − uso −
¨
Qs nso
©
·
¸¸
¹
−3
… (13)
­°§
3σ s pso
®¨¨ V − uso +
Qs nso
¯°©
·
¸¸
¹
−3
½°º
¾»
»
¿°¼
−5
·
§
¸¸ − ¨¨ V
¹
©
… (15)
where we use the boundary conditions that
as φ → 0 , dφ / dη and d 2φ / dη 2 → 0 .
It is interesting to note that Eq. (15) describes the
motion of a pseudo particle of unit mass with velocity
dφ / dη and position φ in a potential ψ (φ ) . The first
term in Eq. (15) can be regarded as the kinetic energy
of the pseudo particle. Since kinetic energy is always
a non-negative quantity ψ (φ ) ≤ 0 for the entire
motion. Thus, zero is the maximum value of ψ (φ ) .
From Eq. (10), we can say that ψ ′(φ ) is the force
acting on the particle at the position φ . Eq. (15) may
also be imagined as an equation of anharmonic
oscillator provided that we interpret φ and Ș as space
and time coordinates, respectively.
For the existence of soliton solution of Eq. (15), the
Sagdeev potential ψ (φ ) must satisfy following the
conditions28:
(i)ψ (φ ) = 0,ψ ′(φ ) = 0 and ψ ′′(φ ) < 0 at φ = 0
…(16)
(ii)ψ (φm ) = 0,ψ ′(φm ) < (>)0 for φm < ( >)0.
…(17)
(iii)ψ (φ ) < 0 for 0 < φ < φm
…(18)
where φm is some extremum value of the potential φ ,
called the amplitude of the soliton. Under the above
conditions, a quasiparticle starting at φ = 0 will roll
back to its initial position making a single transit from
the point φ = φm at which ψ (φ ) causes reflection.
Taking terms up to φ 3 in the Sagdeev potential in
Eq. (13), the soliton solution of Eq. (12) is given as:
§η ·
¸
© W1 ¹
φ1 = φm1 sec h 2 ¨
3
ª
4 2
3
Z
n
1
s
so
S3 = ««(1 + 3β ) + ¦
5
6
s
«¬
2Qs 2 3σ s pso
­°§
3σ s pso
× ®¨ V − uso +
¨
Qs nso
°¯©
2
1 § dφ ·
¨
¸ + ψ (φ ) = 0
2 © dη ¹
where
3σ s pso · ½° º»
− uso −
¸ ¾
Qs nso ¸¹ ° »
¿¼
… (14)
−5
Integrating Eq. (10) with respect to η, we obtain the
so-called energy equation:
φm1 = (3S1 / 2 S 2 )
… (19)
is
the
amplitude
and
W1 = 2 / S1 is the width of the soliton. Note that the
soliton solution is possible for S1 >0. The nature of the
solitary wave i.e. whether compressive or rarefactive,
will depend on the sign of S2. the coefficient of the
cubic non-linear term in the Sagdeev potential in
Eq.(13). If S2 is positive, a compressive solitary wave
is formed. On the other hand, if S2 is negative a
rarefactive solitary wave is formed. Note that S2 does
GHOSH et al.: EFFECT OF NON-THERMAL ELECTRONS ON ION-ACOUSTIC SOLITARY WAVES
not depend on the non-thermal parameter β. This
indicates that the nature of the solitary wave (i.e.
compressive or rarefactive) does not change with the
change in the values of β. For cold ion plasma
(ıi=ıj=0), S2<0. In this case, only compressive soliton
is formed.
Considering higher order non-linear effects after
taking terms up to φ3 we get the higher order M-KdV
solitary wave solution from Eq. (12) as:
φ2 =
6 S1
ª
§ S · º
2 S 2 + 4 S 22 − 18S1 S3 « 2cosh 2 ¨¨ 1 η ¸¸ − 1»
«¬
© 4 ¹ »¼
…(20)
The second order amplitude ( φm 2 ) and width (W2) of
the solitary wave solution are given, respectively by:
φm 2 =
6 S1
2 S 2 + 4 S 22 − 18S1 S3
… (21)
Qs (V − uso )
2
3σ p
− s so
nso
…(25)
Note that the critical value of the non-thermal
parameter depends on ion concentrations, ion
temperatures as well as the ion streaming velocities.
Numerical calculations with typical plasma
parameters show that the value of the critical nonthermal parameter ȕc decreases with the increase in
both the concentration and temperature of negative
ions. The results are shown in Fig. 1. It is to be noted
that the streaming motion of ions has also significant
effects on the critical value of ȕ. Eq. (25) is a
biquadratic equation in V and in general, gives four
values of the soliton velocity V (real or imaginary).
For the special case of cold ions (ıi=ıj=0 ) drifting
with equal speed ( ui 0 = u j 0 = u0 , say), we get two
distinct values of the soliton speed as:
1
1− β
ni 0 + Z 2 n j 0 / Q
… (26)
1
1− β
ni 0 + Z 2 n j 0 / Q
… (27)
and
1
­
½
ª 1.381S
º2 °
°
−1
2
W2 = W1 cosh ® «
+ 1.6905» ¾ … (22)
2
»¼ °
° «¬ 4 S2 − 18S1 S3
¯
¿
Note that the nature, amplitude and width of the
solitary structure depend on plasma parameters i.e.
density of negative ions, drift velocity of ions,
temperature of ions and the non-thermal parameter β
through the coefficients S1, S2 and S3.
4 Existence of Solitary Wave Solution
For localized soliton solution the Sagdeev potential
must satisfy the condition:
ψ ′′(φ ) < 0 at φ = 0
… (23)
This requires that for the existence of soliton, the
plasma parameters must satisfy the following
inequality:
s
s
Z s 2 nso
V = VF = u0 +
and
¦
βc = 1 − ¦
491
Z s 2 nso
2
Qs (V − uso ) −
3σ s pso
nso
<1− β
… (24)
So the critical value (ȕc) of the non-thermal
parameter for the existence of soliton is given by:
V = VS = u0 −
Thus, in the presence of streaming motion of ions
two distinct wave modes (fast and slow) is possible.
5 Results and Discussion
We have investigated the occurrence of ionacoustic sotitary waves in a multi-component warm
drifting plasma having non-thermal electrons. For
numerical analysis, we have considered, for example,
(H+, O−) and (He+, O−) plasmas which are expected to
occur in the D region of the ionosphere. Our analysis
reveals that in the first order both compressive and
rarefactive solitons can exist in these plasmas for
selected range of plasma parameters (Fig. 1).There
exists a critical value of the concentration of negative
ions below which compressive and above which
rarefactive first order silitons are obtained. This
critical value of the concentration of negative ions
depends on temperature, non-thermal parameter, drift
velocity and other plasma parameters. From Fig. 1, it
can be seen that the amplitude of first order solitons
increases with increase in negative ion concentration.
The second order solution shows significant
corrections to the amplitude and width of the first
order compressive soliton (Fig. 2). As is evident from
Fig. 2, the second order theory predicts greater
492
INDIAN J PURE & APPL PHYS, VOL 51, JULY 2013
Fig. 1 — Profiles for first order compressive and rarefactive
solitons for different values of negative ion concentration (nj0)
with ı=0.03, p=1, V=1.6, ui0 =0.07, uj0 =0.45, ȕ=0.24 and Q=4.
Curves a, b, c and d refer to nj0 =0.006, 0.009, 0.033 and 0.036,
respectively
Fig. 3 — Profile for the second order W-type soliton with nj0 =0.033,
ı=0.03, p=1, V=1.6, ui0 =0.07, uj0 =0.45, ȕ=0.24 and Q=4
Fig. 2 — Profiles for first( curve a) and second order (curve b)
compressive solitons with nj0 = 0.006, ı=0.03, p=1, V=1.6, ui0
=0.07, uj0 =0.45, ȕ=0.24 and Q=4
Fig. 4 — Variation of the critical non-thermal parameter ȕ with
negative ion concentration (nj0) for different values of negative
ion temperature with p=1, V=1.6, ui0 =0.14, uj0 =0.14 and Q=4.
Curves labeled a, b and c refer to three different values ı =0.03,
0.06 and 0.10, respectively.
amplitude and width for the compressive soliton as
compared with first order soliton. Above the critical
value of the concentration of negative ions where first
order theory predicts a rarefactive soliton the second
order theory predicts a W-type soliton (Fig. 3).
Somewhat similar results for ion-acoustic solitons
were obtained by Tagare and Reddy29 in the presence
of negative ions and by Ghosh and Das30 even in a
two-component bounded plasma with limited device
dimensions.
The nature of the soliton (i.e. compressive,
rarefactive or W-type) does not change with the
smaller values of the non-thermal parameter. But
there exists a critical value of the non-thermal
parameter (β) above which no soliton solution is
possible. This critical value of β is found to depend on
temperature, negative ion density, drift velocity and
other plasma parameters. The critical value of β
decreases with increase in both the concentration and
temperature of negative ions (Fig. 4). Here it is
interesting to discuss the limiting case with ȕ=0 (i.e.
the absence of non-thermal electrons). It is found that
even in the absence of non-thermal electrons there
exists a critical value of the concentration of negative
ions below which compressive and above which
rarefactive first order silitons are possible.
Thus,
ion-acoustic
solitons,
compressive,
rarefactive or W-type can be formed in a multicomponent plasma system only with certain restricted
values of plasma parameters. Our analysis shows that
GHOSH et al.: EFFECT OF NON-THERMAL ELECTRONS ON ION-ACOUSTIC SOLITARY WAVES
493
References
Fig. 5 — Sagdeev potential profile for different values of the nonthermal parameter ȕ with ı =0.03, nj0 =0.19, p=1, V=1.6, ui0
=0.18, uj0 =0.18 and Q=16. Curves labeled a, b and c refer to three
different values ȕ=0.28, 0.30 and 0.32, respectively
the concentration of negative ions, the temperature of
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(Fig. 5). It is clear from Fig. 5 that both the amplitude
and width of the large amplitude solitons decrease
with increase in the non-thermal parameter. Finally,
we would like to point out that the type of plasma
considered in the present work exists in space and
also can be produced in the laboratory. We suppose
that the present study can be useful to understand
auroral and magnetospheric plasmas.
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