Characteristics of nonlinear dust acoustic waves in a lorentzian dusty plasma with effect of adiabatic and nonadiabatic grain charge fluctuation

Characteristics of nonlinear dust acoustic waves in a Lorentzian dusty plasma with effect of adiabatic and nonadiabatic grain charge fluctuation Characteristics of nonlinear dust acoustic waves in a L[.]

Characteristics of nonlinear dust acoustic waves in a Lorentzian dusty plasma with effect of adiabatic and nonadiabatic grain charge fluctuation

Raicharan Denra, Samit Paul, and Susmita Sarkar

Citation: AIP Advances 6, 125045 (2016); doi: 10.1063/1.4972520

View online: http://dx.doi.org/10.1063/1.4972520

View Table of Contents: http://aip.scitation.org/toc/adv/6/12

Published by the American Institute of Physics

Characteristics of nonlinear dust acoustic waves

in a Lorentzian dusty plasma with effect of adiabatic

and nonadiabatic grain charge fluctuation

Raicharan Denra,aSamit Paul,band Susmita Sarkarc

Department of Applied Mathematics, University of Calcutta, 92, Acharya Prafulla Chandra

Road, Kolkata 700009, India

(Received 11 September 2016; accepted 6 December 2016; published online 21 December 2016)

In this paper, characteristics of small amplitude nonlinear dust acoustic wave have been investigated in a unmagnetized, collisionless, Lorentzian dusty plasma where electrons and ions are inertialess and modeled by generalized Lorentzian Kappa distribution Dust grains are inertial and equilibrium dust charge is negative Both adiabatic and nonadiabatic fluctuation of charges on dust grains have been taken under consideration For adiabatic dust charge variation reductive perturbation anal-ysis gives rise to a KdV equation that governs the nonlinear propagation of dust acoustic waves having soliton solutions For nonadiabatic dust charge variation non-linear propagation of dust acoustic wave obeys KdV-Burger equation and gives rise

to dust acoustic shock waves Numerical estimation for adiabatic grain charge vari-ation shows the existence of rarefied soliton whose amplitude and width varies with grain charges Amplitude and width of the soliton have been plotted for different electron Kappa indices keeping ion velocity distribution Maxwellian For non adi-abatic dust charge variation, ratio of the coefficients of Burger term and dispersion term have been plotted against charge fluctuation for different kappa indices All these results approach to the results of Maxwellian plasma if both electron and ion

kappa tends to infinity © 2016 Author(s) All article content, except where oth-erwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/) [http://dx.doi.org/10.1063/1.4972520]

I INTRODUCTION

Dust grains exist in a wide range of space and astrophysical plasmas like cometary tails, interstellar clouds, earth’s mesosphere, ionosphere, Saturn’s ring, the gossamer ring of Jupiter, and also in laboratory experiments.1,2 Interaction of these dust grains with plasma environment alter the collective plasma behavior and gives birth of new kind of waves In unmagnetized dusty plasma dust ion acoustic and dust acoustic waves are two such low frequency waves whose exper-imental and theoretical studies have become important focus of plasma research for last three decades

In laboratory experiments low frequency fluctuations with a typical frequency of 12 Hz and a wavelength of 0.5 cm were observed by Chu et al3which later interpreted by D’Angelo4as dust acous-tic wave Propagation characterisacous-tics and stability properties of dust acousacous-tic and dust ion acousacous-tic waves were experimentally studied by Merlino et al5in University of Iowa Liang et al6reported exper-imental observation of dust ion acoustic wave propagation down the steep density gradient in an inho-mogeneous diffusive dusty plasma Nonlinear behavior of dust acoustic and dust ion acoustic waves were also investigated by experimental plasma physicists in last few years For example, excitation

a

Electronic mail: raicharanroykolkata@gmail.com

b Electronic mail: samitpaul4@gmail.com

c Electronic mail: susmita62@yahoo.co.in

125045-2 Denra, Paul, and Sarkar AIP Advances 6, 125045 (2016)

and propagation of low frequency finite amplitude solitary waves were investigated by Bandyopad-hyay et al7in Argon plasma impregnated with Kaolin dust particles Nonlinear propagation of dust acoustic shock was experimentally studied by Merlino et al8in a direct current glow discharged dusty plasma But none of these experiments plasma involved suprathermal charge particles Theoretical investigation on linear dust acoustic and dust ion acoustic wave propagation were extensively stud-ied by several authors9 12 in different physical situation even in presence of self gravitating dust grains.13 – 17Extensive theoretical studies have also been reported on several cases of nonlinear prop-agation of dust acoustic and dust ion acoustic waves considering fixed as well as fluctuating gain charges, in case of Maxwellian plasmas.18–26

Space and astrophysical plasmas consist of charged dust particles whose existence was proved by space craft observation.27–31Such plasmas consist of suprathermal charge particles whose velocities follow the generalized kappa distribution32and is known as Lorentzian dusty plasma Characteristics

of dust acoustic and dust ion acoustic wave propagation in Lorentzian plasmas differ from their characteristics in Maxwellian dusty plasmas Few such works have so far done on linear and non linear dust acoustic and dust ion acoustic wave propagation in Lorentzian dusty plasmas considering fixed charges on dust grains.33

In this paper we are interested to study nonlinear propagation of dust acoustic waves in a Lorentzian dusty plasma with effect of grain charge fluctuation Such grain charge fluctuation may

be of adiabatic or nonadiabatic type For adiabatic dust charge variation dust charging frequency is very large compared to dust plasma frequency whereas for nonadiabatic dust charge variation dust charging frequency is comparatively small Thus in space or astrophysical plasmas, the model of adi-abatic dust charge variation is appropriate if dust grains are charged in a very fast time scale(charging frequency is high) On the other hand the model of nonadiabatic dust charge variation is appropriate

if dust grains are charged on slow time scale (charging frequency is low)

The dust grains under our consideration are assumed to be cold and charged by plasma cur-rent As a consequence equilibrium dust charge is negative Since phase velocity of dust acoustic wave is less than electron and ion thermal velocities, both the electrons and ions are consid-ered inertialess, only the dust inertia is taken into consideration Here electron and ion velocities are assumed to follow generalized Kappa distribution as plasma under our consideration is a Lorentzian dusty plasma Small amplitude structures are investigated using the reductive perturbation technique

For adiabatic dust charge variation nonlinear propagation of dust acoustic wave is governed

by KdV equation which has soliton solution To study the nature of this dust acoustic soliton, for simplicity of numerical calculation we have made ion-Kappa index tending to infinity in the analytical results Since ions are heavier than electrons, probability of existence of suprathermal electrons is much higher than of suprathermal ions Thus our assumption is physically consistent Our numerical study shows that in Lorentzian dusty plasma amplitude of dust acoustic soliton is negative and lower

in magnitude than in Maxwellian dusty plasmas On the other hand width of the dust acoustic soliton

is higher in Lorentzian than in Maxwellian dusty plasmas It is also observed that amplitude of this rarefied dust acoustic soliton decreases and width increases with decreasing kappa index, i.e with increasing number of suprathermal electrons

For nonadiabatic dust charge variation nonlinear propagation of Dust Acoustic Wave is governed

by KdV-Burger equation which possesses shock solution Numerical study in this case shows that

in Lorentzian dusty plasma dust acoustic shock wave is dissipation dominated upto certain value of normalized grain charge number and dispersion dominated thereafter As value of the electron kappa index decreases this changeover is prominent

Propagation of such dust acoustic shock wave causes dust density condensation and enhances the gravitational interaction This is an important phenomenon in astrophysical plasmas as it may be

a variable process of star formation.34 , 35

In sectionIIof this paper we have formulated the problem after a brief description of Kappa velocity distribution SectionIIIdeals with the nondimensionalization of basic equations and reductive perturbation analysis of the problem In the same section both adiabatic and non adiabatic dust charge variation have been considered separately Numerical estimation has been reported in section IV Results have been concluded inV

II MATHEMATICAL FORMULATION OF THE PROBLEM

Presence of suprathermal charge particles were confirmed by several spacecraft measure-ments.36 – 41Vasyliunas42suggested that charge particles in natural space environment may be well described by the Kappa (κ) or generalized Lorentzian velocity distribution functions which fits both the thermal as well as the suprathermal parts of the observed energy spectra.42 The conven-tional isotropic, three dimensional form of the Kappa velocity distribution function for electrons in equilibrium may be written as,43

f e0vx, vy, vz = n e

πθ2κe3/2

Γ(κe+ 1)

Γ(κe−1/2)







1+v

2+ v2+ v2

z

κeθ2







−(κe+1)

(1)

where θe=f(2κe−3)Te

κe m e

g1/2

is the thermal velocity, m e , n e , T eare respectively the mass, number density, temperature and ~v= (vx, vy, vz ) is the velocity of electrons, Γ(x) is the Gamma function The value

of the index κedetermines the slope of the energy spectrum of the suprathermal electrons forming the tail of the velocity distribution functions In the limit κe→ ∞, this Kappa distribution reduces to Maxwellian distribution Kappa distributions with 2 < κe< 6 have been found to fit the observations and satellite data in the solar wind, the terrestrial magnetosphere; the magnetosheath, the magneto-sphere of other planets like Mercury, Jupiter, Saturn and Uranus as observed by Ulysses, Cassini and the Hubble Space Telescope.44

In this paper we consider a model of Lorentzian dusty plasma consisting of electrons, ions and negatively charged dust grains Charge neutrality at equilibrium reads as,

where n i0 , n e0 , n d0 are respectively the ion, electron and dust number densities in equilibrium, and z d0

is the unperturbed number of charges residing on the dust grain measured in the unit of electron charge For one-dimensional low-frequency dust acoustic wave motion cold dust grains satisfy the fluid equations

∂n d

∂t +

∂

∂u d

∂t + u d

∂u d

∂x =−

q d

m d

∂φ

and the Poisson equation

∂2φ

∂x2 = 4πe (z d n d + n e−n i) (5)

Here n d , u d , and m d refer to the number density, fluid velocity and mass of the dust grains, q d = −ez d

is the variable dust charge and z d is the variable charge number on dust grains Number densities of non inertial Kappa distributed electrons and ions are,

n e = n e0 1 − 2eφ

m e k eθ2

!−(k e− 1)

and n i = n i0*

,

1+ 2eϕ

m i k iθ2

i

+

-−(k i−1)

(6)

where θe=

r

(k e− 3)

k e

2T e

m e

 , θi=

r (k i− 3)

k i

2T i

m i



with temperatures T e , T iare their thermal velocities,

m e , m iare masses and κe, κiare electron and ion kappa indices respectively The variable dust charge

q dsatisfies the grain charging equation,

dq d

where I e = −πa2en e 8T e

πm e

!1

κe−3 2

!1

Γ(κe−1)

Γ κe−1

2









1 − eΦ d

 κe−3

2



T e







−(κe−1)

(8)

125045-4 Denra, Paul, and Sarkar AIP Advances 6, 125045 (2016)

I i = πa2en i 8T i

πm i

!1

κi−3 2

!1

Γ(κi−1)

Γ κi−1

2











1 − eΦ d(κi−1)

 κi−3

2



T i









(9)

are electron and ion current flowing to the dust surface and Φd denotes the dust surface potential relative to plasma potential φ

III NONDIMENSIONALIZATION AND REDUCTIVE PERTURBATION

In this section all the physical quantities are normalized as follows Electron, ion and dust

number densities n e , n i , n d are normalized by corresponding unperturbed densities n eo , n io and n do whereas the dust charge number z d is normalized by its equilibrium value z d0 The space coordinate

x, time t, electrostatic potential energy eϕ and dust velocity u d are normalized by the Debye length

λDd= T eff

4πz d0 n d0 e2

1 , the inverse of dust plasma frequency ω−1

pd= m d 4πn d0 z2

d0 e2

1 , the electron thermal

energy T e (in eV) and the dust-acoustic speed c d=z d0 T eff

m d

1 respectively Here

T eff = T eαd , with α d= (δ − 1)

 (κe− 1) (κe−3) +

δ(κi− 1)

σ(κi−3)

The set of dimensionless basic equations are then obtained in the form,

∂N d

∂T +

∂

∂V d

∂T + V d

∂V d

∂X =−

Q

αd

∂Φ

∂2Φ

∂X2 = − αd

(δ − 1){(δ − 1) QN d−N e + δN i} (13)

ωpd

υd

∂Q

∂T + V

∂Q

∂X

!

=υ 1

with the dimensionless electron and ion number densities and current expressions,

N e= n e

n e0 =







1 − Φ

κe−3

2









−κe− 1 2



(15)

N i= n i

n i0=







σ(κi−3

2)









−



κi− 1 2



(16)

I e = −πr2

0e

s

8T e

πm e

 κe−3

2

1

Γ(κe−1)

Γ κe−1

2

 n e0 N e









1 − ZQ

 κi−3

2











−(κe−1)

(17)

I i = πr2

0e

s

8T i

πm i

 κi−3

2

1

Γ(κi−1)

Γ κi−1

2

 n i0 N i









1 −ZQ σ

(κi−1)

 κi−3

2











(18) where

Z=e2z d0

r0T e , Q= q d

ez d0, Φd=q d

r0, X=λx

d , T= t

ω−1

pd

, V=u d

c d, Φ=eφ

T e , N d= n d

n d0, σ=T i

T e,

r0is the grain radius, νdis the grain charging frequency and κe, κiare respectively the electron and ion kappa indices Here Z is the normalized grain charge number, effect of whose variation on nonlinear wave propagation in Lorentzian dusty plasma is of prime interest of this paper

A Adiabatic dust charge variation

For the adiabatic dust charge variation, dust charging time is very small and hence dust charging frequency is very large compared to dust plasma frequency, which implies ωυpd d ≈0 Then normalized grain charging equation (14) reduces to,

which after substitution of the expressions of electron and ion currents from (17) and (18) becomes,

πr2

0e

s

8T i

πm i

 κi−3

2

1

Γ(κi−1)

Γ κi−1

2

 n i0





σ(κi−3

2)









−



κi− 1 2











1 − ZQ σ

(κi−1)

 κi−3

2











−πr2

0e

s

8T e

πm e

 κe−3

2

1

Γ(κe−1)

Γ κe−1

2

 n e0





1 − Φ

κe−3

2









−



κe− 1 2











1 − ZQ

 κe−3

2











−(κe−1)

= 0 (20a) This can be further written in the form,

δ







σ(κi−3

2)









−



κi− 1 2











1 −ZQσ

(κi−1)

 κi−32









−A





1 − Φ

κe−3

2









−κe− 1 2











1 − ZQ

 κe−3

2











−(κe−1)

where δ=n i0

n e0 and A= κe−

3 2

1

Γ(κe−1) Γ κi−1

2



 κi−32

1

Γ(κi−1) Γ κe−12

r m i

In equilibrium (Φ= 0, Q = −1) the ion-electron density ratio can therefore be expressed as,

δ = n i0

n e0 = A



1+ Z

(κe−3)

−(κe−1)



1+Z

σ((κκi i−−1)3)

Since dust grains are negatively charged the quasi neutrality condition (2) implies δ > 1.Thus we

have to choose the range of Z keeping δ greater than one Now for the study of small-amplitude

dust acoustic waves in presence of self-consistent adiabatic dust-charge variation, we derive the KdV equation from equations (11–20b) by employing the reductive perturbation technique and using the stretched coordinates ξ= ε1

(X − λT ), and τ= ε32T , where ε is a small parameter and λ is unknown

normalized phase velocity of the linear dust acoustic wave

The variables N d , V d , Φ, and Q are then expanded as,

N d = 1 + εN d1+ ε2N d2+ ε3N d3+ · · · ·

V d = εV d1+ ε2V d2+ ε3V d3+ · · · ·

Φ= εΦ1+ ε2Φ2+ ε3Φ3+ · · · · (22)

Q = −1 + εQ1+ ε2Q2+ ε3Q3+ · · · · Substituting these expansions into equations (11)–(16), (20b) and comparing coefficient of ε from both sides we have the following relations,

λN d1 = V d1 , V d1= − Φ1

α λ, N d1= − Φ1

α λ2, Q1=α1 1 − 1

λ2

!

Φ1, Q1= −βdΦ1 (23)

125045-6 Denra, Paul, and Sarkar AIP Advances 6, 125045 (2016)

where βd=

 (κi− 1) (κi− 3

)



1+Z

σ((κκi i−−1)3

)



1

σ + AB(κe− 1) (κe− 3

)





A(κe−1)

(κe−3)



1 − κe Z

(κe−3)



Z+σ

Z

(κi−1)

(κi−3)

 , B= 1 − (κe−1)

 κe−3

2

 Z +

(κe−1) κe

2 κe−3

2

2Z2

(24)

To the next higher order in, ε i.e ε2we have the following set of equations,

∂N d1

∂τ −λ

∂N d2

∂ξ +

∂V d2

∂ξ +

∂ (N d1 V d1)

∂V d1

∂τ −λ

∂V d2

∂ξ +V d1

∂V d1

∂ξ =

1

αd

∂Φ2

∂ξ +Q1

∂Φ1

∂2Φ1

∂ξ2 = αd N d2−αd Q2+ Φ2+ EΦ2

where

E=λ21α

d

1 − 1

λ2

!

−1 2



















κi− 1 2

 

κi+12



κi− 3 2

 2 δ



κe− 1 2

 

κe+12



κe− 3 2

 2

(κe−1)



κe− 3 2

σ(κi−1)



κi− 3 2



















r d= −r d1

r d2

r d1 = −AB



−1

2  κe+1

2



2 κe−3

2

2 −A(κe−1) κe

2 κe−3

2

2Z2β2

d + A κe−

1 2

 (κe−1)

 κe−3

2

2









1 − κe

 κe−3

2

 Z









Z β d

+





1+ (κi−1)

 κi−3

2



Z

σ









 κi−1

2  κi+1

2



2 κi−3

2

2

1

σ2 −(κi−1) κi−1

2



 κi−3

2

2

Z β d

σ2

r d2= (κi−1)

 κi−3

2



Z

σ +A

(κe−1)

 κe−3

2











1 − κe

 κe−3

2

 Z







 Eliminating all second order terms from equations (25)–(28) we get the KdV equation,

∂Φ1

∂τ +aΦ1

∂Φ1

∂ξ +b

∂3Φ1

where b= 1

2 (1+ αdβd)3

a= αd b







2r d− 3

α2λ4 + βd

αdλ2 −2E

αd







., λ= 1 (1+ αdβd)1

(29b) The solution of equation (29a) can be written as,

which represents soliton of amplitude ϕ1=3M

a and width W= 2

r

b

Here M is the Mach number which is the ratio of the wave velocity and the velocity of sound Clearly

amplitude and width of the soliton depend on electron and ion kappa indices and the grain charge

number Z through the coefficient of nonlinearity a and the coefficient of dispersion b respectively.

B Non adiabatic dust charge variation

The nonadiabaticity of the dust charge variation provides an alternate physical mechanism for generating dissipative effect in dusty plasma In case of nonadiabatic dust charge variation dust charging time is large and hence dust charging frequency is small compared to the case of adiabatic dust charge variation This slow grain charging process captures charge fluctuation induced dissipation which cannot be captured in the fast adiabatic grain charging process In the slow charging process ratio

of the dust plasma frequency ωpdto the grain charging frequency νdis small but finite, i.e., ωpd

υd , 0 With this assumption formation of dust acoustic and dust ion acoustic shock waves were already investigated in Maxwellian dusty plasmas.45,46 In collisionless plasma, however, particle velocity distributions can often depart from being Maxwellian For example, in naturally occurring plasma in the planetary magnetosphere and in solar wind, the particle velocity distribution is observed to have non-Maxwellian (power-law), high-energy tail.43Such distribution may be accurately modelled by a generalized Lorentzian (kappa)distribution.42Recently A Shah et al47have studied the shock wave structure with kappa-distributed electrons and positrons They derived Korteweg-de Vries (KdV)– Burger equation and solved it analytically The effect of plasma parameters on the shock strength and steepness were also investigated Significant effect of nonplanar geometry in the formation of shock waves in adiabatic dusty plasma has also been recently reported.48But in all these above mentioned studies in Lorentzian dusty plasma, effect of charge fluctuation were not taken into account In this paper we are interested to investigate the formation of dust acoustic shock acoustic waves in Lorentzian dusty plasma considering nonadibatic dust charge fluctuation

To study this effect we assume ωpd

νd = ν√ε where ε is small and ν is of the order of unity Then the reductive perturbation technique gives from equation (14) the first and second order dust charge perturbation in the following form

Q1= −βdΦ1and Q2= −βdΦ2+ r dΦ21−µ1∂Φ1

where

C



A(κe−1)

(κe− 3

)



1 − κe Z

(κe− 3

)



z+ z

σ((κκi i−−1)3

)



and

3

2 σ (κi−1) Z

"

1+(κe−1) (κi−1)

( σ(κi−3)+Z(κ i−1)) (

(κe− 3)+Z)

#

In Lorentzian dusty plasma νdνd has been calculated from the expression, νd= −∂(I e +I i)

∂q d in the form

νd= r0

√

2π

ωpi

V thi

Γ(κi)

 κi−3

2

1

Γ κi−1

2











1+(κe−1) (κi−1)

( σ κi−3

2 + Z (κ i−1)) (  κe−32 + Z)









Eliminating all the 2nd order terms from (25)-(27) and (32) we get the KdV- Burger equation,

∂Φ1

∂τ +aΦ1

∂Φ1

∂ξ +b

∂3Φ1

∂ξ3 = µ∂2Φ1

where µ=µ1 λ 3 αd

2 is the coefficient of the Burger term

In hydrodynamic fluid flow the dissipative Burger term arises if viscous effect is present in equation of motion In our problem no such viscous effect has considered The Burger term here

is arising exclusively due to the nonadiabaticity of the dust charge variation which was absent in the adiabatic case This viscous like dissipative effect may be generated due to the relative velocity between the electron fluid and dust fluid layers which is coming into the picture in case of nonadiabatic dust charge variation as it is a slow charging process In case of rapid charging in the adiabatic process this effect is not captured

125045-8 Denra, Paul, and Sarkar AIP Advances 6, 125045 (2016)

IV NUMERICAL ESTIMATION

For numerical estimation we have to first determine the range of the normalized grain charge

number Z



=z d0 e2

r0T e

 satisfying the inequality

δ =n i0

n e0 = A



1+ Z

(κe−3)

−(κe−1)



1+ Z

σ((κκi i−−1)3)

 > 1

The temperature ratio σ=T i

T e is less than one as T e >> T i Figure(1)has been plotted to find the range of

Z for κ e=2,4,6 These values of κehave been considered as in most space and astrophysical plasmas43

kappa index ranges from 2 to 6 For simplicity of numerical calculation we have considered Lorentzian electrons and Maxwellian ions making κi→ ∞in the analytical results Figure1shows that range

of z is different for different electron kappa indices.For lower kappa index it is small.This implies

in presence of very high population of suprathermal electrons charge fluctuation is insignificant But for medium to low population grain charge fluctuation plays important role Our main objective of this paper is to study the effects of suprathermal particles i) on amplitude and width of dust acoustic soliton when grain charge fluctuation is adiabatic and ii) on the nature of dust acoustic shock wave when grain charge fluctuation is nonadiabatic Analytically it has been seen that in case of adiabatic dust charge variation nonlinear propagation of dust acoustic wave obeys KdV equation which has soliton solution Figures2and3have been respectively plotted to show the variation of amplitude and

FIG 1 Plot of δ versus Z considering Lorentzian electrons and Maxwellian ions for different values of κ e.

FIG 2 Plot of soliton amplitude ϕ versus Z for different κ when ions are Maxwellian.

FIG 3 Plot of soliton width W versus Z for different κ ewhen ions are Maxwellian.

width of the dust acoustic soliton with grain charge number Z at κe=2,4,6 Numerical data have been considered22σ = 0.5 and M = 1.1 In both figures results approach to Maxwellian values at κ e→ ∞ Figure(2)shows that amplitude of the dust acoustic soliton is negative and its magnitude decreases

as κedecreases On the otherhand Figure(3)shows that width of the dust acoustic soliton increases with decreasing electron kappa index Thus in case of adiabatic dust charge variation in Lorentzian dusty plasma, dust acoustic solitons are shorter and wider and hence move slower in presence of suprathermal electrons The dust charging time has been estimated in this case is of the order of

10−1s

In case of nonadiabatic dust charging, the estimated dust charging time is of the order of 104s ≈

2hr 47 min It is large compared to the adiabatic dust charging time Since nonadiabatic dust charging process is slower than adiabatic dust charging process, larger time is required to set up equilibrium

in nonadiabatic case

For nonadiabatic dust charge variation, propagation of nonlinear dust acoustic wave is governed

by KdV-Burger equation whose Burger coefficient µ is responsible for an alternative dissipation mechanism in the dusty plasma medium We have plotted the Burger coefficient µ in figure (4), the dispersion coefficient b in Figure(5) and the ratio µb in figure (6)respectively taking the ion-electron temperature ratio σ= 0.5 and the nonadiabaticity parameter ν = 0.5 These numerical data have been taken from Reference22 Figure(4)shows that in presence of suprathermal electrons, µ first increases and then decreases with increasing Z values It is unlike Maxwellian electrons where

µ only increases within this specified range of Z The figure(4)also shows that the maximum of µ is

FIG 4 Plot of the Burger coefficient µ versus Z considering Lorentzian electrons and Maxwellian ions for different values

of κ at ν = 0.5.