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In this paper we have investigated the effect of ion nonthermality on nonlinear dust acoustic wave propagation in a complex plasma in presence of weak secondary electron emission from dust grains. Equilibrium dust charge in this case is negative. Dusty plasma under our consideration consists of inertialess nonthermal ions, Boltzman distributed primary and secondary electrons and negatively charged inertial dust grains. Both adiabatic and nonadiabatic dust charge variations have been taken into account. Our analysis shows that in case of adiabatic dust charge variation, at a fixed non-zero ion nonthermality increasing secondary electron emission decreases amplitude and increases width of the rarefied dust acoustic soliton whereas for a fixed secondary electron yield increasing ion nonthermality increases amplitude and decreases width of such rarefied dust acoustic soliton. Thus shape of the soliton may be retained if strength of both the secondary electron yield and the ion nonthermality are increased. Nonadiabatic dust charge variation shows that, at fixed non-zero ion nonthermality, increasing secondary electron emission suppresses oscillation of oscillatory dust acoustic shock at weak nonadiabaticity and pronounces monotonicity of monotonic dust acoustic shock at strong nonadiabaticity. On the other hand at a fixed value of the secondary electron yield, increasing ion nonthermality enhances oscillation of oscillatory dust acoustic shock at weak nonadiabaticity and reduces monotonicity of monotonic dust acoustic shock at strong nonadiabaticity. Thus nature of dust acoustic shock may also remain unchanged if both secondary electron yield and ion nonthermality are increased.

Dusty plasma has become an important field of plasma research since the last decade of the twentieth century. Such plasma consists of electrons, ions and massive charged dust grains which are commonly observed in space and astrophysical plasmas. These dust grains are charged by different mechanisms. They may be charged by plasma current, secondary electron emission effect, photo electric effect or by some other charging processes. Whether the dust grains are negatively or positively charged depends upon the grain charging mechanism. If the dust grains embedded in a plasma are charged by the flow of plasma current, equilibrium dust charge is negative. Positive equilibrium dust charge may exist if some electrons emit from dust grains. Secondary electron emission is an important dust charging process in dusty plasma where both negative and positive equilibrium dust charge exist [

When primary electrons impacting dust surfaces are energetic enough, they interact with the bulk material and loose energy due to many collisions. This consequently excites material electrons of dust grains. Some of those electrons ultimately leave the material surface and are called secondary electrons. Such secondary electrons have typical energies of few eV with Maxwellian like energy distribution. The secondary electron yield is defined as the average number of secondaries emitted from dust grains per unit incident primary electron. The maximum secondary electron yield and the corresponding maximum energy depend on the dust material. For weak secondary electron emission the secondary electron yield possesses low value and hence the equilibrium dust charge is negative whereas for strong secondary electron emission secondary yield value is high and the equilibrium dust charge is positive [

In most space and astrophysical environment electrons and ions are frequently nonthermal whose existences have been detected by several satellite observations. Nonthermal ions were observed by the Vella satellite from the earth’s bow shock [

Nonlinear evolution of dust acoustic and dust ion acoustic waves in presence of adiabatic and non adiabatic dust charge variation were studied earlier by several authors considering grain charging by the flow of plasma current [

Our present analysis shows that in case of adiabatic dust charge variation, at a fixed non zero ion nonthermality, increasing secondary electron emission decreases amplitude and increases width of the rarefied dust acoustic soliton whereas for a fixed secondary electron yield, increasing ion nonthermality increases amplitude and decreases width of such rarefied dust acoustic soliton. This is opposite to the soliton behaviour when equilibrium dust charge is positive [

For studying the effect of ion nonthermality on nonlinear one dimensional dust acoustic wave propagation in a complex plasma in presence of weak secondary electron emission we have considered a plasma consisting of inertialess nonthermal ions, Boltzman distributed primary and secondary electrons and negatively charged inertial dust grains satisfying the quasineutrality condition

n i o = n e o + n s o + z d 0 n d 0 . (1)

where n i o , n e o , n s o and n d o are equilibrium number densities of ions, primary electrons, secondary electrons and dust grains respectively and z d o is the number of charges on dust grains in equilibrium. The non dimensional number densities N i , N e and N s of nonthermal ions, Boltzmann distributed primary and secondary electrons are

N i = [ 1 + 4 a 1 + 3 a ( Φ σ i + Φ 2 σ i 2 ) ] exp ( − Φ σ i ) (2)

N e = exp ( Φ ) . (3)

N s = exp ( Φ σ s ) (4)

where a is the ion nonthermal parameter, σ i = T i T e and σ s = T s T e are temperature ratios, Φ ( = e ϕ T e ) is the non dimensional electrostatic potential and T i , T e and T s are ion, primary electron and secondary electron temperatures respectively.

Number density N d , velocity V d and charge Q d of dust grains obey the following non dimensional equations,

∂ N d ∂ T + ∂ ∂ X ( N d V d ) = 0 (5)

∂ V d ∂ T + V d ∂ V d ∂ X = − Q d α d ∂ Φ ∂ X (6)

( ω p d ν d ) ( ∂ Q d ∂ T + V d ∂ Q d ∂ X ) = 1 ν d ( I ¯ i + I ¯ e + I ¯ e s z d o e ) (7)

All the above equations are closed if the electrostatic plasma potential F satisfies the Poisson equation,

∂ 2 Φ ∂ X 2 = − 1 1 + δ i σ i + δ s σ s [ δ i { 1 + 4 a 1 + 3 a ( Φ σ i + Φ 2 σ i 2 ) } exp ( − Φ σ i ) − exp ( Φ ) + δ s exp ( Φ σ s ) + ( δ i − δ s − 1 ) Q d N d ] (8)

where α d = δ i − δ s − 1 δ i σ i + δ s σ s + 1 , δ i = n i 0 n e 0 and δ s = n s 0 n e 0 are number density ratios, ω p d = ( 4 π n d 0 z d 0 2 e 2 m d ) 1 / 2 and ν d are the dust plasma frequency and dust charging

frequency, m d is the dust mass and X, T are non dimensional space and time variables.

Here,

I ¯ i = π r 0 2 e 8 T i π m i n i 0 1 + 3 a [ { ( 1 + 24 a 5 ) + 16 a Φ 3 σ i + 4 a Φ 2 σ i 2 } − z Q d σ i ( 1 + 8 a 5 + 8 a Φ 3 σ i + 4 a Φ 2 σ i 2 ) ] exp ( − Φ σ i ) (9)

is the nonthermal ion current when equilibrium dust charge is negative [

F i ( v i ) = F i ( v x , v y , v z ) = n i 0 1 + 3 a ( 1 2 π v t i 2 ) 3 / 2 [ 1 + 4 a ( 1 2 v x 2 v t i 2 + Φ σ i ) 2 ] exp ( − v x 2 + v y 2 + v z 2 2 v t i 2 − Φ σ i ) .(10)

where v t i is the ion thermal velocity and v x , v y , v z are x, y, and z components of ion velocity.

Other current expressions are,

I ¯ e = − π r 0 2 e 8 T e π m e n e 0 exp ( Φ ) ( z Q d ) . (11)

I ¯ e s = 3.7 δ M π r 0 2 e 8 T e π m e n e 0 exp ( Φ + z Q d ) F 5 ( E M 4 T e ) (12)

where z = z d 0 e 2 / r 0 T e , r 0 is the grain radius, δ M is the maximum secondary electron yield, m e is the electron mass and the function F 5 ( x ) is given by [

F 5 ( x ) = x 2 ∫ 0 ∞ u 5 exp [ − ( x u 2 + u ) ] d u and x = E M 4 T e (12a)

Value of δ M is low for weak secondary electron emission and high for strong secondary electron emission. Thus for weak secondary electron emission equilibrium dust charge is negative and for strong secondary electron emission equilibrium dust charge is positive.

The primary electron, secondary electron, ion and dust number densities n e , n s , n i , n d , dust fluid velocity u d , electrostatic potential energy ef, dust charge q d and the independent space, time variables x, t are nondimensionalized here in the following way,

N e = n e / n e 0 ; N s = n s / n s 0 ; N i = n i / n i 0 ; N d = n d / n d 0 ; V d = u d / c d ; Φ = e ϕ T e ; Q d = q d / e z d 0 ; q d o = − z d o e ; X = x / λ d ; T = ω p d t (13)

where λ D = ( T e f f 4 π z d 0 n d 0 e 2 ) 1 / 2 is the dusty plasma Debye length and c d = z d 0 T e f f m d is the dust acoustic speed. The effective temperature T e f f is defined by

1 T e f f = 1 z d o n d o ( n e o T e + n s o T s + n i o T i ) (14)

The grain charging frequency in this case has been calculated in the form,

ν d = − ∂ ( I ¯ i + I ¯ e + I ¯ e s ) ∂ Q d | Q d = − z d e = r 0 2 π ω p i 2 V t h i ( 5 + 8 a 5 ( 1 + 3 a ) ) [ 1 + z + σ i ( 5 + 24 a 5 + 8 a ) ] (15)

The equilibrium current balance equation I ¯ i + I ¯ e + I ¯ e s = 0 gives

δ i = m e i σ i exp ( − z ) α 1 s 5 ( 1 + 3 a ) [ ( 5 + 24 a ) + z σ i ( 5 + 8 a ) ] , α 1 s = 1 − 3.7 δ M F 5 ( E M 4 T e ) (16)

This δ i must be greater than 1 to satisfy the quasi neutrality condition (1).

For the study of small amplitude structures in dusty plasma in presence of weak secondary electron emission, we employ the reductive perturbation technique, using the stretched coordinates ξ = ε 1 / 2 ( X − λ T ) and τ = ε 3 / 2 T where e is a small parameter and l is the wave velocity normalized by c d . The variables N d , V d , Φ and Q d are then expanded as,

N d = 1 + ε N d 1 + ε 2 N d 2 + ⋯ , V d = ε V d 1 + ε 2 V d 2 + ⋯ Φ = ε Φ 1 + ε 2 Φ 2 + ⋯ , Q d = − 1 + ε Q d 1 + ε 2 Q d 2 + ε 3 Q d 3 + ⋯ (17)

Substituting these expansions into Equations (2)-(8) with (9), (11), (12) and collecting the terms of different powers of e we obtain,

λ N d 1 = V d 1 , V d 1 = − Φ 1 λ α d , N d 1 = − Φ 1 λ 2 α d , Φ 1 + α d ( N d 1 − Q d 1 ) = 0 (18)

∂ N d 1 ∂ τ − λ ∂ N d 2 ∂ ξ + ∂ ∂ ξ ( N d 1 V d 1 ) + ∂ V d 2 ∂ ξ = 0 (19)

∂ V d 1 ∂ τ − λ ∂ V d 2 ∂ ξ + V d 1 ∂ V d 1 ∂ ξ = 1 α d ( ∂ Φ 2 ∂ ξ − Q d 1 ∂ Φ 1 ∂ ξ ) (20)

∂ 2 Φ 1 ∂ ξ 2 = P Φ 2 + α d N d 2 − α d Q d 2 + R Φ 1 2 (21)

where P = δ i σ i ( 1 − a 1 + 3 a ) + δ s σ s + 1 δ i σ i + δ s σ s + 1 and R = [ 1 λ 2 ( 1 α a − 1 α d λ 2 ) − 1 2 ( δ i σ i 2 − δ s σ s 2 − 1 ) ( δ i σ i + δ s σ s + 1 ) ] .

In case of adiabatic dust charge variation dust grains are charged in fast time scale. Hence dust charging frequency ν d is very high compared to dust plasma

frequency. With this approximation ω p d ν d ≈ 0 which reduces (7) to,

I ¯ i + I ¯ e + I ¯ e s = 0 (22)

Equating from its both sides the terms containing e and ε 2 , we get Q d 1 and Q d 2 in the following form,

Q d 1 = − β d Φ 1 , Q d 2 = − β d Φ 2 + γ d Φ 1 2 (23)

where

β d = β b z β a ; β a = m e i σ i α 1 s + 5 + 8 a 5 ( 1 + 3 a ) δ i σ i ; β b = m e i σ i α 1 s e − z + δ i σ i ( 1 + 3 a ) { 1 − 8 a 15 + z σ i ( 1 − 16 a 15 ) } and γ d = γ c z β a ; γ c = γ c 1 + γ c 2 + γ c 3 ; γ c 1 = [ δ i ( 1 + 3 a ) σ i 2 { 1 2 − 16 a 15 + z σ i ( 1 2 + 32 a 15 ) } − m e i σ i α 1 s e − z ] ; γ c 2 = [ m e i σ i α 1 s − δ i ( 1 + 3 a ) σ i 2 ( 1 − 16 a 15 ) ] ( z β d ) ; γ c 3 = [ − 0.5 m e i σ i α 1 s ] ( z β d ) 2 λ = 1 P + α d β d , m e i = m i m e . (24)

Eliminating all the second-order terms from Equations (18)-(21) and (23) we get the KdV equation

∂ Φ 1 ∂ τ + A Φ 1 ∂ Φ 1 ∂ ξ + B ∂ 3 Φ 1 ∂ ξ 3 = 0 (25)

where, A = B [ ( δ i σ i 2 − δ s σ s 2 − 1 ) ( δ i σ i + δ s σ s + 1 ) + 2 α d γ d − 3 P λ 2 α d ] , B = λ 3 2 = 1 2 ( P + α d β d ) − 3 / 2 (26)

On transforming to the wave frame η = ξ − M τ , the travelling wave solution

of equation (25) can be written as

number.

In case of nonadiabatic dust charge variation dust grains are charged in comparatively slow time scale so that dust charging frequency

compared to dust plasma frequency. With this approximation

With this assumption, Equation (7) along with expansion (17) give the following first and second order dust charge fluctuation

Eliminating all the second-order terms of Equation (18)-(21) and (27) we get the standard KdV-Burger equation,

where

arising due to nonadiabaticity of dust charge variation. Here all A, B, and m depend on both ion nonthermality parameter a and maximum secondary electron yield

In this section we have drawn different graphs to show the effect of ion nonthermality on nonlinear dust acoustic wave propagation for both adiabatic and non adiabatic dust charge variation when equilibrium dust charge generated by weak secondary electron emission is negative. For numerical estimation, we have considered [_{2}O_{3} dust for which

From

In case of nonadiabatic dust charge variation

consequence increasing secondary electron emission suppresses oscillation of dust acoustic shock when it is oscillatory at weak nonadiabaticity and pronounces monotonicity when it is monotonic at strong nonadiabaticity. This is clear from

pronounces oscillation of the dust acoustic shock at weak (

In this paper we have studied the effect of ion nonthermality on nonlinear dust

acoustic wave propagation in a complex plasma where dust grains are charged by secondary electron emission process. Strength of secondary electron emission has been assumed weak. As a consequence dust grains are negatively charged. Both adiabatic and nonadiabatic dust charge variations have been considered. Analysis shows that adiabatic dust charge variation generates dust acoustic soliton whose amplitude and width both depend on the nonthermal parameter a and the secondary electron yield

dust acoustic soliton whereas for a fixed secondary electron yield, increasing ion nonthermality increases its amplitude and decreases its width.

For nonadiabatic dust charge variation at a fixed nonzero ion nonthermality, increasing secondary electron emission suppresses oscillation of the oscillatory dust acoustic shock at weak nonadiabaticity and pronounces monotonicity of monotonic dust acoustic shock at strong nonadiabaticity. On the other hand at a fixed secondary electron yield

remain unchanged if both ion nonthermality and secondary electron yield are increased.

The results of the present investigation with negative equilibrium dust charge and the investigation of reference [

Bhakta, S. and Sarkar, S. (2018) Ion Nonthermality Induced Nonlinear Dust Acoustic Wave Propagation in a Complex Plasma in Presence of Weak Secondary Electron Emission from Dust Grains. Journal of Modern Physics, 9, 961-975. https://doi.org/10.4236/jmp.2018.95059