The effects of vortex like distributed electron in magnetized multi-ion dusty plasmas

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1 Cent. Eur. J. Phys. 12(9) DOI: /s Central European Journal of Physics The effects of vortex like distributed electron in magnetized multi-ion dusty plasmas Research Article Md. Masum Haider 1,2, Tahmina Ferdous 2, Syed S. Duha 2 1 Department of Physics, Mawlana Bhashani Science and Technology University, Santosh, Tangail-192, Bangladesh 2 Department of Physics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh Received 19 January 214; accepted 19 May 214 Abstract: The nonlinear propagation of small but finite amplitude dust-ion-acoustic solitary waves in a magnetized, collisionless dusty plasma is investigated theoretically. It has been assumed that the electrons are trapped following the vortex-like distribution and that the negatively and positively charged ions are mobile with the presence of charge fluctuating stationary dusts, where ions mass provide the inertia and restoring forces are provided by the thermal pressure of hot electrons. A reductive perturbation method was employed to obtain a modified Korteweg-de Vries (mk-dv) equation for the first-order potential and a stationary solution is obtained. The effect of the presence of trapped electrons, negatively and positively charged ions and arbitrary charged dust grains are discussed. PACS (28): 52.25Xz; Lw; Sb; 94.5.Fg Keywords: trapped electron vortex-like distribution multi-ion dusty plasma Versita sp. z o.o. 1. Introduction The dusty plasma is an ionized gas containing electrons, ions and small micron or sub-micron sized extremely massive charged dust grains. The existence of dusty plasma is widespread in laboratory plasma as well as space and astrophysical plasma system with a large degree of variety. Dust particles are found in plasma processing and plasma crystals where low temperature plasmas are used. On the other hand, dusty plasmas are found in planetary rings, in- masum.phy@gmail.com, masum@mbstu.ac.bd terstellar molecular clouds, protostellar disks, interstellar and circumstellar clouds, asteroid zones, planetary atmospheres, interstellar media, cometary tails, nebula, Earth s ionosphere, etc. [1 9]. Now-a-days electrostatic modes in dusty plasmas has become a field of great interest because of their vital role in understanding the dynamics and fragmentation of molecular clouds, star formation, the galactic structure and its evolution, the magnetic reconnection, the spoke formation of Saturn s rings, to name a few [7 1]. Chow et. al. [11] have theoretically shown that due to the size effect on the secondary emission insulating dust grains in space plasmas can have the opposite polarity; large grains are negatively charged where as small one are positively charged; 71

2 The effects of vortex like distributed electron in magnetized multi-ion dusty plasmas are found in different regions in space, like cometary tails [9, 11, 12], Jupiter s magmetosphere [13], etc. In Dust-ionacoustic (DIA) waves the restoring forces are provided by the thermal pressure of electrons and the inertia, which is due to the ion mass [14, 15]. On the other hand, Cooney et al. [16] presented an experimental observation of a two-dimensional soliton in plasmas which contains both positive and negative ions. Ion acoustic shock waves are formed when the ratio of the negative to positive ion number density exceeds about.9 [17]. There is a potential interactions of negative ions to positive ions not only in natural and technological environments but also in the D- rigion of the Earth s ionosphere, the Earth s mesosphere, the solar photosphere, and in microelectronics plasma processing reactors [18]. In 1998 Sayed et. al. [19] considered non-magnetized dusty plasma mode having positive and negative ions with positive and negative dust and Maxwellian distributed electrons (though the most real plasmas are magnetized, and as a wave propagates, it can change its characteristics according to the wave direction). These modes are only valid if a complete depletion of the background electrons and ions is possible and both positive and negative dust fluids are cold. In practice the hot electrons may not follow a Maxwellian distribution [2, 21] due to the formation of phase space holes caused by the trapping of hot electrons in a wave potential. Accordingly, in most space plasmas, the hot electrons are trapped following the vortex-like distribution [22, 23]. The vortex-like distribution is not a new type of distribution used for thermal particles. Several renown scientist have used this vortex-like distribution for not only thermal electrons but also thermal ions and dusts [22 27]. Rahaman and Manun [28] have explained the effect of trapped electrons in DIA SWs with arbitrarily charged dusts. But they did not consider the presence of magnetic field. So, a magnetized dusty plasma model have considered in present work consisting of negatively and positively charged ion fluid, vortex-like distributed hot electrons, and arbitrary charged stationary dusts. The modified Korteweg-de Vries (mk-dv) equation is derived by applying the reductive perturbation method [29]. The manuscript is organized as follows. The basic equations are given in Sec. 2. The modified Korteweg-de Vries (mk-dv) equation is derived by employing the reductive perturbation method in Sec. 3. The solitary-wave solution of the mk-dv equation is obtained in Sec. 4. A brief discussion is finally given in Sec Basic equations In this paper, a one-dimensional, collisionless magnetized dusty plasma is considered. It is assume that i) the ions (negatively and positively charged) are mobile, ii) the electrons follow the vortex-like distribution, and iii) charge fluctuating stationary dust and iv) an external static magnetic field B acting along the z-direction (B = ˆkB ), where ˆk is the unit vector along the z-direction which is very strong that the electrons and dusts are moving along the magnetic field direction very fast, i.e. the response of electrons and dusts look like as that in the unmagnetized plasma. The nonlinear dynamics of the DIA solitary waves (SWs) propagating in such a multi-component dusty plasma is governed by n s t + (n su s ) =, (1) u n t + (u n )u n = ψ ω cn u n ˆk, (2) u p t + (u p )u p = β ψ + βω cn u p ˆk, (3) 2 ψ = µ n e + n n µ p n p jµ d, (4) where n s (n n /n p ) is the ion number density (negative/positive) normalized by its equilibrium value n s, u n (u p ) is the negative (positive) ion fluid speed normalized by C n = (k B T e /m n ) 1/2, with k B is the Boltzman constant, T e is the temperature of electrons and m n is the rest mass of negative ions. ψ is the DIA wave potential normalized by k B T e /e, with e being the magnitude of the charge of an electron. ω cn is the ion cyclotron frequency (eb /m n c) normalized by plasma frequency ω pn = (4πn n e 2 /m n ) 1/2 with c being the speed of light. The time variable (t) is normalized by ω 1 pn, the space variables are normalized by Debye radius λ D = (k B T e /4πn n e 2 ) 1/2. At equilibrium we have n p + jz d n d = n e + n n, where z d is the number of net positive or negative charged particle residing on dust grains and n d is the dust number density and j = 1, or 1, dependent on the net charge of the dust grain, β = m n /m p. It can also written µ = µ p + jµ d 1, (5) where, µ = n e /n n, µ p = n p /n n and µ d = z d n d /n n. 72

3 Md. Masum Haider, Tahmina Ferdous, Syed S. Duha To model the electron distribution in presence of trapped particles, a vortex-like electron distribution of Schamel [22] have employ, which solves the Vlasov equation. Thus one can write f t (v) = f ef (v) + f et (v), (6) where f ef (v) = 1 [ exp 1 ( v 2 2ψ )] for v > 2ψ, 2π 2 f et (v) = 1 [ exp 1 2π 2 σ ( v 2 2ψ )] for v 2ψ, where m e is the mass of electron, σ is a parameter determining the number of trapped electrons, and its magnitude is defined as the ratio of the free hot electron temperature T ef to the hot trapped electron temperature T et, i.e., σ = T ef /T et. It is noted that the electron distribution function f t (v), as prescribed above, is continuous in velocity space and satisfies the regularity requirements for an admissible Berntein-Greene-Kruskal (BGK) solution [22]. It can be seen from equation (4) that σ = 1(σ = ) represents a Maxwellian (flat-topped) distribution, whereas σ < represents a vortex-like excavated trapped electron distribution corresponding to an under population of trapped electrons. Thus, integrating the electron distribution over the velocity space, the electron number density n e for σ < can be expressed as [22] n e = E (ψ) + 2 π σ W D ( σψ ), (7) where E(x) = [1 erf( x)]e x, with error function erf(x) = 2 x π e y2 dy and Dawson s integral W D (x) = e x2 x ey2 dy. For small amplitude limit one can write Eqn. (7) as [ 4(1 σ) n e = 1 + ψ 3 π ψ3/2 + 1 ] 2 ψ2. (8) 3. Derivation of mk-dv equation To derive a dynamical equation for the nonlinear propagation of the electrostatic waves in a dusty plasma, under consideration, reductive perturbation technique [29] have to employed in Eqns (1)-(4) and (8). To do so, the following stretched coordinates [25, 28, 3 32] are introduced ξ = ε 1/4 (l x x + l y y + l z z V t), (9) τ = ε 3/4 t, (1) where ε is a smallness parameter ( < ε < 1) measuring the weakness of the dispersion, l x, l y and l z are direction cosine along x, y and z direction and V is the Mach number (the phase speed normalized by C n ). n n, u n and ψ can be expanded about their equilibrium values in a power series of ε, viz., n s = 1 + εn (1) s + ε 3/2 n (2) s +, (11) u sx = ε 5/4 u (1) sx + ε 3/2 u (2) sx +, u sy = ε 5/4 u (1) sy + ε 3/2 u (2) sy +, (12) u sz = εu (1) sz + ε 3/2 u (2) sz +, ψ = εψ (1) + ε 3/2 ψ (2) +. (13) Substituting Eqns. (9)-(13) into Eqns. (1)-(4) and (8), one can develop equations in various powers of ε. To the lowest order of ε is u (1) sx = l y ω cn ψ (1), (14) u (1) sy = l x ψ (1) ω cn, (15) u (1) nz = l z V ψ (1), u (1) pz = β l z V ψ (1), (16) n (1) n = l2 z ψ (1), n (1) V 2 p = β l2 z ψ (1). (17) V 2 Using Eqns. (14-17) one can obtain the linear dispersion relation for the DIA solitary waves V = l z 1 + βµ p µ. (18) It can be seen from Eqn. (18) that the presence of the charge fluctuating dust significantly modify the linear dispersion relation, i.e., the presence of the charge fluctuating dust significantly reduces the phase speed of the DIA waves, and introduces a new low phase speed. Following the same procedure, one can obtain the next higher order continuity equations as n (1) s τ V n (2) s u (2) sy +l y + l z + l x u (2) sz u (2) sx =, (19) 73

4 The effects of vortex like distributed electron in magnetized multi-ion dusty plasmas the z-component of momentum equations are u (1) nz τ V u (2) nz l ψ (2) z =, (2) u (1) pz τ V u (2) pz + l zβ ψ(2) =. (21) To the next higher order of ε, i.e. equating the coefficients of ε 3/2, one can express Poisson s equation, and x- and y- components of the momentum equations for both negative and positive ions as 2 ψ = µ ψ (2) 4(1 σ) µ 2 3 [ ] ψ (1) 3/2 + n (2) n µ p n (2) p, π (22) u (2) nx = V u (2) px = V l x ω 2 cn l x βω 2 cn, 2 u(2) ny = V, 2 u(2) py = V l y ω 2 cn l y βω 2 cn 2, (23) 2. (24) Now using Eqns. (14) - (24), one can obtain the modified K-dV equation describing the nonlinear propagation of the DIA SWs in the dusty plasma Ψ m Σ Figure 1. (Color online) Variation of the amplitude of solitary wave (dashed curve for j = 1 and solid curve for j = 1) with σ for U =.1, β =.1, µ p = 1.5 and θ = 45 having the value of µ d =.1 (red), µ d =.2 (green), µ d =.3 (blue) and µ d =.4 (black). Ψ m Θ ψ (1) τ + A ψ(1) ψ (1) + C 3 ψ (1) =, (25) 3 where the nonlinear coefficient A and the dissipation coefficient C are given by Μ p (1 + βµ p ) A = l z (1 σ), (26) πµ [ 1 + βµ p C = l z ] l2 z (1 + βµ 4µ ωcn 2 p ). (27) 4. Solution of mk-dv equation The stationary solitary wave solution of Eqn. (25) can be found by introducing ζ = ξ U τ [where U is the wave speed (in the reference frame) normalized by C n, and ζ is normalized by λ D ], and by imposing the appropriate boundary conditions, namely, ψ (1), dψ (1) /dζ, d 2 ψ (1) /dζ 2 at ζ ±. Thus steady state solution of equation (25), can be expressed as ψ (1) = ψ m sech 4 (ζ/ ), (28) Figure 2. (Color online) Variation of the amplitude of solitary wave with θ and µ p for U =.1, β =.1, σ =.5 and µ d =.1 where upper surface represents the value of j = 1 and lower surface represent the value of j = 1. where ψ m = (15U /8A) 2 and = 16C/U are the amplitude and width of the solitary waves moving with the speed U, respectively. It is seen that the amplitude of the solitary waves is proportional to the soliton speed U whereas the width is inversely proportional to that. Hence the profile of the faster solitary wave will be taller and narrower then slower one. Depending on the value of A the solitary wave might be associated with positive or negative potentials. Eqn. (26) indicate that A is dependent on σ, β, µ p, µ and l z (l z = cosθ, where θ is the angle between the the external magnetic field B and directions of the wave propagation vector), therefore these parameters are responsible for the solitary waves to be rarefactive or compressive. From Eqn. (26) one can say that A is al- 74

5 Md. Masum Haider, Tahmina Ferdous, Syed S. Duha Θ thus narrower soliton profiles. This can be related to the effects of transverse perturbation. The Larmor radius for the ion motions are smaller at larger gyration frequency and then they contrubute less to the nonlinearity of the plasma. Lower dispersion are required for lower nonlinearity as soliton evolves when nonlinearity and dispersion are balanced. This finally leads to the reduction in the soliton width and makes the solitary waves more steeper. The width of the SWs is lower for positive dust grains (j = 1) then negative dust grains (j = 1), thus the positive dust grains makes the soliton profile more steeper. Figure 3. (Color online) Variation of the width of solitary wave (red curve for j = 1 and blue curve for j = 1) with θ for U =.1, β =.1, µ p = 1.5 and µ d =.1 having ω cn =.4 (solid curve), ω cn =.5 (dahsed curve) and ω cn =.6 (dotted curve). ways positive as (1 σ) and others parameters (β, µ p, µ and l z ) are positive. Therefore we can say that the solitary waves always associated with positive potentials as shown in Fig. 1. In the work of Rahman and Mamun [28], they have also found only positive potentials are associate. In this work if negative ions, negative dusts and magnetic field are considered to be absent, the results are exactly similar to the the work of Rahaman and Mamun [28]. Comparing these two works it can be said that presence of magnetic field, negative ions and negative dusts can effect the soliton profile but can not effect the direction. For positive dust grains the amplitude of the SWs increase with increasing dust number density and amplitude decreases with increasing dust number density for negative dusts. The amplitude of the solitary waves goes to zero at µ =, i.e. at this value of µ no solitary waves are found. The amplitude of the SWs increases with temperature ratio of free and trapped electrons as shown in Fig. 1. From the Fig. 2 we have found that the amplitude of the solitary waves increases with both µ p and θ but it is higher for j = 1 then j = 1. That is the amplitude of the SWs is higher for positive dust grains. The amplitude is independent of magnetic field, i.e., there is no effect of external magnetic field in amplitude. The width ( ) of the solitary waves have analyzed numerically as well as amplitude. The width of the solitary waves depends linearly with positive ion concentration and direction cosine and inversely with magnetic field. For lower limit of the angle ( 5 ) the width increases with it and decreases for higher limits of the angle (5 9 ) as shown in Fig. 3 for both positively and negatively charged dust grains. It is also clear that an increase of the external magnetic field leads to a decrease in the potential width, i.e., a stronger magnetic field leads to steeper and 5. Discussion The nonlinear propagation of DIA solitary waves in multi component dusty plasma have been studied, where inertia provided by the positive and negative ions and restoring force provided by the hot electrons which are trapped in the presence of external magnetic field useing reductive perturbation method and derive the modified K-dV equation and its solution. The results can be summarized as follows: 1. The amplitude of the solitary wave are significantly modified by the temperature ratio of free and trapped electron, ratio of number density of positive ion to negative ion (µ p ), ratio of number density of dust to negative ion (µ d ) and propagation angle (θ). The width of the solitary wave depends on ratio of number density of positive ion to negative ion, propagation angle and magnetic field (ω cn ). 2. Only positive SWs associate, i.e. negative ions and negative dusts can not make the soliton profile negative but responsible for lower amplitude of the profile. 3. The magnetic field makes the SWs more spiky but does not effect on amplitude. 4. The temperature ratio of free and trapped electrons are responsible to increase the amplitude but the width does not depend on it. 5. The amplitude of SWs for dust grain having positively charged is higher then that for negatively charged but opposite picture is found in the case of width. It means that SWs associated with negative charged dust grain is narrower. It has also been shown that the basic features (height and thickness) of such DIA solitary structures are completely different from those of the usual ion-acoustic solitary structures. It can be seen that the formation of such 75

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