The effects of Gaussian size distribution dust particles in a complex plasma

Physics Letters A 361 (2007) 368–372 www.elsevier.com/locate/pla

The effects of Gaussian size distribution dust particles in a complex plasma Wen-Shan Duan a , Hong-Juan Yang a,∗ , Yu-Ren Shi a,b , Ke-Pu Lü a a College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, PR China b Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, PR China

Received 17 December 2005; received in revised form 6 June 2006; accepted 22 September 2006 Available online 4 October 2006 Communicated by F. Porcelli

Abstract Investigations for a dusty plasma with many different dust grain species are adopted, which varies self-consistently with the system parameters. Two cases have been studied. One is that the dust size distribution is given by Gaussian distribution. The other is that there is only one kind of dust grains (mono-sized dusty plasma) whose size is the average one of Gaussian distribution. The comparisons between them have been made and it is found that the linear phase velocity is different. © 2006 Published by Elsevier B.V. PACS: 52.35.Sb; 52.25.Vy; 05.45.Yv Keywords: Dust size distribution; Soliton; KdV equation

1. Introduction A dusty plasma is an ionized gas containing small particles of solid matter, which acquire electric charges by collecting electrons and ions from the plasma. The dust particles are usually negatively charged. Charged dust components appear naturally in space environments such as planetary rings, cometary surroundings, interstellar clouds and lower parts of the Earth’s ionosphere [1,2]. In the laboratory, the dust particles appear as impurities and can significantly influence the behaviors of the surrounding plasma [3]. Recently the study of dusty plasma has developed rapidly with many applications in both laboratory and space plasma [4]. Dust acoustic wave (DAW) was first reported in an unmagnetized dusty plasma by Rao et al. [5]. On the other hand, at higher frequency Shukla and Silin [6] showed the existence of dust ion acoustic wave (DIAW). Recent laboratory experiments on dusty plasmas have confirmed the existence of DAW and DIAW [7–9]. Furthermore, lowfrequency electrostatic ion-acoustic and ion-cyclotron waves have been studied in a magnetized dusty plasma [10]. How* Corresponding author.

E-mail addresses: [email protected], [email protected] (H.-J. Yang). 0375-9601/$ – see front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.physleta.2006.09.064

ever, most of the existing studies on dusty plasma have focused attentions only on mono-sized dusty plasma [5,6,10]. Actually, the dust grain sizes for a dusty plasma are different [11–16]. In the laboratory, the dust size distribution is usually different for different cases. However, for simplicity and generality, we use a Gaussian distribution function as an approximation for a real laboratory dusty plasma in order to understand general effects of dust size distribution. This model can be extended to an arbitrary dust size distribution dusty plasma by replacing the Gaussian dust size distribution function with an arbitrary function. For charged dust grains with radii a in a give range [amin , amax ], the differential Gaussian distribution is of the form [11,12] 

¯ da n(a) da = De−μ (a−a) 2

(1)

where amin < a < amax , and D can be determined by amax

Ntot =

n(a) da.

(2)

amin

Ntot is the total number density of dust grains. Outside the limits a < amin and a > amax , we use n(a) = 0. μ can be found by ¯ where a¯ is the avrequiring that n(amin ) = n(amax ) =   n(a),

W.-S. Duan et al. / Physics Letters A 361 (2007) 368–372

erage radii of dust size, for some small value of   . We usually use   = 0.01. Several authors have considered charged dust as a limited number of discrete species, while others have treated the charged dust as a continuous distribution over a limited size range, by using a decreasing power law, as discussed by Verheest and Meuris [17–22]. Brattli et al. considered dust size distributions from a kinetic point of view, and found that Landau damping dominates at short wavelengths [12]. Whereas for larger wavelengths attenuation due to charge variations becomes more important. Changes in dispersion laws have only been given for some of the better-known dusty plasma modes such as dust acoustic waves (DAW) [11,23–25]. Dust acoustic waves are also studied for dusty plasmas containing dust particles with continuous mass or size distributions [26]. For broad size spectra, self-gravitational effects are clearly interwoven with the grain size distribution and the effects of different power law size distributions on the propagation, damping, and instability of low-frequency waves are discussed. Compared to the previous work [1,2] which have been given in detail for space plasma, our work can be used to the laboratory dusty plasma and to the dusty plasma on which the dust particles can be an arbitrary function. Moreover, we obtain the dispersion relation for Gaussian dust size distribution plasma. The relationship between the linear velocity and the dusty plasma parameters are obtained, and it is found that they can influence the dispersion relation, the amplitude and the width of a solitary waves. The Letter is organized in the following fashion. In Section 2 we use the equations of motion for a dusty plasma for N dust grain species and obtain both the linear phase velocity and the KdV equation. In Section 3, we compare the differences between dusty plasmas with Gaussian distribution and monosized one. In Section 4, we give a summary of our conclusions. 2. Governing equation for a dusty plasma with N different dust grain species There are many research works on dust acoustic waves for mono-sized dust grains, the governing equation for long wavelength nonlinear waves is the Korteweg–de Vries equation (KdV equation) [28–30]. Duan studied the dust acoustic wave (DAW) by considering the dust size distribution [27]. For simplicity, the plasma is assumed to be unmagnetized and the waves are assumed to propagate in the x direction. In a dusty plasma, the dusty grains are usually much heavier than both ions and electrons. However, the dust grains have different sizes. The governing equations for a dusty plasma with N different dust grain species are as follows [27] ∂ndj ∂ + (ndj udj ) = 0, ∂t ∂x Zdj ∂φ ∂udj ∂udj + udj = ∂t ∂x mdj ∂x

(3) (j = 1, 2, 3, . . . , N),

(4)

where we assume that the dust fluids are cold, udj is the dust velocity in the x direction, ndj and mdj refer to the number

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density and the mass of the j th dust grains, respectively. Zdj is the charge number of j th dust grain. The Possion equation is given as follows ∂ 2φ  = Zdj ndj + ne − ni ∂x 2 N

(5)

j =1

where the electrons and ions are all assumed Boltzmann dis tributed ni = μe−sφ , ne = νesβ φ , with β  = Ti /Te , s = 1/(μ +  νβ ). Te and Ti denote the temperatures for electrons and ions, respectively. μ and ν are the normalized ion and electron number densities,  respectively.  The average radii of dust size is N a n / given by a¯ = N j dj j =1 j =1 ndj . Note that all the physical quantities are normalized. We now study the linear dispersion relation of the system by assuming ndj = ndj 0 + nj ei(kx−ωt) , udj = uj ei(kx−ωt) , and φ = ψei(kx−ωt) . Inserting these expansions into Eqs. (3), (4) and (5), we obtain the linear dispersion relation as follows ω2 =

N 2 k 2  ndj 0 Zdj . mdj 1 + k2

(6)

j =1

The linear phase velocity (sound speed) can be obtained by  N 1/2 2  ndj 0 Zdj ∂ω c = lim (7) . = k→0 ∂k mdj j =1

It is noted that c = 1 for a mono-sized dusty plasma. Now we use the reductive perturbation method to know the behavior of small, but finite amplitude dust acoustic waves, as was used by many authors [31,32]. The independent variables are stretched as ξ = (x − υ0 t), and τ =  3 t , where  is a small parameter characterizing the strength of nonlinearity, υ0 is the velocity of linear waves as mentioned before. The dependent variables are expanded as follows ndj = nd0j +  2 nd1j +  4 nd2j + · · ·, udj =  2 ud1j +  4 nd2j + · · ·, φ =  2 φ1 +  4 φ2 + · · ·, (j = 1, 2, . . . , N). Inserting the these expansions into Eqs. (3), (4) and (5) and collecting the terms in the different powers of , we obtain the following equations n Zdj Z at the lowest order: nd1j = − d0j φ1 , ud1j = − υ0 mdjdj φ1 , 2

υ0 mdj 2 n Z d0j υ02 = j =1 mdj dj , here υ0 is the linear phase speed at the limit of limk→0 [ ωk ], and the famous KdV equation can be obtained

N

[17,36] at the higher order

∂φ1 ∂ 3 φ1 ∂φ1 + Aφ1 + B 3 = 0, ∂τ ∂ξ ∂ξ

(8)

where A=−

3 N  υ0   2 2 3  nd0j Zdj − νβ s − μs 2 , 3 2 2 2υ0 j =1 mdj

(9)

υ0 (10) . 2 It is well known that the stationary solitary wave solution 2 of the KdV equation (8) can be written √ as: φ1 = φm sech [(ξ − u0 τ )/ω], where φm = 3u0 /A, ω = 2 B/u0 are the amplitude and the width of a soliton respectively. It is noted that both the B=

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soliton amplitude and the width are the functions of an arbitrary constant u0 (the soliton velocity measured in ξ, τ ) at given A and B. However, as one can see from Eqs. (9) and (10), the linear phase speed υ0 , the coefficients of the KdV equation, A and B, are the functions of dusty plasma parameters fixing the plasma composition, namely υ0 = υ0 (nd10 , nd20 , . . . , ndN0 , Zd1 , Zd2 , . . . , ZdN , md1 , md2 , . . . , mdN ), A = A(nd10 , nd20 , . . . , ndN0 , Zd1 , Zd2 , . . . , ZdN , md1 , md2 , . . . , mdN , μ, ν, β  , s), B = B(nd10 , nd20 , . . . , ndN0 , Zd1 , Zd2 , . . . , ZdN , md1 , md2 , . . . , mdN ). Therefore, we note generally that a fast soliton (measured in the coordinates of ξ, τ ) has a larger amplitude and a smaller width, which is a characteristic property of one soliton solution of the KdV equation. On the other hand, for a given soliton propagation speed u0 , the soliton amplitude and width depend on the dusty plasma parameters, such as: ndj 0 , Zdj , mdj (j = 1, 2, . . . , N ) and μ, ν, β  , s. 3. Gaussian dust size distribution dusty plasma When we assume that all dust grain sizes aj  λDd (j = 1, 2, . . . , N ), where λDd is the Debye length (λDd = Teff ) [27], made of same material and placed in iden4πe2 Ntot Z¯ d0 tical plasma potential environments, we can express the masses of the dust particles as follows mdj = km aj3 , where km is a constant. Recently, the dependence of dust particle charge on its size has been reported [34,35]. It was found that the charge dependence on the particle radius is quite nonlinear, as the surface potential increases with the increasing particles size. Indeed, the charge accumulated on a dust particle is always a function of parameters of the surrounding plasma, in particular, the ion and electron number densities as well as ion and electron temperatures. This implies that, in order to investigate the true effect of the particle size on its charging, one needs to consider various size particles under the identical plasma condition. Here in this Letter, following the results reported in Refs. [34,35], we γ γ assume that Qdj ≈ kq aj , or Zdj ≈ kz aj , where kq ≈ 4π0 V0 , kz ≈ 4π0 V0 /e are approximately constants. 0 is the vacuum positivity, and γ is a constant which satisfies 1  γ  2 [34,35]. Based on these assumptions we can easily obtain the following equations for the continuous dust size distribution cases υ02

k2 = z km

A=−

amax

n(a) da, a 3−2γ

(11)

amin amax

3kz3 2 2υ03 km a

 n(a) υ0   2 2 da − νβ s − μs 2 . 2 a 6−3γ

(12)

min

As usually assumed for the Gaussian distribution, given by Eq. (1), we find that n(a) ¯ = D, n(amin ) = n(amax ), i.e., ¯ 2 ] = exp[−μ (amin − a) ¯ 2 ] =   . Therefore exp[−μ (amax − a)

Fig. 1. The variation of υ02 /υ¯ 02 with respect to a for different γ = 1.0, 1.3, 1.5, 1.9.

we obtained that amax − a¯ = a¯ − amin = d0 , and μ =

1 1 ln . d02  

(13)

For the Gaussian distribution, we obtain the following quantities amax  ¯ 2 Ntot = (14) De−μ (a−a) da, amin

amax a¯ = υ02 υ02 υ¯ 02

amin

2

Ntot

k2 = zD km =



¯ da Dae−μ (a−a) amax



(15)

,

¯ e−μ (a−a) da, a 3−2γ 2

(16)

amin

a¯ 3−2γ

amax

−μ (a−a) ¯ 2 1 amin e a 3−2γ amax  (a−a) 2 −μ ¯ da amin e

da

.

(17)

Where υ0 is the linear phase speed for Gaussian distribution dusty plasma, as mentioned before. υ¯ 0 is the linear phase speed for mono-sized dusty plasma with the particle size of average Gaussian distribution dusty plasma, a¯ is the average dust size, Ntot is the total number density of the Gaussian distribution dusty plasma. Fig. 1 shows the variations of υ02 /υ¯ 02 with respect to average dust size a¯ for different γ = 1.0, 1.3, 1.5, 1.9. It can be concluded from Fig. 1 that υ02 /υ¯ 02 > 1, when γ < 1.5 It means that the linear velocity for Gaussian distribution are larger than that of mono-sized dusty plasma with average dust size. We can also see that the values of υ02 /υ¯ 02 decrease as a¯ or γ increases. We also find that when γ = 1.5, υ02 /υ¯ 02 = 1. At this point the linear phase velocity is just the same between Gaussian distribution dusty plasma and that with average mono-sized dusty plasma. If γ > 1.5, we find from Fig. 1 that υ02 /υ¯ 02 < 1. It also indicates that the linear phase velocity of Gaussian distribution dusty plasma is smaller than that with average mono-sized dusty plasma. The values υ02 /υ¯ 02 increase as a¯ increases but

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Fig. 2. The variation of υ0 with respect to γ for different d0 = 1 µm, 2 µm, 3 µm, 3.5 µm.

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Fig. 3. The variation of A/A¯ with respect to a¯ for different γ = 1.0, 1.3, 1.6, 1.9.

decrease as γ increases. Fig. 2 shows that the linear phase velocity, υ0 , increases as γ or d0 increases. The nonlinear coefficient of KdV equation of (8) can be given for the Gaussian distribution as follows A=

amax

−3kz3 2 2υ03 km a

n(a)

1 a 6−3γ

da

−

 υ0   2 νβ − μ s 2 . 2

(18)

min

For mono-sized dusty plasma with average dust grain size, the nonlinear coefficient of KdV equation is A¯ = −

3Ntot kz3 2a 2υ¯ 03 km ¯ 6−3γ

−

 υ¯ 0   2 νβ − μ s 2 . 2

(19)

For a typical dusty plasma [33], ni0 ∼ 105 –1010 cm−3 , Ti ∼ Te ∼ 0.1 eV, nd0 ∼ 105 cm−3 , Qd ∼ (−104 e)–(−105 e), md ∼ 10−15 –10−12 g ∼ 109 –1012 mi , ρd ∼ 1 g/cm3 , a ∼ 0.1–1 µm. We estimated that the first terms of both A and A¯ are about 1020 –1025 , while for the second terms are about 10−5 –104 . Therefore we can neglect the second terms and obtain A υ¯ 03 a¯ 6−3γ D = A¯ υ03 Ntot

amax



¯ e−μ (a−a) da. a 6−3γ

Fig. 4. The variation of A with respect to γ for different d0 = 1.1 µm, 1.2 µm, 1.3 µm, 1.4 µm.

4. Conclusion

2

(20)

amin

Fig. 3 shows the ratio of A/A¯ as a function of average dust size a, ¯ d0 and γ . In order to compare the soliton amplitude and other parameters, we let the arbitrary constant u0 be a constant, we then conclude from Fig. 3 that the soliton amplitude is smaller for Gaussian distribution than that of mono-sized dusty plasma for the same condition. The soliton amplitude ra¯ increases as average dust size a¯ increases, tio, φm /φ¯ m = A/A, and it decreases as γ increases. Fig. 4 shows that A increases as γ or d0 increases. The soliton amplitude φm = 3u0 /A decreases as γ increases but increases as d0 increases. We can conclude that if there is a soliton or other nonlinear waves in this system the perturbation velocities among different dust grains are different and the propagation distances for different grains are also different.

In this Letter, we investigated the more real dusty plasma with many different dust grain species. We have compared two cases. One is that the dust size is given by Gaussian distribution. The other is mono-sized dusty plasma whose size is that of average Gaussian distribution dust particles. We find that the linear phase velocity for Gaussian distribution is larger than that of mono-sized dusty plasma with average dust size when γ < 1.5, but they are the same when γ = 1.5. The linear phase velocity of Gaussian distribution dusty plasma is smaller than that with average mono-sized dusty plasma when γ > 1.5. Refs. [33,35] reported that γ = 1.69–1.87, therefore, for this case the linear phase velocity of Gaussian distribution dusty plasma is smaller than that with average mono-sized dusty plasma. It is reported that γ varies with the pressure and other parameters of dusty plasma. Our results are just extension of previous approximated work [14,24,27], on which the surface potentials for every dust particles are approximately assumed to be the same, and there-

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fore γ = 1. Actually, the surface potentials are different with different dust grains, therefore γ depends on the many dust parameters. So the results in this Letter possibly are more realistic and useful to the variety of the dusty plasmas. Acknowledgements Work supported by the National Natural Science Foundation of PR China (Grant Nos. 10575082, 10247008), the Scientific Research Foundation (SRF) for the Returned Overseas Chinese Scholars (ROCF), State Education Ministry (SEM). References [1] C.K. Goertz, Rev. Geophys. 27 (1989) 271. [2] O. Havnes, F. Melandso, C. La Hoz, T. Aslaksen, T.W. Hartquist, Phys. Scr. 45 (1992) 433. [3] S. Benkadda, V.N. Tsytovich, A. Verga, Comments Plasmas Phys. Cont. Fusion 16 (1995) 321. [4] F. Verheest, Space. Sci. Rev. 77 (1996) 267. [5] N.N. Rao, P.K. Shukla, M.Y. Yu, Planet. Space Sci. 38 (1990) 543. [6] P.K. Shukla, V.P. Silin, Phys. Scr. 45 (1992) 508. [7] A. Barkan, R.L. Merlino, N. D’Angelo, Phys. Plasmas 2 (1995) 2563. [8] G.E. Morfill, H. Thomas, J. Vac. Sci. Technol. A 14 (1996) 490. [9] R.L. Merlino, A. Barkan, C. Thompson, N. D’Angelo, Planet. Space Sci. 38 (1990) 1143. [10] N. D’Angelo, Planet. Space Sci. 38 (1990) 1143. [11] P. Meuris, Planet. Space Sci. 45 (1997) 449. [12] A. Brattli, O. Havres, F. Melandso, J. Plasma Phys. 58 (1997) 691.

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