Nonplanar quantum dust ion-acoustic Gardner double layers

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2012 J. Phys. A: Math. Theor. 45 125501

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J. Phys. A: Math. Theor. 45 (2012) 125501 (8pp) doi:10.1088/1751-8113/45/12/125501

Nonplanar quantum dust ion-acoustic Gardner doublelayers

M M Hossain and A A Mamun

Department of Physics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh

E-mail: manir_phys@yahoo.com and mamun_phys@yahoo.co.uk

Received 25 October 2011, in final form 17 January 2012Published 5 March 2012Online at stacks.iop.org/JPhysA/45/125501

AbstractNonplanar (cylindrical and spherical) double layers (DLs) in a quantum dustyplasma (composed of inertial ions, Fermi electrons and negatively chargedimmobile dust particles) are studied by employing the reductive perturbationmethod. The modified Gardner equation describing the nonlinear propagationof the quantum dust ion-acoustic (QDIA) waves is derived, and its nonplanar DLsolutions are numerically analyzed. The parametric regimes for the existenceof the DLs, which are found to be associated with both positive and negativepotentials, are obtained. It has been found that the existence of small butfinite-amplitude electrostatic DLs depends on μ = Zdnd0/ni0 (where Zd is thenumber of electrons residing onto the dust grain surface, and nd0 and ni0 are,respectively, the dust and ion number density at equilibrium) as well as thequantum diffraction parameter H. It has also been found that the propagationcharacteristics of nonplanar QDIA DLs are significantly different from thoseof the planar ones.

PACS numbers: 52.27.Lw, 52.35.−g, 52.35.Fp, 52.35.Mw

(Some figures may appear in colour only in the online journal)

1. Introduction

Quantum plasmas have attracted a great deal of attention because of their potential applicationsin dense plasma, particularly in different astrophysical and cosmological systems [1–3] (e.g.,interstellar or molecular clouds, planetary rings, comets, the interior of Jupiter and massivewhite dwarfs, magnetars, neutron stars, etc), in nanostructures [4], in a microelectronic device[5] and in the next-generation intense laser [6]. About 50 years ago, Pines [7] studied theproperties of the high-density and low-temperature quantum plasmas. Many authors include thequantum corrections to the quantum plasma echoes [8], the self-consistent dynamics of Fermigases [9], quantum beam instabilities [10], wave interactions in quantum magnetoplasmas[11], classical and quantum kinetics of the Zakharov system [12], quantum corrections tothe Zakharov equations [13], expansion of quantum electron gas into vacuum [14], quantum

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J. Phys. A: Math. Theor. 45 (2012) 125501 M M Hossain and A A Mamun

ion-acoustic waves [15], quantum Landau damping [16], magnetohydrodynamics of quantumplasmas [17], etc. Quantum plasmas have extremely high plasma number densities and lowtemperatures. At extremely low temperatures, the thermal de Broglie wavelength becomescomparable to the interelectron distance, the electron temperature becomes comparable to theelectron Fermi temperature (TFe) and the electrons follow the Fermi–Dirac distribution law.In this condition, quantum mechanical effects are expected to play a significant role in thebehavior of charged particles [18–26].

As electrons are lighter than ions, the quantum behavior of electrons is reached fasterthan ions. The electron gas in an ordinary metal is a good example where both plasma andquantum effects coexist. At room temperature and standard metallic densities, quantum effectscannot be ignored, so that the electron gas constitutes a true quantum plasma. Another possibleapplication of quantum plasmas arises from semiconductor physics. Even though the electrondensity in semiconductors is much lower than in metals, the great degree of miniaturizationof today’s electronic components is such that the de Broglie wavelength of the charge carrierscan be comparable to the spatial variation of the doping profiles. Hence, typical quantummechanical effects, such as tunneling, are expected to play a central role in the behaviorof electronic components. In astrophysical compact objects, the density of charged particlesis extremely high (some ten orders of magnitude larger than that of ordinary solids). Theproperties of matter existing under such ultradense plasmas possess strong quantum effectsand exhibit fluid and crystal properties in a quantum sea of electrons [2]. The dust particlesare also quite common in various plasma systems. The presence of static charged dust inelectron–ion plasmas leads to the introduction of a new mode. Shukla and Silin [27] firsttheoretically showed the existence of low-frequency dust ion-acoustic (DIA) waves in a dustyplasma, which was later observed in laboratory experiments [28, 29]. The nonlinear waves,associated with the DIA and quantum DIA (QDIA) waves, particularly solitary waves (SWs)[30, 31], shock waves [32] and double layers [33], etc have received a great deal of interest inunderstanding the basic properties of localized electrostatic perturbation in space [34–36] andlaboratory dusty plasmas [37, 38].

The formation of double layers (DLs) is also possible in different dense plasmaenvironments [39–41] where the electron quantum statistical pressure and tunneling forceassociated with the Bohm potential play important roles in the propagation of quantum ion-acoustic waves [40], quantum dust-acoustic waves [41], quantum electron-acoustic waves[42, 43], etc. Most of these studies are limited to one-dimensional (planar) geometry whichmay not be a realistic situation in space and laboratory devices, since the waves observed inspace (laboratory devices) are certainly not infinite (unbounded) in one dimension.

Very recently Hossain et al [45] have considered a plasma system composed of inertialions, massless quantum electrons and negatively charged immobile dust particles and havederived a higher order nonlinear equation, known as the modified Gardner equation (MGE)[43–46]. They have studied the nonlinear features of the finite-amplitude nonplanar QDIA SWsbeyond the Korteweg–de Vries (K–dV) limit. However, they have not studied the cylindricaland spherical QDIA Gardner DLs. Therefore, in this paper, we attempt to study the basicfeatures of cylindrical and spherical QDIA DLs by deriving the MGE [45], which is valid nearthe critical value of a certain parameter, say μ (μ = Zdnd0/ni0 = 2/3, where Zd is the numberof electrons residing onto the dust grain surface, and nd0 and ni0 are, respectively, the dust andion number density at equilibrium), in a quantum dusty plasma containing inertial ions, Fermielectrons and negatively charged immobile dust.

The paper is organized as follows. The model equations and MGE are provided insection 2. The numerical analysis of the MGE (for DLs) and a brief discussion are presentedin section 3.

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J. Phys. A: Math. Theor. 45 (2012) 125501 M M Hossain and A A Mamun

2. Model equations and the MGE

We consider the nonlinear propagation of an unmagnetized quantum dusty plasma systemcomposed of inertial ions, massless Fermi electrons and negatively charged immobile dust.Thus, at equilibrium we have ni0 = ne0 + Zdnd0. The nonlinear dynamics of the QDIAwaves, whose phase speed is much smaller (larger) than the electron (ion) thermal speed, in anonplanar (cylindrical or spherical) geometry is governed by

∂ni

∂t+ 1

rν

∂

∂r(rνniui) = 0, (1)

∂ui

∂t+ ui

∂ui

∂r= −∂φ

∂r, (2)

∂φ

∂r− 2

5ne

∂n5/3e

∂r+ β

∂

∂r

(1√ne

∂2

∂r2

√ne

)= 0, (3)

1

rν

∂

∂r

(rν ∂φ

∂r

)= −ρ, (4)

ρ = ni − (1 − μ)ne − μ, (5)

where ν = 0 for 1D planar geometry; ν = 1(2) for a nonplanar cylindrical (spherical)geometry; ni(ne) is the ion (electron) number density normalized by its equilibrium valueni0(ne0); ui is the ion fluid speed normalized by the quantum ion-acoustic speed Ci =(kBTFe/mi)

1/2 (with mi being the ion rest mass, kB being the Boltzmann constant and TFe beingthe Fermi temperature of the electron gas); φ is the electrostatic wave potential normalizedby kBTFeφ/e (with e being the magnitude of the charge of an electron); ρ is the normalizedsurface charge density, μ = Zdnd0/ni0; β = δH2/2 with δ = ni0/ne0; H = �ωpe/kBTFe

is the ratio between the plasmon energy and the electron Fermi energy. The time variable(t) is normalized by ω−1

pi = (mi/4πni0e2)1/2 and the space variable r is normalized byλDi = (kBTFe/4πni0e2)1/2. We note that in order to obtain the second term of (3), we haveused the following Fermi pressure law for the electron species [47, 48]:

Pe = 1

5

meV 2Fen5/3

e

n2/3e0

, (6)

where VFe = (2kBTFe/me)1/2 is the electron Fermi speed at temperature TFe. To study QDIA

DLs by analyzing the ingoing solutions of equations (1)–(5), we first introduce the stretchedcoordinates [43]

ζ = ε(r − Vpt), τ = ε3t, (7)

where Vp is the QDIA wave phase speed (ω/k) and ε is a smallness parameter measuring theweakness of the dispersion (0 < ε < 1). We then expand ni, ne, ui, φ and ρ in power seriesof ε,

ni = 1 + εn(1)i + ε2n(2)

i + ε3n(3)i + · · ·, (8)

ne = 1 + εn(1)e + ε2n(2)

e + ε3n(3)e + · · ·, (9)

ui = 0 + εu(1)i + ε2u(2)

i + ε3u(3)i + · · ·, (10)

φ = 0 + εφ(1) + ε2φ(2) + ε3φ(3) + · · ·, (11)

ρ = 0 + ερ(1) + ε2ρ(2) + ε3ρ(3) + · · ·, (12)

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J. Phys. A: Math. Theor. 45 (2012) 125501 M M Hossain and A A Mamun

and develop equations in various powers of ε. After some assumptions and mathematicalcalculations [45], we obtain an equation of the form

∂ψ

∂τ+ ν

2τψ + c2sψ

∂ψ

∂ζ+ α1ψ

2 ∂ψ

∂ζ+ α2

∂3ψ

∂ζ 3= 0, (13)

where

c2 = 1

2c1V

3p , Vp =

√2

3(1 − μ), α1 = 1

2V 3

p

{15

2V 6p

− 81(1 − μ)

16

},

α2 = 1

2V 3

p

(1 − 9H2

16

).

Equation (13) is known as the MGE. The detailed calculations and assumptions can beseen in [45]. The modification is due the extra term (namely ν

2τψ), which arises due to the

effects of the nonplanar geometry. It is important to note that if we neglect the ψ3 term andput c2s = V 3

p A/2 = 9V 3p (1 − μ)(2 − 3μ)/8, the MGE reduces to a modified K-dV equation

which can be derived by using a lower order stretching, namely ζ = ε1/2(r − Vpt), τ = ε3/2t.However, in this modified K-dV equation, the nonlinear term vanishes at μ = μc and is notvalid near the critical value μ = μc = 2/3. But the MGE derived here is valid for μ near itscritical value.

It should be noted here that Hossain et al [45] derived this MGE and discussed thepossibility of the SWs solution beyond the K-dV limit. But in our present work, we attemptto find out the DL solution. It is well known that the basic properties of DLs are completelydifferent from those of SWs.

3. Numerical analysis and discussion

We have already mentioned that ν = 0 corresponds to a 1D planar geometry which reducesequation (13) to a standard Gardner equation (SGE). Our aim now is to numerically analyzethe MGE. However, for clear understanding, we first briefly discuss the stationary DL solutionof this SGE (i.e. equation (13) with ν = 0). The stationary DL solution of this SGE is obtainedby considering a moving frame (moving with the speed U0) ξ = ζ − U0τ , and imposingall appropriate boundary conditions for DL solutions, including ψ(1) → 0, dψ/dξ → 0,d2ψ/dξ 2 → 0 at ζ → −∞. Thus, one can express the stationary DL solution of this SGE as

ψ = ψm

2

[1 + tanh

(ξ

�

)], (14)

where the amplitude (ψm) and the width (�) of the DLs are, respectively,

ψm = 6U0

c2s, � = 2√−ψ2

mγ, (15)

with γ = α1/6α2.Equation (14) represents a DL solution when α1 and α2 maintain the opposite sign, i.e.

α1/α2 < 0 to make γ < 0. Thus, DLs are formed (i) for α1 > 0 and α2 < 0, i.e. for μ < 0.56and H > 4/3 or (ii) for α1 < 0 and α2 > 0, i.e. for μ > 0.56 and H < 4/3. Around μc, DLsare formed for α1 < 0 and α2 > 0. This means that for μ > 0.56 and H < 4/3, DLs can beformed near μc in the dusty plasma system under consideration.

It may be noted here that if we would neglect the higher order nonlinear term (namelythe fourth term of (13) or the term containing ψ3), but would keep the lower order nonlinearterm (namely the third term of (13) or the term containing ψ2), we would obtain the solitary

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J. Phys. A: Math. Theor. 45 (2012) 125501 M M Hossain and A A Mamun

-200

0

200-30

-25

-20

-15

-10

-5

0

0.1

0.2

-200

0

200

Figure 1. Showing the effects of cylindrical geometry on QDIA positive DLs for μ = 0.66,U0 = 0.05 and H = 0.3.

-200

0

200-30

-25

-20

-15

-10-5

-0.2-0.15-0.1

-0.050

-200

0

200

Figure 2. Showing the effects of cylindrical geometry on QDIA negative DLs for μ = 0.67,U0 = 0.05 and H = 0.3.

structures that are due to the balance between nonlinearity (associated with ψ2 only) anddispersion [49–51]. On the other hand, in our present work, we have kept both the termscontaining ψ2 and ψ3, and have obtained the DL structures which are formed due to thebalance between the nonlinearity (associated with ψ2 and ψ3) and dispersion. It may be addedhere that the dissipation (which is usually responsible for the formation of the shock-likestructures [52, 53]) is not essential for the formation of solitary and DL structures [38, 54–56].The stationary DL solution of the SGE equation and the conditions for the existence of DLsclearly imply that the DL structures predicted in our present investigation are not due to thedissipation (which has been neglected in our present investigation), but are due to the balancebetween the harmonic nonlinearities and the dispersion caused by density correlation due toquantum fluctuation and charge separation effect.

We now turn to equation (13) with the term (ν/2τ )ψ , which is due to the effects ofthe nonplanar (cylindrical or spherical) geometry. An exact analytic solution of equation (13)is not possible. Therefore, we have numerically solved equation (13) and have studied theeffects of cylindrical and spherical geometries on time-dependent QDIA DLs. The results aredepicted in figures 1–4. The initial condition, that we have used in our numerical analysis, is

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J. Phys. A: Math. Theor. 45 (2012) 125501 M M Hossain and A A Mamun

-200

0

200-30

-25

-20

-15

-10

-5

0

0.2

0.4

0.6

-200

0

200

Figure 3. Showing the effects of spherical geometry on QDIA positive DLs for μ = 0.66,U0 = 0.05 and H = 0.3.

-200

0

200-30

-25

-20

-15

-10

-5

-0.4

-0.2

0

-200

0

200

Figure 4. Showing the effects of spherical geometry on QDIA negative DLs for μ = 0.67,U0 = 0.05 and H = 0.3.

in the form of the stationary solution of equation (13) without the term (ν/2τ )ψ . Figures 1and 2 show how the effects of a cylindrical geometry modify the QDIA positive (negative)DLs. On the other hand, figures 3 and 4 show how the effects of a spherical geometry modifythe QDIA positive (negative) DLs.

The numerical solutions of equation (13) (displayed in figures 1–4) reveal that for a largevalue of τ , the spherical and cylindrical DLs are similar to 1D structures. This is becausefor a large value of τ ( e.g. τ = −30), the term (ν/2τ )ψ , which is due to the effects of thecylindrical or spherical geometry, is no longer dominant. However, as the value of τ decreases,the term (ν/2τ )ψ becomes dominant, and both spherical and cylindrical DL structures differfrom the 1D ones. It is found that as the value of τ decreases, the amplitude (the magnitudeof the amplitude) of these localized pulses increases. It is also found that the amplitude ofcylindrical QDIA DLs is larger than those of the 1D ones, but smaller than that of the sphericalones. From equation (15), it is clear that the amplitude of positive and negative DLs does notvary with the quantum diffraction parameter, but the width of the both positive and negativeDLs varies with it.

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J. Phys. A: Math. Theor. 45 (2012) 125501 M M Hossain and A A Mamun

To compare the properties of QDIA cylindrical/spherical DL structures with those ofQDIA cylindrical/spherical solitary structures [45], we note that (i) the solitary structuresare hump- or dip-shaped nonlinear waves of permanent profile, whereas DLs can be thoughtof as an internal sheath separating plasmas at different potentials within a plasma system;(ii) the other properties (such as amplitude, width, etc) of the DLs (shown in figures 1–4) alsosignificantly differ from those of the SWs.

To summarize, we have investigated cylindrical (ν = 1) and spherical (ν = 2) QDIA DLsin quantum dusty plasma by deriving the MGE which is valid for μ ∼ 2/3. It is found thatthe quantum dusty plasma system under consideration supports finite-amplitude DLs, whosebasic features (polarity, amplitude, width, etc) depend on the ion and dust number densitiesas well as the quantum diffraction (tunneling) parameter H. It is also found that at μ < 2/3,positive DLs exist, whereas at μ > 2/3, negative DLs exist. We have used a wide rangeof the quantum dusty plasma parameters (namely μ = 0.1–0.9 and H = 0.1–1.33) in ournumerical analysis. Thus, the dusty plasma parameters are within the appropriate ranges forboth space environments and laboratory devices. To conclude, we hope that our results maybe useful in understanding the localized electrostatic disturbances in both space environmentsand laboratory devices.

Acknowledgments

The Third World Academy of Science (TWAS) Research Grant for research equipment isgratefully acknowledged.

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