Physics Letters A 373 (2009) 2940–2943

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Physics Letters A

www.elsevier.com/locate/pla

Rayleigh–Taylor/gravitational instability in dense magnetoplasmas

S. Ali a,b,∗, Z. Ahmed c, Arshad M. Mirza d, I. Ahmad e

a National Centre for Physics, Quaid-i-Azam University Campus, Islamabad, Pakistanb IPFN, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugalc COMSATS Institute of Information Technology, Department of Physics, Wah Campus, Pakistand Theoretical Plasma Physics Group, Physics Department, Quaid-i-Azam University, Islamabad 45320, Pakistane COMSATS Institute of Information Technology, Department of Physics, Islamabad Campus, Pakistan

a r t i c l e i n f o a b s t r a c t

Article history:Received 10 September 2008Received in revised form 25 May 2009Accepted 9 June 2009Available online 13 June 2009Communicated by F. Porcelli

The Rayleigh–Taylor instability is investigated in a nonuniform dense quantum magnetoplasma. For thispurpose, a quantum hydrodynamical model is used for the electrons whereas the ions are assumed tobe cold and classical. The dispersion relation for the Rayleigh–Taylor instability becomes modified withthe quantum corrections associated with the Fermi pressure law and the quantum Bohm potential force.Numerically, it is found that the quantum speed and density gradient significantly modify the growthrate of RT instability. In a dense quantum magnetoplasma case, the linear growth rate of RT instabilitybecomes significantly higher than its classical value and the modes are found to be highly localized. Thepresent investigation should be useful in the studies of dense astrophysical magnetoplasmas as well asin laser-produced plasmas.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

It is well known that the quantum plasmas which are com-posed of electrons and ions, and have the characteristics of highparticle number densities and low temperatures, in contrast to theclassical plasmas. Recently, a great deal of interest has been devel-oped in the context of collective modes, instabilities, and nonlin-ear structure formation in quantum plasmas. The main reasons ofattraction are: the new developments in the manufacturing elec-tronics and nano-technology [1], the discovery of ultracold plas-mas [2,3], the existence of quantum effects in ultrasmall electronicdevices [4], in superdense astrophysics [5,6], and in strong laser-produced plasmas [7], as well as in Rydberg systems (in whichquantum plasma oscillations has been detected [8]). In all thesedense quantum systems, the Fermi–Dirac distribution is importantto use rather than the classical Maxwell–Boltzmann type of dis-tribution. Accordingly, the quantum scales such as the time, thelength, and the thermal speeds of the charged particles are quitedifferent from the classical plasmas. The quantum effects beginto play a role [9], when the de Broglie wavelength (λB) becomescomparable to the average distance d (= n−1/3

0 ) among the chargeparticles, i.e. n0λ

3B � 1. Here λB roughly represents the spatial ex-

tension of the charged particle wave function due to quantumuncertainty.

* Corresponding author at: National Centre for Physics, Quaid-i-Azam UniversityCampus, Islamabad, Pakistan.

E-mail address: shahid.ali@ncp.edu.pk (S. Ali).

0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2009.06.021

For modeling dense quantum plasmas [9], the most fundamen-tal models used are the Wigner–Poisson and the Schrödinger–Poisson which demonstrate the statistical and hydrodynamic be-havior of the plasma particles. However, the quantum hydrody-namical (QHD) model describing the transport of charge, mo-mentum, and energy, can be derived from the Wigner–Poissonand the Schrödinger–Poisson systems, are usually employed to ad-dress the issues of the resonant tunnelling, and negative differ-ential resistance [10] in semiconductor physics. Numerous stud-ies [11–19] have been carried out with quantum effects, for in-stance, the Debye screening in quantum plasmas, Langmuir andion-acoustic waves, quantum surface waves, quantum drift-waves,quantum magnetohydrodynamic waves, quantum dark solitons andvortices, quantum Bernstein–Greene–Kruskal equilibria, etc. Thecollective modes become unstable due to the availability of freeenergy sources. The latter could be due to the density gradient, thetemperature gradient or the plasma particle streaming, etc. Con-sequently, the amplitudes of the collective modes either grow ordamp exponentially with respect to time depending upon the sig-nature of the imaginary part of the wave frequency. The collectivemodes and instabilities have also been investigated [20–24] in thedust-contaminated quantum plasmas.

A quantum multistream model has been introduced [25] tostudy the dispersion properties of one-and two-stream plasmainstabilities by using the Schrödinger–Poisson system and the two-stream instability is found to be enhanced for small quantumeffects. Anderson et al. [26] have done the same to consider thestatistical view by using the Wigner–Poisson system. A Landau-likedamping was found to suppress the one- and two-stream instabili-

S. Ali et al. / Physics Letters A 373 (2009) 2940–2943 2941

ties. Furthermore, Haas et al. [27] extended the work for the stabil-ity of the three-stream quantum plasmas. The linear and nonlinearproperties of the electrostatic ion-acoustic waves [28] have re-cently been investigated accounting for the quantum statistics andquantum Bohm potential. Later, a QHD model has been formulated[29] to including the magnetic field effects for a three-dimensionalquantum plasma. Bychkov et al. [30] derived the dispersion rela-tion for the internal waves and the Rayleigh–Taylor (RT) instabilityin a nonuniform unmagnetized quantum plasma with a constantgravitational field. They have shown that the quantum effects al-ways play a stabilizing role for the RT wave instability. Cao et al.[31] studied the RT instability incorporating the quantum magne-tohydrodynamic equations and solved the second-order differentialequation under different boundary conditions with quantum set-tings.

In this Letter, we present the RT wave instability in a nonuni-form quantum magnetoplasma by employing the so-called driftapproximation and by assuming that ω2

ci � (ω − kUi0)2, where

ωci (= eB0/mic) is the ion gyrofrequency. While employing thequantum hydrodynamical equations and assuming that the ionsare cold and classical, the inertialess hot electrons are treated asquantum mechanically. It is found that the growth rate and thereal wave frequency are significantly modified with the Fermi dis-tribution and quantum Bohm potential effects.

The manuscript is organized as follows. In Section 2, we con-sider the quantum hydrodynamical equations to obtain new dis-persion relation for the RT wave instability and the real wavefrequency which are affected significantly by the quantum correc-tions. Section 3 summarizes the results of our present investiga-tions with possible applications.

2. RT instability in quantum magnetoplasma

We consider a dense quantum magnetoplasma whose con-stituents are the electrons and ions. The quasineutrality conditionat equilibrium is given as ne0 = ni0 = n0. We assume that theplasma is embedded in an external magnetic field B0 = B0z, whereB0 is the strength of external magnetic field. In equilibrium, thedensity gradient and the gravitational field are assumed to be inthe opposite direction, i.e. ∇n0 = −|∇n0|x and g = gx, where xand z are the unit vectors along the x- and z-directions, respec-tively. The electric field and propagation vectors are assumed to liein the y-direction, i.e., E1 = (0, E y,0) and k = (0,ky,0). To investi-gate the problem of Rayleigh–Taylor instability in a dense quantumplasma, we may start with the following quantum hydrodynamic(QHD) model:

∂ni1

∂t+ ∇ · (ni1Ui0) + ∇ · (ni0Ui1) = 0, (1)

mini0

(∂

∂t+ Ui0 · ∇

)Ui1 = eni0

(E1 + Ui1 × B0

c

), (2)

∂ne1

∂t+ ∇ · (ne1Ue0) + ∇ · (ne0Ue1) = 0, (3)

mene0

(∂

∂t+ Ue0 · ∇

)Ue1 = −ene0

(E1 + Ue1 × B0

c

)

− 2T F e∇ne1 + h2

4me∇∇2ne1, (4)

and

ni1 = ne1, (5)

where U j1 (U j0) is the perturbed (unperturbed) fluid velocity ofthe jth species ( j equals to e for the electrons and i for the ions),n j1 is the perturbed number density with the equilibrium valuen j0, m j is the mass, h is the Planck constant divided by 2π , e is the

electronic charge, and c is the speed of light in vacuum. Eqs. (1)–(5) are only valid in the long wavelength limit, i.e., kλF e � 1,where “λF e” is the electron Fermi length. We have ignored thequantum effects of the cold ions due to their large mass comparedto the electrons. We have also neglected the term ne1(∇ · Ue0) inEq. (3) by assuming that a constant electron-streaming velocity.The inertialess hot electrons are assumed to obey the Fermi pres-sure law with the electron Fermi temperature T F e . The electronquantum mechanical effects are appearing in the last two terms of(4) showing the quantum Fermi statistics and quantum correlationof density fluctuations. Here we are assuming that the phase speedof the mode under consideration is much smaller than the elec-tron Fermi speed (ω/k � V F e) and it is much larger than quantumBohm potential speed (V F e � hk/me) [32].

Assuming a plane wave solution of the form exp(iky − iωt) toall the perturbed quantities and using the drift approximation forwhich ω2

ci � (ω−kUi0)2 into Eq. (2), we readily obtain the various

components of the ion fluid velocity

Uix = E y

B0and Uiy = −i

ω − kUi0

ωci

E y

B0. (6)

Here Uiy represents the ion-polarization drift. Thus, by insertingEq. (6), the ion continuity equation (1) becomes

(ω − kUi0)ni1 + iE y

B0

(∂ni0

∂ X+ ni0(ω − kUi0)k

ωci

)= 0. (7)

From Eq. (4), various components of the electron fluid velocity canbe written as

Uex = E y

B0+ i

(2T F e + h2k2

4me

)k

eB0

ne1

ne0and Uey = 0. (8)

It is important to note that the electron polarization drift vanishesfor me/mi � 1. Substituting (8) into (3) we obtain

E y

B0= iω

∂ne0/∂ Xne1 − i

(2T F e + h2k2

4me

)k

eB0

ne1

ne0. (9)

In deriving (9), we have assumed that the electron streaming speedUe0 → 0.

Eliminating E y/B0 from (7) and (9) and using the plasma ap-proximation, i.e., ni0 = ne0 = n0 and Eq. (5), we obtain the follow-ing resultant equation

ω(ω − kUi0)

= −knUi0ωci +(

2T F e

mi+ h2k2

4memi

)kn

(kn + k(ω − kUi0)

ωci

),

(10)

where kn = (∂n0/∂ X)/n0 is the inverse of density scale length.Eq. (10) can also be rewritten as

ω2 + ωk

(g

ωci− knU 2

q

ωci

)− gkn

(1 + knU 2

q

g+ k2U 2

q

ω2ci

)= 0, (11)

where g(= −Ui0ωci) is the constant gravitational field. Eq. (11) isthe desired modified dispersion relation for the RT mode of in-stability in a dense quantum magnetoplasma. Here, Uq = (C2

i +h2k2/4memi)

1/2 represents a new quantum speed which appearsdue to the presence of quantum Fermi pressure and the quantumBohm potential. Ci = (2T F e/mi)

1/2 is the quantum ion-acousticspeed using the drift approximation. One can retrieve the earlierresults of Ref. [33] by ignoring the quantum effects, i.e. by takingh → 0.

2942 S. Ali et al. / Physics Letters A 373 (2009) 2940–2943

Fig. 1. The normalized growth rate (γ /ωci ) of RT instability is plotted against the normalized wavenumber (k/kn) in a cold magnetoplasma with (a) classical case (b) quan-tum case. Here, we have taken different values of the density gradient scale lengths, i.e. kn = k/10 (green dashed curve), and kn = k/8 (blue dashed curve) in case (b).(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)

The solutions of Eq. (11) can be written as

ω = −k

2

(g

ωci− knU 2

q

ωci

)

±[

k2

4

(g

ωci− knU 2

q

ωci

)2

+ gkn

(1 + knU 2

q

g+ k2U 2

q

ω2ci

)]1/2

.

(12)

Thus, the Rayleigh–Taylor mode of instability would develop underthe following condition, such that

gkn

(1 + knU 2

q

g+ k2U 2

q

ω2ci

)>

k2

4

(g

ωci− knU 2

q

ωci

)2

. (13)

It is evident from Eq. (12) that RT instability would develop if gand kn are in the opposite direction.

Expressing the quantum speed in terms of quantum param-eter (He) and the electron Fermi length (λF e), as Uq = Ci(1 +H2

e k2λ2F e/4)1/2, where He (= hωpe/2T F e) is ratio of the plasmon

energy to the Fermi energy, and λF e = (2T F e/4πn0e2)1/2. FromEq. (12), we obtain the growth rate and real part of the wave fre-quency for RT mode of instability as

γ = Im(ω)

≈[−gkn

{1 + k2ρ2

s

(1 + H2

e k2λ2F e

4

)}

+ k2nC2

i

(1 + H2

e k2λ2F e

4

)]1/2

, (14)

and

Re(ω) = −k

2

{g

ωci− knC2

i

ωci

(1 + H2

e k2λ2F e

4

)}. (15)

Here ρs (= Ci/ωci) is the ion-sound gyroradius. It is worth men-tioning here that Eqs. (14) and (15) are in the agreement to the ap-proximation kλF e � 1. For numerical analysis, we express Eq. (11)by employing the normalized parameters such as γ = γ /ωci , g =gkn/ω2

ci , kn = kn/kn , ωci = ωci/ωci , k = k/kn , and Uq = Uqkn/ωci .Fig. 1 displays the variation of the normalized growth rate of RT in-stability versus the normalized wavenumber representing both theclassical (Fig. 1(a)) and quantum (Fig. 1(b)) cases. It is evident fromFig. 1(b) that the variation of the density gradient leads to increasethe normalized quantum speed and as a result the growth rateof RT instability significantly increases. Furthermore, for a densequantum plasma system, the mode is highly localized.

3. Summary

To summarize, we have presented a model to estimate thegrowth rate of Rayleigh–Taylor instability in a nonuniform densequantum magnetoplasma whose constituents are the electrons andions. The quantum hydrodynamical equations have been used byassuming that the ions are cold and classical while the electronsare inertialess and thus treated as quantum mechanically. New dis-persion relation for the RT instability has been derived under thedrift approximation and analyzed both analytically as well as nu-merically. It is found that the variation of the quantum speed anddensity gradient modifies the RT instability and the growth rate ofinstability has significantly been increased in a very dense quan-tum magnetoplasma system. Finally, our results should be usefulto understand the underlying physics of nonuniform dense astro-physical quantum magnetoplasmas.

Acknowledgements

Useful discussions with Dr. Waqas Masood are gratefully ac-knowledged. This work was supported by the Quaid-i-Azam Uni-versity, Research Fund, under the project URF (2008-09).

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