Physics Letters A 372 (2008) 4827–4830

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Physics Letters A

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Potential distributions in dense plasmas composed of degenerate electronsand positive nanoparticles

S. Ali a,b,∗, P.K. Shukla a,c

a Institut für Theoretische Physik IV, Fakultät für Physik und Astronomie, Ruhr-Universität Bochum, D-44780 Bochum, Germanyb Department of Physics, Government College University, Lahore 54000, Pakistanc School of Physics, University of KwaZulu-Natal, Durban 4000, South Africa

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

Article history:Received 15 April 2008Accepted 19 April 2008Available online 22 May 2008Communicated by V.M. Agranovich

The test charge potential involving the electron dust acoustic oscillations is computed in a two-component plasma whose constituents are hot electrons and positive nanoparticles. The hot degenerateinertialess electrons are assumed to follow the Thomas–Fermi distribution, while positive nanoparticlesare inertial. The expressions for the Debye–Hückel and wake potentials due to a moving test chargeare obtained. Furthermore, the effects of the Fermi energy, the number density of the hot degenerateelectrons, and the test charge speed on the potential profiles are numerically examined.

© 2008 Published by Elsevier B.V.

The electrostatic potential and energy loss of a test particlehave been studied due to their applications in many areas suchas in wake-field accelerators, in inertial confinement fusion (ICF),and in the context of crystallization of dust particles in dustyplasmas. The test charge propagation produces the possibility ofa far-field potential [1], in addition to the Debye–Hückel poten-tial. However, when the speed of the test charge equals the phasespeed of plasma oscillations, an oscillatory wake-field [2] is formedbehind the test charge. The wake-field contains both the positiveand negative potential regions. In the negative potential region thepositive ions can be trapped, and there appears an attractive forcebetween the same polarity charges (e.g., the electrons and negativedust grains [3]).

Quantum plasmas are quite common in environments like insemiconductor devices [4], quantum wells, quantum dots, andquantum nanowires [5], carbon nanotubes [6], quantum diodes [7],biophotonics [8], ultracold plasmas [9], and microplasmas [10], insuperdense white dwarfs and neutron stars [11], and in intenselaser-solid density plasma experiments [12–14], etc. The physicsof such plasmas can be described by the Wigner–Poisson andthe Schrödinger–Poisson systems [15] describing the kinetic andfluid behaviors of the plasma particles at quantum scales. Fordense quantum plasmas, the quantum mechanical effects [16] be-come relevant when the de-Broglie wavelength of the charge car-ries is equal to or greater than the average interparticle distance

* Corresponding author at: Department of Physics, Government College Univer-sity, Lahore 54000, Pakistan.

E-mail addresses: shahid_gc@yahoo.com (S. Ali), ps@tp4.rub.de (P.K. Shukla).

0375-9601/$ – see front matter © 2008 Published by Elsevier B.V.doi:10.1016/j.physleta.2008.04.072

d = n−1/3e0 , that is ne0λ

3Be � 1, where ne0 is the electron equilibrium

number density and λBe is the de-Broglie wavelength.

Numerous investigations have been made [17,18] for collec-tive modes with quantum corrections. Haas et al. [19] studied thelinear and nonlinear properties of the ion-acoustic waves (IAWs)in an unmagnetized quantum plasma. Shukla and Ali [20] havederived a linear dispersion relation for the dust acoustic wavesin a Fermi dusty plasma consisting of electrons, ions, and dustparticulates. Later, they extended their study for planar [21] andnonplanar [22] dust acoustic solitary waves, dust acoustic dou-ble layers [23], as well as investigated the ion and dust stream-ing instabilities [24] in an unmagnetized, collisionless dense dustyplasma.

Long ago, the test charge potential was computed without [25]and with ion dynamics [26] in an electron–ion classical plasma.Nambu and Akama [2] were the first who introduced the conceptof the wake-field potential in an electron–ion plasma. Some effortshave been made towards the multiple test charges (projectiles) ar-ranged into different geometries [27] to examining the correlationeffects upon the test charge potential and energy loss. However,the potential of a slowly moving test charge is calculated by Shuklaet al. [28] in an electron–hole Fermi plasma incorporating the lon-gitudinal perturbations. The study has further been extended [29]to solve numerically the wake-potential due to resonant interactionbetween the test charge and IAWs in a quantum plasma. Recently,Shukla and Eliasson [30] obtained the analytical expressions of theDebye and wake potentials at nanoscale.

In this Letter, we consider a two-component dense plasmaconsisting of hot degenerate inertialess electrons and positivenanoparticles. Such a situation is common in metallic plasmas withdoped nanoparticles which may be charged positively due to UV

4828 S. Ali, P.K. Shukla / Physics Letters A 372 (2008) 4827–4830

radiation. By using the dielectric constant for our two-componentdense plasma, we obtain the electrostatic potential around a mov-ing test particle. The effects of the Fermi energy, the electron num-ber density, and the test charge speed on the wake-field potentialprofile are examined.

We consider a dense metallic plasma containing hot degener-ate inertialess electrons and mobile positively charged nanopar-ticles. At equilibrium, the charge-neutrality condition is Zd0nd0 ≈nh0, where Zd0 is the charging state of nanoparticles, n j0 is theequilibrium number density of the jth species ( j equals d forthe nanoparticles and h for hot degenerate electrons). In sucha plasma, there appears the electro-dust-acoustic (EDA) waveswhose phase speed lies between the Fermi speeds of the nanopar-ticles and hot electrons, i.e. V f d � ω/k � V f h . We also considera test particle of charge qt propagating through such a degenerateplasma with a constant velocity Vt along the z-axis. The dynam-ics and dispersion properties of the EDA waves can be describedby the following equations:

The equation of continuity for the nonparticles is

∂nd1

∂t+ nd0∇ · Ud1 = 0, (1)

the equation of motion for the inertial nanoparticles is

∂Ud1

∂t= − Zd0e

md∇ϕ1, (2)

the Poisson equation with a single test charge is

∇2ϕ1 = 4πe(nh1 − Zd0nd1) − 4πqtδ(r − Vtt), (3)

the number density of the hot degenerate inertialess electrons [16]is

nh = nh0

(1 + eϕ1

ε f h

)3/2

. (4)

Here ε f h is the Fermi energy of the hot degenerate inertialess elec-trons. Eq. (4) can be obtained by assuming the hot electrons tobe inertialess and degenerate following the pressure law [16] in athree-dimensional system as

ph = 2

5

ε f h

n2/3h0

n5/3h , (5)

where nh is the number density with its equilibrium value nh0and the perturbed value nh1, ε f h = me V 2

f h/2 is the Fermi energy,

V f h = (2kB T f h/me)1/2 is the Fermi speed, T f h is the Fermi tem-

perature of the hot electrons, δ is a three-dimensional Dirac deltafunction, and r is an arbitrary position vector from the test charge.The perturbed hydrodynamic nanoparticle velocity is denoted byUd1 and the linearized electrostatic potential by ϕ1. Eqs. (1)–(5)describe the dynamics of the EDA waves in a degenerate collision-less unmagnetized plasma. The quantum statistical effects appearthrough Eq. (5), representing the behavior of the degenerate hotelectrons, can be important in the investigation of superdense flu-ids. Applying space–time Fourier transformations to Eqs. (1)–(5),we obtain the Fourier transformed test charge potential and theFourier transformed perturbed number densities in ω − k space,as

k2ϕ1 = −4πe(nh1 − Zd0nd1) + 8π2qtδ(ω − k · Vt), (6)

where

nd1 = Zd0ek2ϕ1

ω2mdnd0, (7)

and

nh1 = 3

2

eϕ1

εnh0. (8)

f h

Here ω and k are the angular wave frequency and the wavevector,respectively.

By taking the inverse space–time Fourier transformations,Eq. (6) will reduce to the result [31]

ϕ1(r, t) = qt

2π2

∫exp[ik · (r − Vtt)]

ε(k,k · Vt)

dk

k2. (9)

Eq. (9) is the electrostatic potential calculated for a single testcharge at an arbitrary position r and time t . The dielectric con-stant ε(k,k · Vt) is now modified with the Fermi energy and canbe expressed as

ε(k,k · Vt) = 1 + 3

2

k2f h

k2− ω2

pd

(k · Vt)2, (10)

where ωpd = (4πnd0 Z 2d0e2/md)

1/2 is the plasma frequency for thenanoparticles and k f h = (4πnh0e2/ε f h)1/2 is the Fermi wavenum-ber associated with the Fermi energy, describing the inverse of theFermi shielding length.

To study the Debye and wake potentials caused by a test chargein the presence of the EDA waves in a degenerate plasma, we canexpress the inverse of the dielectric constant as

1

ε(k,k · Vt)= k2

k2 + 32 k2

f h

(1 + ω2

k

(k · Vt)2 − ω2k

), (11)

where ωk = Cedak/(1+2k2/3k2f h)1/2 represents the dispersion rela-

tion of the EDA waves in a degenerate collisionless unmagnetizedplasma, Ceda = (2/3)1/2ωpdλ f h is the EDA speed, and λ f h is theFermi shielding length.

Expressing the propagation vector k, the position vector r, andthe velocity vector Vt into the spherical polar coordinates [27], wesubstitute (11) into (9) and obtain the Debye–Hückel potential in amoving frame, as

ϕD(X, Y , ξ, t)

= qt

2π2

∞∫0

1∫−1

2π∫0

exp(ikX cosϕk + kY sinϕk + ikμZ)

k2 + 32 k2

f h

k2 dk dμdϕk.

(12)

Here X = ρ√

1 − μ2 cosϕr , Y = ρ√

1 − μ2 sinϕr , and Z = ξ =rz − Vt t represent the separation of a test charge from the positionvector r. Moreover, ρ = r sin θr , rz = r cos θr , μ = cos θk = cos(k, Vt),and sin θk = √

1 − μ2.After performing the integrations, we obtain

ϕD(r, t) = qt

rexp

(− 3r

2λ f h

), (13)

where r = √ρ2 + ξ2 is the distance of a test charge from the

position vector r, ρ and ξ are the radial and axial distancesfrom the test charge. Eq. (13) represents the expression for theDebye–Hückel potential in an unmagnetized collisionless degen-erate Thomas–Fermi metallic plasma incorporating the quantumstatistical effects.

To study the wake potential caused by a test charge in a degen-erate plasma, we substitute (11) into (9) and solve for the wakepotential in the cylindrical coordinates by using the standard tech-nique [2,27]. The result is

ϕW (ρ, ξ, t) = qt

πλ f h

∫J0(K⊥ρ/λ f h)exp(iK‖ξ/λ f h)

32 [ 2

3 (K 2⊥ + K 2‖ ) + 1] K⊥ dK⊥ dK‖

×( C2

eda(K 2⊥ + K 2‖ )/V 2t

(K 2 − K 2 )(K 2 + K 2 )

). (14)

‖ + ‖ −

S. Ali, P.K. Shukla / Physics Letters A 372 (2008) 4827–4830 4829

Fig. 1. (Color online.) The normalized wake potential (eϕW /ε f h) is plotted against the normalized axial distance (ξ/λ f h) from a test charge for different values of theFermi energies ε f h = 8.863 × 10−12 erg, solid line; ε f h = 8.877 × 10−12 erg, line of long dashes; ε f h = 8.891 × 10−12 erg, line of small dashes, with fixed Fermi speedV f h ∼ 1.39 × 108 cm/s and test charge speed Vt ∼ 1080 cm/s. Other parameters used in our numerical calculations are: nh0 ≈ 5.9 × 1022 cm−3, nd0 ≈ 7.5 × 1020 cm−3,ωpd = 1.45 × 1011 s−1, T f h = 6.4 × 104 K, λ f h = 9.0 × 10−9 cm, and Zd0 = 79.

(a) (b)

Fig. 2. (Color online.) Shows the dependence of the number density (nh0) and the test charge speed (Vt ) on the cosine function having fixed values of (a) an axial distance−7 −6 22 −3

ξ ∼ 10 cm and the test charge speed Vt ∼ 1080 cm/s, (b) ξ ∼ 8.9 × 10 cm and nh0 ≈ 5.9 × 10 cm . All other parameters are the same as in Fig. 1.

Here J0(K⊥ρ/λ f h) is the zeroth order Bessel function, K‖ (K⊥) isthe parallel (perpendicular) component of normalized wave vectorK = kλ f h and ξ = r‖ − Vtt . For small argument, i.e. K⊥ρ/λ f h < 1,we assume the zeroth order Bessel function to be unity. Thus, wecalculate residues at the poles K‖ = ±K+ where K 2± in (14) isgiven by

K 2± = ∓3

4

(1 + 2

3K 2⊥ − C2

eda

V 2t

)

+ 3

4

√√√√(1 + 2

3K 2⊥ − C2

eda

V 2t

)2

+ 8

3

C2eda K 2⊥V 2

t

. (15)

Under the limit K 2⊥ < 1, K 2± can be approximated from (15) asK 2+ ≈ C2

eda K 2⊥/V 2t and K 2− ≈ 3

2 (1 − C2eda/V 2

t ). Performing the inte-gration over K‖ , we obtain from Eq. (14)

ϕW (ξ, t) = −8qt Ceda

9Vtλ f h

(1 + C2

eda

V 2t

)(1 − C2

eda

V 2t

)−1

×∞∫

0

K 2⊥ sin

(Ceda K⊥Vtλ f h

ξ

)dK⊥. (16)

Integrating over K⊥ , we obtain

ϕW (ξ, t) = 8

9

qt

ξ

(1 + 2ω2

pdλ2f h

3V 2t

)(1 − 2ω2

pdλ2f h

3V 2t

)−1

× cos

(√2

3

ωpd

Vtξ

). (17)

Eq. (17) gives the wake potential due to a test charge propagat-ing with a constant speed along the z-direction through a de-generate metallic plasma. For Vt � ωpdλ f h and cos θ < 0 [where

θ = (2/3)1/2ξωpd/Vt ], one can have an attractive wake potentialas studied in Refs. [2,27]. The effects of Thomas–Fermi distributioncan be examined numerically on the wake potential for the limitVt ∼ Ceda .

For numerical analysis, we choose some typical parameters:nh0 ≈ 5.9 × 1022 cm−3, T f h = 6.4 × 104 K, and λ f h = 9.09 ×10−9 cm from Ref. [15]. In order to satisfy the charge-neutralitycondition Zd0nd0 = nh0, the equilibrium number density of thenanoparticles has been assumed nd0 ≈ 7.5 × 1020 cm−3 and thecharging state of the gold Zd0 = 79. We have numerically solvedthe wake potential obtained from Eq. (17) in a degenerate plasmainvolving the EDA waves.

Fig. 1 displays the normalized wake potential (eϕW /ε f h)

against the normalized axial distance (ξ/λ f h) of a test chargefor different Fermi energies due to the hot degenerate electronsε f h (= 8.863 × 10−12, 8.877 × 10−12, 8.891 × 10−12) erg, with afixed test charge speed Vt ∼ 1080 cm/s, the dust plasma frequencyωpd = 1.45×1011 s−1, and the Fermi speed V f h ∼ 1.39×108 cm/s.We are interested in examining the effects due to the Fermi ener-gies and the excitations produced by the test charge involving theEDA waves. The amplitudes of the wake potential decrease withthe increase of Fermi energy of the hot degenerate electrons andvice versa in case of test charge speeds. For Vt ∼ Ceda , an oscilla-tory wake-field becomes pronounced behind the test charge andis slowly damped against the direction of propagation of the testcharge. The negative wake-field potential plays an important rolefor providing the possibility of attraction for the similar polaritycharges. We have noted that the excitation of the wake-field occursin a degenerate plasma when the test charge speed is comparableto the EDA speed. To show wether the field potential is attrac-tive or repulsive, we plot the cosine function (cos θ ) against theequilibrium number density of the hot degenerate electrons (nh0)

and the test charge speed. The attractive and repulsive field re-

4830 S. Ali, P.K. Shukla / Physics Letters A 372 (2008) 4827–4830

gions can be determined by employing the numerical values of nh0and Vt as depicted in Fig. 2. For example, for nh0 ∼ 4 × 1022 cm−3

(∼ 6 × 1022 cm−3), we may have an attractive (repulsive) potentialas can be seen from cos θ ∼ −0.92(∼ 0.95), respectively. Similarly,for test charge speeds Vt(∼ 1003,1005) cm/s, we obtain negativeand positive values of cos θ(∼ −0.99,0.47) under the approxima-tion λ f h � ξ . We have also observed from (17) that 1/ξ is playinga role in the slowly damped behavior of the wake potential whichis completely absent from Fig. 2.

To summarize, we have presented the linearized test charge po-tential around a test charge propagating with a constant speedalong the z-axis in a two-component degenerate plasma. The lat-ter is consist of the hot degenerate inertialess electrons follow-ing the Thomas–Fermi distribution and the cold inertial positivenanoparticles. Numerical and analytical calculations reveal thatan oscillatory wake field is pronounced when the test chargespeed equals the EDA speed; the latter is significantly affectedby the variation of the Fermi energy and the number densityof the hot degenerate electrons. The present results can be offundamental importance in the context of repulsive and attrac-tive fields due to the degenerate electrons in a gold metallicplasma.

Acknowledgements

S.A. acknowledges the financial support from the EU-ProjectHPRN-2001-00314 “Turbulent Boundary Layers in Geospace Plas-mas”.

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