(2008) 74 (5) ISSN

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1 Eliasson, Bengt and Shukla, Padma (008) Ion solitary waves in a dense quantum plasma. Journal of Plasma Physics, 74 (5). pp ISSN , This version is available at Strathprints is designed to allow users to access the research output of the University of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may not engage in further distribution of the material for any profitmaking activities or any commercial gain. You may freely distribute both the url ( and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge. Any correspondence concerning this service should be sent to the Strathprints administrator: strathprints@strath.ac.uk The Strathprints institutional repository ( is a digital archive of University of Strathclyde research outputs. It has been developed to disseminate open access research outputs, expose data about those outputs, and enable the management and persistent access to Strathclyde's intellectual output.

2 J.PlasmaPhysics(008), vol. 74, part 5, pp c 008 Cambridge University Press doi: /s x Printed in the United Kingdom 581 Letter to the Editor Ion solitary waves in a dense quantum plasma B.ELIASSONandP.K.SHUKLA Theoretische Physik IV, Ruhr Universität Bochum, D Bochum, Germany (bengt@tp4.ruhr-uni-bochum.de) (Received17March008andaccepted7May008,firstpublishedonline13June008) Abstract. The existence of localized ion waves in a dense quantum plasma is established. Specifically, ion solitary waves are stationary solutions of the equations composed of the nonlinear ion continuity and ion momentum equations, together with the Poisson equation and the inertialess electron momentum equation in which the electric force is balanced by the quantum force associated with the Bohm potential that causes electron tunneling at nanoscales. The solitary ion waves are characterized by a large-amplitude electrostatic potential and ion density maxima and smaller amplitude minima on the flanks of the solitary waves. We identify the speed interval for the existence of the ion solitary waves around a quantum Mach numberthatisoftheorderofunity. Dense quantum plasmas are ubiquitous in microelectronics and nanotechnologies(e.g. semiconductors[1] and micromechanical systems[], quantum dots and nanowires[3], resonant tunneling[4], quantum diodes[5] and nanoelectron tubes/ nanotriode [6]), in intense laser solid density plasma experiments [7], as well as in compact astrophysical objects [8 10] (e.g. giant planetary interiors, neutron stars/magnetars, massive white dwarfs, supernovae). In quantum plasmas, the electrons and anti-electrons are degenerate since the interparticle distance is comparable to the de Broglie wavelength. Here quantum mechanical effects(e.g. electron and positron tunneling through the Bohm potential barrier[11]) play an important role. The effect of quantum force associated with the Bohm potential has been incorporated into studies of linear and nonlinear electrostatic waves [1 15] in quantum plasmas. In this letter, we present a theory for arbitrary large-amplitude nonlinear ion waves in an unmagnetized quantum plasma. For our purposes, we shall use the nonlinear ion continuity and momentum equations, together with the Poisson equation and the equation of motion for the inertialess electrons. We assume that the quantum force acting on electrons dominates over the quantum statistical pressure,whichamountstoassumingthat k B T Fe n e ( /4m e ) n e,wherek B isthe Boltzmannconstant, T Fe isthefermielectrontemperature, istheplanckconstant dividedby π, m e istheelectronmass,and n e istheelectronnumberdensity.the latter is then obtained from the inertialess electron momentum equation e φ + m e ( ) n e =0, (1) ne

3 58 B.EliassonandP.K.Shukla which dictates that the electrostatic force is balanced by the quantum force(minus thegradientofthebohmpotential).here eisthemagnitudeoftheelectroncharge and φ is the electrostatic potential. The electrons are coupled with ions through the space charge electric field( φ). The ion dynamics is governed by the ion continuity and ion momentum equations and n i t + (n iv i )=0, () [ ] vi m i t +(v i )v i = e φ, (3) where n i istheionnumberdensity,v i istheionfluidvelocityperturbation,and m i istheionmass.thesystemofequations(1) (3)isclosedbythePoissonequation φ =4πe(n e n i ). (4) Letusnowassumeaslabgeometrywithspatialvariationsonlyalongthe x-axis, so that = x / xand v i = xu i,where x is the unit vector along the x-axis in acartesiancoordinatesystem.furthermore,welookforstationarynonlinearion wave structures moving with a constant speed u 0. Hence, all unknowns depend onlyonthevariable ξ = x u 0 t[16,17].defining n e ψ,wenotethat(1)can be integrated once to obtain ψ + eφψ =0, (5) m e ξ where for localized disturbances we have used the boundary conditions [16,17] φ =0and / ξ =0at ξ =. Equations()and(3)canbeintegratedoncewiththeboundaryconditions n i = n 0 and u i =0at ξ =,andtheresultscanbecombinedtogive n 0 u 0 n i =. (6) u 0 eφ/m i Inserting(6) into(4) we obtain ( φ ξ =4πe ψ n 0 u 0 u 0 eφ/m i ). (7) Equations(5)and(7)arethedesiredequationsforthestudyofnonlinearionwaves in dense quantum plasmas. It is convenient to introduce dimensionless quantities(see below(9)) into(5) and (7),andrewritethemas Ψ X + ΦΨ =0, (8) and Φ X M Ψ + =0, (9) M Φ wherewehavenormalizedthespacevariableasx = k q ξ,theelectronwavefunction as Ψ= n 0 ψ,andthepotentialas Φ=eφ/m i c q.herec q = ω pi /k q isthequantum ionwavespeedand k q =(m e ω pe / ) 1/ isthequantumwavenumber.thequantum Machnumber isdefinedas M = u 0 /c q.furthermore, ω pe =(4πn 0 e /m e ) 1/ and

4 Ion solitary waves in a dense quantum plasma 583 Figure1.Theprofilesofthepotential Φ (toppanel)andtheelectrondensity Ψ (bottom panel)asafunctionof X,fordifferentvaluesoftheMachnumber: M =1.1(dashedcurves), M =0.9(solidcurves),and M =0.75(dottedcurves). ω pi =(4πn 0 e /m i )aretheelectronandionplasmafrequencies,respectively,where n 0 istheequilibriumelectronnumberdensity. We note that the coupled Equations(8) and(9) admit a conserved quantity ( Ψ H = X ) + 1 ( ) Φ ΦΨ M( M X Φ M)=0, (10) where we have used the boundary conditions Φ= Φ/ X = Ψ/ X =0,and Ψ=1at X =.Forasymmetricsolitaryionwavestructure,wecanassume that Φ=Φ max and Ψ=Ψ max,aswellas Ψ/ X = Φ/ X =0at X =0.Hence, at X =0,(10)yields Φ max Ψ max + M( M Φ max M)=0. (11) In the wave-breaking limit, where M = (Φ max ) 1/, we find that Φ max Ψ max + Φ max =0,orΨ max =.Accordingly,theelectrondensitywilllocallyrisetotwice the background density at wave breaking. We have numerically solved (8) and (9), and the resulting profiles of the electrostatic potential and electron number densities for different values of M are displayed in Fig. 1. We see that both the electrostatic potential and the electron density have localized and strongly peaked maxima and an oscillatory tail. The latter is in sharp contrast to the classical(non-quantum) case, where the ion acoustic solitary waves have a monotonic profile, which in the small-amplitude limit, where the system is governed by the Korteweg de Vries equation[18], assumes a secant hyperbolicusshape[16].weobservefromfig.1that M =1.1isclosetothewavebreaking limit above which there do not exist solitary wave solutions. Our numerical investigationalsosuggeststhatthereisalowerlimitof M(slightlylowerthan0.75), below which the solitary wave solution vanishes. In the numerical solution of(8) and (9), we used a centered second-order difference scheme to approximate the

5 584 B.EliassonandP.K.Shukla secondderivatives.attheboundariesat X = ±0weused Φ=0and Ψ=1.The resulting nonlinear system of equations was solved with Newton s method. To summarize, we have studied fully nonlinear ion waves in an unmagnetized quantum plasma, in which the electrostatic and quantum forces acting on the electrons are in balance. The solitary ion waves arise due to a balance between nonlinearities(coming from the divergence of the ion flux and ion advection) and the dispersion originating from the quantum electron tunneling effect. The associated electrostatic potential and density profiles have localized large-amplitude maxima, and smaller amplitude minima on each side of the maximum. The solitary waves existinalimitedvelocityintervalaroundthequantummachnumber M =1.Itis expected that the ion solitary wave, as found here, may be observed at nanoscales in dense plasmas. Acknowledgement This research was partially supported by the Swedish Research Council(VR). References [1] Markowich,P.A.,Ringhofer,C.A.andSchmeister,C.1990SemiconductorEquations. Berlin:Springer. [] Berggren,K.F.andJi,Z.L.1996Chaos6,543. [3] Shpatakovskaya,G. V. 006JETP 10, 466. [4] Kluksdahl,N.C.,Kriman,A.M.,Ferry,D.K.andRinghofer,C.1989Phys.Rev.B39, 770. [5] Lau,Y.Y.,Chernin,D.,Colombant,D.G.andHo,P.-T.1991Phys.Rev.Lett.66,1446. Ang,L.K.,Kwan,J.T.andLau,Y.Y.003Phys.Rev.Lett.91, Ang,L.K.andZhang,P.007Phys.Rev.Lett.98, [6] Driskill-Smith,A. A. G.,Hasko, D. G. andahmed, H.1999Appl.Phys.Lett.75, 845. [7] Marklund,M.andShukla,P.K.006Rev.Mod.Phys.78,591. Glenzer,S.H.etal.007Phys.Rev.Lett.98, Malkin,V.M.,Fisch,N.J.andWurtle,S.J.007Phys.Rev.E75, [8] Chabrier, G., Douchin, F. and Potekhin, Y. 00 Condes.Matter14, Chabrier,G.,Saumon,D.andPotekhin,A.Y.006J.Phys.A:Math.Gen.39,4411. [9] Harding,A.K.andLai,D.006Rep.Prog.Phys.69,631. [10] vanhorn,h.m.1991science5,384. [11] Gardner,C.L.andRinghofer,C.1996Phys.Rev.E53,157. Ancona,M.G.andIafrate,G.1989J.Phys.Rev.B39,9536. Manfredi,G.andHaas,F.001Phys.Rev.B64, [1] Pines,D.J.1961Nucl.EnergyC:PlasmaPhys.,5. [13] Haas,F.,Garcia,L.G.,Goedert,J.andManfredi,G.003Phys.Plasmas10,3858. [14] Manfredi, G. 005 Fields Inst. Comm. 46, 63. [15] Shukla,P.K.andEliasson,B.006Phys.Rev.Lett.96, Shukla,P.K.andEliasson,B.007Phys.Rev.Lett.99, Shaikh,D.andShukla,P.K.007Phys.Rev.Lett.99,1500. [16] Sagdeev, R. Z Reviews of Plasma Physics 4 (ed. M. A. Leontovich). New York: ConsultantsBureau,p.3. Sagdeev,R.Z.1979Rev.Mod.Phys.51,11. [17] Schamel,H.,Yu,M.Y.andShukla,P.K.1977Phys.Fluids19,186. Yu,M.Y.andShukla,P.K.1977PlasmaPhys.19,889. Shukla,P.K.andStenflo,L.1984Phys.Rev.A30,110. Shukla,P.K.andYu,M.Y.1078J.Math.Phys.19,506. [18] Washimi,H.andTaniuti,T.1966Phys.Rev.Lett.17,996.