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Electrostatic mode envelope excitations in warm pair ion plasmas with a small fraction of stationary positive ions – application in e-p-i and doped fullerene plasmas

Electrostatic mode envelope excitations in warm pair ion plasmas with a small fraction of stationary positive ions – application in e-p-i and doped fullerene plasmas

A.  Esfandyari-Kalejahi¤, I. Kourakis and P. K. Shukla ¤¤

¤Azarbaijan University of Tarbiat Moallem, Faculty of Science, Department of physics, 51745-406, Tabriz, Iran.

Ruhr-Universität Bochum, Institut für Theoretische Physik IV, D-44780 Bochum, Germany.

¤¤Ruhr-Universität Bochum, Institut fürTheoretische Physik IV, D-44780 Bochum, Germany.

Abstract. The nonlinear propagation of electrostatic wave packets in electron-positron-ion (e-p-i) plasmas, or pair- (e.g. fullerene) ion plasmas in the presence of a small fraction of uniform and stationary positive ions is studied. A two-fluid plasma model is employed. Two distinct electrostatic modes are obtained, namely a quasi-ion-thermal lower mode and a Langmuir-like optic-type upper one, as in pure pair plasmas, in agreement with previous experimental observations and theoretical studies of equal-temperature pair plasmas. The basic set of model equations is reduced to a nonlinear Schrödinger equation for the slowly varying electric field perturbation amplitude. The analysis reveals that the stability range of lower (acoustic) mode increases as the positive-to- negative-ion (or positron-to-electron) density ratio increases, so this quasi-thermal mode may propagate in the form of a dark-type envelope soliton (i.e. a potential dip, or a void) modulating a carrier wave packet for small wave-numbers, for a fixed value of the positive-to-negative-ion (or positron-to-electron) temperature ratio. On the other hand, the upper mode is modulationally unstable, and may thus favor the formation of bright- type envelope soliton (pulse) modulated wave-packets in the same wave-number region.

 

Keywords: Pair plasma,Electron-Positron-Ion Plasma, Modulational Instability, Envelope soliton

 

PACS: 52.27.Ep, 52.27.Aj, 52.35.Mw

 

THE MODEL EQUATIONS

 

The present study is devoted to an investigation of the nonlinear amplitude modulation of electrostatic modes [1] propagating parallel to the external magnetic field, in e-p-i plasmas, which is an extension to our previous work on pure pair plasma [2]. Recently, the production of pair fullerene-ion plasmas in laboratory [3, 4] has enabled experimental studies of pair plasmas rid of intrinsic problems involved in electron-positron plasmas, namely pair recombination processes and strong Landau damping. Here, we consider the nonlinear propagation of electrostatic wave packets in e-p-i plasmas or pair- (e.g. fullerene) ion plasmas in the presence of a small fraction of uniform and stationary positive ions, by employing a two-fluid plasma model. The two-fluid plasma-dynamical (moment) equations for our three-component plasma include the two density (continuity) equations

 

THE PERTURBATIVE ANALYSIS.

 

In order to obtain an explicit evolution equation describing the propagation of modulated EA envelopes, from the model Eqs (1)-(3), we shall employ the standard reductive perturbation (multiple scales) technique [5]. The independent variables x and t are stretched as x = e(x ¡lt) and t = e2t , where e is a small (real) parameter; here, l is a free (real) parameter, which is to be later determined as the wave’s group velocity by compatibility requirements. The dependent variable vector Sa is expanded as

 

 

 

NUMERICAL ANALYSIS

 

We have seen that two distinct electrostatic modes, namely a quasi-thermal lower mode and a Langmuir-like optic-type upper one, may propagate in our plasma system in the linear approximation; see Eqs. (7) and (8). Now, We may investigate the numerical value of the quantities PQ and P=Q in terms of the relevant physical parameters, namely the positron-to-electron (or positive-to-negative ion) density and temperature ratio(s), b and s, respectively, for these modes. The results of the calculations for the lower and higher modes are shown in Figs. 1 and 2 respectively. We conclude that the lower (acoustic) mode is generally stable, for realistic large wavelength situations (cf. Fig. 2) and may propagate in the form of a dark-type envelope soliton (i.e. a potential dip, a void). On the other hand, the upper (Langmuir-like) mode is modulationally unstable (cf. Fig. 3), and may favor the formation of bright-type envelope soliton (pulse) modulated wave packets at low wave-numbers. Fig.1 reveals that the stability range of the lower (acoustic) mode increases as the positive ion (or positron) to negative ion (or electron) ion density ratio b increases. Furthermore, careful inspection of Figs. 1 and 2 shows that the temperature ratio s is an important factor, from the point of view of stability, for both modes. In specific, one may anticipate that a local coexistence of positrons with a colder (warmer), say, population of negative electrons, viz. s < 1 (s > 1), may critically affect the stability profile of electrostatic modes, for instance by stabilizing the lower mode, or by destabilizing the upper mode.

 

Acknowledgments I.K. acknowledges partial support by the Deutsche Forschungsgemeinschaft (Bonn, Germany) through the Sonderforschungsbereich (SFB) 591.

 

 

REFERENCES

 

 

1.  T. STIX, “Waves in plasmas” (New York: American Institute of Physics, 1992)

 

2.  I. KOURAKIS, A. ESFANDYARI-KALEJAHI, M. MEHDIPOOR AND P.K. SHUKLA, Modulated electrostatic modes in pair plasmas, submitted to Physics of Plasmas, (2006).

 

3.  W. OOHARA AND R. HATAKEYAMA, Phys. Rev. Lett, 91, 205005 (2003).

4.  W. OOHARA, D. DATE AND R. HATAKEYAMA, Phys. Rev. Lett, 95, 175003 (2005).

5.  T. TANIUTI AND N. YAJIMA, J. Math. Phys, 10, 1369 (1969).

6.  P. SULEM AND C. SULEM , “Nonlinear Schrödinger Equation” (Speringer, Berlin, 1999).

7.  R. FEDELE, Phys. Scr, 65, 502 (2002).

8.  I. KOURAKIS AND P.K. SHUKLA, Nonlinear Proc. Geophys, 12, 407 (2005).

 

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