Stimulated Raman Scattering in a Magnetized Electron-Positron-Ion Plasma

Stimulated Raman Scattering in a Magnetized Electron-Positron-Ion Plasma

N. Tinakiche1,2, R. Annou2

1Departement of Physics, Faculty of Science, U.M.B.B, Boumerdes, ALGERIA

2Faculty of physics, U.S.T.H.B, Algiers-ALGERIA

Abstract – Electron-positron plasmas that may be found in numerous environments, such as, active galactic nuclei, pulsar magnetosphere, solar flares, ultra-short laser-matter interaction and interplanetary space, have dielectric properties appreciably different from those of classical electron-ion plasmas since modes occurring on the ion timescale disappear due to a mass symmetry effect. These properties as well as excited instabilities have been studied earlier, whereas in the present work, we focus on stimulated Raman scattering instability (SRS) in a magnetized electron-positron plasma containing a fraction of ions (e-p-i plasma), where the background magnetic field is parallel to the pump electric field.

1. INTRODUCTION

   Electron-positron (e-p) plasmas can be found in the early universe, in astrophysical objects such as pulsars, active

   galactic nuclei, supernovae remnants, in -raybursts and at

   the center of the Milky Way galaxy 1,2. These plasmas are created by collisions between particles that are accelerated by electromagnetic and electrostatic waves and/or by gravitational forces. High energy laser-plasma interactions and fusion devices can be used to produce e-p plasmas. Electron-positron plasmas are also created in large tokamaks3 through collisions between Mev electrons and thermal particles. Due to the mass symmetry in e-p plasmas, there are fewer spatial and temporal scales on which collective effects (e.g, electrostatic and electromagnetic waves as well as their instabilities and coherent nonlinear structures, etc) can occur 1,2. For instance, Iwamoto 4 has given an elegant description of the linear modes in a non

   relativistic pair plasma that is magnetized. In addition, Zank

   stimulated Brillouin scattering if the excited plasma wave is an ion-acoustic wave. Furthermore, while the unmagnetized plasma may reveal three wave modes, namely, Langmuir, ion-acoustic and electromagnetic modes, the magnetized plasma may support a great number of new wave modes 10. The basic modes of magnetized plasmas such as those that are found in molecular clouds, cometary plasmas and stellar atmospheres are of a great interest 11 , for instance, Alfvén waves are of importance to the understanding of many basic plasma phenomena 12,13. It has been shown also that the magnetic field generated in laser- produced plasmas are strong enough to alter the spectrum of electrostatic modes in the plasma but not strong enough to modify the characteristics of propagation of the incident and scattered electromagnetic modes14. In the present paper, the stimulated Raman scattering is studied in a magnetized e-p plasma containing a fraction of ions (e-p-i plasma). As a matter of fact, astrophysical and laboratory plasmas contain in general a fraction of ions, e.g, electron-positron-ion plasmas are encountered in the magnetosphere of neutron stars, and also in the solar atmosphere15-18. Indeed, these plasmas received some attention from numerous authors19-

   21. The maximum growth rate of the stimulated Raman scattering process is calculated and compared to a previous study. The paper is organized as follows: in Sec.2, the

   problem is exposed and solved and we conclude in Sec.3

2. THEORY

   B0

   We consider an (e-p-i) plasma embedded in a uniform background magnetic field . We consider a large

   5 amplitude electromagnetic pump wave with an electric field

   and Greaves studied the linear properties of various

   parallel to , where,

   electrostatic and electromagnetic waves in unmagnetized and magnetized pair plasmas. Recently, Annou and Bahamida6 have studied the parametric coupling in electron- positron plasma and they obtained the nonlinear dispersion

   Et B0

   t t0 t

   E 2E cos kt . x t

   , (1)

   relation and the nonlinear growth rate of the parametrically generated modes in this kind of plasmas. On the other hand,

   The equilibrium state contains electrons, positrons and ions. The electrons and positrons oscillating respectively with the

   Bornatici et al.7, Liu and Rosenbluth8and Drake9studied also

   – +

   the parametric interaction of an intense coherent electromagnetic wave with collective modes in an electron- ion plasma. If the excited waves are both electrostatic, they may be absorbed in the plasma. In contrast, if one of the excited plasma waves is electromagnetic, it can be scattered. This process is called stimulated Raman scattering if the excited plasma wave is a Langmuir wave and it is called

   velocitiesV and V given by,

   te te

   V – Et0 sin kt . x t t , (2)

   2e

   te mt

   2e

   V + Et0 sin kt . x t t

   perturbation whereas, N

   0e and N0e

   • are respectively,

     te mt (3)

     the equilibrium electron and positron densities. At the

     equilibrium state, we have

     N0e N 0i

     • N0e , where

       where, m is the electron and positron mass. The ions are

       considered to be immobile because of their inertia. In the SRS process where electrons as well as positrons

       N is the equilibrium ions density. The velocities

       –

       V

       0i t e

       +

       t , kt

       participate, the electromagnetic pump wave decays

       and

       V which represent respectively the response of

       t e

       into an electromagnetic sideband wave

       and an

       electrons and positrons to the sideband are given by,

       t, kt

       electrostatic modified Langmuir wave

       with the

       – 2e t0

       tt ,

       following constraints,

       1, k1

       V E

       t

       te m

       sin

       kt . x

       (10)

       t t 1 (4)

       and,

       and,

       2e

       V + Et0 sin kt. x tt ,

       kt kt + k1 (5)

       te mt

       (11)

       Equations (4) and (5) when used with the dispersion relations of the modes involved prove to encompass both the forward and backward Raman processes as the characteristics of the parametrically generated mode are sensitive to the scattering angle. Let us perturb this

       where the field t 2 t0 cos . x t t represents the sideband electromagnetic wave.

       E E kt

       To obtain the equation (9) we have used the fact that

       equilibrium and study the time development of this perturbation using the linearized fluid equations and Maxwells equations which are given by (c.f.Ref. [14]),

       n

       V

       V

       +

       te te

       and,

       – , (12)

       1e N

       –

       +

       –

       t 0e

       .V 0

       1e

       (6)

       V V

       t e t e

       (13)

       n

       Taking the divergence of the equations (8) and (9), and

       +

       1e N

       t 0e

       .V 0

       1e

       (7)

       using the following equation,

       3KT

       V –

       e 1

       . E1

       4 e n

       n

       (14)

       1e

       e n

       E1 c V

       – B0

       1e 1e

       t m N0e 1e m 1e

       where, n

       n0 1t and

       cos k1. x

       V

       – V –

       1e 1e

       te t e

       n n0

       +

       V

       (8)

       3KT e

       1

       We find the following set of equations,

       1e

       1e cos

       k1. x 1

       1e

       e n E1 c V + B0

       t m N0e 1e m 1e

       2n

       3K T

       2 2 2

       V

       – V –

       1e e k1 n

       • pe n

       • n

         ce n

         te te

         (9)

         t2

         N

         0e

         m 1e

         V

         V

         – –

         1e 1e 1e

         T , T being respectively the electron and positron

         te t e

         V –

         V +

         E1

         e e (15)

         temperatures, n ,

         1e

         n , ,

         1e 1e 1e

         and

         are

         respectively the electron density perturbation, the positron density perturbation, the electron velocity perturbation, the positron velocity perturbation and the electric field

         2n 3K T

         On the other hand, the sideband

         2 m 1 1e

         pp 1e

         1e

         ce 1e

         E 2E cos kt . x

         t

         V

         N

         0e

         – V –

         equation,

         2

         te t e

         1 Et

         4

         (16)

         Et 2 2

         2 e (n

         )V – e( n )V +

         c t c

         t

         1e te

         1e te

         where, 2

         4 N e2

         0e

         ,2

         4 N e2

         0e

         and

         By using Eq.(12), Eq.(23) may be cast as follows,

         (23)

         pe m

         2 eB0 2

         pp m

         1 2

         ce

         . The propagation has been considered

         Et Et

         4 e

         n

       • n

         V –

         (24)

         mc

         perpendicular to

         . By using the following relationship,

         c2 t2

         c2 t 1e

         1e te

         B0

         n n

         1e 1e

         (17)

         The scattered electromagnetic wave satisfies then the following equation,

         and,

         t Et 4 e 1 n V

         – ,

         2 2

         t2

         t 1e te

         N0e N0e (18)

         With,

         where,

         (25)

         2 k2 c2 1 2

         0e

         N

         t t pe

         (26)

         0e

         N

         (19)

         where, Eq.(18) may be imposed by experiment whereas Eq.

         In other hand, we have,

         e2k 2

         (17) is derived by way of conservation equations ( and

         V – V – 2 1 Et . Et

         are constants), the subtraction of Eq.(16) from Eq. (15) leads to the following differential equation,

         and,

         te t e

         m tt

         (27)

         2 u n

         N

         V

         – V –

         et

         2

         t

         2

         1e

         1

         1 0e

         te t e

         n V

         – n

       • Et

       (20)

       t 1e

       te

       mt 1e

       (28)

       To obtain Eq. (20), we have used the fact that,

       T T Te

       Replacing these results in Eqs. (20) and (25), we find,

       2 u

       2 2

       e e (21)

       t

       n

       1e

       • Et Et

       u

       In Eq. (20), 2 is given by,

       (29)

       1

       3KT

       2 2 E

       E n

       (30)

       2

       2 1 2

     • e k2

   2 t t

   t 1e

   u 1

   ce pe m 1

   t

   (22)

   where,

   2 1 2 3K Te k 2 , being the dispersion relation 2 2

   1 pe m 1

   1 pe k1

   of the modified Langmuir wave.

   1

   4 m ut .ut,

   t t

   and,

   (31)

   2

   D

   We plot on figures 1and 2, the variation of the growth rates

   1 t

   t

   pe

   N0e

   ut .ut , (32)

   ratio

   maxe pi

   / maxeion

   in terms of k1

   * and

   respectively in the case

   k 2 c2 2

   for an interstellar

   since and

   can be written as,

   t pe

   Et Et

   plasma, where maxe pi / maxeion is given by,

   Et Et ut ,

   and,

   (33)

   maxe pi

   / maxeion

   2 1/ 4

   2 ce ,

   E E u ,

   1 2

   t t t

   (34)

   1

   1 3 / 4 pe

   1

   u

   and u specify the direction of polarization of the fields

   1 ce 3 Te k *

   t t

   2 *

   2

   2 1 D

   pe T

   Et and Et .

   D

   Using the Fourier transformation and after some algebra,

   (38)

   one finds (c.f. Ref.[14]) the maximum growth rate

   max of

   where *

   is the Debye length corresponding to

   D

   the stimulated Raman scattering process,

   T* 0.01eV . On figure 1, is plotted

   maxe pi

   maxe pi

   / maxeion

   versus k1

   * for three

   1 2 k

   u .u

   4 1 1

   1

   pe 1 t t

   1/ 2 Et0

   different temperatures, where pe 10 s

   1

   , ce 10 s ,

   1 2 2 3K Te

   2 1/ 2

   0.3 , Te 10

   eV , 1eV , and 10eV . On figure 2, it is

   4t t N0e m ce 1 pe

   k1

   1

   m

   shown the variation of

   maxe pi / maxeion in

   (35)

   terms of where ce 10 s1, pe 104 s1 , Te 1eV

   The maximum growth rate of the instability is proportional to the amplitude of the electromagnetic pump wave Et0 .

   On the other hand, the maximum growth rate of the SRS process calculated for an e-ion plasma in Ref.[14] is given by,

   and k1

   * 0.2 .

   D

   0,58

   2

   0,57

   maxeion

   pe k1 ut .ut Et0

   1/ 2

   2 2

   3K Te

   2 1/ 2

   0,56

   Te=10-1eV

   Te=1 eV Te=10 eV

   4t N0 m ce pe

   m k1

   t ' sh

   where,

   , (36)

   0,55

   0,54

   max(e-p-i) max(e-ion)

   2 k 2 c2 2 ,

   D

   0

   2

   t ' sh t pe

   (37)

   k1 *

   D

   By setting

   0 (no positron in the plasma) in the

   expression of maxe pi (c.f.Eq.(35))we get the

   Fig.1 : Ratio

   maxe pi / maxeion

   versus k1

   * for

   expression of the maximum growth rate an (e-i) plasma given by Eq.(36).

   maxeion for

   different temperatures Te .

   max(e-p-i) max(e-ion)

   1

   0

   0,0 0,2 0,4 0,6 0,8 1,0

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      e pi / maxeion

3. CONCLUSION

versus .

ICPDP, AIPCP 649, Edited by

R. Bharuthram et al., South Africa (2002).

1. L. N. Tsintsadze, N. L. Tsintsadze, P. K .Shukla and L. Stenflo, Astrophys. Space Sci.,

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This paper is devoted to the study of the stimulated Raman scattering in a magnetized electron-positron-ion plasma. The maximum growth rate of this parametric process is calculated and compared to the maximum growth rate obtained in a previous study by Shivamoggi 14 in the case of a magnetized e-i plasma. It is found that the ratio

maxe pi / maxeion increases for increasing

1. P. K. Shukla, G. Brodin, M. Marklund, and L. Stenflo, Wake Field Generation and Non-Linear Evolution in a Magnetized Electron-Positron-Ion Plasma, e-print arXiv: 0805.1617 vl [physics. Plasm-ph], 12 May2008; Phys. Plasmas 15, 082305 (2008).

2. S. Bahamida, K. Annou, and R. Annou, in 34th EPS Conference on Plasma Physics, Warsaw, 2-6 July 2007, ECA 31F, P-4.139.

3. V. K. Valiulina,A.E.Dubinov, Astrophys Space Sci. 337,

   *

   k1 D

   , whereas for a given wavelength, the ratio increases

   201-207 (2012).

4. N. Roy, S. Tasnim and A . A. Mamun, Phys.Plasmas 19,

with temperature. We find also that the normalized

033705 (2012).

maximum growth rate

maxe pi / maxeion

decreases for increasing values of the parameter . This tendency has been revealed in an unmagnetized electron- positron-ion plasma (c.f.Ref.[2]).

AKNOWLEDGMENT: R.A thanks Pr.V.K.Tripathi for interesting discussions.

Figure caption

1/ Fig.1 : Ratio maxe pi / maxeion versus

*

k1 D

for different temperatures Te .

2/Fig.2 : Ratio maxe pi / maxeion versus .

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