 Open Access
 Total Downloads : 210
 Authors : N. Tinakiche, R. Annou
 Paper ID : IJERTV3IS21495
 Volume & Issue : Volume 03, Issue 02 (February 2014)
 Published (First Online): 08032014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Stimulated Raman Scattering in a Magnetized ElectronPositronIon 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, AlgiersALGERIA
Abstract – Electronpositron plasmas that may be found in numerous environments, such as, active galactic nuclei, pulsar magnetosphere, solar flares, ultrashort lasermatter interaction and interplanetary space, have dielectric properties appreciably different from those of classical electronion 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 electronpositron plasma containing a fraction of ions (epi plasma), where the background magnetic field is parallel to the pump electric field.

INTRODUCTION
Electronpositron (ep) 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 laserplasma interactions and fusion devices can be used to produce ep plasmas. Electronpositron plasmas are also created in large tokamaks3 through collisions between Mev electrons and thermal particles. Due to the mass symmetry in ep 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 ionacoustic wave. Furthermore, while the unmagnetized plasma may reveal three wave modes, namely, Langmuir, ionacoustic 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, AlfvÃ©n 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 ep plasma containing a fraction of ions (epi plasma). As a matter of fact, astrophysical and laboratory plasmas contain in general a fraction of ions, e.g, electronpositronion plasmas are encountered in the magnetosphere of neutron stars, and also in the solar atmosphere1518. 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

THEORY
B0
We consider an (epi) 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 eion 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=101eV
Te=1 eV Te=10 eV
4t N0 m ce pe
m k1
t ' sh
where,
, (36)
0,55
0,54
max(epi) max(eion)
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 (ei) plasma given by Eq.(36).
maxeion for
different temperatures Te .
max(epi) max(eion)
1
0
0,0 0,2 0,4 0,6 0,8 1,0

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e pi / maxeion


CONCLUSION
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This paper is devoted to the study of the stimulated Raman scattering in a magnetized electronpositronion 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 ei plasma. It is found that the ratio
maxe pi / maxeion increases for increasing

P. K. Shukla, G. Brodin, M. Marklund, and L. Stenflo, Wake Field Generation and NonLinear Evolution in a Magnetized ElectronPositronIon Plasma, eprint arXiv: 0805.1617 vl [physics. Plasmph], 12 May2008; Phys. Plasmas 15, 082305 (2008).

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k1 D
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with temperature. We find also that the normalized
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maximum growth rate
maxe pi / maxeion
decreases for increasing values of the parameter . This tendency has been revealed in an unmagnetized electron positronion 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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