- Email: [email protected]
Ion acceleration by the space charge electric force arising from the radiation pressure in a magnetized electron–positron plasma
Physics Letters A 373 (2009) 3547–3549
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Ion acceleration by the space charge electric force arising from the radiation pressure in a magnetized electron–positron plasma P.K. Shukla 1 Institut für Theoretische Physik IV, Ruhr-Universität Bochum, D-44780 Bochum, Germany
a r t i c l e
i n f o
Article history: Received 9 July 2009 Accepted 19 July 2009 Available online 4 August 2009 Communicated by V.M. Agranovich PACS: 52.35.Ep 52.35.Mw
a b s t r a c t It is shown that ions can be accelerated by the space charge electric force arising from the separation of electrons and positrons due to the ponderomotive force of the magnetic field-aligned circularly polarized electromagnetic (CPEM) wave in a magnetized electron–positron–ion plasma. The ion acceleration critically depends on the external magnetic field strength. The result is useful in understanding differential ion acceleration in magnetized electron–positron–ion plasmas, such as those in magnetars and in some laboratory experiments that aim to mimic astrophysical environments. © 2009 Elsevier B.V. All rights reserved.
In electron–positron (hereafter referred to as the e–p or pair) plasmas, the electrons and positrons have the same mass but opposite charge. Such pair plasmas occurred in the early Universe [1,2], and are frequently found in bipolar outflows (jets) in active-galactic nuclei [3,4], in micro-quasars [5], in pulsar magnetospheres [6–10], in magnetars [11], in cosmological gamma ray fireballs [12], in solar flares [13], and at the center of our galaxy [14,15]. Multiterawatt and petawatt laser pulses interacting with solid density matters can create pair plasmas as well [16–19]. The e–p plasmas at the surface of fast rotating neutron stars and magnetars are held in strong magnetic fields, while super-strong magnetic fields are created in intense laser-plasma interaction experiments. Accordingly, the understanding of collective phenomena in a strongly magnetized e–p plasma has been a topic of significant interest [20–23]. Specifically, it is to be noted that in a magnetized pair plasma, we have new wave modes whose counterparts do not exist in an electron–ion magnetoplasma [24]. However, a magnetized pair plasma may also contain a fraction of ions, which can significantly affect the linear and nonlinear propagation of electrostatic and electromagnetic waves in a pair-ion magnetoplasma [25–36]. In this Letter, we consider acceleration of ions by the space charge electric force in an electron–positron–ion (e–p–i) magnetoplasma. The space charge electric force arises due to the separation
E-mail address: [email protected]. Also at the Department of Physics, Umeå University, SE-90187 Umeå, Sweden; Scottish Universities Physics Alliance Department of Physics, University of Strathclyde, Glasgow G4 ONG, United Kingdom; Centre for Fundamental Physics, Rutherford Appleton Laboratory, Chilton, Didcot, UK; GoLP/Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico, 1049-001 Lisboa, Portugal; School of Physics, University of KwaZulu-Natal, 4000 Durban, South Africa. 1
0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.07.084
of the electrons and positrons on account of the ponderomotive force of a large amplitude magnetic field-aligned circularly polarized electromagnetic (CPEM) wave [37] in an e–p–i magnetoplasma. Let us consider a uniform e–p–i magnetoplasma in the presence of a large amplitude right-hand CPEM wave propagating along zˆ B 0 , where zˆ is the unit vector along the z axis in a Cartesian coordinate system and B 0 is the strength of the external magnetic field. The right-hand CPEM wave electric field is E⊥ = i (ω/c )A, where A = A ⊥ (ˆx + iy) is the perpendicular component of the vector potential, x (y) is the unit vector along the x ( y ) axis, and c is the speed of light in vacuum. The wave frequency ω (assumed to be much larger than the ion gyrofrequency) is obtained from the dispersion relation [35,36,38]
k2 c 2 + ω2pi
ω2
=1−
ω2pe ω2pp − , ω(ω − ωc ) ω(ω + ωc )
(1)
where ω pi = (4π ni0 Z i2 e 2 /mi )1/2 is the ion plasma frequency, ni0 is the unperturbed ion number density, Z i is the ion charge state, e is the magnitude of the electron charge, mi is the ion mass, and k is the magnetic field-aligned wave-number [for the left-hand (LH) CPEM wave, ω − ωc in Eq. (1) is replaced by ω + ωc [37,39]]. We have denoted ω pe = (4π ne0 e 2 /m)1/2 , ω pp = (4π n p0 e 2 /m)1/2 and ωc = e B 0 /mc, where ne0 (n p0 ) is the unperturbed electron (positron) number density, and m is the rest mass of electrons/positrons. Eq. (1) can be rewritten as [36]
ω2 −
ω(ωω2p + ωc ωi2 ) − k2 c 2 − ω2pi = 0, ω2 − ωc2
(2)
ω p = (ω2pe + ω2pp )1/2 ≡ (4π n0 e 2 /m)1/2 , n0 = ne0 + n p0 , and ωi = (4π Z i ni0 e 2 /m)1/2 . In the absence of the ions, we have ωi = 0 where
3548
P.K. Shukla / Physics Letters A 373 (2009) 3547–3549
and ω pi = 0. However, according to (2), we have new branches of the CPEM waves when ions are present. For ω ωc , the solution of (2) is
1
ω = − Ω ± Ω 2 + 4γ 2
where Ω =
2 1/2
(3)
,
ωc ωi2 /ω2H , ω H = (ω2p + ωc2 )1/2 , and γ = ωc (k2 c 2 +
ω2pi )1/2 /ω H . The electrons and positrons are pushed by the ponderomotive force of the RH-CPEM wave. The ponderomotive force acting on the electrons and positrons is given by [40]
Fj =
kωc j ∂ ωe 2 ∂ | A ⊥ |2 , − (ω − ωc j )mc 2 ∂ z ω(ω − ωc j ) ∂ t
(4)
where j equals e for the electrons and p for positrons, ωce = ωc , and ωcp = −ωc . We note that the non-stationary ponderomotive force (the second term in the right-hand side of (4)) was ignored in Ref. [41]. However, as we will see below, the non-stationary ponderomotive force equally contributes to the space charge electric field and ion acceleration. The ponderomotive force pushes both electrons and positrons opposite to each other along the external magnetic field direction. Consequently, electrons and positrons are separated, resulting into the magnetic field-aligned space charge electric field E s . The balance between the space charge electric and ponderomotive forces yields for the electrons and positrons, respectively,
∂ ωe 2 kωc ∂ | A ⊥ |2 , −e E s = − (ω − ωc )mc 2 ∂ z ω(ω − ωc ) ∂ t and
eEs =
(5)
∂ ωe 2 kωc ∂ | A ⊥ |2 . + (ω + ωc )mc 2 ∂ z ω(ω + ωc ) ∂ t
(6)
The space charge electric field is obtained by subtracting (5) from (6), yielding
Es =
e ωωc
(ω − ω 2 c
2 )mc 2
∂| A ⊥ |2 ek ωc (ω2 + ωc2 ) ∂| A ⊥ |2 + , ∂z ∂t mc 2 (ω2 − ωc2 )2
(7)
which indicates that enhanced space charge electric fields in our e–p–i magnetoplasma by the ponderomotive force of the CPEM wave is created only if the external magnetic field is present. In the absence of the external magnetic field (e.g. ωc = 0), the space charge electric field E s is zero. The magnetic field-aligned ion acceleration (the rate of change of the ion velocity u) caused by the space charge electric force is given by
mi
∂u = Zie Es, ∂τ
(8)
where we are assuming that the time scale of ion acceleration is much longer than the temporal variation of the CPEM vector potential envelope. Eliminating E s from (8) by using (7) we obtain for the ion acceleration
∂u Z i e 2 ωωc ∂| A ⊥ |2 = 2 ∂τ mi (ωc − ω2 )mc 2 ∂ z +
ωc (ω2 + ωc2 ) ∂| A ⊥ |2 . ∂t mi mc 2 (ω2 − ωc2 )2 Z i e2k
(9)
Eq. (9) dictates that ions with different Z i /mi distributions are accelerated at different rates by the CPEM wave ponderomotive force.
When the CPEM wave frequency have from (9)
ω is much smaller than ωc , we
∂u Z i e ω ∂| A ⊥ |2 Z i ek ∂| A ⊥ |2 = + , ∂τ B 0 mi c ∂ z B 0 mi c ∂ t
(10)
which exhibits that the rate of change of the ion speed is inversely proportional to the external magnetic field strength. In summary, we have presented an investigation of the magnetic field-aligned ion acceleration by the space charge electric force in a magnetized e–p–i plasma. It is shown that the ponderomotive force of the CPEM wave in a magnetized e–p–i plasma pushes electrons and positrons in different directions along the magnetic lines of force, and consequently there appear separation of electrons and positrons and associated space charge electric fields. The space charge electric force, in turn, would accelerate ions along the external magnetic field direction in our magnetized e–p–i plasma. The present result should be useful in understanding the origin of differential ion acceleration in the presence of the electromagnetic wave in magnetized e–p–i plasmas, such as those in magnetars and in forthcoming laboratory experiments designed for studying the nonlinear physics of pair plasmas. Acknowledgements This work was partially supported by the Deutsche Forschungsgemeinschaft through the project SH21/3-1 of the Research Unit 1048. References [1] W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation, Freeman, San Francisco, 1973. [2] M.J. Rees, in: G.W. Gibbons, S.W. Hawking, S. Siklas (Eds.), The Very Early Universe, Cambridge University Press, Cambridge, 1983. [3] M.C. Begelman, R.D. Blandford, M.J. Rees, Rev. Mod. Phys. 56 (1984) 255. [4] H.R. Miller, P.J. Witta, in: Active Galactic Nuclei, Springer, Berlin, 1987, p. 202. [5] R. Fender, Annu. Rev. Astron. Astrophys. 42 (2004) 317. [6] P. Goldreich, W.H. Julian, Astrophys. J. 157 (1969) 869; P.A. Sturrock, Astrophys. J. 164 (1971) 529; V.L. Ginzburg, Sov. Phys. Usp. 14 (1971) 83; M.A. Ruderman, P.G. Sutherland, Astrophys. J. 196 (1975) 51. [7] F.C. Michel, Rev. Mod. Phys. 54 (1982) 1. [8] P.K. Shukla, M.Y. Yu, N.N. Rao, N.L. Tsintsadze, Phys. Rep. 138 (1984) 1. [9] F.C. Michel, Theory of Neutron Star Magnetosphere, Chicago University Press, Chicago, 1991. [10] V.S. Beskin, A.V. Gurevich, Ya.N. Istomin, Physics of the Pulsar Magnetosphere, Cambridge University Press, Cambridge, 1993. [11] M. Marklund, P.K. Shukla, Rev. Mod. Phys. 78 (2006) 591. [12] T. Piran, Phys. Rep. 314 (1999) 575; T. Piran, Rev. Mod. Phys. 76 (2004) 1143. [13] E. Tandberg-Hansen, A.G. Emshie, The Physics of Solar Flares, Cambridge University Press, Cambridge, 1988, p. 124. [14] M.L. Burns, in: M.L. Burns, A.K. Harding, R. Ramaty (Eds.), Positron–Electron Pairs in Astrophysics, AIP, New York, 1983. [15] R. Schlickeiser, P.K. Shukla, Astrophys. J. 599 (2003) L57. [16] V.I. Berezhiani, D.D. Tskhakaya, P.K. Shukla, Phys. Rev. A 46 (1992) 6608. [17] E.P. Liang, et al., Phys. Rev. Lett. 81 (1998) 4887. [18] C. Gahn, et al., Appl. Phys. Lett. 77 (2000) 2662. [19] S.C. Wilks, et al., Astrophys. Space Sci. 298 (2005) 347. [20] J.G. Lominadze, et al., Phys. Scr. 26 (1982) 455; M.E. Gedalin, et al., Astrophys. Space Sci. 108 (1985) 393. [21] M.Y. Yu, P.K. Shukla, L. Stenflo, Astrophys. J. 309 (1986) L63. [22] N. Iwamoto, Phys. Rev. E 47 (1993) 604. [23] G. Brodin, M. Marklund, B. Eliasson, P.K. Shukla, Phys. Rev. Lett. 98 (2007) 125001. [24] P.K. Shukla, L. Stenflo, Phys. Rev. A 30 (1984) 2110. [25] F.B. Rizzato, J. Plasma Phys. 40 (1988) 289. [26] V.I. Berezhiani, L.N. Tsintsadze, P.K. Shukla, J. Plasma Phys. 48 (1992) 139; V.I. Berezhiani, L.N. Tsintsadze, P.K. Shukla, Phys. Scr. 46 (1992) 35. [27] V.I. Berezhiani, S.M. Mahajan, Phys. Rev. Lett. 73 (1994) 1110. [28] V.I. Berezhiani, S.M. Mahajan, Phys. Rev. E 52 (1995) 1968. [29] S.I. Popel, et al., Phys. Plasmas 2 (1995) 716. [30] P.K. Shukla, L. Stenflo, R. Fedele, Phys. Plasmas 10 (2003) 310. [31] J.B. Kim, et al., Phys. Lett. A 329 (2004) 464.
P.K. Shukla / Physics Letters A 373 (2009) 3547–3549
[32] T. Cattaert, I. Kourakis, P.K. Shukla, Phys. Plasmas 12 (2005) 012319. [33] M. Marklund, P.K. Shukla, L. Stenflo, G. Brodin, M. Servin, Plasma Phys. Control. Fusion 47 (2005) 25. [34] C.R. Stark, D.A. Diver, A.A. da Costa, E.W. Laing, Astron. Astrophys. 476 (2007) 17. [35] P.K. Shukla, et al., Phys. Plasmas 15 (2008) 082305. [36] P.K. Shukla, Plasma Phys. Control. Fusion 51 (2009) 024013.
[37] P.K. Shukla, M.Y. Yu, K.H. Spatschek, Phys. Fluids 18 (1975) 265; P.K. Shukla, L. Stenflo, Phys. Plasmas 12 (2005) 084502. [38] L. Stenflo, Phys. Scr. 14 (1976) 320. [39] G. Brodin, L. Stenflo, Phys. Scr. 37 (1988) 89. [40] V.I. Karpman, H. Washimi, J. Plasma Phys. 18 (1977) 173; L. Stenflo, P.K. Shukla, M.Y. Yu, Astrophys. Space Sci. 117 (1985) 303. [41] P.K. Shukla, J. Plasma Phys. 72 (2006) 159.
3549
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Ion acceleration by the space charge electric force arising from the radiation pressure in a magnetized electron–positron plasma P.K. Shukla 1 Institut für Theoretische Physik IV, Ruhr-Universität Bochum, D-44780 Bochum, Germany
a r t i c l e
i n f o
Article history: Received 9 July 2009 Accepted 19 July 2009 Available online 4 August 2009 Communicated by V.M. Agranovich PACS: 52.35.Ep 52.35.Mw
a b s t r a c t It is shown that ions can be accelerated by the space charge electric force arising from the separation of electrons and positrons due to the ponderomotive force of the magnetic field-aligned circularly polarized electromagnetic (CPEM) wave in a magnetized electron–positron–ion plasma. The ion acceleration critically depends on the external magnetic field strength. The result is useful in understanding differential ion acceleration in magnetized electron–positron–ion plasmas, such as those in magnetars and in some laboratory experiments that aim to mimic astrophysical environments. © 2009 Elsevier B.V. All rights reserved.
In electron–positron (hereafter referred to as the e–p or pair) plasmas, the electrons and positrons have the same mass but opposite charge. Such pair plasmas occurred in the early Universe [1,2], and are frequently found in bipolar outflows (jets) in active-galactic nuclei [3,4], in micro-quasars [5], in pulsar magnetospheres [6–10], in magnetars [11], in cosmological gamma ray fireballs [12], in solar flares [13], and at the center of our galaxy [14,15]. Multiterawatt and petawatt laser pulses interacting with solid density matters can create pair plasmas as well [16–19]. The e–p plasmas at the surface of fast rotating neutron stars and magnetars are held in strong magnetic fields, while super-strong magnetic fields are created in intense laser-plasma interaction experiments. Accordingly, the understanding of collective phenomena in a strongly magnetized e–p plasma has been a topic of significant interest [20–23]. Specifically, it is to be noted that in a magnetized pair plasma, we have new wave modes whose counterparts do not exist in an electron–ion magnetoplasma [24]. However, a magnetized pair plasma may also contain a fraction of ions, which can significantly affect the linear and nonlinear propagation of electrostatic and electromagnetic waves in a pair-ion magnetoplasma [25–36]. In this Letter, we consider acceleration of ions by the space charge electric force in an electron–positron–ion (e–p–i) magnetoplasma. The space charge electric force arises due to the separation
E-mail address: [email protected]. Also at the Department of Physics, Umeå University, SE-90187 Umeå, Sweden; Scottish Universities Physics Alliance Department of Physics, University of Strathclyde, Glasgow G4 ONG, United Kingdom; Centre for Fundamental Physics, Rutherford Appleton Laboratory, Chilton, Didcot, UK; GoLP/Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico, 1049-001 Lisboa, Portugal; School of Physics, University of KwaZulu-Natal, 4000 Durban, South Africa. 1
0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.07.084
of the electrons and positrons on account of the ponderomotive force of a large amplitude magnetic field-aligned circularly polarized electromagnetic (CPEM) wave [37] in an e–p–i magnetoplasma. Let us consider a uniform e–p–i magnetoplasma in the presence of a large amplitude right-hand CPEM wave propagating along zˆ B 0 , where zˆ is the unit vector along the z axis in a Cartesian coordinate system and B 0 is the strength of the external magnetic field. The right-hand CPEM wave electric field is E⊥ = i (ω/c )A, where A = A ⊥ (ˆx + iy) is the perpendicular component of the vector potential, x (y) is the unit vector along the x ( y ) axis, and c is the speed of light in vacuum. The wave frequency ω (assumed to be much larger than the ion gyrofrequency) is obtained from the dispersion relation [35,36,38]
k2 c 2 + ω2pi
ω2
=1−
ω2pe ω2pp − , ω(ω − ωc ) ω(ω + ωc )
(1)
where ω pi = (4π ni0 Z i2 e 2 /mi )1/2 is the ion plasma frequency, ni0 is the unperturbed ion number density, Z i is the ion charge state, e is the magnitude of the electron charge, mi is the ion mass, and k is the magnetic field-aligned wave-number [for the left-hand (LH) CPEM wave, ω − ωc in Eq. (1) is replaced by ω + ωc [37,39]]. We have denoted ω pe = (4π ne0 e 2 /m)1/2 , ω pp = (4π n p0 e 2 /m)1/2 and ωc = e B 0 /mc, where ne0 (n p0 ) is the unperturbed electron (positron) number density, and m is the rest mass of electrons/positrons. Eq. (1) can be rewritten as [36]
ω2 −
ω(ωω2p + ωc ωi2 ) − k2 c 2 − ω2pi = 0, ω2 − ωc2
(2)
ω p = (ω2pe + ω2pp )1/2 ≡ (4π n0 e 2 /m)1/2 , n0 = ne0 + n p0 , and ωi = (4π Z i ni0 e 2 /m)1/2 . In the absence of the ions, we have ωi = 0 where
3548
P.K. Shukla / Physics Letters A 373 (2009) 3547–3549
and ω pi = 0. However, according to (2), we have new branches of the CPEM waves when ions are present. For ω ωc , the solution of (2) is
1
ω = − Ω ± Ω 2 + 4γ 2
where Ω =
2 1/2
(3)
,
ωc ωi2 /ω2H , ω H = (ω2p + ωc2 )1/2 , and γ = ωc (k2 c 2 +
ω2pi )1/2 /ω H . The electrons and positrons are pushed by the ponderomotive force of the RH-CPEM wave. The ponderomotive force acting on the electrons and positrons is given by [40]
Fj =
kωc j ∂ ωe 2 ∂ | A ⊥ |2 , − (ω − ωc j )mc 2 ∂ z ω(ω − ωc j ) ∂ t
(4)
where j equals e for the electrons and p for positrons, ωce = ωc , and ωcp = −ωc . We note that the non-stationary ponderomotive force (the second term in the right-hand side of (4)) was ignored in Ref. [41]. However, as we will see below, the non-stationary ponderomotive force equally contributes to the space charge electric field and ion acceleration. The ponderomotive force pushes both electrons and positrons opposite to each other along the external magnetic field direction. Consequently, electrons and positrons are separated, resulting into the magnetic field-aligned space charge electric field E s . The balance between the space charge electric and ponderomotive forces yields for the electrons and positrons, respectively,
∂ ωe 2 kωc ∂ | A ⊥ |2 , −e E s = − (ω − ωc )mc 2 ∂ z ω(ω − ωc ) ∂ t and
eEs =
(5)
∂ ωe 2 kωc ∂ | A ⊥ |2 . + (ω + ωc )mc 2 ∂ z ω(ω + ωc ) ∂ t
(6)
The space charge electric field is obtained by subtracting (5) from (6), yielding
Es =
e ωωc
(ω − ω 2 c
2 )mc 2
∂| A ⊥ |2 ek ωc (ω2 + ωc2 ) ∂| A ⊥ |2 + , ∂z ∂t mc 2 (ω2 − ωc2 )2
(7)
which indicates that enhanced space charge electric fields in our e–p–i magnetoplasma by the ponderomotive force of the CPEM wave is created only if the external magnetic field is present. In the absence of the external magnetic field (e.g. ωc = 0), the space charge electric field E s is zero. The magnetic field-aligned ion acceleration (the rate of change of the ion velocity u) caused by the space charge electric force is given by
mi
∂u = Zie Es, ∂τ
(8)
where we are assuming that the time scale of ion acceleration is much longer than the temporal variation of the CPEM vector potential envelope. Eliminating E s from (8) by using (7) we obtain for the ion acceleration
∂u Z i e 2 ωωc ∂| A ⊥ |2 = 2 ∂τ mi (ωc − ω2 )mc 2 ∂ z +
ωc (ω2 + ωc2 ) ∂| A ⊥ |2 . ∂t mi mc 2 (ω2 − ωc2 )2 Z i e2k
(9)
Eq. (9) dictates that ions with different Z i /mi distributions are accelerated at different rates by the CPEM wave ponderomotive force.
When the CPEM wave frequency have from (9)
ω is much smaller than ωc , we
∂u Z i e ω ∂| A ⊥ |2 Z i ek ∂| A ⊥ |2 = + , ∂τ B 0 mi c ∂ z B 0 mi c ∂ t
(10)
which exhibits that the rate of change of the ion speed is inversely proportional to the external magnetic field strength. In summary, we have presented an investigation of the magnetic field-aligned ion acceleration by the space charge electric force in a magnetized e–p–i plasma. It is shown that the ponderomotive force of the CPEM wave in a magnetized e–p–i plasma pushes electrons and positrons in different directions along the magnetic lines of force, and consequently there appear separation of electrons and positrons and associated space charge electric fields. The space charge electric force, in turn, would accelerate ions along the external magnetic field direction in our magnetized e–p–i plasma. The present result should be useful in understanding the origin of differential ion acceleration in the presence of the electromagnetic wave in magnetized e–p–i plasmas, such as those in magnetars and in forthcoming laboratory experiments designed for studying the nonlinear physics of pair plasmas. Acknowledgements This work was partially supported by the Deutsche Forschungsgemeinschaft through the project SH21/3-1 of the Research Unit 1048. References [1] W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation, Freeman, San Francisco, 1973. [2] M.J. Rees, in: G.W. Gibbons, S.W. Hawking, S. Siklas (Eds.), The Very Early Universe, Cambridge University Press, Cambridge, 1983. [3] M.C. Begelman, R.D. Blandford, M.J. Rees, Rev. Mod. Phys. 56 (1984) 255. [4] H.R. Miller, P.J. Witta, in: Active Galactic Nuclei, Springer, Berlin, 1987, p. 202. [5] R. Fender, Annu. Rev. Astron. Astrophys. 42 (2004) 317. [6] P. Goldreich, W.H. Julian, Astrophys. J. 157 (1969) 869; P.A. Sturrock, Astrophys. J. 164 (1971) 529; V.L. Ginzburg, Sov. Phys. Usp. 14 (1971) 83; M.A. Ruderman, P.G. Sutherland, Astrophys. J. 196 (1975) 51. [7] F.C. Michel, Rev. Mod. Phys. 54 (1982) 1. [8] P.K. Shukla, M.Y. Yu, N.N. Rao, N.L. Tsintsadze, Phys. Rep. 138 (1984) 1. [9] F.C. Michel, Theory of Neutron Star Magnetosphere, Chicago University Press, Chicago, 1991. [10] V.S. Beskin, A.V. Gurevich, Ya.N. Istomin, Physics of the Pulsar Magnetosphere, Cambridge University Press, Cambridge, 1993. [11] M. Marklund, P.K. Shukla, Rev. Mod. Phys. 78 (2006) 591. [12] T. Piran, Phys. Rep. 314 (1999) 575; T. Piran, Rev. Mod. Phys. 76 (2004) 1143. [13] E. Tandberg-Hansen, A.G. Emshie, The Physics of Solar Flares, Cambridge University Press, Cambridge, 1988, p. 124. [14] M.L. Burns, in: M.L. Burns, A.K. Harding, R. Ramaty (Eds.), Positron–Electron Pairs in Astrophysics, AIP, New York, 1983. [15] R. Schlickeiser, P.K. Shukla, Astrophys. J. 599 (2003) L57. [16] V.I. Berezhiani, D.D. Tskhakaya, P.K. Shukla, Phys. Rev. A 46 (1992) 6608. [17] E.P. Liang, et al., Phys. Rev. Lett. 81 (1998) 4887. [18] C. Gahn, et al., Appl. Phys. Lett. 77 (2000) 2662. [19] S.C. Wilks, et al., Astrophys. Space Sci. 298 (2005) 347. [20] J.G. Lominadze, et al., Phys. Scr. 26 (1982) 455; M.E. Gedalin, et al., Astrophys. Space Sci. 108 (1985) 393. [21] M.Y. Yu, P.K. Shukla, L. Stenflo, Astrophys. J. 309 (1986) L63. [22] N. Iwamoto, Phys. Rev. E 47 (1993) 604. [23] G. Brodin, M. Marklund, B. Eliasson, P.K. Shukla, Phys. Rev. Lett. 98 (2007) 125001. [24] P.K. Shukla, L. Stenflo, Phys. Rev. A 30 (1984) 2110. [25] F.B. Rizzato, J. Plasma Phys. 40 (1988) 289. [26] V.I. Berezhiani, L.N. Tsintsadze, P.K. Shukla, J. Plasma Phys. 48 (1992) 139; V.I. Berezhiani, L.N. Tsintsadze, P.K. Shukla, Phys. Scr. 46 (1992) 35. [27] V.I. Berezhiani, S.M. Mahajan, Phys. Rev. Lett. 73 (1994) 1110. [28] V.I. Berezhiani, S.M. Mahajan, Phys. Rev. E 52 (1995) 1968. [29] S.I. Popel, et al., Phys. Plasmas 2 (1995) 716. [30] P.K. Shukla, L. Stenflo, R. Fedele, Phys. Plasmas 10 (2003) 310. [31] J.B. Kim, et al., Phys. Lett. A 329 (2004) 464.
P.K. Shukla / Physics Letters A 373 (2009) 3547–3549
[32] T. Cattaert, I. Kourakis, P.K. Shukla, Phys. Plasmas 12 (2005) 012319. [33] M. Marklund, P.K. Shukla, L. Stenflo, G. Brodin, M. Servin, Plasma Phys. Control. Fusion 47 (2005) 25. [34] C.R. Stark, D.A. Diver, A.A. da Costa, E.W. Laing, Astron. Astrophys. 476 (2007) 17. [35] P.K. Shukla, et al., Phys. Plasmas 15 (2008) 082305. [36] P.K. Shukla, Plasma Phys. Control. Fusion 51 (2009) 024013.
[37] P.K. Shukla, M.Y. Yu, K.H. Spatschek, Phys. Fluids 18 (1975) 265; P.K. Shukla, L. Stenflo, Phys. Plasmas 12 (2005) 084502. [38] L. Stenflo, Phys. Scr. 14 (1976) 320. [39] G. Brodin, L. Stenflo, Phys. Scr. 37 (1988) 89. [40] V.I. Karpman, H. Washimi, J. Plasma Phys. 18 (1977) 173; L. Stenflo, P.K. Shukla, M.Y. Yu, Astrophys. Space Sci. 117 (1985) 303. [41] P.K. Shukla, J. Plasma Phys. 72 (2006) 159.
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