REVISTA COLOMBIANA DE F´ ISICA, VOL.36, No.2, 2004 ELECTRON-POSITRON PLASMA UNDER AN EXTERNAL ELECTROMAGNETIC FIELD
a
B.A. Paeza , N.C. Forerob , L. Casta˜edac n Institut f¨r Physik, Technische Universit¨t Chemnitz, D-09107, Germany u a b Physics Department, Universidad Nacional de Colombia, Bogot´ a c Observatorio Astron´mico Nacional, Bogot´, Colombia o a
ABSTRACT
An ensamble based on electrons and positrons under the influence of an external electromagnetic field is studied by means of the Fokker-Planck (F-P) formalism in the difussion approximation. Local and no local effects of the ensamble due to its interaction with field is also discussed considering mainly the collision term in the F-P equation, to compute the zeroth order velocity moment than allows to determine the continuity equation and then to verify the conservation of the number of particles
Keywords: Fokker-Planck
equation, electron-positron plasma, Landau operator.
Introduction
Recently, the nonlinear propagation of electromagnetic (EM) waves in electronpositron plasma has attracted the interest of researchers due to the fact that the electron-positron plasmas are found near the polar cap of a pulsar, in the active galactic nuclei, as well as in the early universe.[1] Also in the solid state physics field, this plasma has been used as spectroscopy technique to study structural properties in materials, since the positrons often annihilated preferentially at defect or vacany sites.[3-5] Although, there are several applications of this plasma system, still underlies basic aspects releted to interaction with external fields, i.e. when a high intensity laser illuminates a plasma, large scale nonuniformities in the laser spatial profile will cause differential heating in the plasma, arising steep temperature gradients, nonlocal heat transport, heat flux inhibition, and non-Maxwellian inverse bremsstrahlung heating especially in laser hot spots.[6] Together with these effects is induced a non maxwellian distribution.[2]
Dynamical Equations
To describe the electron-positron plasma dynamics and the effect of the electromagnetic field, the distribution functions of both electrons (e) and positrons (p), fe,p (x, p, t), are considered. In this way, the dynamics of the system is given by ∂fe ∂fe dfe = + ve · q fe + (−eE + Je ) × B) · p fe = (1) dt ∂t ∂t c dfp ∂fp = + vp · dt ∂t
q fp
+ (eE + Jp × B) ·
p fp
=
∂fp ∂t
(2)
c
where v and p are velocity and momentum of the particles and are related by ve,p = pe,p (m2 e,p 222 + p2 ) 2 e,p
1
(3)
REVISTA COLOMBIANA DE F´ ISICA, VOL.36, No.2, 2004 with me,p the rest mass of each particle. In eq.(1), the right hand side term describes the interactions of the species, and is referred as the collision term. In general, the collsion term depends on the distribution functions and couples the dynamical equations. The fields E and B contain basically contributions from the plasma and external sources, i.e. densities of charge and current. These fields can be determined in a selfconsistent way by solving the Maxwell equations, ×E =− ×B =
e,p
∂B ∂t ∂E ∂t
(4) (5)
ee,p
d3 p ve,p fe,p + Jext + ee,p d3 p fe,p + 4πρext
· E = 4π
e,p
(6) (7)
·B =0
Wave equation
Another coupling in the distribution functions is given by the electromagnetic fields, responsible of the forces in the plasma. To verify this the vectorial A and scalar φ potentials can be used is such way that the electromagnetic equations are now written as ∂A E=− − φ (8) ∂t B = ×A (9) and also considering Coulomb’s gauge, · A = 0. From these equations is easy to find the inhomogeneous wave equation relating both potentials and the density current induced by the external electromagnetic field ∂2A ∂ φ − 2A + =J (10) ∂t2 ∂t this result describes the plasma behavior among an electromagnetic field. The solution is determined also selfconsistently. As an example let a circular polarized laser field be the external electromagnetic interaction with the plasma. In this way the vector potential A can be given by [1] A = A(z, t)(ˆ + iˆ) exp (−iϕ) + const x y (11)
where ϕ = ωo t − ko z being ωo the frequency of the excitation energy and ko the wave vector. Additionally, considering in eq.(10) the term involving the scalar potential (φ) constant, eq.(10) yields to
2 2 2 (ωo − ko )A + ∂z A + 2iωo ∂t A + 2iko ∂A = J + const
(12) (13)
being J = e(np vp − ne ve ) with the previous considerations, this wave equation can be splitted in two parts. One concerning the homogeneous situation and the other, a linear dependence of the vectorial potential with the density current, J. 223
REVISTA COLOMBIANA DE F´ ISICA, VOL.36, No.2, 2004
Continuity equation
The continuity equation is found by direct integration of the Boltzmann equation. Before doing this, some important properties are useful to consider ∂fe,p ∂t c (14) the first term in the left hand side is the rate of change of electrons and positrons ∂fe,p + ∂t d3 pve,p ·
q fe,p +
d3 p
d3 p (−eE + Je,p × B)·
p fe,p
=
d3 p
d3 p
∂fe,p ∂ = ∂t ∂t
d3 pfe,p =
∂Ne,p ∂t
(15)
the second term can be expressed as d3 pve,p ·
q fe,p
=
q
·
d3 pve,p fe,p
(16)
which is just the density current of each specie. Finally, the last term d3 pΦ ·
p fe
=
d3 p
p
· (fe Φ) −
d3 pfe
p
·Φ
(17)
considering here the divergence theorem in the momenta space, and taking into account that fe vanishes at infinity, is found the first term in the right side is zero. In eq.(14) the Φ term is defined as Φ = (−eE + Je,p × B) (18)
When the electric field does not explicity depend on the momentum, and given that in the second term the cross product in each pi component is none dependent on this component. Hence the integral is zero. Therefore, the effect of electromagnetic fields is leaving unmodified the number of particles. On the other hand, the collision operator when the number of particles are conserved obeys dp ∂fe ∂t =0
c
(19)
A first inference on this term is the dynamical movement of shells in the p space, but is possible to generalize it when mumber of particles is not conserved [5] .
The collision operator
In this paper only the two body relativistic collisions are considered, in general in hot plasmas the particle number density is not conserved [5], the collision operator for non relativistic Coulomb collision was obatined by Landau [7] and its generalization has been studied in several papers [6], the general relativistic Fokker Planck collision operator Q(fαβ ) can be written as Q(fαβ ) = ∂ ∂fα · Dαβ · − Fαβ fα ∂u ∂u (20)
here the subscripts α and β are referred to the sort of species involve in the collision process. The first term in the right side in eq.(20) is the diffusion term and the 224
REFERENCIAS
REVISTA COLOMBIANA DE F´ ISICA, VOL.36, No.2, 2004
other one is the friction term. The particles described by the distribution function fα in the collision operator, interact by binary collisions with particles described through fβ , in this case u is the momentum per rest mass and in general the operators D and F are functios of one kinematical function defined as[7] U (u, u ) =
1
r2 /γγ (r2 − 1) 2
3
(r2 − 1)I − uu − u u + r(uu + u u)
(21)
with γ = (1 + |u2 |) 2 and r = γγ − uu is the relativistic correction factor for the relative velocity between the two particles, I is the diade identity. The operators in the F-P aproximattion are Dαβ (u) =
2 2 qα qβ lnΛαβ 8π 2 m2 0 α
U(u, u )fβ du ∂ · U(u, u ) fβ du ∂u
(22)
Fαβ (u) = −
2 2 qα qβ lnΛαβ 8π 2 mα mβ 0
(23)
the factor lnΛ is the Coulomb logarithm, in the particular when the electronpositron plasma is considered this factor easily is computed. Considering the Lorentz interaction of the electron-positron plasma with the external electromagnetic field, no changes in the number of particles is observed. The equations (22, 23) of the electron-positron plasma are sketched in the relativistic Fokker-Planck method as an alternative to those proposed elsewhere [5]. Not only the coupling in the distribution functions makes the problem more realistic but also introduces additional difficulties to solve the dynamical equations. The advantage of this formulation allows to study high energetic processes, i.e inelastic relativistic collisions.
Referencias
[1] V. I. Berezhiani, M. Y. El-Ashry and U. A. Mofiz. Phys. Rev. E 50, 448 (1994) [2] S. Brunner and E. Valeo. Phys. Plasmas 9, 923 (2002). [3] P. A. Sterne, J. H. Kaiser. Phys. Rev. B 43, 13892-13898 (1991) [4] M. J. Puska, M. Sob, G. Brauer and T. Korhonen. Phys. Rev. B 49, 10947 (1994). [5] M. Gedalin. Phys. Rev. Lett. 76, 3340 (1996). [6] B. B. Afeyan, A. E. Chou, J. P. Matte, R. P. J. Town, and W. J. Kruer. Phys. Rev. Lett. 80, 2322-2325 (1998) [7] L.D. Landau. Phys.Z. Sowjetunion 10, 154 (1936). [8] B. J. Braams, Karney Charles. Phys. Rev. Lett. Vol 59, N16,1817-1820 (1987)
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