Serb. Astron. J. } 186 (2013), 15 - 23 UDC 52–726–355.3
DOI: 10.2298/SAJ130218002D Original scientific paper
CORRELATION FUNCTION AND ELECTRONIC SPECTRAL
LINE BROADENING IN RELATIVISTIC PLASMAS
S. Douis and M. T. Meftah
Department of Physics, Faculty of Sciences, LRPPS Laboratory, University of Kasdi Merbah Ouargla, 30000, Algeria
E–mail: mewalid@yahoo.com
(Received: November 9, 2012; Accepted: February 18, 2013)
SUMMARY: The electrons dynamics and the time autocorrelation function CEE(t) for the total electric microfield of the electrons on positive charge impu-rity embedded in a plasma are considered when the relativistic dynamic of the electrons is taken into account. We have, at first, built the effective potential gov-erning the electrons dynamics. This potential obeys a nonlinear integral equation that we have solved numerically. Regarding the electron broadening of the line in plasma, we have found that when the plasma parameters change, the amplitude of the collision operator changes in the same way as the time integral ofCEE(t).
The electron-impurity interaction is taken first as screened Deutsh interaction and second as Kelbg interaction. Comparisons of all interesting quantities are made with respect to the previous interactions as well as between classical and relativistic dynamics of electrons.
Key words. plasmas – relativistic processes – line: profiles
1. INTRODUCTION
The nonlinear behavior of electric charges around an impurity charge of the same sign is a prob-lem that has been studied for a long time due to its great importance in many techniques (Holtsmark 1919, Hooper 1966, Iglesias et al. 1983, Boerker et al. 1987, Berkovsky et al. 1996). It is worth to mention that for fully ionized plasma composed of electrons and positive ions, the hypothesis of one component plasma (OCP) allows us to ignore the ef-fects of ions motion with respect to those of electrons because the mass ratio is about me/mi ≈ 1/2000. So, the system is composed of only one kind of mobile charges (electrons), whereas the species of the opposite charge (ions) are modeled by the con-tinuous background which provides electrical
neu-trality. Coulomb forces between point charges are purely repulsive and charges approach very close to each other only rarely whatever the plasma condi-tions. Concerning the ion-electron interaction, it is clear that it requires a quantum mechanical descrip-tion. In this case, the Coulomb potential is replaced by a finite and regularized potential at the origin (Deutsh 1977, Deutsh et al. 1978, Minoo et al. 1981) VSD
ie (r)(Screened−Deutsh) = −(Ze2/r)(1− exp(−r/λT))exp(−r/λD) or Kelbg interaction
(Fili-nov et al. 2003)VieK(r) =−Ze
2
r√π(1−exp(−r
2/λ2
T) +
√π r
λT(1−erf(
r
λT))) whereλD= (kBT /(4πnee
2))1/2, λT = (2π¯h2/(mekBT))1/2, and ne is the density of
continu-ous positive background. We note that many recent works on statistical properties of electrons in plas-mas exist. For example, we find in Dufour et al. (2005) and Dufty et al. (2003) that the electrons are considered as classical particles moving by the first Newton law (d−→v /dt = −m−1
e −→∇.V(r) where me is the electron rest mass). In addition, the interaction between the electron and the continuous background is taken as purely Coulombic. In our work, we con-sider the relativistic motion of the electron around the impurity to compute CEE(t). This task passes
through two steps: - The first step: computation of the effective potential V(r) in which the electron moves, - The second step: we solve the relativistic equation of motion for the electron in the effective potential (d−→p /dt=−−→∇.V(r) where−→p =m−→v and m=me/(1−v2/c2)1/2). Furthermore, when we com-pute the effective potential V(r), we consider that the electron interacts with the test charge and with the continuous background positive charge via a regu-larized potential, whereas it interacts with the elec-trons via the Debye potential. In Section 2 we con-struct the equation for the effective potential which yields the subsequent results of this paper. We also solve this equation and present some discussions on its solutions. The dynamics properties, that is to say, the time autocorrelation function of the elec-tron microfield, are presented in Section 3. Section 4 applies the results of Section 3 to the electronic broadening of the line shape in plasmas. At the end, we close this paper by Conclusion. Before starting the second Section, let us recall the relevant param-eters for our study: the charge number Z, the av-erage distance between electrons a = (3/4πne)1/3, the electron coupling constant Γ = e2/(kT a), the electron de Broglie thermal lengthλT, the degree of
quanticityη =λT/a and the Debye lengthλD. The
cases considered in this work are (for Z=2, Z=4 and Z=8): the coupling parameter Γ = 0.1, the quan-ticity parameter η= 0.177, the dimensionless Debye length η′ = λD/a = 1.826. These parameters
cor-respond to the electron density ne = 2.×1020cm−3 and to the temperature T = 1.6×105K. In this re-gion of temperature and electron density, the non-relativistic treatment of the plasma becomes invalid (Mihajlov et al. 2011). We shall then treat the elec-tron motion around the impurity in the framework of the relativistic classical mechanics.
2. INTEGRAL EQUATION FOR THE EFFECTIVE ENERGY POTENTIAL 2.1. Construction of integral equation for
the effective energy potential
Let us consider a medium consisting of elec-trons and a continuous background of neutralizing positive electrical charges. This is the so called model of the one component plasma (OCP). At first, the distribution of the electrons is that of Maxwell-Boltzmann governing the equilibrium state of the
electrons’ system. If we place a positive ion of charge Ze (called test charge or impurity) at the coordi-nates origin the system is disturbed and, after a cer-tain timet, it will reach a new equilibrium state de-scribed by a new distribution of electrons over the space around the charge Ze. The latter is deter-mined through the potential energy of an electron located at distancer from the test chargeZe when the system reaches this new equilibrium state. This potential energy is built as a sum of three contribu-tions:
V(r) =Vie(r) +Vee(r) +Vef(r) (1)
where Vie(r) is the potential energy of ion-electron interaction (the ion is the test charge),Vee(r) is the interaction energy of the electron with all other elec-trons andVef(r) is the interaction energy of the elec-tron with the continuous neutralizing background of ions (Kalman et al. 2002, Talin et al. 2002). The ion-electron interaction is taken in a way that we can consider the quantum effects at short distances; we represent it here by the following pseudo-potential (Deutsh 1977, Deutsh et al. 1978, Minoo et al. 1981):
VieSD(r) =− Ze2
r (1−e
−r/λT)e−r/λD (2)
or Kelbg interaction:
VieK(r) =− Ze2 r√π.(1−e
−r2
λ2
T) +r
√
π λT
(1−erf( r λT
))) (3) We will first investigate the case of screened Deutsh potential, while the results for the Kelbg po-tential are then straightforward. Most previous stud-ies (Talin et al. 2002, Dufty et al. 2003, Dufour et al. 2005) took the Coulomb interaction like elec-trons interactions with themselves and with uniform neutralizing background of positive electric charge. To approach to the reality and taking into account the effect of screening in our study, we will take the electron-electron interaction to be that of the Debye potential energye2e−r/λD/rsuch as the potential
en-ergyVee(r) in the mean field so that approximation is equal to:
Vee(r) =e2 Z
f(−→r′,−→p) e
−|−→r−
− →
r′|
λD
|−→r −−→r′| d −
→p3d−→r′3, (4)
where:
f(−→r ,−→p) = N Ω(
mβ 2π)
3/2e−β(−→p
2
2m+ V(r)) (5)
VefSD(r) =−nee2
Z (1−e−|−→r−
− →
r′|
λT )e−
|−→r−−→r′|
λD
|−→r −−→r′| d − →
r′3
(6) In this formula, we have introduced the screened Deutsh interaction between the electron and the continuous background of positive charge. Then, the potential interaction energy of the elec-tron, with all the plasma components, satisfies the following nonlinear integral equation:
VSD(r) =VSD ie (r)+
e2R R
f(−→r′,−→p) e
−|−→r−
− →r′
|
λD
|−→r−−→r′|
d−→p3d−→r′3
−nee2R (1−e−
|−→r−−→r′|
λT )e−
|−→r−−→r′|
λD
|−→r−−→r′|
d−→r′3
(7)
Using spherical coordinates and some basic calculations, the last integral equation is transformed into the following:
VSD(r) =VieSD(r) + 2πnee2 R∞
0
r′dr′
r λD.
(e−(r+r′)/λ
D−e−|r−r′|/λD)(1−e−βVSD(r′))
−2πnee2R∞ 0
r′dr′
r λ(e−
r+r′
λ −e−
|r−r′ |
λ )
(8)
where λ=λDλT/(λT +λD).
In order to deal with an dimensionless equa-tion, we put YSD(x) = −aVSD(r)/(Ze2) , a = (4πne/3)−1/3, x=r/a, η=λT/a,η′ =λD/aand
ξ =ηη′/(η+η′). After that, we obtain the desired
dimensionless integral equation:
YSD(x) = 1
x(1−e−
x/η)e−x/η′
− 3 2Z
R∞ 0
t x[η′(e
−x+t η′ −e−
|x−t|
η′ )(1−eZΓYSD(t))
−ξ(e−x+ξt −e−
|x−t|
ξ )]dt
(9)
The same calculations for the case of Kelbg interaction give:
YK(x) =YieK(x)+ 3
2Z
R∞ 0
t x[η′(e
−x+t η′
−e−
|x−t|
η′ )eZΓYK(t)
−η√1π(F(t+x)−F(|x−t|))]dt.
(10)
where:
YK
ie(x) =x√1π[1−e−
(x η)2+x
√π
η (1−erf(
x
η))]
(11) and
F(x) =−x(η+x√2π +η2e−(xη)
2
)+ η2√π
2 erf(
x η)(
3 2+ (
x η)
2) (12)
It should be noted here that Shukla et al. (2008) has also studied the electron dynamics around an impurity by considering the hot and degener-ate electrons. For this the quantum distribution of Thomas-Fermi was used.
2.2. Numerical solution of the integral equation for the potential energy
We can solve the nonlinear integral equation (9) by the method of successive iterations (fixed point method FPM) starting with the initial func-tionY0(x) =YSD
ie (x) = x1(1−e−
x/η)e−x/η′
. We can also solve this integral equation by transforming it into a second order nonlinear differential equation and then use the method of Runge-Kutta (RKM) to solve it. The numerical solution of the nonlinear inte-gral equations (9-10), in the case Γ = 0.1,η= 0.177, η′ = 1.826 and Z = 2 and Z = 8 by the iterative
method gives the potential energy as shown in Figs. 1-2.
✁ ✂ ✄
✂
☎
✆
x Y(x
)
x
Y
(x
)
✝✟✞✠☛✡ ☞✟✌ ✍ ✎✏
Γ=0.1 η=0.177 η✑ ✎✓✒✔✕✏✖
✁ ✂ ✄
✂
☎
✆
✝✘✗✙✡ ☞✟✌ ✍✓✎✏
Γ=0.1 η=0.177 η✑ ✎✓✒✔✕✏✖
✁ ✂ ✄
✂
☎
✆
Y
(x
)
x x
Y
(x
)
✝✟✞✠☛✡✌☞✎✍ ✏✒✑ ✓
Γ=0.1 η=0.177 η✔ ✑ ✕✖✓✗✘
✁ ✂ ✄
✙
✚
✛
✜
✝✣✢✤✡☞✎✍ ✏✒✑✓
Γ=0.1 η=0.177 η✔ ✑ ✕✖✓ ✗✘
Fig. 2. Effective potential energy of the electron for Z=8 in Deutsh and Kelbg cases.
In Figs. 1-2, we notice that using the Kelbg potential as initial potential and interaction potential between the electron and the continuous background of positive charge in the integral equation, gives a solution decreasing more weakly than when screened Deutsh potential is used. This means that Deutsh solution is strongly screened, that is to say that its range is also inferior to that of Kelbg solution. The RKM allows us to solve, equivalently to the integral equation, the nonlinear differential equation for the Deutsh case. In this way we have solved the equa-tion:
Y′′+2
rY′ =
3
Z(eZΓY(r)−1 + ( ξ η′)2) +
1
η′2Y(r)
−rη1(
2
η′ +1η)e−
r(1
η′+1η)
(13) with initial conditions:
Y(0)≈1/η and Y′(0)≈ −(η′/2η+ 1)/ηη′ (14)
Fig. 3 shows this equivalence, for the Deutsh case, with the effective potential YSD(r) when Z=8, Γ = 0.1,η= 0.177,η′= 1.826.
✁ ✂ ✄
✁
✂
✄
☎
✆
✝
✞✠✟✡
Γ=0.1 η=0.177 η☛ ✟✌☞✍✡✎✏
✑
✒✓
✔
✕
✖✘✗✙✛✚ ✕✢✜✤✣✌✥✠✦
✖
✗✙
✚
✕✢✜✤✧✩★✪✦
Fig. 3. Effective potential energy of the electron for Z=8 in the Deutsh case computed with the Fixed Point Method (* * *) and Runge-Kutta Method (—).
Fig. 4. Schematic map of electron distribution around the impurities in the plasma.
We have also found that the RKM (differential equation) is faster than the fixed point method (inte-gral equation). The drawback of the RKM is that it is very sensitive to the initial conditions. Conversely, the FPM, despite that it requires a lot of time for computation, it has more guarantees that the result converges towards an exact solution. Another draw-back of the FPM is that it is adaptable only for the shielded initial potential.
What has just been said suggests that the numeri-cal integration of the integral equation or the equa-tion differential must be truncated to the size of this neighborhood. When the screening is weak (that is to say thatλDis very large) the convergence towards the solution of the integral Eq. (9) becomes slow and the solution coincides at large distance with Kelbg solution, see Figs. 1-2.
3. THE DYNAMICAL PROPERTIES OF THE ELECTRONS
3.1. Relativistic electron trajectories in plasma
The calculation of real trajectories of relativis-tic electrons in a hot plasma is a necessary step to cal-culate several dynamical properties such as the time autocorrelation function, the diffusion coefficient and the electric permittivity... So, the purpose of this sec-tion is the calculasec-tion of relativistic trajectories of an electron in a plasma, and then we compare between classical and relativistic trajectories for few cases of potential energy. The relativistic force acting on an electron in the plasma is equal to the derivative of the momentum −→P:
− →F =d−→P
dt = dm
dt −→v +m d−→v
dt (15)
and the derivative of the mass is:
dm
dt =me
d dt
³ 1/p
1−v2/c2´ =mec12−→v · −→γ
¡
1−v2/c2¢−3/2 (16)
where −→γ is the relativistic acceleration, me is the
rest mass of the electron and cthe speed of light in vacuum. Therefore, this force is equal to:
−→F =me
c2 ¡
1−v2/c2¢−3/2
(−→v · −→γ)−→v+p me 1−v2/c2−→γ
(17) and the force on the other hand equals:
− →F = d−→P
dt =−
−
→∇V(r) (18)
whereV(r) represents the potential energy (8) of an electron in the plasma at positionrfrom the coordi-nate origin.
We write the Eqs. 17) and (18) in cartesian co-ordinates, and equating member to member we find the following system of equations:
meω3
c2 vx
³
(vx+ c
2
ω2vx)γx+vyγy+vzγz
´ =−xr
∂V(r)
∂r meω3
c2 vy
³
vxγx+ (vy+ c
2
ω2vy)γy+vzγz
´ =−yr
∂V(r)
∂r meω3
c2 vz
³
vxγx+vyγy+ (vz+ c
2
ω2vz)γz
´ =−z
r ∂V(r)
∂r
(19)
where: ω = 1/q1−¡ v2
x+vy2+v2z
¢
/c2 = 1/p
1−v2/c2
Solving this system of equations gives the ex-pression for the acceleration as follows:
− →γ :
γx=ζ£+¡c2−vx2
¢
x−yvxvy−zvxvz¤
γy =ζ£−xvxvy+¡c2−vy2
¢
y−zvyvz¤
γz=ζ£−xvxvz−yvyvz+¡c2−vz2
¢ z¤
(20)
where:ζ=− 1
mec2ω
1
r ∂V(r)
∂r and, now, we can use the
following Taylor formula to find the trajectories of relativistic electrons:
−
→r(t+ ∆t) = 2−→r(t)− −→r(t−∆t) + (∆t)2−→
γ (t) (21) In Figs. 5-6, we present the difference between clas-sical and relativistic trajectories of an electron in plasma governed by the effective potential (9) and (10). These figures show that the motion of elec-trons in a cold plasma around the impurity center can be bound, but the electrons in the hot plasma are free due to their high velocities and the bounded trajectories do not appear in relativistic motion.
= Γ=0.1η=0.177
.HOEJ3RWHQWLDO
Y
X
&ODVVLFDOWUDMHFWRU\
5HODWLYLVWLFWUDMHFWRU\
Fig. 5. Classical and relativistic trajectories for the initial conditions: r(0)=0.7 and v(0)=0.485.
= Γ=0.1η=0.177η 6FUHHQHG'HXWVK3RWHQWLDO
Y
X &ODVVLFDOWUDMHFWRU\ 5HODWLYLVWLFWUDMHFWRU\
3.2. The microfield autocorrelation function.
The total electric microfield due to the elec-trons on the impurity centered at the coordinate ori-gin is given by:
− →E =XN
k=1
−→
eie(rk) (22)
The dimensionless electric field auto-correlation function is given by Talin et al. (2008):
CEE(t) = a
4
e2 D−→
E(t)·−→EE
= a 4
e2 Z
d−→r1d−→v1....d−→rNd→−vN−→E ·−→E(−t)ρe
=a 4
e2 Z
d−→r1d−→v1−→e(r1)N Z
d−→r2d−→v2....d−→rNd−→vN−→E(−t)ρe
= a 4
e2 Z
d−→r1d−→v1−→e(r1)Ψ(−→r1,−→v1, t) (23)
where ρe is the equilibrium canonical ensemble and
e(rα) is the single particle field. The integrations
over degrees of freedom 2...N in the second equality define a reduced function Ψ(r1, v1, t), which is the first member of a set of such functions
Ψ(−→r1,−→v1....→−rs,−→vs, t) =
Ns Z
d−→rs+1d−→vs+1....d−→rNd−→vN−→E(−t)ρe. (24)
It is straightforward to verify that these functions satisfy the BBGKY hierarchy (Van Kampen et al. 1967):
(∂t+−→v ·−→∇r−
(1−v2 c2)1
/2
me {
− →∇
r[Vie(r) +Vef(r)]}
·{−→∇v−−→vc (
− →v.−→∇v
c )})Ψ(−→r ,−→v , t)
= 1
meω
R
d−→r1d−→v1[−→∇rVee(−→r − −→r1)]
·{−→∇v−−→vc (
− →v.−→∇v
c )}Ψ
(2)(−→r ,−→v;−→r1,−→v1, t) (25) wheremeis the electron mass at the rest andcis the speed of light. Recognizing this linear relationship, the basic approximation for weak coupling among the electrons is to neglect all of their correlations at all times:
Ψ(2)(−→r1,−→v1,−→r2,−→v2, t)→f(−→r2,−→v
2)Ψ(−→r1,−→v1, t) +f(−→r1,−→v1)Ψ(−→r2,−→v2, t)
(26) Use of (26) in the first hierarchy equation (25) gives directly the kinetic equation:
(∂t+L) Ψ(−→r ,−→v , t) =
1
meω[
−→
∇v−−→cv2(−→v .∇−→v)]f(−→r ,−→v)·
− →∇
rRd−→r2Vee(−→r − −→r2)R d−→v2Ψ(−→r2,−→v2, t) (27)
where:
L=−→v .−→∇ − 1
meω
h−→
∇rV(r)
i
.(−→∇v−−→cv2(−→v .−→∇v)
(28) We limit ourselves to the solution of the homoge-neous Eq. (27) which is given by:
Ψ(−→r ,−→v , t) =f(−→r ,→−v)−→emf(−→r(t)) (29)
where:
−
→emf(−→r) = 1
Ze
−
→∇V(r) (30)
andf(r, v) is the Maxwell-Juttner-Boltzmann distri-bution given by:
f(r, v) = exp(−(mc
2+V(r))/kT) m3
ec3K2(mec2/kT)
. (31)
Here m = meω and K2(x) is the modified Bessel function. Replacing (29) in (23) we find:
CEE(t) =
a4 e2
Z
f(−→r ,−→v)−→e (r)· −→emf(r(t))d−→r d−→v
(32) where −→r(t) is the time-dependent position vector. To get it for any time t, we have solved numeri-cally (using Verlet algorithm) the equation of motion d−→P /dt=−e.−→emf(r) where→−P =me−→v /√(1−v2/c2)
is the relativistic momentum of the electron. In the calculation ofCEE(t) the average on the velocities is
done over the relativistic distributionf(r, v). Regarding the function CEE(t) in Fig. 7, we
found: - when one moves away from t = 0, the rel-ativistic effect is manifested more clearly. - In Fig. 8, we note that when we increase the charge number Z, the covarianceC(0) also increases, but all the rel-ativistic curves in Kelbg and Deutsh cases decrease more quickly and have the same behavior for large values of time (vanishe quickly and never cross the Oxaxis).
4. APPLICATION TO THE ELECTRONIC BROADENING IN PLASMAS
✁✂✁✄☎✁✆✝✁✞☎✁✟✡✠✁
✄
✆
✞
✟
C
(t
)
C
(t
)
ωpt ωpt
☛✌☞✍
Γ=0.1 η=0.177 η✎ ☞ ✏✑✒ ✍✓
Classical C(t)SD Relativity C(t)SD
✁✂✁✄☎✁✆✝✁✞☎ ✁✟✡✠✁
✄
✆
✞
✟
☛✌☞✍
Γ=0.1 η=0.177 η✎ ☞ ✏✑✒ ✍✓
Classical C(t)K Relativity C(t)K
Fig. 7. Classical and relativistic electric field auto-correlation function for Z=2 in Deutsh and Kelbg cases.
✁✂✁✄☎✁✆☎✁✝✂✁✞✠✟✁
✡
✟
✟✡
✄
✄✡
☛
☛✡
☞✌✄ ☞ ✌✆
☞ ✌✞ Relativity C(t)
SD
for Z=2, 4 and 8
Γ=0.1 η=0.177 η✍✎✑✏ ✒✓✔ ✕
ωpt
✁☎✁✄✂✁✆✖✁✝✂✁✞✠✟✁
✡
✟
✟✡
✄
✄✡
☛
☛✡
C
(t
)
C
(t
)
☞ ✌✞
☞ ✌✆
☞✌✄
ωpt
Γ=0.1 η=0.177 η✍✎✗✏✒✓✔✕
Relativity C(t)K
for Z=2, 4 and 8
Fig. 8. Relativistic electric field auto-correlation function for different Z in Deutsh and Kelbg cases.
V(t) =−→d .−→e(t) =−e−→R .−→e(−→r(t)) (33) where −→R is the position vector of the emitter elec-tron and −→e(t) is the individual electric field on the impurity due to the electron located at −→r. Now we want to separate the purely atomic part and the part that depends on the details of the collision process. This leads to defining two quantities (Alexiou 1994):
φd=−2π
nee2 3¯h2
Z
vF(v)dv Z
ρdρ
Z ∞
−∞
dt1
Z t1 −∞
dt2eiω1t1eiω2t2−→e(t1).−→e(t2) (34)
and:
φint=−2π
nee2 3¯h2
Z
vF(v)dv Z
ρdρ
Z ∞
−∞
dt1
Z ∞
−∞
dt2eiω1t1eiω2t2−→e(t1).−→e(t2) (35)
whereF(v) is the Maxwell distribution of the veloc-ities. The integral:
2πne Z
vF(v)dv Z
ρdρ (36)
defines the average in the phases space of the particle positions ~ri and the particle momentum
~
pi. The distribution is that of Maxwell-Boltzmann
F(v)exp(−V /kT), then we can transformφd as:
φd=−
e2 3¯h2
Z ∞
−∞
dt1 Z t1
−∞
dt2eiω1t1+iω2t2
<−→E(t1).−→E(t2)>can (37)
CEE(t1−t2) =<−→E(t1).−→E(t2)>can (38)
we can check that:
φd=−
e4 3¯h2a4
Z ∞
0
CEE(t)dt (39)
(in s−1 unit) where C
EE(t) is given by (32). The
same formula is used by (Nguyen et al. 1967) for the collision operator:
φ=−−→Rn.−→Rn( e
4
3.¯h2a4) Z +∞
0
CEE(t)dt≡−→Rn.−→Rnφd
(40) In computing the collision operatorφ, conversely to Alexiou (1994) and his predecessors, the plasma elec-tron (the perturber) moves in the effective field cre-ated by the entire plasma. Moreover, this electron creates a field(Deutsh or Kelbg) at the impurity ion. Then we call φd the amplitude of the collision
op-erator because it is this quantity that contains the plasma parameters through the correlation function CEE(t). This contains all the information regarding
the density ne, the temperature T, and the charge number Z of ions. We present in Table 1 the ratio of the amplitudes of the collision operator between the classical and relativistic case for different plasma conditions such as ne,T and Z. We find that when Z is increased, the relativistic effect increases.
Table 1. The ratio φclassical
d /φrelativisticd .
Z Deutsh Kelbg 1 0.9052 0.9154 2 0.8102 0.8263 4 0.5767 0.6554 8 0.3256 0.3396
From this table, we see that the difference be-tween the classical and relativistic collision operator becomes more important when the number of charge Z increases.
5. CONCLUSION
The electrons dynamics and the time autocor-relation function CEE(t) for the total electric
mi-crofield of the electrons on positive charge impurity embedded in a plasma are considered when the rel-ativistic dynamics of the electrons is taken into ac-count. We have, first, built the effective potential governing the electrons dynamics. This potential obeys a nonlinear integral equation which we solved numerically. We found, for fixed Γ = 0.1 and fixed density ne, that the relativistic effect becomes im-portant (forZ = 8,φclassical/φrelativistic≃1/3) when the charge numberZ increases. The collision opera-tor, responsible for electronic broadening in plasma,
is investigated. The result is that when the plasmas parameters change, the amplitude of the collision op-erator becomes important. The electron-impurity in-teraction is firs taken to be the screened Deutsch interaction and then the Kelbg interaction. Com-parisons of all relevant quantities are made for these interactions as well as between classical and relativis-tic dynamics of electrons. The relativisrelativis-tic trajecto-ries of the plasma electrons around the impurity are more complicated the classical trajectories as it can be seen in Figs. 5-6. This fact has a direct effect on the behavior of the electric auto-correlation function. Indeed, when we move away from t= 0, the differ-ence between the classical CEE and the relativistic
CEE becomes more obvious.
Acknowledgements– During the preparation of this paper we have been financially supported by the Min-istry of Education and Science and Research (PNR project) and LRPPS Laboratory at Ouargla Univer-sity of the Republic, Democratic and Popular of Al-geria for which we present our full and high acknowl-edgements.
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KORELACIONA FUNKCIJA I ELEKTRONSKO XIRENjE
LINIJE U RELATIVISTIQKIM PLAZMAMA
S. Douis and M. T. Meftah
Department of Physics, Faculty of Sciences, LRPPS Laboratory, University of Kasdi Merbah Ouargla, 30000, Algeria
E–mail: mewalid@yahoo.com
UDK 52–726–355.3
Originalni nauqni rad
Razmatraju se dinamika elektrona i vremenski zavisna autokorelaciona funkcija
CEE(t)za ukupno elektriqno mikro-polje
elek-trona na pozitivno naelektrisanoj primesi uronjenoj u plazmu, kada se uzima u obzir relativistiqka dinamika elektrona. Najpre smo konstruisali efektivni potencijal koji upravlja dinamikom elektrona. Ovaj potenci-jal se pokorava nelinearnoj integralnoj jed-naqini koju smo rexili numeriqki. Xto se tiqe elektronskog xirenja linije u plazmi,
naxli smo da kada se parametri plazme me-njaju, amplituda sudarnog operatora se menja na isti naqin kao i integral po vremenu od
CEE(t). Interakcija elektron-primesa je
