relativistic electronic broadening of the degenerated … · relativistic electronic broadening of...

Global Journal of Pure and Applied Mathematics.

ISSN 0973-1768 Volume 13, Number 1 (2017), pp. 51-62

© Research India Publications

http://www.ripublication.com

Relativistic Electronic Broadening of the Degenerated

Isolated Spectral Line in Plasma

A. Naama*, L. Benmebroukb and M. T. Meftahb

aUniv kasdi Merbah Ouargla, Fac. Faculty of Applied Science, Lab. Rayonnement et Plasmas et Physique de Surface, Ouargla 30 000, Algeria.

bUniv kasdi Merbah Ouargla, Fac. Faculty of Mathematics and Matter Sciences, Lab. Rayonnement et Plasmas et Physique de Surface, Ouargla 30 000, Algeria

The impact collision operator for the relativistic electron broadening of degenerate

lines in plasmas, are calculated in the semiclassical approximation. The impact

approximation is used to treat the elastic interaction between the radiating or

absorbing hydrogen atom or hydrogenic ion) and the relativistic perturbing electron.

In this work, we considered that the trajectory of electrons is a straight line, taking

into account the relativistic effects. On the other hand, we included the relativistic

effects in the velocity distribution taking those of Maxwell-Juttner. The results are

compared with the non relativistic case.

The spectroscopy Spectral line Stark broadening in dense plasma is primarily due to

interactions of emitting atoms, ions, or molecule with the perturbing plasma

constituents. For accurate spectroscopic observations, it is necessary to studies this

phenomenon, it can in addition yield useful information about the concentrations and

conditions in the plasma. In astrophysics plasmas, the alternative methods are not

possible, the line shapes and shifts affords a important and sensitive diagnostic for

determining plasma characteristics such as particle density. According to Anderson,

Baranger, Griem, and Kolb,1-3 the electron contribution to the broadening can be

calculated by using a recently developed impact theory. The perturber electrons can

be treated as point charges moving along their classical trajectories (classical path

approximation).

http://dx.doi.org/10.1063/1.4968803

http://dx.doi.org/10.1063/1.4968803

52 A. Naam, L. Benmebrouk and M. T. Meftah

In high-temperature ionized gases (number of hot astrophysical plasmas: Stellar

interior, Solar photosphere ...) where pressure broadening could dominate, the much

thought has given to using this pressure as a tool for measuring temperatures and

densities inside the plasma. In the other hand, the perturber electrons may become

relativistic due to the extreme temperatures. Then, it must taken account the

modification to the pressure broadening by relativistic effects. The relativistic

collision operator calculated by4 where the trajectory of perturbed electron is

considered hyperbola.

Instead of the hyperbolic trajectories, we can use the straight classical path for

perturber electron to calculate the impact broadening of hydrogenic ion lines.5 In the

present work, the trajectory of the perturber electron is taken straight lines. The

influence of the external field (due to the ion or atom) on the perturber electron

trajectory is negligible.

In this study, we use the region corresponding to the particular condition of plasma:

very high temperature and high density. In this region, the elastic electron- emitter

collision will be binary, and the dynamics of perturber electrons will be treated

relativistically.

We developed the electron collision operator in the impact approximation. We revisit

the standard semiclassical collision of the degenerate isolated line shapes and take

into account relativ-istic effects.

We consider a radiant hydrogen atom (or hydrogenic ion), motionless in the middle of

the gas formed by perturber electrons. Electrons are fast particle have very low

masses, the collision delay is too small then the time interval with which the

correlation function are calculated.2,6 So they are treated by the impact approximation.

The validity condition of the impact approximation2,6-8 can be written as:

ωτ<<1 (1)

where ω is the line width.

By using the estimation of elastic collision delay τ and the line width ω, the validity

condition of the impact theory2 is written as:

1)2(2

63

2/33

nhN

TKmZ

e

Be (2)

where em is the rest mass of electron, BK is the Boltzmann constant.

Relativistic Electronic Broadening of the Degenerated Isolated Spectral Line 53

This condition is well satisfied for the high density and high plasma temperature. In

this range of densities and temperatures, we shall use the statistical classical

mechanics (not the quantum statistical mechanics as in the Fermi-Dirac distribution)

because the De Broglie th thermal length is inferior to 3/1

eN :

3/1

2

eBe

th NTKm

h

(3)

We consider a system of plane polar coordinates ( , )r , the perturper electron

movement is plane. The vector of the angular momentum of the system M is

constant; its modulus is given by:

2mrM (4)

In our case, the emitter is neutral or ion neglecting the influence of its field on the

perturber electron trajectory. The perturber electron trajectory is considered to be a

straight path. Then, the total energy E of the system is maintained:

22

22 mvEmve (5)

where ev the velocity of the perturber electron, defined by:

2 2 2

ev r r (6)

and:

vve (7)

with v is the initial velocity of the perturber electron.

By using expressions of energy and kinetic moment conservations (4) and (5), we get

the time t and the polar angle :

CteEdrt

rmM

m

22

22 (8)


CtemE

dr

rM

rM

2

2

2

2 (9)

We perform an elementary integration, we get:

mvrM

arccos (10)

One can choose the origin of the angle so that the constant is zero and introduce the

formula of angular momentum:

mvMM 0 (11)

where is impact parameter.

Including the formula of the straight path of perturber electron can be written as:

cosr (12)

One can also express the parametric equations and the Cartesian coordinates ),( YX of

the electron is obtained:

coshr x (13)

sinht x

v

(14)

cosr X (15)

sinhY x (16)

with x real parameter, varies between and .

Within the impact approximation, electronic collision width of the isolated spectral

line can be expressed in terms of the operator matrix elements 3 ,9,10:


ddvvvfeNe

max

min02

220

211

1 tEtEdtdtt

(17)

where eN is the electron density, ( )E t is the electric micr-

owave field which can be defined as:

3

0

( )4

e rE tr

(18)

)(vf is the Maxwell distribution of velocity:

)2

exp(2

42

2

2/3

TKmvv

TKmvf

BB

(19)

Using parametric equations polar r and Cartesian X and Y coordinate, the

broadening operator for degenerate is-olated lines is written as (Sahal-Bréchot,1969a):

max

min

24

2 00

2 10

3 4

e f vN e ddvv

1 1 2

1 2 2 2

1 2

1 sinh sinh

cosh cosh

t x xdx dxx x

(20)

We can consider two functions )0(1G , )0(2G :

xdxG

21cosh

1)0(

(21)

xxdxG

22cosh

sinh)0(

(22)

Then the formula (20) can be written as:

dvvvfeNe

0

2

0

2

4

4

1

3

20


)]0()0([ 2

2

2

1

max

min

GGd

(23)

After integration over the parameter x , we find functions )0(1G , )0(2G :

2

cosh

sinh

cosh

1)0(

21

xx

xdxG (24)

0

cosh

sinh)0(

22

xxdxG (25)

what we allow to deduce the following formula:

ddvvvfNe

max

min0

2

0

2 4

1

3

)4exp(80

(26)

Now integrating over the impact parameter, we get:

min

max

0

2

0

2ln

4

1

3

)4exp(80

dv

vvfNe

(27)

Replacing the Maxwell distribution of velocity by its formula, we obtain11:

3/24

22

0

8 1 20

3 4

e

B

N e mK T

min

max2

0ln)

2exp(

dv

TKmvv

B

(28)

It should normally use the Debye length D to choose the maximum impact

parameter max . The maximum impact parameter is introduced to account for Debye

shielding3 reported a similar cut to max 1.1 D .

On the other hand,S. Alexiou12 uses the value D 68.0max . One can also choose:

D max (29)


In this work, we use a simplified determination of )(min v based on the average

velocity v of the perturber electron12:

mZv

v )(min (30)

where Z is the atomic number of the emitter.

Bearing the expression (29) and (30) into (28) we have:

3/24

22

0

8 1 20

3 4

e

B

N e mK T

2

0exp( ) ln

2

D

B

mZmvv dv vK T

(31)

By integrating over the initial electron velocity v , we find the formula electronics

collision operator: TK

meNB

e

2

4

1

3

40

2

0

2

4

e

DB ZTmK

)

2ln(

2

22

(32)

with e the Euler-Mascheroni constant:

577.0e (33)

Note that this operator depends on the electron temperature T and the electron density

eN , and also related to the atomic number of the emitter Z .

In this section, we take into account the relativistic effect of perturber electron in the

calculation of the electronic collision operator. It is assumed that all the electron-

electron interactions are neglected. Then we are in the image of a set of binary

collisions (atom- electron and ion-electron). To establish the expression of the

relativistic electronic collision operator, It is necessary to study the relativistic motion

of the electron.


Just as the problem in the non-relativistic case, we consider a system of plane polar

coordinates ( r , ). It is easy, expressed in terms of polar coordinates the relativistic

kinetic momentum M :

2mrM (34)

with:

21

1

e

(35)

and:

cvee /

(36)

where c the velocity of the light in vaccum.

We neglect the influence of the emitter on the trajectory of perturber electron,

introducing the conservation laws of relativistic total energy E and relativistic

momentum M of the system:

11 0

22 mcEmc (37)

and:

00

2 mvMmrM (38)

then:

vve ,0 (39)

Using the equations (37) and (38), we obtain:

)1(

1)1(

2

222

2

2

2

mcE

rcmM

mcEc

r

(40)


from where:

Cte

c

drt

rcmM

mcE

mcE

222

2

2

2

1)1(

)1(

2 (41)

Cte

dr

rcmM

mcE

cmrM

222

2

2

2

1)1( 2 (42)

By substituting equation (42), we find:

r1

arccos

(43)

So the formula of relativistic trajectory of the perturber elect-ron is:

cosr (44)

Note that in the relativistic case, the perturber electron describes the same non-

relativistic straight path. Relativity does not affect the straight path. Relativity does

not affect the straight path, therefore, the Cartesian and the polar coordinates of the

electron are the same in the expressions (13), (14), (15) and (16).

Now, we use the path formula and parametric equations of linear motion to get the

relativistic electron collisions operator. The calculations are developed as part of the

semi-classical theory. The average effect of disturbing electrons is calculated using

the relativistic Maxwell- Juttner distribution of velocity electrons:

d

kTmc

Kdf )exp(

)/1()( 0

2

2

5

0

2

(45)

with:

2mcTKB

)64(

where )(2 xK is the modified Bessel function of t-he second kind of imaginary order.


Determining the influence of relativistic correction in the relativistic distribution, on

the electronic collision operator, the formula (27) be-comes:

2

4

2

0 2

8 1 10

3 4 (1/ )

eN ec K

1

5

0 00

exp( / ) ln DmZcd (47)

Using the approximation 2

12

0

, the formula (47) can be written as:

2

4 1

2

0 2

exp( )8 10

3 4 (1/ )

eN ec K

21

5

00

exp ln2

d a

(48)

We introduce the following notations:

mZcab D

,

2

1 (49)

We obtain:

24 1

2

0 2

exp( )8 10

3 4 (1/ )

eN ec K

2

15

00

exp ln2

d a

(50)

To get the expression of the relativistic electron collision operator, we calculate the

integral over :

24 1

2

0 2

6 5 4 3 2

6 5 4 3 2

1 6

5 4 3 2 1

6

exp( )10

48 4 (1/ )

[ 274 500 440 240 80

8(15 30 30 20 10

4 ) ln Ei(1, ) 2 ln 6(40

120 180 180 135 81 )

ln exp( ) (274 654

e

e

N ec K

b b b b bb b b b b

b b b a b

b b b b ba b b b

5 4747b

3 2519 211 )exp( )]b b b (51)


It is clear that the expression of the relativistic electron collision operator depends on

the temperatureT , electron density eN , and the velocity of light c .

To validate our results we take the limit :

2

4

200

4 1 2lim 0

3 4

eN e mKT

2 2

2

2ln 0D

eKTm Z

(52)

This is the case where the collision operator relativistic reduces to the non-relativistic

case given by the formula (32). From a mathematical point of view, the expression in

the non-relativistic case is the limited development of second order of relativistic

expression.

The fig. (1) presents the variation of the operator of electron relativistic and non-

relativistic collision, as a function of the electron temperature. Note that due to the

shorter collision duration for fast collision, both operators decrease with increasing

temperature, with relativistic operator exhibiting a slower decrease.

In fig. (2) shows the variation of (0) / (0) ( )0( is the difference between the

relativistic and non-relativistic *(0) (0) ) as a function on the temperature in the

case of Lyman- transition with electron density 2010 cm-3 . We remark that

(0) / (0) increases to reach .91%35 at a temperature of 910 K .

0 2 4 6 8 10

0

5

10

15

20

25

,

( 1

032 H

z/m

2)

T ( 108 K )

Ne=10

20 cm

-3

Z=56

Figure 1. Relativistic and non-relativistic collision operators versus temperature for

Lyman- transition in the case of 56Z (Barium).


0 2 4 6 8 10

0

5

10

15

20

25

30

35

40

()

*10

0/

T ( 108 K )

Ne=10

20 cm

-3

Z=56

Figure 2. Variation of )0(/)0( versus temperature for Lyman- transition in the

case of 56Z (Barium).

REFERENCES

[1] P.W. Anderson, Phys. Rev 76, 647(1949).

[2] M. Baranger, Phys. Rev 111, 494(1958b).

[3] H.R.Griem, A. Kolb and Y. Shen, Phys. Rev 116, 4(1959).

[4] A.Naam, M. T.Meftah, S. Douis, S. Alexiou, ASR 54,1242(2014).

[5] H. R. Griem, Y. Shen, Phys. Rev 122, 1490(1961).

[6] M. Baranger, Phys. Rev 111, 481(1958a).

[7] M. Baranger, Phys. Rev 112, 855(1958c).

[8] S. Sahal-Bréchot, Astron. Astrophys 2, 91(1969a).

[9] H. R. Griem, M. Baranger , A.C. Kolb, G. Oertel, Phys.Rev 125, 177(1962).

[10] S.Sahal-Bréchot, Astron. Astrophys 2, 322(1969b).

[11] H.R. Griem, Astrophys 148, 547(1967).

[12] S.Alexiou, Phys. Rev 49(1),106(1994).

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