Path Integral Formulation for Ionic Broadening in Plasmas: Lyman-<i>α</i> with Fine Structure and Dynamical Effects

doi:10.4236/jmp.2012.330205

Paper Menu >>

Journal Menu >>

Journal of Modern Physics, 2012, 3, 1678-1682

http://dx.doi.org/10.4236/jmp.2012.330205 Published Online October 2012 (http://www.SciRP.org/journal/jmp)

Path Integral Formulation for Ionic Broadening in

Plasmas: Lyman-α with Fine Structure and

Dynamical Effects

N. Bedida, M. T. Meftah

Laboratoire de Rayonnement et Plasmas et Physique des Surfaces (LRPPS), Université Kasdi Merbah Ouargla,

Ouargla, Algérie

Email: [email protected], [email protected]

Received August 19, 2012; revised September 17, 2012; accepted September 24, 2012

ABSTRACT

Using the path integral formalism, the fine structure and dynamics effects are taken into account for the broadening of

spectral lines in a plasma. A compact expression of the dipolar autocorrelation function for an emitter in the plasma is

derived for Lyman alpha lines with fine structure. The expression of the dipolar autocorrelation function takes into ac-

count the dynamics effects, which are represented by the time microfield autocorrelation function.

Keywords: Path Integrals; Autocorrelation Function; Electric Dipole; Fine Structure; Dynamical Effects

1. Introduction

The spectral line shapes of radiative atoms and ions in

the plasma provide valuable diagnostic tools for a num-

ber of physical quantities, such as the density and tem-

perature of charged particles, the transported radiative

energy, and possibly the determination of electric fields

[1]. The shape of lines in a plasma results from the inter-

actions between the radiator and all constituents (neutrals,

electrons and ions) of the plasma. With variable contri-

butions depending on plasma conditions, causes of broad-

ening are the Doppler effect, which is produced by the

movement of the radiator, natural broadening, due to the

finite lifetime of the atomic excited state, and what will

be the focus of this paper, the Stark broadening which is

due to the interaction between the radiator and the elec-

tric field of the two kind of perturbers (ions-electrons) [2].

This problem has been widely studied using the standard

Hamiltonian approach of quantum mechanics. It started

with the work of Baranger [3], and Kolb and Griem [4].

In these classic papers on Stark broadening, the electrons

are treated within the impact theory, and the ions in the

quasi-static approximation. Both kind of particles having

a Coulomb interaction with the radiator, the difference

between ions and electrons is merely due to their velocity

difference. For many plasma conditions, ions are slow

enough to justify the use of a quasi-static approximation,

but for hydrogen plasmas. In our investigation, we intro-

duce an alternative method able to take into account the

fine structure and the dynamics effects. This method is

based on the Feynman path integral formalism [5,6]

which deals with electrons and ions on the same physical

basis. The general frame for this formalism has been pre-

viously developped [7,8], but has then only be applied to

the static ion case. Using this formalism, one can treat

time-independent and time-dependent problems on the

same footing, which is a real advantage over the standard

Hamiltonian approach when solving time-dependent pro-

blems.

In this paper we retrieved the formula of the dipolar

auto-correlation function common in the line broadening

theory. Our derivation uses the Feynman path integral

formalism. Since the mean time of the electron-emitter

collision is negligible compared with the ion-emitter one,

we shall replace the electron-emitter collision effects by

a standard collision operator, whereas the ion-emitter

collisions effects, via the dipole approximation, will be

treated in the perturbative approach using the path inte-

gral formalism. Section 2 is concerned by with rather low

density, and/or high temperature, this static approxima-

tion may however no longer be valid. The formulation of

the dipolar auto-correlation function and in Section 3, we

apply earlier results to the Lyman alpha line with fine

structure in time-dependent electric microfield. Conclu-

sion and perspectives are given in Section 4.

2. The Spectral Line Shape in the Path

Integrals Theory

We start here by the time dipolar autocorrelation function

N. BEDIDA, M. T. MEFTAH 1679

of the radiator from which the spectral line shapes are

generally deduced. The emitter is perturbed by ions and

electrons treated as charged particles moving on classical

paths. For a description of the radiator perturber interac-

tion, it is usually sufficient to keep only the first term in

the multipole expansion, using the so-called dipolar ap-

proximation. As quoted before, the effect of the electrons

is usually treated with the impact theory by a collision

operator. Our path integral approach could be applied to

both electrons and ions. The electric microfield appear-

ing in our formalism could thus be created by the elec-

trons, the ions or both kind of particles. The usual start of

spectral line shape theory is the general formula giving

the radiation power [9]:

 

e d

1Re

Ct t











 (1)

where is the auto-correlation function of the dipo-

lar momentum of the emitter given by:

 



,0 ,0

CtT tTt





 









dd (2)

where



As we are concerned in this work by the Lyman structure,

we have in this case that the lower state ,





is degen-

erate, and we can, after using the representation





 and







are the upper and lower states

respectively of the emitter including the spin states, and



stands for a statistical average over the perturbers.

njm , transform





as:



jj mm

ja j

njmmnj mmnjmTnj m

























dd



(3)



Using the Wigner-Eckart theorem, we obtain in the

njm



representation:



21 2

CtnjnjmTnj m











(4)



where is the reduced matrix element, 1

the en-

ergy corresponding to







1,1 2nj



and the matrix

element of the evolution operator T in upper state a.

3. Application to Lyman-α Broadening

In the case of Lyman alpha line with fine structure, the

autocorrelation function can be written as:

  



 



121 2

ФФ

1234 56

1313 31111 1

e2d1e222d1e22

4222 22222 2

11 1

2d1e

22 2

e e

jj j

ja jja j

jm mm

CtmTmmTm

t Ct Ct CtCt Ct





 

 





 





121

13 1

e2d1e

42 2

ћC





 

 

 

 









 















(5)

where

 

 

123

456

333331313 13 1

222222

222222222 22 2

3333111111 11

2222 22

2 22 222222 22 2

aa a

aaa

CtT CtTCtT

CtTCtT CtT

























(6)

and are the collision operators.

i1,2

Ф

The matrix element of the evolution operator in the upper state is:













,0d d,,0

TtK t



 

 

rrrr rr



(7)

where



,,0Kt







a is the Feynman propagator de-

scribing the emitter evolution in the surrounding ion

plasma.



are the eigen functions of the Dirac op-

erator relative to the free hydrogen atom. Then, to calcu-

late the dipolar auto-correlation function





Ct it is

useful to evaluate the Feynman propagator



,,0



rr as follows:

 









_e.d

,,0 e

Kt D















E

rr r

rrr η (8)

where 0

L is the Dirac Lagrangian for the free hydrogen

atom, and (







rE ) is the interaction between the hy-

drogen atom and the surrounding plasma in the dipole

approximation.





It is possible to develop the propagator ,,0



N. BEDIDA, M. T. MEFTAH

1680

as a perturbation series knowing the free propagator 0

relative to the free hydrogen atom:



 



 

11 22

1 1

1121

,; ,0

ddddee

exp exp

exp.exp0 .

k k

kk kkk

kkkk

Krtr

ћћ





 

 

 

 





















 



 

 

ΛΛ

ΛΛrrEr Errr

rr rr



1, 2lj ljl



 











(9)

where and nj is the spectra of the

free atom including the fine structure.

The eigenfunctions of Dirac Hamiltonian in Coulom-

bian field [10] are given by:







 

jlm

















jl m





(10)

where

i. j

lm jlm









 





σ (11)











are the spherical spinors, r

1, 2lj l jl

 are a

radial functions and 



Knowing that



depend on the spherical har-

monics as:

,,l ,l,mm

Ω

llm







 



 



 

 

 



 

 





(12)

where

 





π2π

θ00

,, cos

for1 and

2123

dfor1 and

2121

0otherwise

lm lmllmm

lmlm

YY llmm



 





 





sin d

























(13)

Let us compute first the matrix element which

can be written as:

   



3ar





1333

22 22

tdd,;,0CKt





 rr rrr

r



(14)

Replacing the propagator by its expression, integrating

over and , and using the orthogonality of the

wave functions we get :





 

111

,0 00

1 3

222

1331

22 2

Ctd d

de e

exp

ktt

kkkkk

ћk

 























 



ΛΛ

rEr Err



t



r (15)

Let us examine the structure of the first terms in this

expansion. Calling 1 the successive terms in the sum

over k in Equation (14), we can write the first terms as:

 term 0k



k



 term 1k



 

11133 1133 1

022 22

ie dd 0

Uћ





















Er rr r

where we have used the selection rule for the total mo-

ment .

 term 2k



 

112133111

00 22

22332

222

1ieddd

Uћ



 















 

 

























Errr r

Er rrrΛ

(17)

N. BEDIDA, M. T. MEFTAH 1681

In this formula, neglecting the coupling effect of the

states 2,,

njm with the state 31

2, ,

, we obtain

that:

 

1 1

222



































r r

3 1

222











Er r

(18)

By using the properties of the spherical harmonics and

the selection rules for the different terms of order , the

result of this component is :









1C t

 

i 2,3,5,6k

Ct (19)

If we proceed similarly for the other components:

, we show that the odd terms of

vanish, and only even terms contribute, and their results

may be written as:



 

cos

Ct Ct



Dht

















where

 

2, ,





2, ,

edDrr

Summarizing (19) and (20), we can write the formula

(5) for the dipolar auto-correlation function





Ct as

[11]:



 





2,1 21,12

13 1i

2d1 exp

42 2

2cos cos

11 1

2d1 exp

222

icos

cos

Ct Ф

Фt

ћћ

Dht











 



































2,3 21,12

ћћ

tht































(21)

where

 







00 00

dd dd

tt tt

htE E



 





 EE

Ф2

Фe



(22)

e and are the electronic collision operators

relative to



1ps

2 and



12 12

21ps





transi-

tions respectively, and the symbol

means as said

earlier the statistical average over the ionic perturbers.

Making the integral over



 in (22), leads to express





ht as a function of the time microfield auto-correla-

tion function







EE . The latter represents, as known,

the dynamical effect of the electric microfield on the

emitters radiative properties :

 





2d10

2d1

ht t

Et C





























(23)

where t means the time in the inverse of the electronic

4πe













plasma frequency unit .

Equation (21) gives the time dipolar autocorrelation

function regardless of the nature of charged particles, we

can apply it according to different interests, either to ions

or to electrons.

4. Conclusions

Using the path integral formalism, we derive an expres-

sion for the Lyman alpha line shape retaining the fine

structure and the effect of ion dynamics. Our main ap-

proximation is a reduction to pair correlation functions of

a cluster expansion in the electric microfield.

This allows to sum all the terms appearing in the stan-

dard perturbative solution in the path integral point of

view, and to express the time dipolar autocorrelation

function in a compact expression involving the electric

field autocorrelation function. In particular, we would

like to use the ability of the path integral point of view

for the description of a full quantum emitter-perturber

interaction. Interesting applications of a full quantum

approach exist in high temperature plasmas such as

found in fusion devices, for a modelling of the emission

of multicharged emitters perturbed by electrons.

REFERENCES

[1] H. R. Griem, M. Blaha and P. Kepple, “Stark-Profile Cal-

culations for Resonance Lines of Heliumlike Argon in

Dense Plasmas,” Physical Review A, Vol. 41, 1990, pp.

5600-5609.

[2] M. Baranger, “Atomic and Molecular Processes,” Acade-

mic Press Inc., New York, 1962.

[3] M. Baranger, “Simplified Quantum-Mechanical Theory

of Pressure Broadening,” Physical Review, Vol. 111, No.

2, 1958, pp. 481-491. doi:10.1103/PhysRev.111.481

[4] A. C. Kolb and H. R. Griem, “Theory of Line Broadening

in Multiplet Spectra,” Physical Review, Vol. 111, 1958,

pp. 514-521.

N. BEDIDA, M. T. MEFTAH

1682

[5] R. Feynman and A. R. Hibbs, “Quantum Mechanics and

Path Integrals,” McGraw- Hill, New York, 1965.

[6] H. Kleinert, “Path Integrals in Quantum Mechanics Sta-

tistics and Polymer Physics,” World Scientific, Singapore,

1990.

[7] Is. Chihi, M. T. Meftah and H, Kleinert, “Path Integral

Approach in the Plasma Radiation Theory,” Journal of

Plasma Physics, Vol. 70, 2004, pp. 553-559.

doi:10.1017/S0022377803002794

[8] H. Bouguettaia, Is. Chihi, K. Chenini, M. T. Meftah, F.

Khelfaoui and R. Stamm, “Application of Path Integral

Formalism in Spectral Line Broadening: Lyman-α in Hy-

drogenic Plasma,” Journal of Quantitative Spectroscopy

and Radiative Transfer, Vol. 94, No. 3-4, 2005, pp. 335-

346. doi:10.1016/j.jqsrt.2004.09.015

[9] H. R. Griem, “Spectral Line Broadening by Plasma,”

McGraw-Hill, New York, 1964.

[10] L. Landau and E. Lifchitz, “Théorie Quantique Relati-

viste,” MIR, Moscou, 1972.

[11] N. Bedida, M. T. Meftah, D. Boland and R. Stamm, “Path

Integral Formalism for Spectral Line Shape in Plasmas,”

Proceedings of the 19th International Conference on Spe-

ctral Line Shapes, Valladolid, 15-20 June 2008, pp. 100-

101.