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: n_bedida@yahoo.fr, mewalid@yahoo.com

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

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