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
C
opyright © 2012 SciRes. JMP