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
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]:
 
i
0
e d
t
1Re
π
I
Ct t

Ct
(1)
where is the auto-correlation function of the dipo-
lar momentum of the emitter given by:
 

,0 ,0
ba
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
Λ
stands for a statistical average over the perturbers.
,,
j
njm , transform
Ct

as:

i1
12
*
e
11
11
22
t
jj
jj
ћ
jj mm
ja j
m
Ct
njmmnj mmnjmTnj m

dd
(3)
Using the Wigner-Eckart theorem, we obtain in the
,,
j
njm

representation:

1
12
i
2
1
e1
d1
21 2
t
j
ћ
j
aj
jm
CtnjnjmTnj m
j
1
d
(4)
where is the reduced matrix element, 1
12
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:
  

 

i1
121 2
2
31
22
22
ФФ
11
31
22
2
Ф
1
1234 56
1313 31111 1
e2d1e222d1e22
4222 22222 2
11 1
2d1e
22 2
t
e e
jj j
e
tt
ћ
ja jja j
jm mm
t
CtmTmmTm
t Ct Ct CtCt Ct

 
 

 

i1
121
2
Ф
1
13 1
e2d1e
42 2
t
et
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










Ta

(6)
and are the collision operators.
i1,2
e
Ф
The matrix element of the evolution operator in the upper state is:
*
'
,0d d,,0
qa
TtK t

 
 
rrrr rr
(7)
where

,,0Kt
rr

r
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
a
Kt
rr as follows:
 



0
0
i
_e.d
0
,,0 e
tD
tL
a
Kt D


E
rr r
rr
rrr η (8)
where 0
D
L is the Dirac Lagrangian for the free hydrogen
atom, and (
e
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
a
Kt
rr
Copyright © 2012 SciRes. JMP
N. BEDIDA, M. T. MEFTAH
1680
as a perturbation series knowing the free propagator 0
K
relative to the free hydrogen atom:







 
 

 
11
11
11 22
1 1
*
11
0
**
1121
*
11
,; ,0
ddddee
ii
exp exp
ii
exp.exp0 .
kk
k
k k
a
k
kk kkk
k
kkkk
tt
oo
Krtr
kk
t
ћћ
ћћ



 
 
 
 






 
 
 
Λ
ΛΛ
Λ
ΛΛrrEr Errr
rr rr
rr


1, 2lj ljl
 
21
21
1
i1
!
.
kk




(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:

 
j
jlm
ll
nj
f
g






r
r
r
1
2
1j
jl m

(10)
where
i. j
ll
j
j
lm jlm
r

 


r
σ (11)

f
r,
g
j
j
lm
are the spherical spinors, r
1, 2lj l jl
are a
radial functions and 
Knowing that
j
lm
j
depend on the spherical har-
monics as:
11
m
22
l
11
11
,,l ,l,mm
22
22
l
m1
Y
22
m1
Y
22
jj
jj
jj
jj
llm
jj
j
jj
j
jj



 

 
 
 
 

 
 
 



m
l
m
l
jm
Y
jm
Y
(12)
where
 

π2π
**
θ00
,, cos
11
for1 and
2123
dfor1 and
2121
0otherwise
mm
ll
lm lmllmm
ll
lmlm
YY llmm
ll
 

 
sin d









1
Ct
(13)
Let us compute first the matrix element which
can be written as:
   
3ar
*
1333
22
22 22
tdd,;,0CKt


 rr rrr
r
r

(14)
Replacing the propagator by its expression, integrating
over and , and using the orthogonality of the
wave functions we get :


 
23
2
111
,0 00
*
1 3
222
*
1331
22
22 2
i1
Ctd d
!
de e
i
exp
k
k
ktt
k
kkkkk
ћk
ћ
 









 
Λ
Λ
ΛΛ
Λ
rEr Err
rr
k
U
3
3
d
0
k
t
r
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
:0
1
U1
k
term 1k
:
 
1*
11133 1133 1
22
022 22
ie dd 0
t
k
Uћ










Er rr r
j
where we have used the selection rule for the total mo-
ment .
term 2k
:
 
 
2
2
2
2
2*
112133111
2
00 22
*
22332
222
1ieddd
2
d
tt
k
Uћ
 







 
 











Errr r
Er rrrΛ
(17)
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N. BEDIDA, M. T. MEFTAH 1681
In this formula, neglecting the coupling effect of the
states 2,,
j
njm with the state 31
2, ,
22
, we obtain
that:
 
 
2
*
1 1
33
222
d0














r r
r r
k
2
13
3 1
222
*
d

Er r
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 :
14
1C t
 
i 2,3,5,6k
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:

i
Ct
 
 
25
36
cos
cos
Ct Ct
Ct Ct


Dht
ћ
Dht
ћ






where
 
*
11
2, ,
22


rr
13
1
2, ,
22
edDrr
Summarizing (19) and (20), we can write the formula
(5) for the dipolar auto-correlation function
Ct as
[11]:


 



2
11
2
12
2,1 21,12
13 1i
2d1 exp
42 2
2cos cos
11 1
2d1 exp
222
icos
cos
e
e
Ct Ф
ћ
ht
Фt
D
ћћ
Dht
ћ










2,3 21,12
tt
DD
ht
ћћ
tht











(21)
where
 



00 00
1
dd dd
3
tt tt
htE E
 


 EE
1
Ф2
Фe
(22)
e and are the electronic collision operators
relative to

12
1ps
32
2 and

12 12
21ps
transi-
tions respectively, and the symbol
Λ
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
Ct

EE . The latter represents, as known,
the dynamical effect of the electric microfield on the
emitters radiative properties :
 


0
2
0
2d10
3
2d1
3
t
t
EE
p
t
ht t
Et C
t










EE
(23)
where t means the time in the inverse of the electronic
2
4πe
e
p
e
N
m




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.
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N. BEDIDA, M. T. MEFTAH
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