ls0">
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)
Copyright © 2012 SciRes. JMP
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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