Journal of Quantitative Spectroscopy &Radiative Transfer 75 (2002) 723–739
www.elsevier.com/locate/jqsrt
Development of an analytical potential to includeexcited congurations
R. Rodr guez
a
;
∗
, J.G. Rubiano
a
;
b
, J.M. Gil
a
;
b
, P. Martel
a
;
b
,E. Mnguez
a
;
b
, R. Florido
a
a
Departamento de Fsica de la Universidad de Las Palmas de Gran Canaria, 35017 Las Palmas de Gran Canaria, Spain
b
Instituto de Fusi on Nuclear, Universidad Polit ecnica de Madrid, 28006 Madrid, Spain
Received 16 November 2001; accepted 11 March 2002
Abstract
Excited congurations are very important in dense plasma physics. In this work we propose a new analytical potential for excited congurations obtained from another one for ground conguration. With this potential several atomic magnitudes have been calculated for ions in excited congurations analyzing whatkind of excited congurations introduce more inuences in those magnitudes. Using this potential, atomic datagenerated are satisfactorily compared with those obtained using other analytical potential using sophisticatedselfconsistent codes, and with others available in the bibliography.
?
2002 Elsevier Science Ltd. All rights reserved.
1. Introduction
In several scientic areas such as astrophysics, radiation physics or laser–plasma interaction isnecessary to deal with an enormous amount of atomic data. In particular, to study the radiative properties of hot dense matter the knowledge of the atomic structure of dierent ions in the plasmas,energy levels for ground and excited congurations, line transitions and plasmas interactions arerequired.Some computer codes [1–3] based on numerical solution of Schrodinger or Dirac equations for highly ionized atoms are currently available to calculate these atomic data, by using selfconsistentmethods. However, these methods need an iterative procedure, which is a drawback when the number of atomic data is huge as it happens when we consider many excited congurations per ion, becausethe computer time and the complexity increase considerably.
∗
Corresponding author. Fax: +073428452922.
Email address:
rafael@cicei.ulpgc.es (R. Rodr guez).00224073/02/$see front matter
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2002 Elsevier Science Ltd. All rights reserved.PII: S00224073(02)000389
724
R. Rodrguez et al./Journal of Quantitative Spectroscopy & Radiative Transfer 75 (2002) 723–739
To deal with this problem, some computer codes including analytical expressions for eective potentials have been developed [4–6]. It is an interesting choice since they avoid the selfconsistent procedure and they allow detailed calculations of plasma optical properties such as opacities to be made with considerable saving of computation time. This fact is useful when the number of congurations is large, as it happens when we consider many excited congurations, that will makea selfconsistent detailed conguration calculation unapproachable. However, although there are manyanalytical potentials for ions in ground state in the bibliography [7–14], it does not occur for excitedcongurations wherein there are few contributions [4,16,17], this being an important task becauseexcited congurations are specially important in plasmas at LTE and NLTE conditions,.In this work we propose, rst, a method to obtain an analytical potential for ions in excitedcongurations as well as a particular expression for this potential and this expression is obtained by two dierent ways. With both expressions of the potential we have calculated atomic magnitudes specially important in plasma physics such as total energies, energy levels, transition energiesand oscillator strengths. These calculations allow us to determine which are the excited congurations that introduce more eects on the atomic magnitudes mentioned above. Finally, and withthe aim of checking the potential proposed in this work, we have compared our results with othersobtained by using analytical or selfconsistent potentials and with experimental results when it is possible.
2. Method to obtain an analytical potential for ions in excited congurations
Within the independent particle model (IPM) [1] an excited conguration is obtained from an ionin ground state promoting an electron from its initial energy level
k
(with quantum numbers
n; l; j
)to a nal one
k
(
n
; l
; j
), which is higher than the initial. Let
U
(
r
) be the central analytical potential which describes an ion in ground state. We propose for the potential for the ion in theexcited conguration the following expression:
U
e
(
r
) =
U
(
r
) +
U
(
r
) (1)
U
e
(
r
) being the analytical potential for the excited conguration and
U
(
r
) the correction term tothe ground state potential,
U
(
r
).Considering the expression of the electrostatic potential created by an electron in all the space isgiven by (in atomics units)
U
u
(
r
) =

’
u
(
r
u
)

2

r
−
r
u

d
r
u
;
(2)where
’
u
(
r
u
) is the electron wave function, we propose for
U
(
r
) the following expression:
U
(
r
) =
−
U
k
(
r
) +
U
k
(
r
) =
−

’
k
(
r
k
)

2

r
−
r
k

d
r
k
+

’
k
(
r
k
)

2

r
−
r
k

d
r
k
:
(3)
R. Rodrguez et al./Journal of Quantitative Spectroscopy & Radiative Transfer 75 (2002) 723–739
725
U
k
(
r
) and
U
k
(
r
) being the electrostatic potential created by the promoted electron when it is locatedin its initial and nal levels, respectively. By using in (3) the expansion of 1
=

r
−
r
k

in sphericalharmonics, for each term we obtain
U
u
(
r
) =

’
u
(
r
u
)

2
d
r
u
∞
l
=0
4
2
l
+ 1
r
l¡
r
l
+1
¿l
m
=
−
l
Y
lm
(
;
)
Y
lm
(
k
;
k
)
; u
=
k;k
:
(4)As we can see, Eq. (4) depends on the electron wave function. We have used a relativistic onegiven by
’
u
(
r
u
) = 1
r
u
i
l
u
A
u
(
r
u
)
l
u
j
u
m
u
i
l
u
B
u
(
r
u
)
l
u
j
u
m
u
:
(5)In last equation
A
u
and
B
u
are the radial wave functions of large and small components, respectively, and
are the spherical bispinors. Inserting (5) into (4) and separating the large and smallcomponents and the integrals over angles and radial variables, we obtain the following expression:
U
u
(
r
)=
∞
l
=0
4
2
l
+ 1
1
=
2
r
l¡
r
l
+1
¿
A
∗
u
(
r
u
)
A
u
(
r
u
)d
r
ul
m
=
−
l
Y
lm
(
;
)
C
(
l
u
ll
u

000)
×
=
±
1
=
2
C
2
(
l
u
;
1
=
2
;j
u

m
u
−
;;m
u
)
C
(
l
u
ll
u

m
u
−
mm
u
−
)
×
∞
l
=0
4
2
l
+ 1
1
=
2
r
l¡
r
l
+1
¿
B
∗
u
(
r
u
)
B
u
(
r
u
)
dr
ul
m
=
−
l
Y
lm
(
;
)
C
(
l
u
ll
u

000)
×
=
±
1
=
2
C
2
(
l
u
;
1
=
2
;j
u

m
u
−
;;m
u
)
C
(
l
u
ll
u

m
u
−
mm
u
−
)
;
(6)where an integration over the angles has also been realized and
C
denotes a Clebsch–Gordan coecient and
the two spin components. A central analytical potential can be obtained if we make anangular average in Eq. (6) using the spherical bispinor associated with the electron
u
. In this way,we obtain for each term in (3)
U
u
(
r
)
=
2
l
u
l
=0
;l
even
D
(
l;l
u
)
1
r
l
+1
∞
0
A
∗
u
(
r
)
A
u
(
r
)
r
lu
d
r
u
+
r
l
∞
r
A
∗
u
(
r
)
A
u
(
r
) 1
r
l
+1
u
d
r
u
+
2
l
u
l
=0
;l
even
D
(
l;l
u
)
1
r
l
+1
r
0
B
∗
u
(
r
)
B
u
(
r
)
r
lu
d
r
u
+
r
l
∞
r
B
∗
u
(
r
)
B
u
(
r
) 1
r
l
+1
u
d
r
u
;
(7)where
D
(
l;l
u
) = 2
l
k
+ 12
l
+ 1
C
2
(
l
k
l
k
l

000)
:
(8)
726
R. Rodrguez et al./Journal of Quantitative Spectroscopy & Radiative Transfer 75 (2002) 723–739
Finally, expression (3) for the corrector term proposed by us for excited congurations is given by
U
(
r
)=
−
2
l
u
l
=0
;l
even
D
(
l;l
u
)
1
r
l
+1
r
0
A
∗
u
(
r
)
A
u
(
r
)
r
lu
d
r
u
+
r
l
∞
r
A
∗
u
(
r
)
A
u
(
r
) 1
r
l
+1
u
d
r
u
−
2
l
u
l
=0
;l
even
D
(
l;l
u
)
1
r
l
+1
r
0
B
∗
u
(
r
)
B
u
(
r
)
r
lu
d
r
u
+
r
l
∞
r
B
∗
u
(
r
)
B
u
(
r
) 1
r
l
+1
u
d
r
u
+
2
l
u
l
=0
;l
even
D
(
l;l
u
)
1
r
l
+1
r
0
A
∗
u
(
r
)
A
u
(
r
)
r
lu
d
r
u
+
r
l
∞
r
A
∗
u
(
r
)
A
u
(
r
) 1
r
l
+1
u
d
r
u
+
2
l
u
l
=0
;l
even
D
(
l;l
u
)
1
r
l
+1
r
0
B
∗
u
(
r
)
B
u
(
r
)
r
lu
d
r
u
+
r
l
∞
r
B
∗
u
(
r
)
B
u
(
r
) 1
r
l
+1
u
d
r
u
(9)and for a given analytical potential for ground state,
U
(
r
), the analytical potential for excited congurations is given by (1) where the corrector term,
U
(
r
), is given by (9).The asymptotic behavior of the corrector term (9) is the following: for short distances of thenucleus (i.e.
r
→
0) we obtainlim
r
→
0
U
(
r
) =
−
1
r
1
+
1
r
2
(10)
1
=r
i
being the expected value of
r
i
. Therefore, this is a constant for each conguration in particular.On the other hand, for long distances from the nucleus we obtainlim
r
→∞
U
(
r
) = 0
:
(11)This fact implies that the behavior for long distances is the same for both the potentials (groundand excited congurations) and this obeys the fact that the changes in the ion structure are notconsiderable at long distances.
3. Determination of an expression of an analytical potential for excited congurations
In the previous section we have presented a method to obtain an analytical potential for ionsin excited congurations. By using that method in the following we present an expression for this potential. Our starting point is an analytical potential for ions in ground state, given by [14]
U
g
(
r
) =
−
1
r
{
(
N
−
1)
(
r
) +
Z
−
N
+ 1
}
;
(12)where
Z
is the nuclear charge,
N
the number of bound electrons and
(
r
) a screening functiongiven by
(
r
) =
e
−
a
1
r
a
3
if
N
¿
12
;
(1
−
a
2
r
) if 8
6
N
6
11 or
N
= 2
;
3
;
e
−
a
1
r
if 4
6
N
6
7
:
(13)
R. Rodrguez et al./Journal of Quantitative Spectroscopy & Radiative Transfer 75 (2002) 723–739
727
The parameters
a
1
;a
2
;
and
a
3
were determined by tting Eq. (13) to a selfconsistent potentialcomputed by using the code DAVID [3] giving as a result a fourthdegree polynomial in
Z
:
a
k
=
c
1
k
Z
4
+
c
2
k
Z
3
+
c
3
k
Z
2
+
c
4
k
Z
+
c
5
k
:
(14)The coecients in (14),
c
ik
, were obtained for the ground state of isolated ions from He to Ulikeions [17]. With this potential the analytical one that we propose for ions in excited congurationsis given by the following expression:
U
e
(
r
) =
U
g
(
r
) +
U
(
r
)
;
(15)where
U
g
(
r
) and
U
(
r
) are given by Eqs. (12) and (9), respectively. From Eq. (9), we can observethat the potential proposed depends on the radial wave functions of the initial and nal levelsinvolved in the electron promotion responsible for the generation of the excited conguration. Wehave used two ways to obtain them:(1) Wave functions calculated by using the potential for ground state,
U
g
(
r
). These wave functionsare numerically determined and in the following we will denote the potential obtained with thisfunctions as
U
e
A
(
r
).(2) Wave functions obtained by using a method based on a screening hydrogenic model (SHM)[18]. These functions are given by
A
k
(
x
) =
D
k
4
a
(
−
)
x
e
−
x=
2
[
f
1
xL
2
+1
n
−
j
−
3
=
2
(
x
) +
f
2
L
2
−
1
n
−
j
−
1
=
2
(
x
)]
; B
k
(
x
) =
D
k
4
a
(
−
)
x
e
−
x=
2
[
g
1
xL
2
+1
n
−
j
−
3
=
2
(
x
) +
g
2
L
2
−
1
n
−
j
−
1
=
2
(
x
)]
;
(16)where
=
(
j
+
12
)
2
−
2
;
=
Q
nlj
c ; a
=
(
n
−
j
−
1
=
2 +
)
2
+
2
; x
= 2
acr; D
nlj
=
c
1
=
2
2
a
2
(
−
)(
n
−
j
−
1
=
2)![
a
(
n
−
j
−
1
=
2 +
)]
(
n
−
j
−
1
=
2 + 2
)
;f
1
=
a
2
a
(
n
−
j
−
1
=
2 +
)
−
; f
2
=
−
; g
1
=
f
1
f
2
; g
2
=
:c
is the speed of the light and
Q
nlj
is the screened charge of the level
k
(
n; l; j
), which is calculated by means of the following equation:
Q
nlj
=
Q
(
r
nlj
) =
−
[(
Z
−
N
+ 1) + (
N
−
1)(
(
r
nlj
)
−
r
(
r
nlj
))] (17)
being the screening function in Eq. (13) and
its derivate. In the following we will denotethis expression for the potential as
U
eS
(
r
). This way to obtain our potential has the advantage, withrespect to the aforementioned, that the wave functions are analytical and hence computing time issaved. Moreover, for elements with
Z ¡
30, in which
can be neglected with respect to (
j
+ 1
=
2)