Development of an analytical potential to include excited configurations

Journal of Quantitative Spectroscopy & Radiative Transfer 75 (2002) 723 – 739

www.elsevier.com/locate/jqsrt

Development of an analytical potential to include excited con(gurations R. Rodr*+gueza; ∗ , J.G. Rubianoa; b , J.M. Gila; b , P. Martela; b , E. M*+ngueza; b , R. Floridoa a

Departamento de F sica 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 con(gurations are very important in dense plasma physics. In this work we propose a new analytical potential for excited con(gurations obtained from another one for ground con(guration. With this potential several atomic magnitudes have been calculated for ions in excited con(gurations analyzing what kind of excited con(gurations introduce more in=uences in those magnitudes. Using this potential, atomic data generated are satisfactorily compared with those obtained using other analytical potential using sophisticated self-consistent codes, and with others available in the bibliography. ? 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction In several scienti(c areas such as astrophysics, radiation physics or laser–plasma interaction is necessary 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 diAerent ions in the plasmas, energy levels for ground and excited con(gurations, line transitions and plasmas interactions are required. Some computer codes [1–3] based on numerical solution of SchrDodinger or Dirac equations for highly ionized atoms are currently available to calculate these atomic data, by using self-consistent methods. 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 con(gurations per ion, because the computer time and the complexity increase considerably. ∗

Corresponding author. Fax: +07-34-28-452-922. E-mail address: [email protected] (R. Rodr*+guez).

0022-4073/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 0 2 ) 0 0 0 3 8 - 9

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To deal with this problem, some computer codes including analytical expressions for eAective potentials have been developed [4 – 6]. It is an interesting choice since they avoid the self-consistent 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 con(gurations is large, as it happens when we consider many excited con(gurations, that will make a self-consistent detailed con(guration calculation unapproachable. However, although there are many analytical potentials for ions in ground state in the bibliography [7–14], it does not occur for excited con(gurations wherein there are few contributions [4,16,17], this being an important task because excited con(gurations 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 excited con(gurations as well as a particular expression for this potential and this expression is obtained by two diAerent ways. With both expressions of the potential we have calculated atomic magnitudes specially important in plasma physics such as total energies, energy levels, transition energies and oscillator strengths. These calculations allow us to determine which are the excited con(gurations that introduce more eAects on the atomic magnitudes mentioned above. Finally, and with the aim of checking the potential proposed in this work, we have compared our results with others obtained by using analytical or self-consistent 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 con(guration is obtained from an ion in 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 the excited con(guration the following expression: Ue (r) = U (r) + U (r)

(1)

Ue (r) being the analytical potential for the excited con(guration and U (r) the correction term to the ground state potential, U (r). Considering the expression of the electrostatic potential created by an electron in all the space is given by (in atomics units)
Uu (r) =

|’u (ru )|2 dru ; |r − ru |

(2)

where ’u (ru ) is the electron wave function, we propose for U (r) the following expression:
U (r) = −Uk (r) + Uk
(r) = −

|’k (rk )|2 drk + |r − rk |




|’k
(rk
)|2 drk
: |r − rk
|

(3)

R. Rodr guez et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 75 (2002) 723 – 739

725

Uk (r) and Uk
(r) being the electrostatic potential created by the promoted electron when it is located in its initial and (nal levels, respectively. By using in (3) the expansion of 1=|r − rk | in spherical harmonics, for each term we obtain
Uu (r) =

2

|’u (ru )| dru

∞  l=0

l l  4 r¡ Ylm (; ) Ylm (k ; k ); l+1 2l + 1 r¿ m=−l

u = k; k
:

(4)

As we can see, Eq. (4) depends on the electron wave function. We have used a relativistic one given by  lu  1 i Au (ru )lu ju mu ’u (ru ) = : (5) ru il
u Bu (ru )l
ju mu u

In last equation Au and Bu 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 small components and the integrals over angles and radial variables, we obtain the following expression: 
l l ∞    r¡ ∗ 4 1=2 Uu (r) = A (r )A (r ) dr Ylm (; )C(lu llu |0 0 0) u u u l+1 u u 2l + 1 r ¿ l=0 m=−l ×



C 2 (lu ; 1=2; ju |mu − ; ; mu )C(lu llu |mu − mmu − )

=±1=2


l ∞  l   r¡ ∗ 4 1=2 B (r )B (r )dr Ylm (; )C(l
u ll
u |0 0 0) × u u u l+1 u u 2l + 1 r ¿ l=0 m=−l ×



C 2 (l
u ; 1=2; ju |mu − ; ; mu )C(l
u ll
u |mu − mmu − );

(6)

=±1=2

where an integration over the angles has also been realized and C denotes a Clebsch–Gordan coeOcient and  the two spin components. A central analytical potential can be obtained if we make an angular average in Eq. (6) using the spherical bispinor associated with the electron u. In this way, we obtain for each term in (3)  
∞
∞ 2lu  1 1 ∗ l l ∗ Uu (r) = D(l; lu ) l+1 Au (r)Au (r)ru dru + r Au (r)Au (r) l+1 dru r ru 0 r l=0;l even


+

2lu  l=0;l even




D(l; lu )



1 r l+1


0

r

∗

Bu (r)Bu (r)rul

dru + r

l


r

∞

∗

Bu (r)Bu (r)

1 rul+1

 dru ;

(7)

where D(l; lu ) =

2lk + 1 2 C (lk lk l|0 0 0): 2l + 1

(8)

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R. Rodr guez et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 75 (2002) 723 – 739

Finally, expression (3) for the corrector term proposed by us for excited con(gurations is given by  
r
∞ 2lu  1 1 ∗ l l ∗ D(l; lu ) l+1 Au (r)Au (r)ru dru + r Au (r)Au (r) l+1 dru U (r) = − r ru 0 r l=0;l even


−

2lu 

D(l; l
u )



2lu 

 D(l; lu )




+

l=0;l even




D(l; lu )

1



1 r l+1

r

0




r l+1

l=0;l even 2lu 




r l+1

l=0;l even

+

1

r

0


0

r

Bu∗ (r)Bu (r)rul dru + r l A∗u (r)Au (r)rul dru + r l

∗

Bu (r)Bu (r)rul

dru + r

l




∞

r




∞

r


r

∞

Bu∗ (r)Bu (r)

1 rul+1

A∗u (r)Au (r)

∗

Bu (r)Bu (r)

1 rul+1 1

rul+1

 dru  dru  dru

(9)

and for a given analytical potential for ground state, U (r), the analytical potential for excited con(gurations 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 the nucleus (i.e. r → 0) we obtain

1 1 lim U (r) = − + (10) r →0 r1 r2 1=ri  being the expected value of ri . Therefore, this is a constant for each con(guration in particular. On the other hand, for long distances from the nucleus we obtain lim U (r) = 0:

r →∞

(11)

This fact implies that the behavior for long distances is the same for both the potentials (ground and excited con(gurations) and this obeys the fact that the changes in the ion structure are not considerable 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 ions in excited con(gurations. 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] 1 U g (r) = − {(N − 1)(r) + Z − N + 1}; (12) r where Z is the nuclear charge, N the number of bound electrons and (r) a screening function given by  a  if N ¿ 12; e − a1 r 3   (13) (r) = (1 − a2 r) if 8 6 N 6 11 or N = 2; 3;    − a1 r e if 4 6 N 6 7:

R. Rodr guez et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 75 (2002) 723 – 739

727

The parameters a1 ; a2 ; and a3 were determined by (tting Eq. (13) to a self-consistent potential computed by using the code DAVID [3] giving as a result a fourth-degree polynomial in Z: ak = c1k Z 4 + c2k Z 3 + c3k Z 2 + c4k Z + c5k :

(14)

The coeOcients in (14), cik , were obtained for the ground state of isolated ions from He to U-like ions [17]. With this potential the analytical one that we propose for ions in excited con(gurations is 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 observe that the potential proposed depends on the radial wave functions of the initial and (nal levels involved in the electron promotion responsible for the generation of the excited con(guration. We have used two ways to obtain them: (1) Wave functions calculated by using the potential for ground state, U g (r). These wave functions are numerically determined and in the following we will denote the potential obtained with this functions as UAe (r). (2) Wave functions obtained by using a method based on a screening hydrogenic model (SHM) [18]. These functions are given by Ak (x) =

Dk 2!−1 x! e−x=2 [f1 xLn2!+1 −j−3=2 (x) + f2 Ln−j−1=2 (x)]; 4a!(" − !)

Bk (x) =

Dk 2!−1 x! e−x=2 [g1 xLn2!+1 −j−3=2 (x) + g2 Ln−j−1=2 (x)]; 4a!(" − !)

where !=



(j + 12 )2 − %2 ;

2a2 Dnlj = c1=2 % f1 =



%=

Qnlj ; c

(16)

%

a= 

(n − j − 1=2 + !)2 + %2

;

x = 2acr;

(" − !)(n − j − 1=2)![a"(n − j − 1=2 + !)] ; '(n − j − 1=2 + 2!)

a%2 ; a"(n − j − 1=2 + !) − %!

f2 = " − !;

g1 =

f 1 f2 ; %

g2 = %:

c is the speed of the light and Qnlj is the screened charge of the level k (n; l; j), which is calculated by means of the following equation: Qnlj = Q(rnlj ) = −[(Z − N + 1) + (N − 1)((rnlj ) − r
(rnlj ))]

(17)

 being the screening function in Eq. (13) and 
its derivate. In the following we will denote this expression for the potential as USe (r). This way to obtain our potential has the advantage, with respect to the aforementioned, that the wave functions are analytical and hence computing time is saved. Moreover, for elements with Z ¡ 30, in which % can be neglected with respect to (j + 1=2)

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R. Rodr guez et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 75 (2002) 723 – 739

in the evaluation of !, we have obtained an approximate but fully analytical expression for this potential since each term of the correction term for excited con(gurations is given by the following equation: Uu (r) = −

2lu 

C(l; lu )V [f12 I + 2f1 f2 I2 + f22 I3 ]

l=0;l even

−

2lu 

C(l; lu )V [g12 I + 2g1 g2 I2 + g22 I3 ];

(18)

l=0;l even

where n−j −3=2 n−j −3=2

I1 =





r=0

p=0





r=0

p=0

and





r=0

p=0

 V=

Du 4a!(" − !)



s!xr+p+2!+1−s ;

(19)

e−x E(r)D(p)

r+p+2!+l+1



s!xr+p+2!−s ;

(20)

s=r+p+2!−l

n−j −1=2 n−j −1=2

I3 =

r+p+2!+l+2 s=r+p+2!+1−l

n−j −3=2 n−j −1=2

I2 =

e−x D(r)D(p)

e−x E(r)E(p)

r+p+2!+l



s!xr+p+2!−1−s :

(21)

s=r+p+2!−l−1

2

;

(22)

D(t) =

(−1)t (n − j − 5=2 + 2!)! (n − j − 3=2 − t)!(2! − 1 + t)!t!

t = r; p;

(23)

E(t) =

(−1)t (n − j − 3=2 + 2!)! (n − j − 1=2 − t)!(2! − 1 + t)!t!

t = r; p:

(24)

4. Inuence of excited congurations on the atomic magnitudes With the potential proposed for excited con(gurations we have realized a study to know the importance of excited con(gurations on the potential, energy levels, wave functions and total energies. We have decided to analyze these magnitudes because of their importance in plasma physics: wave functions and energy levels in optical properties determination and total energies in ionic abundance calculations. This study will allow us to determine which are the behaviors of these magnitudes in excited con(gurations and, at the same time, to know what kind of con(gurations introduce more eAects. The main eAect of an excited con(guration is to decrease the electronic shielding of the nuclear charge due to the promotion of an electron from an initial level belonging to the ground state to a (nal more external one in the excited con(guration. This fact implies that the eAective charge

-2 0

-2 0

-2 5

-2 5

-3 0

-3 0

rU e ef (r)

rU ef (r)

R. Rodr guez et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 75 (2002) 723 – 739

-3 5

-4 0

729

-3 5

-4 0 Exc. Anal. Pot. Ground Conf. Excited Conf.

-4 5

-5 0 0 .0 0 0 1

0 .0 0 1

0 .0 1

(a)

0 .1

1

10

A S

-4 5

-5 0 0 .0 0 0 1

100

r (a .u .)

0 .0 0 1

0 .0 1

-2 0

0 .1

1

10

100

1

10

100

r (a .u .)

(a) -1 2

-2 5

Exc. Anal. Pot. A S

-1 6

rU e ef (r)

rU ef (r)

-3 0

-3 5

-2 0

-4 0 -2 4 Ground Conf. Excited Conf.

-4 5

-5 0 0 .0 0 0 1

(b)

0 .0 0 1

0 .0 1

0 .1

1

10

-2 8 0 .0 0 0 1

100

r (a .u .)

Fig. 1. EAective charge (in a.u.) for Fe-like Ag ion for ground state and the excited con(gurations: (a) 2p3=2 → 3d5=2 and (b) 1s1=2 → 7p1=2 .

(b)

0 .0 0 1

0 .0 1

0 .1

r(a.u .)

Fig. 2. EAective charges (in a.u.) corresponding to UAe (r) (A) and USe (r) (S) for: (a) Fe-like Ag ion for the excited con(guration 1s1=2 → 7p1=2 ; (b) Al-like Fe ion for the excited con(guration 3p1=2 → 5p1=2 .

that each electron perceives is greater than in ground state situation. The potential proposed veri(es this result. As an example, we have plotted in Fig. 1 the eAective charge calculated for ground con(guration and two excited con(gurations for Fe-like Ag ion. With the aim of estimating the quantitative eAect of the excited con(gurations on the potential we have de(ned the following relative diAerence “.” between the analytical potentials for ground

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R. Rodr guez et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 75 (2002) 723 – 739

Table 1 Values of . (in percent) for several excited con(gurations for Fe-like ions Con(guration

Z = 47

Z = 79

Z = 90

1s1=2 → 3d5=2 2s1=2 → 3d5=2 2p3=2 → 3d5=2 2p1=2 → 3d5=2 1s1=2 → 7p1=2 2p1=2 → 7p1=2 1s1=2 → 5p1=2 3s1=2 → 3d5=2 3d3=2 → 3d5=2 3d5=2 → 10s1=2 3d5=2 → 7p1=2

0.48 0.20 0.22 0.23 1.10 0.83 0.61 0.02 0.01 0.88 0.63

0.25 0.10 0.10 0.12 0.51 0.36 0.28 0.01 0.01 0.36 0.26

0.22 0.09 0.09 0.11 0.43 0.32 0.35 0.01 0.01 0.30 0.21

state and for excited con(guration: ND 1  |U g (i) − U e (i)| .= ND i=1 |U g (i)|

(25)

ND being the total number of points of the radial mesh. As . increases the eAect of the corresponding excited con(guration is greater. For a (xed ion, we have found that the excited con(gurations that introduce more changes on the potential are those that satisfy some of the following conditions or both simultaneously: (a) The principal quantum number of the electron starting energy level is lower. (b) The principal quantum number of the electron ending energy level is larger. This result obeys the fact that these con(gurations are those that lead to a greater diminution of the electronic shielding. We have also obtained that excited con(gurations generated by promoting the electron from an initial level belonging to the last shell occupied in ground state to a not very outer (nal level can be modeled with the potential for ground state given by Eqs. (12) and (13) instead of the potential for excited con(gurations without loss of accuracy. To illustrate all these conclusions, in Table 1 we have listed . in percent for several excited con(gurations of Fe-like ions. In this table we can also observe that for (xed excited con(guration and number of bound electrons, . decreases (i.e. the eAect of the excited con(guration decreases) as the atomic number increases (i.e. the ionicity increases). This is a normal result because the relative importance of an electronic excitation on the electronic shielding has to decrease as ionicity becomes greater. As we said in Section 3, we have developed two expressions for the analytical potential proposed, depending on the way of obtaining the wave functions of the two levels involved in the excitation, which have been denoted as UAe (r) and USe (r). In our analysis we have concluded that both potentials lead to very close results. This allows us to use any of them to calculate the atomic magnitudes, which is an advantage because USe (r) is a fully analytical potential. In Fig. 2, the eAective charge for two

R. Rodr guez et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 75 (2002) 723 – 739

731

0 .0 8

0 .0 0 5

Energy levels relative shifts

Configurations

0 .0 0 4

δ Ue

0 .0 0 3

0 .0 0 2

Conf. 1 Conf. 2 Conf. 3

0 .0 6

Conf. 4

0 .0 4

0 .0 2 0 .0 0 1

0 0 .0 0 0 1

0 .0 0 1

0 .0 1

0 .1

1

10

r (a.u.)

(a)

0 .0 0

100

1

2

3

4

5

6

7

8

9

10

Levels ns

(a) 0 .0 8

0 .0 1

Configurations

Energy levels relative shifts

0 .0 0 8

δ U e(r)

0 .0 0 6

0 .0 0 4

Conf. 1 Conf. 2 Conf. 3

0 .0 6

Conf. 4

0 .0 4

0 .0 2

0 .0 0 2

0 0 .0 0 0 1

(b)

0 .0 0 0 .0 0 1

0 .0 1

0 .1

1

10

100

r (a.u.)

Fig. 3. Relative diAerence, in absolute value, between rUAe and rUSe for: (a) Fe-like Ag ion in excited con(guration 1s1=2 → 7p1=2 ; (b) Al-like Fe ion in excited con(guration 3p1=2 → 5p1=2 .

(b)

2

3

4

5

6

Levels np

7

8

9

10

1/2

Fig. 4. Energy levels relative shifts for Ag-like Fe ion for the excited con(gurations: (1) 2p3=2 → 3d5=2 , (2) 1s1=2 → 7p1=2 , (3) 2p1=2 → 7p1=2 and (4) 3d5=2 → 7p1=2 . (a) Levels ns1=2 and (b) levels np1=2 .

ions in two excited con(gurations corresponding to the two expressions for the above-mentioned potential has been plotted and the Fig. 3 shows the relative diAerences between them in each situation. Since the monoelectronic magnitudes of an ion, such as energy levels and wave functions, depend on the eAective potential with which they have been calculated, the eAects established above for the

732

R. Rodr guez et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 75 (2002) 723 – 739

Table 2 Energy levels (in atomic units) for Al-like Fe ion in ground state and two excited con(gurations, and the absolute shifts (/) (in a.u.)

Level

Ground state

1s1=2 → 7p1=2

/

3p1=2 → 7p1=2

/

1s1=2 2s1=2 2p1=2 2p3=2 3s1=2 3p1=2 3p3=2 4s1=2 3d3=2 3d5=2 4p1=2 4p3=2 5s1=2 4d3=2 4d5=2 5p1=2 5p3=2 6s1=2 5d3=2 5d5=2 6p1=2 6p3=2 6d3=2 6d5=2 7s1=2 7p1=2 7p3=2 8s1=2 8p1=2 10s1=2 10p1=2

−270.6596 −44.3895 −40.5673 −40.0968 −15.3258 −14.1136 −14.0221 −7.8038 −12.1394 −12.1234 −7.3333 −7.0331 −4.7373 −6.6156 −6.6093 −4.5191 −4.4931 −3.1812 −4.1667 −4.1635 −3.0539 −3.0449 −2.8636 −2.8618 −2.2833 −2.2052 −2.1998 −1.7183 −1.6671 −1.0742 −1.0488

−286.4205 −48.9937 −45.8561 −45.3013 −17.0119 −15.8871 −15.7768 −8.5853 −13.7853 −13.7640 −8.1391 −8.0992 −5.1438 −7.3708 −7.3626 −4.9262 −4.9072 −3.4021 −4.2805 −4.5559 −3.2808 −3.2703 −3.0781 −3.0758 −2.4060 −2.3323 −2.3258 −1.7932 −1.7455 −1.1094 −1.0860

−15.7609 −4.6042 −5.2888 −5.2045 −1.6861 −1.7735 −1.7547 −0.7815 −1.6459 −1.6406 −0.8058 −1.0661 −0.4065 −0.7552 −0.7533 −0.4071 −0.4141 −0.2209 −0.1138 −0.3924 −0.2269 −0.2254 −0.2145 −0.2140 −0.1227 −0.1271 −0.1260 −0.0749 −0.0784 −0.0352 −0.0372

−272.3354 −45.7921 −42.0150 −41.5410 −16.3520 −15.1510 −15.0551 −8.3693 −13.2115 −13.1944 −7.8961 −7.8599 −5.0443 −7.1851 −7.1781 −4.8131 −4.7956 −3.3483 −4.4724 −4.4689 −3.2191 −3.2094 −3.0298 −3.0277 −2.3739 −2.2953 −2.2894 −1.7728 −1.7219 −1.0996 −1.0746

−1.6758 −1.4026 −1.4477 −1.4442 −1.0262 −1.0374 −1.0330 −0.5655 −1.0721 −1.0710 −0.5628 −0.8268 −0.3070 −0.5695 −0.5688 −0.2940 −0.3025 −0.1671 −0.3057 −0.3054 −0.1652 −0.1645 −0.1662 −0.1659 −0.0906 −0.0901 −0.0896 −0.0545 −0.0548 −0.0254 −0.0258

potential will determine the behavior of these quantities. Thus, for the energy levels, by solving the Dirac equation [c
· p + c2 0 + Uef (r)] k (r) = Ek

k (r);

(26)

we have obtained that the main eAect of the excited con(gurations on this quantity is to introduce negative shifts on the energy levels. It is due to the electronic shielding diminution, these shifts being more important for those excited con(gurations and as we have stated above, produce more changes on the potential. This result is illustrated in Table 2 which includes the energy levels for Al-like Fe ions, calculated for the ground state and two excited con(gurations, and, also, the absolute shifts with respect to the ground state energy levels.

R. Rodr guez et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 75 (2002) 723 – 739

733

0 .1 Configurations Conf. 1 Conf. 2

Energy levels relative shifts

0 .0 8

Conf. 3 Conf. 4

0 .0 6

0 .0 4

0 .0 2

0

3

4

5

6

Levels nd 3/2

Fig. 5. Energy levels nd3=2 relative shifts for Ag-like Fe ion for the excited con(gurations: (1) 2p3=2 → 3d5=2 , (2) 1s1=2 → 7p1=2 , (3) 2p1=2 → 7p1=2 , and (4) 3d5=2 → 7p1=2 .

Fig. 6. Absolute diAerences (in a.u.) between the radial large component wave functions for the ground sate and two excited con(gurations: (1) 1s1=2 → 3p3=2 and (2) 1s1=2 → 10s1=2 . Al-like Ag ion: (a) 1s1=2 wave function and (b) 7s1=2 wave function.

In Table 2 we can observe that, for a (xed con(guration, the energy levels shifts do not depend on the total angular momentum, j, i.e. levels with the same quantum numbers n and l but diAerent j present similar energy shifts. This is a general conclusion. For levels with the same principal quantum number n, but diAerent l (levels belonging to the same shell) we have concluded that the

734

R. Rodr guez et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 75 (2002) 723 – 739

Table 3 Relative diAerence in percent of the total energy for several excited con(gurations with respect to the ground state energy for Fe-like ions Con(guration

Z = 47

Z = 79

Z = 90

1s1=2 → 3d5=2 2s1=2 → 3d5=2 2p3=2 → 3d5=2 2p1=2 → 3d5=2 1s1=2 → 7p1=2 2p1=2 → 7p1=2 1s1=2 → 5p1=2 3s1=2 → 3d5=2 3d3=2 → 3d5=2 3d5=2 → 10s1=2 3d5=2 → 7p1=2

18.34 2.51 2.20 2.33 19.01 3.18 18.88 0.23 0.01 0.72 0.66

17.59 2.74 2.20 2.62 18.57 3.71 18.35 0.25 0.02 1.07 0.97

17.60 2.85 2.17 2.75 18.61 3.75 18.37 0.29 0.03 1.11 1.00

energy levels shifts are more important for l=1 and 2 (i.e. p and d) than for l=0(s). Finally, varying n (levels of diAerent shells), we have obtained that the shifts rise as n increases until a maximum value, which depends on the ion and the excited con(guration considered, and then decrease for greater values of n. As an example, we have plotted in Figs. 4 and 5 the relative shifts in energy levels for several excited con(gurations in Fe-like Ag ion. Obviously, if the mean eAect of an excited con(guration is to decrease the electronic shielding this fact will imply that the monoelectronic wave functions, and consequently the electronic density charge, will be shifted towards the nucleus with respect to the ground state situation. We have reproduced this behavior with the potential proposed in this work. Moreover, we have obtained that, (xing the excited con(guration and the ion, the shifts are greater as the wave function principal quantum number increases. As an example, we have plotted in Fig. 6 the absolute diAerences between the wave functions, for several levels, calculated in ground state and some excited con(gurations situations. All of the preceding quantities are monoelectronic properties and, consequently, depend strongly on the potential with which they have been calculated. However, the total energy of an ion is a global magnitude and although its value also depends on the potential, this dependence is less important than for the monoelectronic magnitudes. We have found that the excited con(gurations that lead to more changes in total energies, with respect to the ground state situation, are those in which the starting level of the promoting electron is the 1s1=2 level. This result is shown in Table 3, where we have listed the relative diAerence in percent of the total energy calculated for several excited con(gurations with respect to the ground state energy. As we can see, the diAerences increase as the initial level principal quantum number of the promoting electron decreases, having a maximum for n = 1. We can explain this fact easily taking into account how the total energy is obtained. In this work, total energies have been calculated by means of an expression established in the density functional theory [19] given by



 1 2(r)2(r
) Z E= dr dr
+ Exc [2] !k Ek − 2(r)Uef (r) dr − 2(r) dr + (27) r 2 |r − r
| k

R. Rodr guez et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 75 (2002) 723 – 739

735

0.04

δU rel (r)

0.03

0.02

0.01 A R

0 0.0001

0.001

0.01

0.1

1

10

100

r (a.u.)

(a) 0.012

0.010

δU rel (r)

0.008

0.006

0.004 A R

0.002

0.000 0.0001

(b)

0.001

0.01

0.1

1

10

100

r (a.u.)

Fig. 7. Relative shifts (in a.u.) between the potentials UAe (r) (A) and URe (r) (R) for the excited con(guration 1s1=2 → 3p1=2 and their respective potentials for ground state. (a) Al-like Fe ion and (b) Al-like Au ion.

!k being the occupation of level k; 2(r) is the bound electrons density and Exc [2] is the exchange and correlation energy, which we assume is given by
(28) Exc [2] = 2(r)/xc (2(r)) dr; where /xc is the exchange and correlation contribution to the total energy by each electron in an non-interaction electron homogenous gas with density 2 [20]. Although Eq. (27) was established at (rst for ions in ground state, we still have been using it for excited con(gurations introducing

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R. Rodr guez et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 75 (2002) 723 – 739

Table 4 Relative diAerence in percent of the total energy for several excited con(gurations with respect to the ground state energy for Fe-like Ag

ET

EKIN

E e –n

E e –e

Eexc

Ground state

−5067.869

5315.123

−11575.830

1311.483

−118.645

1s1=2 → 7p1=2

−4104.251

4336.619 18.41%

−9512.896

1175.595 10.36%

−103.569

2s1=2 → 7p1=2

−4906.549

5159.699 2.92%

−11178.670

1225.556 6.55%

−113.135

19.01% 3.18%

17.82% 3.43%

12.70% 4.64%

their eAects through the energy levels and bound electronic density (both of them calculated using the potential for excited con(gurations proposed in this work), and through the levels occupation. If we analyze the changes on each term in (27) when we consider excited con(gurations we can explain the behavior that we have advanced above. When an electron is removed from the level 1s1=2 this leads to a very important variation on each term with respect to the changes that other excited con(gurations introduce, these variations being drastically greater for the kinetic energy and the electron–nucleus potential energy (as we can see, as example, in Table 4). In the kinetic energy it is due to the fact that this magnitude depends fundamentally on the level occupation through the summation of eigenvalues. Since the eigenvalue of the 1s1=2 level is the largest (it could be ten times greater than the second one in importance) its elimination yields a diminution much more important than any other. In the electron–nucleus potential energy, since the level 1s1=2 is the nearest to the nucleus, the reduction in the electronic shielding when the electron is promoted is greater than for the other levels and then it leads to more changes in this quantity.

5. Comparison with other models To check the quality of the potential proposed in this work we have compared the results obtained by others who calculated using an analytical potential developed by Rogers et al. [15] for the OPAL code and a self-consistent potential developed in the DAVID code [3]. Moreover, we have compared our results with data available in the bibliography. In all comparisons we have obtained good agreements. As an example, we have represented in Fig. 7 the relative shifts introduced by the excited con(gurations, obtained as the diAerence between the potential proposed by us and the analytical potential cited above, which has been denoted by URe (r), with respect to their potentials for ground state. As we can see in each (gure, the behavior of the shifts is similar for both potentials, presenting a maximum value located at same distances from the nucleus, approximately. We also show the diminution in the shifts when the ionicity increases, as we have established above. Moreover, we have obtained very similar results with the advantage respect URe (r) that our potential does not need to determine new parameters for each excited con(guration considered.

R. Rodr guez et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 75 (2002) 723 – 739

737

Table 5 Energy levels (in atomic units) for Al-like Au ion in the excited con(guration 1s1=2 → 3p1=2 calculated by using URe (r)(/R ); UDe (r)(/D ) and UAe (r)(/A ) Levels

/A

/D

/R

1s1=2 2s1=2 2p1=2 2p3=2 3s1=2 3p1=2 3p3=2 4s1=2 3d3=2 3d5=2 4p1=2 4p3=2 5s1=2 4d3=2 4d5=2 5p1=2 5p3=2 6s1=2 5d3=2 5d5=2 6p1=2 6p3=2 6d3=2 7s1=2 7p1=2 8s1=2 8p1=2 10s1=2 10p1=2

−3224.6730 −737.7733 −723.0051 −650.3122 −299.6676 −293.7468 −274.3168 −161.1003 −265.3600 −260.7484 −158.5504 −150.7817 −100.3651 −147.1841 −145.2813 −99.0766 −95.2089 −68.4483 −93.4363 −92.4689 −67.7135 −65.5154 −64.5141 −49.6396 −49.1825 −37.6340 −37.3308 −23.7604 −23.6077

−3216.6130 −736.5270 −720.9335 −648.8935 −299.5029 −293.4654 −274.1052 −160.9916 −265.4173 −260.8293 −158.3676 −150.6610 −100.3042 −147.0907 −145.2038 −98.9817 −95.1399 −68.4130 −93.3891 −92.4269 −67.6603 −65.4748 −64.4892 −49.6177 −49.1500 −37.6195 −37.3096 −23.7532 −23.5972

−3191.5010 −737.7397 −721.4994 −651.1448 −301.2546 −295.3081 −276.0978 −161.6598 −268.6074 −263.9595 −159.0983 −151.4009 −100.5789 −148.2972 −146.3835 −99.2793 −95.4581 −68.5539 −93.9043 −92.9371 −67.8115 −65.6432 −64.7602 −49.7007 −49.2386 −37.6727 −37.3663 −23.7790 −23.6246

In Table 5, we have listed energy levels calculated by using the potential proposed and those obtained by using URe (r) and UDe (r). As we can see in this table the diAerences between the results are not bigger than 0.5% in the worst case. This result is very important because of the transition energies and so the position of the line transitions depends on this quantity. Finally, in Table 6 we have listed total energies for some Li-like ions obtained by using our potential, the Rogers analytical potential, DAVID code self-consistent potential and results provided by the bibliography obtained with the combined CI-Hylleraas method using 1157 correlated con(gurations or a full core plus correlation method [21]. In this table we can observe again excellent agreement between the total energies calculated by us and the others provided for other models. Obtaining good total energies is an important task because the LTE ionic populations depend on them.

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R. Rodr guez et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 75 (2002) 723 – 739

Table 6 Excited con(gurations total energies (in a.u.) for Li-like obtained using the potential proposed (A0), DAVID code (D), Rogers potential (R), and results obtained by Wang et al. (W) and Pipin (P) [21]

Z

Conf.

A0

D

R

W

P

6

1s2 2p21=2 1s2 3p21=2 1s2 4p21=2 1s2 5p21=2

−33.8725 −32.7601 −32.3694 −32.1904

−33.4552 −32.0617 −32.6162 −32.4173

−32.8376 −32.4026

−34.4821 −33.3179 −32.9159 −32.7312

−34.4821 −33.3179

9

1s2 2p21=2 1s2 3p21=2 1s2 4p21=2 1s2 5p21=2

−80.9586 −77.5311 −76.3285 −75.7746

−80.5309 −77.3057 −76.9720 −75.3738

−77.6631 −76.3825

−81.8208 −78.3025 −77.0834 −76.5221

−81.8208 −78.3025 −77.0834

10

1s2 2p21=2 1s2 3p21=2 1s2 4p21=2 1s2 5p21=2

−101.1786 −96.6991 −95.1281 −94.4041

−101.4963 −96.0167 −95.2857 −94.5078

−96.8496 −95.1893

−102.1012 −97.5198 −95.9309 −95.1991

−102.1012 −97.5198 −95.9309

6. Conclusions In this paper we have proposed an analytical potential for excited con(gurations. This potential allows to consider a large number of excited con(gurations with a considerable time saving because it does not need iterative procedures or to introduce new parameters with respect to the potential for the ground state, as it occurs in other analytical potentials. We have also obtained a fully analytical expression for the potential in low-Z elements. With our potential we have reproduced the expected behaviours in atomic magnitudes for ions in excited con(gurations. Moreover, we have checked our results with others provided by diAerent ways obtaining excellent agreements in all cases. Acknowledgements This work was supported in part by the European Communities in the framework of the European Commission, within the “keep-in touch” activities and also by the project number FTN2001-2643C02-01 of the Spanish Science and Technology OOce. References [1] [2] [3] [4] [5] [6]

Cowan RD. The theory of atomic structure and spectra. Berkeley: University of California Press, 1981. Grant IP, Mckenzie BJ, Norrington PH, Mayos DF, Pyper NC. Comput Phys Commun 1986;21:207. Liberman DA, Zangwill A. Comput Phys Commun 1984;32:75. Klapisch M. Comput Phys Commun 1971;2:239. Koening E. Physica 1972;62:393. Rogers FJ, Iglesias CA. Astrophys J 1992;79:507.

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