Spectroscopy of forbidden transitions in ions of astrophysical interest

Journal of Molecular Structure (Theochem) 621 (2003) 75–85 www.elsevier.com/locate/theochem

Spectroscopy of forbidden transitions in ions of astrophysical interest E. Charro*, I. Martı´n Departamento de Quı´mica Fı´sica, Universidad de Valladolid, 47011 Valladolid, Spain Accepted 11 February 2002

Abstract The spectroscopic study of forbidden transitions in ions of astrophysical interest has been performed using an extension of the well-established relativistic quantum defect orbital (RQDO) method. The method has so far proved to be a very useful tool for predicting a large body of electron transition probabilities, and other related properties. The RQDO method has been applied to the study of E2 transitions in the spectra of ions relevant in astrophysics, such as Ti XII, Fe XVI, Co XVII and Ni XVIII. q 2002 Elsevier Science B.V. All rights reserved. Keywords: Spectroscopy study; Atomic spectra and related properties; Relativistic quantum defect orbital

Resumen El estudio espectrosco´pico de transiciones prohibidas en iones de intere´s astrofı´sico ha sido realizado usando una extensio´n reciente del metodo Relativista de Orbitales de Defecto Cua´ntico (RQDO). Este me´todo, extensamente probado con anterioridad, resulta ser una herramienta muy u´til a la hora de predecir un gran nu´mero de probabilidades de transicio´n y otras propiedades con ellas relacionadas. En este trabajo, se ha aplicado el me´todo RQDO al estudio de transiciones de tipo E2 en los espectros de iones que son relevantes en el campo de la astrofı´sica, tales como Ti XII, Fe XVI, Co XVII y Ni XVIII. q 2002 Elsevier Science B.V. All rights reserved.

1. Introduction The prediction of atomic spectra, using different theoretical methods, is an area of continuous development. The most thoroughly studied atomic

* Corresponding author. Tel.: þ34-983-423272; fax: þ 34-983423013. E-mail address: [email protected] (E. Charro), lmnieto@ wamba.cpd.uva.es (E. Charro).

transitions have been those, which take place through the electric dipole or E1 mechanism, as these are usually responsible for the strongest lines in atomic spectra. The calculation of transition probabilities, expressed as S-, f- or A-values, is a subject of considerable interest in many fields. For example, these quantities are very important in astrophysical studies as they play an important role in the determination of atomic abundances. However, it is now realized that under conditions which prevail in astrophysical and low-density laboratory tokamak

0166-1280/03/$ - see front matter q 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 - 1 2 8 0 ( 0 2 ) 0 0 5 3 5 - 3

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plasmas, transitions that are forbidden through the electric dipole mechanism, i.e. electric quadrupole (E2) and magnetic dipole (M1) transitions, gain in intensity and can be used to infer information about plasma temperature and dynamics. The importance of forbidden lines in astrophysical and fusion studies has been pointed out by several authors [1]. Particularly relevant is the study of the electric quadrupole (E2) forbidden lines, given that they are one of the bases for reliable electron density and/or temperature diagnosis in various astronomical objects or in laboratory tokamak plasmas [2]. A number of such transitions have been observed in the ultraviolet spectrum of the solar corona; moreover, many gaseous nebulae exhibit in their spectra forbidden lines of low excitation energy belonging to alkali-like ions. The detailed study of these astronomical objects requires the knowledge of electric quadrupole transition probabilities. Additionally, the determination of electric quadrupole oscillator strengths can help in assessing the contribution of electric quadrupole radiation to multiphoton transitions. The availability of intense tunable radiation sources has made it possible to measure the transition probabilities of some E2 transitions in the laboratory. Yet, an accurate assignment requires theoretical data. It is desirable, thus, to have a consistent set of accurate theoretical quadrupole Einstein coefficients for these forbidden lines. This, in turn, implies that accurate wavefunctions must be used for both initial and final states of a given transition. The relativistic quantum defect orbital method (RQDO) [3] is a simple but reliable analytical method based on a model Hamiltonian. It has the great advantage of the computational effort not being increased as the atomic system dealt with becomes heavier. The convenience of employing exactly solvable model potentials for calculating atomic transition probabilities manifests itself not only from a practical point of view but also because of the involved physical implications, when they are capable of achieving a good balance between computational effort and accuracy of results. The RQDO formalism has been extensively applied to the calculation of atomic data corresponding to ions of different degree of complexity, many of them of astrophysical importance ([4 – 9]). However, only very recently the RQDO procedure has been

extended in such a way it now allows us the calculation of atomic data concerning forbidden transitions [10,11]. In the present work, the intensities of the electric quadrupole (E2) transitions in the spectrum of several astrophysically important ions (Na-like Ti, Fe, Co and Ni) have been computed with the RQDO formalism and compared with the data available in the literature.

2. The RQDO method A formulation of the quantum defect theory, known as the quantum defect orbital (QDO) method, was proposed by Simons [12] and Martı´n and Simons [13,14]. This approach, unlike previous models based on the quantum defect, provides exact eigenfunctions of a model Hamiltonian. The resulting orbitals are also valid in the core region retaining approximate core-valence orthogonality. The relativistic formulation of the QDO formalism, as proposed by Martı´n and Karwowski [3] and Karwowski and Martı´n [15] was based on the decoupling of the Dirac second-order equation, and the interpretation of the resulting solutions, that had been carried out by Karwowski and Kobus [16,17]. These authors generalised the decoupling procedure of Biedenharn [18] in such a way that it may be applied to the case of a general atomic spherical potential. They also showed that both the large and small components of the Dirac wavefunction might be recovered without any computational effort. The decoupled system of equations reads [14,15] " # d2 sðs ^ 1Þ 2Zð1 þ a2 EÞ 2 2 þ 2 f^ k r r2 dr ¼ Eð2 þ a2 EÞf^ k ;

ð2:1Þ

Its solutions are related to the large and small components of the Dirac wave function in the form ! ! Gk fþ k 21 ¼A ; ð2:2Þ Fk f2 k where k is the relativistic angular momentum quantum number,   ð2:3Þ k ¼ ^ j þ 12

E. Charro, I. Martı´n / Journal of Molecular Structure (Theochem) 621 (2003) 75–85

when j¼l^

1 2

ð2:4Þ

and a is the fine structure constant. Two interpretations may be given to the solutions of Eq. (2.1): first, they give us two components of the relativistic wave function. Second, they give us two solutions (corresponding to k and to 2 k ) of a single scalar equation for a scalar [quasi-relativistic (QR)] wave function ck : A QR theory is not only considerably simpler than the relativistic one, but, first of all, it closely resembles the Schro¨dinger formulation. As a consequence, one may rather easily implement the QR formalism in the majority of the methods being developed for the non-relativistic theory. The last equation is formally very similar to the radial hydrogenic equation and passes into it in a trivial manner in the non-relativistic limit of a ! 0: The QDOs are solutions of the Schro¨dinger equation [10 – 12] " # d2 xðx þ 1Þ Znet QD 2 2þ 2 ck ¼ 2EQD cQD ð2:5Þ k ; r r2 dr

as "

# d2 LðL þ 1Þ 2Z 0net RD 2 ck ¼ 2eRD cRD 2 2þ k ; ð2:8Þ r r2 dr

The RQDO is expressed in terms of Kummer’s functions,   0 Lþ1 2Z 0net r 2Z net r ck ðrÞRD ¼ C exp p h hp   2Z 0 r £ F 2ðhp 2 L 2 1Þ; 2L þ 2; net hp ð2:9Þ and C is the normalization constant,  0 1=2

2Z net Gðhp þ L þ 1Þ 1=2 1 C¼ 2 hp Gð hp 2 L Þ hp Gð2L þ 2Þ ð2:10Þ with hp ¼ h 2 d0 ; being h the relativistic principal quantum number given by

h ¼ n 2 lkl þ lsl;

k ¼ ^ðj þ 1=2Þ

x ¼ n 2 d þ c;

when

where d is the quantum defect and c is an integer chosen to ensure the correct number of nodes and the normalization of the radial wave function. The eigenvalue EQD in Eq. (2.5) depends only upon the non-integral part of x and, hence, is independent of c. The quantum defect is empirically obtained from the following equation: E

QD

2

ðZnet Þ ¼E ¼2 ; 2ðn 2 dÞ2 x

ð2:7Þ

where Ex is the experimental energy. A comparison of Eqs. (2.1) and (2.5) demonstrates that the formal mathematical structures of the QDO theory and the scalar relativistic theory are the same. This formal similarity allowed Martı´n and Karwowski to reinterpret the QDO theory so that it would account for a major part of relativistic effects [3,15]. Analogously to the non-relativistic case, Eq. (2.5), the RQDO equation was written by the above authors

ð2:11Þ

where k is the relativistic angular momentum quantum number,

where Znet is the nuclear charge on the active electron at large r and ð2:6Þ

77

j ¼ l ^ 1=2

ð2:12Þ

ð2:13Þ

and 2 =k2 Þ1=2 ; s ¼ kð1 2 a2 Znet

ð2:14Þ

with a as the fine structure constant. Other parameters are defined by Z 0net ¼ Znet ð1 þ a2 Ex Þ;

ð2:15Þ

and,

L ¼ s 2 1 2 d0 þ c;

ð2:16Þ

when j ¼ l þ 1=2; or

L ¼ 2s 2 d0 þ c;

ð2:17Þ

when j ¼ l 2 1=2: The relativistic quantum defect is determined empirically. We have eRD ¼ 2

ðZ 0net Þ2 2ðh 2 d0 Þ2

ð2:18Þ

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E. Charro, I. Martı´n / Journal of Molecular Structure (Theochem) 621 (2003) 75–85

and it has been proven [3] that the value of d0 may be obtained from: 2

ðZ 0net Þ2 ð1 þ a2 Ex =2Þ ¼ Ex : 0 2 ð1 þ a2 Ex Þ2 2ðh 2 d Þ

ð2:19Þ

It should be stressed that this formulation is ‘exact’ in the sense that is equivalent to a four-component formulation based on the standard first-order form of the Dirac equation. All matrix elements, in particular the transition moments, may be expressed in a simple way, using the solutions of the second-order equation. A set of recurrent formulas which are fulfilled by the radial integrals [16,17,19,20] allows the formalism to be very simple and compact. Karwowski and Martı´n [15] have remarked that the relativistic density distribution approximates very well the exact one at large values of r. At small distances, the quality of the density deteriorates, as happens when the nonrelativistic QDO densities are compared with the exact non-relativistic ones [12]. Fortunately, the consequences of this drawback are very seldom reflected in the quality of the QDO and RQDO transition probabilities, because the strongest contribution to radial matrix elements comes, in most cases, from large radial distances. The most important difference between the RQDO and QDO equations is the explicit dependence of the former on the total angular momentum quantum number k. As a consequence, values of the relativistic quantum defect are determined from the fine structure energies rather than from their centers of gravity. The corresponding RQDOs are different for each component of a multiplet and, if c ¼ 0; they retain the nodal structure of the large components of the hydrogenic Dirac wave function. 3. Electric quadrupole transitions Let us consider electric multipole radiation. The operator responsible for this radiation is the electric 2j-pole moment introduced by the general expression X j  4p 1=2 m ð jÞ Yj ð3:1Þ Q ¼ e rk 2j þ 1 or using the Racah tensor Cmj we have X QðjÞ ¼ e rkj CmðjÞ

ð3:2Þ

In the particular case of j ¼ 2; the transitions are taking place via the electric quadrupole mechanism, E2. In this context, the electric quadrupole line strength for a transition between two states withinthe LSJ-coupling, which is the coupling scheme followed throughout in this work, in the notation of Condon and Shortley [21], is given by the equation Sð2Þ nlj;n0 l0 j0 ¼

2 3

lkaJkQð2Þ ka0 J 0 ll2

ð3:3Þ

where the matrix-elements have the form kaJkQð2Þ ka0 J 0 l pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð2J þ 1Þð2J 0 þ 1ÞWðSJL0 2; LJ 0 ÞkaLkQð2Þ ka0 L0 ldSS0 ð3:4Þ where the Kronecker delta, dSS0 ; occurs because the electric-n-pole operators do not depend on spin. Thus, we define a line factor Rline by Rline ðSLJ; S0 L0 J 0 Þ ¼ ð2J þ 1Þ1=2 ð2J 0 þ 1Þ1=2 WðSJL0 2; LJ 0 Þ

ð3:5Þ

where WðSJL0 2;LJ 0 Þ are the Racah coefficients which can be described in terms of 6j-symbols ( ) S J L 0 0 SþJþL0 þ2 WðSJL 2;LJ Þ ¼ ð21Þ ð3:6Þ 2 L0 J 0 The line strength then becomes 0 2 Sð2Þ nlj;n0 l0 j0 ¼ 3 ð2J þ 1Þð2J þ 1Þ

½WðSJL0 2; LJ 0 Þ2 lkaLkQð2Þ ka0 L0 ll2 ¼ ½Rline ðSLJ; S0 L0 J 0 Þ2 lkaLkQð2Þ ka0 L0 ll2

ð3:7Þ

As mentioned earlier, the relevant operator is independent of spin, and hence spin does not change during the transition. The following selection rules apply to E2 transitions between LSJ-coupling states: DJ ¼ ^2;^1; 0; DS ¼ 0 and DL ¼ ^2; ^1;0; with the exception of nsj ! n0 sj0 and np1=2 ! n0 p1=2 : Thus, the reduced matrix element kaLkQð2Þ ka0 L0 l provides the relative strengths of different multiplets. We can write this reduced matrix element as the product of a single-electron reduced matrix element knLkQð2Þ kn0 L0 l; which depends on only the quantum numbers of the jumping electron, and a factor that we

E. Charro, I. Martı´n / Journal of Molecular Structure (Theochem) 621 (2003) 75–85

shall call the multiplet factor, Rmult : kaLkQð2Þ ka0 L0 l ¼ Rmult ðaL; a0 LÞ £ knLkQð2Þ kn0 L0 l ð3:8Þ where knLkQð2Þ kn0 L0 l ¼ kLkCð2Þ kL0 lkRnlj lQð2Þ lRn0 l0 j0 l

ð3:9Þ

in Eq. (3.9), kRnlj lQð2Þ lRn0 l0 j0 l is the transition integral, kLkCð2Þ kL0 l is the pertinent reduced matrix element, which can be evaluated using 3j-symbols by the following expression ! L 2 L0 ð2Þ 0 0 kLkC kL l ¼ ½L;L  ð3:10Þ 0 0 0

Rmult ðaL; a0 L0 Þ ¼ ð2L þ 1Þ1=2 ð2L0 þ 1Þ1=2 WðLc Ll0 2; lL0 Þ

ð3:11Þ

with the last symbol in Eq. (3.11) being, ( ) Lc L l 0 0 Lc þLþl0 þ2 : WðLc Ll 2; lL Þ ¼ ð21Þ 2 l0 L 0

£ kLkC ð2Þ kL0 lkRnlj lQð2Þ lRn0 l0 j0 ll2

ð3:13Þ

The transition integral, kRnlj lQð2Þ lRn0 l0 j0 l; is analytically solved in the RQDO method, being the expression kRnlj lQð2Þ lRn0 l0 j0 l 2 Ni

Li þ2

"

Gðhpj þ Lj þ 1ÞGðhpi þ Li þ 1Þ £ Gðhpj 2 Lj ÞGðhpi 2 Li Þ £

" XX

#1=2

k1 2 hpj þ Lj lm k1 2 hpi þ Li ln Gðð2Lj Þ þ 2 þ mÞGðð2Li Þ þ 2 þ nÞ 0

Z mþm Nim Nj n GðA þ 5Þ £ Gð1 þ mÞGð1 þ nÞ ðNi þ Nj Þmþn

ð3:15Þ

The relationships between the line strength Sð2Þ (in atomic units, e2 a4o ), the oscillator strength f ð2Þ (dimensionless), and the transition probability Að2Þ (in s21) for E2 transitions are given by [22] g0 Að2Þ ¼ ð8p2 "a=ml2 Þgf ð2Þ ¼ ð6:6703 £ 1015 =l2 Þgf ð2Þ ð3:16Þ g0 Að2Þ ¼ ð32p5 aca4o =15l5 ÞSð2Þ ð3:17Þ

˚ ), g and g0 where l is the transition wavelength (in A are the degeneracies of the lower and upper state, respectively, and a is the fine-structure constant.

ð3:12Þ

0 0 0 0 0 2 Sð2Þ nlj;n0 l0 j0 ¼ 3 lRline ðSLJ; S L J ÞRmult ðaL; a L Þ

!Lj þ2 

A ¼ Lj þ Li þ m þ n

4. E2 transitions in Na-like Ti, Fe, Co and Ni

Finally, the line strength takes the form

1 2 ¼ 0 0 4Z i Z j Nj

where the symbols ‘k l’ are the Pochhamer functions and A is the expression,

¼ ð1:11995 £ 1018 =l5 ÞSð2Þ

and Rmult may be expressed as follows:

79

#

! ð3:14Þ

Because of their nuclear stability, iron groups elements (especially the even-Z ones without nuclear spin, such as Ti) are frequently encountered in stellar spectra. Many of the emission lines from the chromosphere of A-type stars belong to iron group in high degrees of ionization. In the Sun, Ti figures in third place in terms of number of lines [23]. The relatively high cosmic abundances of the iron group elements in stellar objects stimulates interest in these elements. Cobalt, being an odd-Z number element, is far less abundant than the even-Z elements of the group: Cr, Fe, and Ni. However, a number of lines of cobalt have been identified in astrophysical studies. Spectral data for all ions of iron are of a particular significance in astrophysics because of their abundance in both solar and stellar spectra [24]. Emission lines arising from transitions in Ni XVIII are frequently observed in the spectra of astrophysical objects, such as the solar transition region, Corona [25,26]. It, thus, appears that the theoretical determination of radiative transition probabilities or line strengths (S-values) for the estimation of stellar chemical composition is vital. It has been found that, at least for the abundant elements, the lines in the stellar spectra have rather small oscillator strengths and/or

E. Charro, I. Martı´n / Journal of Molecular Structure (Theochem) 621 (2003) 75–85

80

rather high excitation energies and are, thus, often difficult to measure experimentally. The energy data required as input in our procedure have been taken from the critical compilation by Kelly [27]. For all the transitions studied here, the integer c in Eq. (2.6) has been chosen to be zero for both, initial and final states, which leads to RQDO wavefunctions that possess the same number of radial nodes as their hydrogenic counterparts. Table 1 Line strengths for the np ! n0 p fine-structure transitions of Ti XII Transition

Multiplet

J

J0

RQDOa

Compilationb

3p ! 4p

2

1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2

3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2

0.164 0.173 0.169 0.154(21)c 0.157(21) 0.157(21) 0.168(þ 1) 0.177(þ 1) 0.173(þ 1) 0.136 0.138 0.138 0.358(21) 0.360(21) 0.360(21) 0.147(21) 0.147(21) 0.147(21) 0.988(þ 1) 0.104(þ 2) 0.102(þ 2) 0.721 0.728 0.728 0.178 0.178 0.178 0.412(þ 2) 0.423(þ 2) 0.423(þ 2) 0.281(þ 1) 0.284(þ 1) 0.284(þ 1) 0.144(þ 3) 0.144(þ 3) 0.144(þ 3)

0.164 0.164 0.164 0.15(21) 0.15(21) 0.15(21) 0.170(þ 1) 0.170(þ 1) 0.170(þ 1) 0.132 0.132 0.132 0.34(21) 0.14(21) 0.14(21) 0.14(21) 0.14(21) 0.14(21) 0.100(þ 2) 0.100(þ 2) 0.100(þ 2) 0.71 0.71 0.71 0.172 0.172 0.172 0.417(þ 2) 0.417(þ 2) 0.417(þ 2) 0.277(þ 1) 0.277(þ 1) 0.277(þ 1) 0.142(þ 3) 0.142(þ 3) 0.142(þ 3)

P – 2P

3p ! 5p

2

P – 2P

4p ! 5p

2

P – 2P

4p ! 6p

2

P – 2P

4p ! 7p

2

P – 2P

4p ! 8p

2

P – 2P

5p ! 6p

2

P – 2P

5p ! 7p

2

P – 2P

5p ! 8p

2

P – 2P

6p ! 7p

2

P – 2P

6p ! 8p

2

P – 2P

7p ! 8p

a b c

2

P – 2P

Relativistic quantum defect orbital method, this work. Ref. [28]. In this and the remaining tables, AðBÞ denotes A £ 10ðBÞ :

Table 2 S-values for E2 transitions of Ti XII Transition

Multiplet

J

J0

RQDOa

Compilationb

3s ! 3d

2

S – 2D

3s ! 4d

2

S – 2D

3s ! 5d

2

S – 2D

3s ! 6d

2

S – 2D

4s ! 4d

2

S – 2D

4s ! 5d

2

S – 2D

4s ! 6d

2

S – 2D

4s ! 7d

2

S – 2D

4s ! 8d

2

S – 2D

5s ! 5d

2

S – 2D

5s ! 6d

2

S – 2D

5s ! 7d

2

S – 2D

5s ! 8d

2

S – 2D

6s ! 6d

2

S – 2D

6s ! 7d

2

S – 2D

6s ! 8d

2

S – 2D

7s ! 7d

2

S – 2D

7s ! 8d

2

S – 2D

3d ! 4s

2

D – 2S

4d ! 5s

2

D – 2S

4d ! 6s

2

D – 2S

4d ! 7s

2

D – 2S

5d ! 6s

2

D – 2S

5d ! 7s

2

D – 2S

6d ! 7s

2

D – 2S

3p ! 4f

2

P – 2F

1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 3/2 5/2 3/2 5/2 3/2 5/2 3/2 5/2 3/2 5/2 3/2 5/2 3/2 5/2 1/2 3/2

3/2 5/2 3/2 5/2 3/2 5/2 3/2 5/2 3/2 5/2 3/2 5/2 3/2 5/2 3/2 5/2 3/2 5/2 3/2 5/2 3/2 5/2 3/2 5/2 3/2 5/2 3/2 5/2 3/2 5/2 3/2 5/2 3/2 5/2 3/2 5/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 5/2 5/2

0.335 0.504 0.145 0.217 0.257(21)c 0.386(21) 0.879(22) 0.132(21) 0.510(þ1) 0.766(þ1) 0.983 0.147(þ1) 0.171 0.257 0.578(21) 0.866(21) 0.266(21) 0.399(21) 0.363(þ2) 0.545(þ2) 0.436(þ1) 0.653(þ1) 0.739 0.111(þ1) 0.243 0.365 0.172(þ3) 0.258(þ3) 0.152(þ2) 0.227(þ2) 0.248(þ1) 0.371(þ1) 0.625(þ3) 0.937(þ3) 0.429(þ2) 0.644(þ2) 0.189 0.284 0.269(þ1) 0.404(þ1) 0.612(21) 0.916(21) 0.116(21) 0.173(21) 0.185(þ2) 0.278(þ2) 0.357 0.533 0.869(þ2) 0.131(þ3) 0.862 0.250

0.322 0.483 0.162 0.244 0.28(21) 0.41(21) 2 0.14(21) 0.50(þ1) 0.76(þ1) 0.108(þ 1) 0.162(þ 1) 0.179 0.269 0.60(21) 0.89(21) 0.27(21) 0.41(21) 0.364(þ 2) 0.55(þ2) 0.468(þ 1) 0.70(þ1) 0.78 0.116(þ 1) 0.252 0.379 0.162(þ 3) 0.242(þ 3) 0.162(þ 2) 0.244(þ 2) 0.260(þ 1) 0.390(þ 1) 0.570(þ 3) 0.850(þ 3) 0.468(þ 2) 0.70(þ2) 0.170 0.254 0.252(þ 1) 0.377(þ 1) 0.60(21) 0.89(21) 0.11(21) 0.17(21) 0.179(þ 2) 0.269(þ 2) 0.360 0.54 0.86(þ2) 0.128(þ 3) 0.79 0.225

E. Charro, I. Martı´n / Journal of Molecular Structure (Theochem) 621 (2003) 75–85 Table 2 (continued) Transition

Multiplet

3p ! 5f

2

3p ! 6f

4p ! 4f

4p ! 5f

4p ! 6f

4p ! 7f

4p ! 8f

5p ! 5f

2

2

2

2

2

2

2

P – 2F

P – 2F

P – 2F

P – 2F

P – 2F

P – 2F

P – 2F

P – 2F

5p ! 6f

2

P – 2F

5p ! 7f

2

P – 2F

5p ! 8f

2

P – 2F

6p ! 6f

2

P – 2F

6p ! 7f

2

P – 2F

6p ! 8f

2

P – 2F

7p ! 7f

2

P – 2F

7p ! 8f

2

P – 2F

8p ! 8f

2

P– 2F

4f ! 5p

2

F – 2P

81

Table 2 (continued) J

J0

RQDOa

Compilationb

3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 5/2

7/2 5/2 5/2 7/2 5/2 5/2 7/2 5/2 5/2 7/2 5/2 5/2 7/2 5/2 5/2 7/2 5/2 5/2 7/2 5/2 5/2 7/2 5/2 5/2 7/2 5/2 5/2 7/2 5/2 5/2 7/2 5/2 5/2 7/2 5/2 5/2 7/2 5/2 5/2 7/2 5/2 5/2 7/2 5/2 5/2 7/2 5/2 5/2 7/2 5/2 5/2 7/2 1/2

0.150(þ 1) 0.451(21) 0.127(21) 0.761(21) 0.708(22) 0.194(22) 0.117(21) 0.523(þ 1) 0.150(þ 1) 0.898(þ 1) 0.557(þ 1) 0.163(þ 1) 0.974(þ 1) 0.518 0.148 0.887 0.121 0.340(21) 0.204 0.436(21) 0.122(21) 0.732(21) 0.497(þ 2) 0.142(þ 2) 0.854(þ 2) 0.229(þ 2) 0.671(þ 1) 0.402(þ 2) 0.249(þ 1) 0.716 0.429(þ 1) 0.638 0.181 0.109(þ 1) 0.261(þ 3) 0.746(þ 2) 0.448(þ 3) 0.731(þ 2) 0.214(þ 2) 0.129(þ 3) 0.846(þ 1) 0.244(þ 1) 0.146(þ 2) 0.998(þ 3) 0.285(þ 3) 0.171(þ 4) 0.206(þ 3) 0.589(þ 2) 0.353(þ 3) 0.309(þ 4) 0.884(þ 3) 0.530(þ 4) 0.105(þ 1)

0.135(þ 1) 0.45(21) 0.13(21) 0.78(21) – – 0.14(21) 0.50(þ 1) 0.143(þ 1) 0.86(þ 1) 0.51(þ 1) 0.146(þ 1) 0.87(þ 1) 0.493 0.141 0.85 0.116 0.33(21) 0.199 0.43(21) 0.12(21) 0.73(21) 0.477(þ 2) 0.136(þ 2) 0.82(þ 2) 0.216(þ 2) 0.62(þ 1) 0.370(þ 2) 0.234(þ 1) 0.67 0.402(þ 1) 0.60 0.172 0.103(þ 1) 0.250(þ 3) 0.71(þ 2) 0.430(þ 3) 0.69(þ 2) 0.197(þ 2) 0.118(þ 3) 0.80(þ 1) 0.228(þ 1) 0.137(þ 2) 0.970(þ 3) 0.280(þ 3) 0.1700(þ4) 0.192(þ 3) 0.55(þ 2) 0.330(þ 3) 0.2900(þ4) 0.830(þ 3) 0.5000(þ4) 0.113(þ 1)

Transition

Multiplet

4f ! 6p

2

F – 2P

4f ! 7p

2

F – 2P

5f ! 6p

2

F – 2P

5f ! 7p

2

F – 2P

5f ! 8p

2

F – 2P

6f ! 7p

2

F – 2P

6f ! 8p

2

F – 2P

7f ! 8p

2

F – 2P

a b c

J

J0

RQDOa

Compilationb

5/2 7/2 5/2 5/2 7/2 5/2 5/2 7/2 5/2 5/2 7/2 5/2 5/2 7/2 5/2 5/2 7/2 5/2 5/2 7/2 5/2 5/2 7/2 5/2 5/2 7/2

3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2 1/2 3/2 3/2

0.294 0.177(þ1) 0.351(21) 0.100(21) 0.603(21) 0.728(22) 0.208(22) 0.125(21) 0.113(þ2) 0.316(þ1) 0.190(þ2) 0.337 0.963(21) 0.578 0.654(21) 0.187(21) 0.112 0.622(þ2) 0.178(þ2) 0.107(þ3) 0.174(þ1) 0.498 0.299(þ1) 0.255(þ3) 0.728(þ2) 0.438(þ3)

0.324 0.194(þ 1) 0.38(21) 0.11(21) 0.65(21) – – 0.13(21) 0.119(þ 2) 0.340(þ 1) 0.204(þ 2) 0.353 0.101 0.61 0.68(21) 0.20(21) 0.117 0.67(þ2) 0.190(þ 2) 0.114(þ 3) 0.181(þ 1) 0.52 0.310(þ 1) 0.270(þ 3) 0.77(þ2) 0.462(þ 3)

Relativistic quantum defect orbital method, this work. Ref. [28]. In this and the remaining tables, AðBÞ denotes A £ 10ðBÞ :

The S-values for the electric quadrupole (E2) transitions for Na-like Ti are displayed in Tables 1 and 2. Two sets of S-values are given, the first entry corresponds to the RQDO calculations, and the second entry includes results from the critical compilation of Martin et al. [28]. The S-values by Martin et al. [28] have been derived from Bie´mont and Godefroid [29] by using LS-coupling rules on the multiplet gf-values calculated using a fully variational Hartree – Fock approach with no allowance for configuration mixing. These values compare satisfactorily with those of the present calculation for Ti XII. A general view of the quality of our results for Co XVII and Ni XVIII are shown on Fig. 1(A) and (B), respectively. A comparison between the RQDO multiplet line strengths with the theoretical results reported by Tull et al. [30] is made for both ions. There are only two and three points in the graphs of

82

E. Charro, I. Martı´n / Journal of Molecular Structure (Theochem) 621 (2003) 75–85

Fig. 1. Comparison of the RQDO multiplet data for Na-like Co XVII (a) and Na-like Ni XVIII (b) with the results of Tull et al. [30]. Deviations of line strengths SRQDO =SHF are plotted versus log SRQDO :

Fig. 2. Regularities in the RQDO line strength along the 3s2 S1=2 2 n d2 D3=2 (a) and 3s2 S1=2 2 nd2 D5=2 (b) spectral series in Fe XVI, Co XVII and Ni XVIII.

E. Charro, I. Martı´n / Journal of Molecular Structure (Theochem) 621 (2003) 75–85

83

Fig. 3. The predicted spectra of Ti XII in the UV region. The simulation only includes the RQDO intensities for E2 lines: (a) contains all the lines, and (b) for the same region, the transition 3p 2 4f has been omitted.

Co XVII and Ni XVIII, respectively, which show discrepancies. These values correspond to 3p2 P0 2 n f 2 F 0 transitions (with n ¼ 6 and 7 in Fig. 1(A), and n ¼ 6 – 8 in Fig. 1(B)). The rest of the results are very close to the perfect value of zero. The source of the discrepancies may be the different atomic energies in

the two different procedures coresponding to the highly excited nf 2 F 0 levels, which imply sizable differences in the corresponding wavefunctions. As examples of the regular behaviour complied with by the present results, we have plotted in Fig. 2(A) and (B), the RQDO line strengths for Fe

84

E. Charro, I. Martı´n / Journal of Molecular Structure (Theochem) 621 (2003) 75–85

Fig. 4. The predicted spectra of Fe XVI in the UV region. The simulation only includes the RQDO intensities for E2 lines: (a) contains all the lines, and (b) for the same region, the transition 3p 2 4f has been omitted.

XVI, Co XVII and Ni XVIII, corresponding to the 3s2 S ! nd2 D fine-structure E2-lines, multiplied by the cube of the effective quantum number of the final state, against n0 p: It is apparent that the trend followed by our data is such that a constant value is reached as

soon as the upper state is suficiently excited to acquire a near-hydrogenic character. The first value plotted (in all the cases), corresponds to the 3s2 S ! 3d2 D transition, and gives a lower value in Y-axis than was expected. It may be due to some configuration

E. Charro, I. Martı´n / Journal of Molecular Structure (Theochem) 621 (2003) 75–85

mixing suffered by the 3d level. The rest of the spectral series, in the three ions, appear to be free from perturbations, thus leading to a regular behaviour. On the basis of the general good agreement between the RQDO data and the comparative ones, as well as of the compliance of the present results with the expected systematic trends along different spectral series, we have done a simulation of the E2 spectra of Ti XII (Fig. 3(A) and (B)) and Fe XVI (Fig. 4(A) and (B)). We have plotted the theoretical weighted transition probabilities (gA-values in s21 ) as a ˚ ). The spectra include function of the wavelength (in A only the intensities of the RQDO E2 transitions calculated in the present work, where only the region with the strongest lines is shown.

[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

Acknowledgements This work has been supported by the DGES of the Spanish Ministry of Education within Project No. PB97-0399-C03-01.

[21] [22] [23] [24]

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