- Email: [email protected]
On the use of the Cowan’s code for atomic structure calculations in singly ionized lanthanides
Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646
Note
On the use of the Cowan's code for atomic structure calculations in singly ionized lanthanides P. Quinet *, P. Palmeri , E. BieH mont Astrophysique et Spectroscopie, Universite& de Mons-Hainaut, Place du Parc, 20, B-7000 Mons, Belgium Institut d+Astrophysique, Universite& de Lie% ge, B-4000 Lie% ge, Belgium Received 19 June 1998
Abstract From a detailed comparison of Relativistic Hartree-Fock calculations with recent laser lifetime measurements in Tm II, it is shown that Cowan's code, which has been widely used in the past for atomic structure calculations in light elements, is adequate also for providing accurate radiative data in heavy ions and particularly in the astrophysically important singly ionized lanthanides provided con"guration interaction and core-polarization e!ects are consistently taken into account in the adopted physical model. 1999 Elsevier Science Ltd. All rights reserved. PACS: 32.70; 32.80 Keywords: Cowan's code; Lanthanides; Atomic structure calculations
1. Introduction Although they are characterized by low cosmic abundances, singly ionized lanthanides play an important role in astrophysics because they are observed in the photospheric solar spectrum [1,2] and in the spectra of some chemically peculiar stars [3]. In stars, Tm can be produced by both the r- and s-processes. Tm II has been identi"ed in one FO Ib star by Reynolds et al. [4] and in Ap stars of the Si subgroup by Poli et al. [5] and by Cowley and Crosswhite [6]. Tm I has been observed in Ap stars of the Cr}Eu}Sr subgroup by Jaschek and Brandi [7] and by Adelman [8] and also in S-type stars by Bidelman [9]. Interest in this research "eld is stimulated by new technologies (large-aperture telescopes, new spectrometers and advanced CCD detector arrays) that allow recordings of the stellar spectra with high resolution and better signal-to-noise ratio
* Corresponding author. Tel.: 0032 65 37 37 27; fax: 0032 65 37 30 54; e-mail: [email protected] 0022-4073/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved PII: S 0 0 2 2 - 4 0 7 3 ( 9 8 ) 0 0 1 2 7 - 7
626
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646
than previously available. In addition, the determination of improved chemical compositions of stars, made possible by the recent high-resolution Hubble Space Telescope observations, relies heavily upon accurate spectroscopic data and more particularly atomic oscillator strengths. Rare-earth elements are also increasingly used in metal-halide arc lamps. These devices, which consist basically of high-pressure mercury arc lamps with metal-halide salts [10], provide highquality light characterized by an improved color-rendering index and a high luminous e$ciency. The line-rich spectrum of thulium and the relatively high vapor pressure of salts of Tm are very advantageous for improving the capabilities of such lamps. Consequently, spectroscopic data of rare earths are needed for further developments in the "eld. Atomic transition probabilities of Tm II have been little considered up to now. Arc measurements have been performed by Corliss and Bozman [11] more than 30 years ago but, in the past, their results have been frequently shown to be in error for many transitions. Radiative lifetime determinations include works by Curtis et al. [12], Andersen and Sorensen [13], Blagoev et al. [14] and Blagoev and Komarovskii [15] but they concern a rather limited number of levels. Very recently, accurate radiative lifetime values measured by laser-induced #uorescence technique and transition probabilities obtained from combination of these lifetimes with branching fractions deduced from Fourier transform spectra have been published [16,17]. This provides a unique opportunity to test the adequacy of the semi-empirical relativistic Hartree-Fock (HFR) approach described by Cowan [18] for predicting energy levels and line intensities in the case of the &&complex'' con"gurations encountered in the lanthanide ions. This method has been widely used in the past for predicting, with success, radiative properties of light elements but has been little considered up to now in the case of heavy elements or ions if we except some isolated attempts (see e.g. Ref. [19]). The case of singly ionized thulium is considered in the present paper and, more speci"cally, the e!ects of con"guration interaction and of core-polarization on the transitions originating from the 4f6s, 4f5d, 4f6s6p and 4f5d6p odd con"gurations and the 4f6p, 4f6s, 4f5d6s and 4f5d even con"gurations are discussed in detail.
2. The calculations For complex systems, accurate calculations of atomic structure should allow for both intravalence and core-valence correlation. Simultaneous treatment of both e!ects within con"guration interaction (CI) scheme is very complex and reliable only if enough con"guration mixing is considered as shown, for example, by Quinet and Hansen [20] in the case of the iron group elements. However, in practice, even a large computer imposes rather severe limitations on the number of interacting con"gurations that can be considered simultaneously and, in particular, the inclusion of core-polarization e!ects through the consideration of a huge number of con"gurations with open inner shells is often prevented by computer limitations. Migdalek and Baylis [21] have suggested an approach in which most of the intravalence correlation is represented within CI scheme while core-valence correlation is represented approximately by a core-polarization model potential, < , given by . !a r , (1) < (r)" . 2(r#r)
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646
627
where a is the static dipole polarizability of the core and r is the cut-o! radius which is arbitrarily chosen as a measure of the size of the ionic core. This latter parameter is usually taken as the expectation value of r for the outermost core orbitals. When including the core-polarization in the one-electron Hamiltonian, the dipole-moment operator in the transition matrix element has also to be modi"ed for consistency since these two corrections appear in the same order of the perturbation theory. The dipole-moment operator, d"!r, of the valence electron has to be replaced by ar . d"!r# (r#r)
(2)
In the present work, we used the pseudo-relativistic Hartree}Fock (HFR) method described by Cowan [18] and Cowan and Gri$n [22] in which we have included expressions (1) and (2) given above to make allowance for core-polarization e!ects. Con"guration interaction was retained among the con"gurations 4f6s, 4f7s, 4f5d, 4f6d, 4f7d, 4f6s6p, 4f5d6p, 5p4f6s, 5p4f5d, 5p4f6s6p, 5p4f5d6p for the odd parity and 4f6p, 4f7p, 4f6s, 4f5d6s, 4f5d, 5p4f6p, 5p4f6s, 5p4f5d6s, 5p4f5d for the even parity. The dipole polarizability of the ionic core, a , was chosen equal to 7.62a for the 4fnl con"gurations and to 5.60a for the 4fnln l con"gurations, which corresponds to the values reported by Fraga et al. [23] in the cases of Tm III and Tm IV, respectively. For the cut-o! radius, r , we used the average value 1r2 for the outermost core orbitals (5p) as calculated by the Cowan's code, i.e. r "1.484a . Using a well-established least-squares "tting procedure [18], the radial parameter values were adjusted to obtain the best agreement between calculated and observed energy levels. The "tted parameters were the center-of-gravity energies (E ), the single-con"guration direct (FI) and exchange (GI) Coulomb-interaction integrals, the spin}orbit integrals (f) and some CI (RI) integrals related to the con"gurations studied experimentally. For the remaining con"gurations, the FI, GI and RI integrals were arbitrarily scaled down by a factor 0.85 while the ab initio values of the spin-orbit parameters, f, computed by the Blume}Watson method, were used without scaling. All the known experimental odd levels below 60 000 cm\ compiled by Martin et al. [24] have been introduced in the "tting procedure if we except the J"2 level at 59135.89 cm\ which has no spectroscopic designation. In fact, the levels situated above 60 000 cm\ are fragmentarily known and therefore some of the designations appear dubious. Moreover, some of these levels overlap unknown levels belonging to higher con"gurations such as 4f6d and 4f7d for which no experimentally determined energy levels are available. Consequently, these levels have not been included in the "t. All the parameters of the con"gurations 4f6s, 4f5d, 4f6s6p and 4f5d6p have been adjusted. For 4f7s, the average energy only was "tted in view of the scarcity of the experimental levels below 60 000 cm\. In addition, the con"guration interaction integrals, RI, between 4f6s6p and 4f5d6p were also adjusted. All the observed even levels have been "tted if we except those situated at 25219.51, 40056.32, 40446.40, 40505.88, 40545.25, 40959.37, 41634.85 and 41682.46 cm\ which have no designations in Ref. [24]. All the parameters, including the CI integrals, describing the 4f6p, 4f6s, 4f5d6s and 4f5d con"gurations have been adjusted except the small GI integrals of the 4f6p con"guration which have been left at 85% of their ab initio values. The parameter values adopted in the "nal calculation are given in Table 1 (odd parity) and Table 2 (even parity).
628
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 1 Adopted parameters (in cm\) for odd con"gurations in Tm II. The ratios between "tted and ab initio values are also given
Fixed parameter.
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 2 Adopted parameters (in cm\) for even con"gurations in Tm II. The ratios between "tted and ab initio values are also given
Fixed parameter.
629
630
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646
Table 3 Comparison between calculated energy levels and LandeH g-factors obtained in the present work and experimental values from the NBS compilation [24] below 60 000cm\ for the odd parity in Tm II. *E"E !E . All the energies are given in cm\
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 3. Continued.
631
632
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646
Table 4 Comparison between calculated energy levels and LandeH g-factors obtained in the present work and experimental values from the NBS compilation [24] below 45 000cm\ for the even parity in Tm II. *E"E !E . All the energies are given in cm\
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 4. Continued.
633
634
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 4. Continued.
3. Results and discussion The calculated energy levels and LandeH g-factors are compared with experiment, when available, in Tables 3 and 4 for odd and even parities, respectively. In all cases, the agreement between observed and calculated values is very good. The standard deviations, according to the de"nition given by Cowan [18] are equal to 128 cm\ for the odd parity (141 "tted levels with 45 variable parameters) and 118 cm\ for the even parity (157 "tted levels with 45 adjustable parameters). Calculated HFR radiative lifetimes obtained in the present work, q , are compared to the recent laser-induced #uorescence measurements [16], q , in Tables 5 and 6, respectively, for odd parity levels below 55 000 cm\ and even-parity levels below 42 000 cm\. The calculated lifetime values greater than 1.5 ls (resulting from very weak transitions) are not reported in these tables. For the odd-parity levels, the agreement between the theory and experiment is very good for all the levels. In fact, the mean ratio q /q is, for 19 levels, 1.158$0.145 where the uncertainty represents the standard deviation of the mean. In the case of the even-parity levels, the same ratio for 69 levels is 1.070$0.292 but if we exclude from the mean 13 levels for which the ratio is larger than 2 or smaller than 0.5. For 6 of these levels at 26579, 30841, 35773, 38094, 39637 and 40232 cm\ characterized by a large discrepancy, the HFR lifetimes are considerably larger than experiment indicating probable cancellation e!ects in the calculation of the line strength. For the other 7 levels, no clear explanation exist for explaining the discrepancies. One could possibly be
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646
635
Table 5 Calculated and experimental radiative lifetimes for odd-parity levels below 55 000 cm\ in Tm II
From Ref. [24]. Present work. From Ref. [16].
a wrong designation of the levels in Ref. [24] but no evidence has been found for justifying such an explanation. The calculated weighted HFR oscillator strengths, gf , obtained in the present work are reported in Table 7 alongside the lower and upper observed energy levels of the transition and the
636
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 6 Calculated and experimental radiative lifetimes for even-parity levels below 42 000 cm\ in Tm II
From Ref. [24]. Present work. From Ref. [16].
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 7 Calculated and experimental oscillator strengths, gf, for transitions in Tm II. Only transitions with j410 000 As , upper level below 50 000 cm\and calculated gf-values greater than 0.01 are listed
637
638
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 7. Continued.
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 7. Continued.
639
640
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 7. Continued.
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 7. Continued.
641
642
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 7. Continued.
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 7. Continued.
643
644
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 7. Continued.
Air wavelengths deduced from the experimental levels of Ref. [24]. From Ref. [24]. Present work. From Ref. [17]. (e) Even parity. (o) Odd parity.
air wavelengths in A> . These wavelengths were derived from the Tm II experimental levels compiled by Martin et al. [24]. Only transitions with j410 000 As , the upper level below 50 000 cm\ and calculated gf-values greater than 0.01 are listed in Table 7. The complete table is available upon request to the authors. Experimental weighted oscillator strengths, gf , when available, are also given in Table 7 for comparison. These values were obtained by Wickli!e and Lawler [17] from the combination of branching fractions deduced from Fourier transform spectra with the radiative lifetimes measured by laser-induced #uorescence technique [16]. The comparison between calculated and experimental gf-values is illustrated in Figs. 1 and 2 for transitions with odd and even upper levels, respectively. As seen from these "gures, the general agreement between both sets of results is better in the former case than in the latter one re#ecting the comparison of the lifetimes reported in Tables 5 and 6. In fact, for the transitions involving odd upper levels, the mean di!erence between our oscillator strengths and experimental values is equal to 23% and is even reduced to 20% when excluding from the mean, the lines at 3673.14, 5114.55, 5128.08 and 5342.40 As for which our calculated gf-values are 50}60% smaller than the measurements. For the many transitions for which no experimental data are available, the predictive power is expected to be similar providing for the most intense transitions gf-values with errors reaching typically a few percent. For the strongest transitions with even upper levels, the general agreement between experimental and calculated oscillator strengths is also good. Indeed, for such lines characterized by gf '0.5, the mean di!erence between gf - and gf -values is equal to 19% while, for the lines with 0.14gf 40.5, the mean discrepancy reaches 22% if we exclude from the mean the transitions from the even levels situated at 26 579, 26 838 (J"3), 30 377, 30 509, 30 841 (J"4), 31 927, 33 037 (J"5) and 35 274 (J"2) cm\ for which large di!erences (reaching a factor of two) are observed. These extensive comparisons show that in complex ions like the singly ionized lanthanides, the HFR method is adequate for providing good quality oscillator strengths provided con"guration interaction and polarization e!ects are carefully considered in the calculations. One should also emphasize that such comparisons are themselves dependent upon the uncertainties of the experimental results which are directly related to the di$culty to measure branching fractions for weak transitions. In addition, wrong assignments of some levels cannot be entirely ruled out.
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646
645
Fig. 1. Comparison between the calculated HFR oscillator strengths obtained in the present work, log gf , and the experimental results, log gf , published by Wickli!e and Lawler [17] for Tm II transitions with odd upper levels.
Fig. 2. Comparison between the calculated HFR oscillator strengths obtained in the present work, log gf , and the experimental results, log gf , published by Wickli!e and Lawler [17] for Tm II transitions with even upper levels.
646
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646
Acknowledgements EB and PQ are respectively Research Director and Research Fellow of the Belgian Fund for Scienti"c Research (FNRS). Financial support from this organisation is acknowledged.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
Grevesse N, Blanquet G. Solar Phys 1969;8:5. Gorshkov VN, Komarovskii VA. Sov. Astron.-A.J. 1986;30:333. Jaschek C, Jaschek M. The behaviour of chemical elements in stars. Cambridge: Cambridge University Press, 1995. Reynolds SE, Hearnshaw JB, Cottrell PL. Mon Not Roy Astron Soc. 1988;235:1423. Poli AA, Bord DJ, Cowley R. PASP 1987;99:623. Cowley CR, Crosswhite HM. PASP 1978;90:108. Jaschek M, Brandi E. Astron Astrophys 1972;20:233. Adelman S. Astrophys J Suppl 1973;26:1. Bidelman WP. Astrophys J 1953;117:377. Waymouth JF. Electric discharge lamps. Cambridge, MA: MIT Press, 1971:206}306. Corliss CH, Bozman WR. Experimental transition probabilities for spectral lines of seventy elements. Nat Bur Stand (US) Monography 1962:53. [12] Curtis LJ, Martinson I, Buchta R. Nucl Inst and Methods 1973;110:391. [13] Andersen T, Sorensen G. Sol Phys 1974;38:343. [14] Blagoev KB, Komarovskii VA, Penkin NP. Opt Spectrosc (USSR) 1978;44:130. [15] Blagoev KB, Komarovskii VA. At Data Nucl Data Tables 1994;56:1. [16] Anderson HM, Den Hartog EA, Lawler JE. J Opt Soc Am 1996;B13:2382. [17] Wickli!e ME, Lawler JE. J Opt Soc Am 1997;B14:737. [18] Cowan RD. The theory of atomic structure and spectra. Berkeley: University of California Press, 1981. [19] BieH mont E, Quinet P, Phys Scripta 1996;54:36. [20] Quinet P, Hansen JE. J Phys 1995;B28:L213. [21] Migdalek J, Baylis WE. J Phys 1978;B11:L497. [22] Cowan RD, Gri$n DC. J Opt Soc Am 1976;66:1010. [23] Fraga S, Karwowski J, Saxena KMS. Handbook of atomic data. Amsterdam: Elsevier, 1976. [24] Martin WC, Zalubas R, Hagan L. Atomic energy levels*the rare-earth elements. Natl Stand Ref Data Series, 1978;NBS-60:261.
Note
On the use of the Cowan's code for atomic structure calculations in singly ionized lanthanides P. Quinet *, P. Palmeri , E. BieH mont Astrophysique et Spectroscopie, Universite& de Mons-Hainaut, Place du Parc, 20, B-7000 Mons, Belgium Institut d+Astrophysique, Universite& de Lie% ge, B-4000 Lie% ge, Belgium Received 19 June 1998
Abstract From a detailed comparison of Relativistic Hartree-Fock calculations with recent laser lifetime measurements in Tm II, it is shown that Cowan's code, which has been widely used in the past for atomic structure calculations in light elements, is adequate also for providing accurate radiative data in heavy ions and particularly in the astrophysically important singly ionized lanthanides provided con"guration interaction and core-polarization e!ects are consistently taken into account in the adopted physical model. 1999 Elsevier Science Ltd. All rights reserved. PACS: 32.70; 32.80 Keywords: Cowan's code; Lanthanides; Atomic structure calculations
1. Introduction Although they are characterized by low cosmic abundances, singly ionized lanthanides play an important role in astrophysics because they are observed in the photospheric solar spectrum [1,2] and in the spectra of some chemically peculiar stars [3]. In stars, Tm can be produced by both the r- and s-processes. Tm II has been identi"ed in one FO Ib star by Reynolds et al. [4] and in Ap stars of the Si subgroup by Poli et al. [5] and by Cowley and Crosswhite [6]. Tm I has been observed in Ap stars of the Cr}Eu}Sr subgroup by Jaschek and Brandi [7] and by Adelman [8] and also in S-type stars by Bidelman [9]. Interest in this research "eld is stimulated by new technologies (large-aperture telescopes, new spectrometers and advanced CCD detector arrays) that allow recordings of the stellar spectra with high resolution and better signal-to-noise ratio
* Corresponding author. Tel.: 0032 65 37 37 27; fax: 0032 65 37 30 54; e-mail: [email protected] 0022-4073/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved PII: S 0 0 2 2 - 4 0 7 3 ( 9 8 ) 0 0 1 2 7 - 7
626
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646
than previously available. In addition, the determination of improved chemical compositions of stars, made possible by the recent high-resolution Hubble Space Telescope observations, relies heavily upon accurate spectroscopic data and more particularly atomic oscillator strengths. Rare-earth elements are also increasingly used in metal-halide arc lamps. These devices, which consist basically of high-pressure mercury arc lamps with metal-halide salts [10], provide highquality light characterized by an improved color-rendering index and a high luminous e$ciency. The line-rich spectrum of thulium and the relatively high vapor pressure of salts of Tm are very advantageous for improving the capabilities of such lamps. Consequently, spectroscopic data of rare earths are needed for further developments in the "eld. Atomic transition probabilities of Tm II have been little considered up to now. Arc measurements have been performed by Corliss and Bozman [11] more than 30 years ago but, in the past, their results have been frequently shown to be in error for many transitions. Radiative lifetime determinations include works by Curtis et al. [12], Andersen and Sorensen [13], Blagoev et al. [14] and Blagoev and Komarovskii [15] but they concern a rather limited number of levels. Very recently, accurate radiative lifetime values measured by laser-induced #uorescence technique and transition probabilities obtained from combination of these lifetimes with branching fractions deduced from Fourier transform spectra have been published [16,17]. This provides a unique opportunity to test the adequacy of the semi-empirical relativistic Hartree-Fock (HFR) approach described by Cowan [18] for predicting energy levels and line intensities in the case of the &&complex'' con"gurations encountered in the lanthanide ions. This method has been widely used in the past for predicting, with success, radiative properties of light elements but has been little considered up to now in the case of heavy elements or ions if we except some isolated attempts (see e.g. Ref. [19]). The case of singly ionized thulium is considered in the present paper and, more speci"cally, the e!ects of con"guration interaction and of core-polarization on the transitions originating from the 4f6s, 4f5d, 4f6s6p and 4f5d6p odd con"gurations and the 4f6p, 4f6s, 4f5d6s and 4f5d even con"gurations are discussed in detail.
2. The calculations For complex systems, accurate calculations of atomic structure should allow for both intravalence and core-valence correlation. Simultaneous treatment of both e!ects within con"guration interaction (CI) scheme is very complex and reliable only if enough con"guration mixing is considered as shown, for example, by Quinet and Hansen [20] in the case of the iron group elements. However, in practice, even a large computer imposes rather severe limitations on the number of interacting con"gurations that can be considered simultaneously and, in particular, the inclusion of core-polarization e!ects through the consideration of a huge number of con"gurations with open inner shells is often prevented by computer limitations. Migdalek and Baylis [21] have suggested an approach in which most of the intravalence correlation is represented within CI scheme while core-valence correlation is represented approximately by a core-polarization model potential, < , given by . !a r , (1) < (r)" . 2(r#r)
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646
627
where a is the static dipole polarizability of the core and r is the cut-o! radius which is arbitrarily chosen as a measure of the size of the ionic core. This latter parameter is usually taken as the expectation value of r for the outermost core orbitals. When including the core-polarization in the one-electron Hamiltonian, the dipole-moment operator in the transition matrix element has also to be modi"ed for consistency since these two corrections appear in the same order of the perturbation theory. The dipole-moment operator, d"!r, of the valence electron has to be replaced by ar . d"!r# (r#r)
(2)
In the present work, we used the pseudo-relativistic Hartree}Fock (HFR) method described by Cowan [18] and Cowan and Gri$n [22] in which we have included expressions (1) and (2) given above to make allowance for core-polarization e!ects. Con"guration interaction was retained among the con"gurations 4f6s, 4f7s, 4f5d, 4f6d, 4f7d, 4f6s6p, 4f5d6p, 5p4f6s, 5p4f5d, 5p4f6s6p, 5p4f5d6p for the odd parity and 4f6p, 4f7p, 4f6s, 4f5d6s, 4f5d, 5p4f6p, 5p4f6s, 5p4f5d6s, 5p4f5d for the even parity. The dipole polarizability of the ionic core, a , was chosen equal to 7.62a for the 4fnl con"gurations and to 5.60a for the 4fnln l con"gurations, which corresponds to the values reported by Fraga et al. [23] in the cases of Tm III and Tm IV, respectively. For the cut-o! radius, r , we used the average value 1r2 for the outermost core orbitals (5p) as calculated by the Cowan's code, i.e. r "1.484a . Using a well-established least-squares "tting procedure [18], the radial parameter values were adjusted to obtain the best agreement between calculated and observed energy levels. The "tted parameters were the center-of-gravity energies (E ), the single-con"guration direct (FI) and exchange (GI) Coulomb-interaction integrals, the spin}orbit integrals (f) and some CI (RI) integrals related to the con"gurations studied experimentally. For the remaining con"gurations, the FI, GI and RI integrals were arbitrarily scaled down by a factor 0.85 while the ab initio values of the spin-orbit parameters, f, computed by the Blume}Watson method, were used without scaling. All the known experimental odd levels below 60 000 cm\ compiled by Martin et al. [24] have been introduced in the "tting procedure if we except the J"2 level at 59135.89 cm\ which has no spectroscopic designation. In fact, the levels situated above 60 000 cm\ are fragmentarily known and therefore some of the designations appear dubious. Moreover, some of these levels overlap unknown levels belonging to higher con"gurations such as 4f6d and 4f7d for which no experimentally determined energy levels are available. Consequently, these levels have not been included in the "t. All the parameters of the con"gurations 4f6s, 4f5d, 4f6s6p and 4f5d6p have been adjusted. For 4f7s, the average energy only was "tted in view of the scarcity of the experimental levels below 60 000 cm\. In addition, the con"guration interaction integrals, RI, between 4f6s6p and 4f5d6p were also adjusted. All the observed even levels have been "tted if we except those situated at 25219.51, 40056.32, 40446.40, 40505.88, 40545.25, 40959.37, 41634.85 and 41682.46 cm\ which have no designations in Ref. [24]. All the parameters, including the CI integrals, describing the 4f6p, 4f6s, 4f5d6s and 4f5d con"gurations have been adjusted except the small GI integrals of the 4f6p con"guration which have been left at 85% of their ab initio values. The parameter values adopted in the "nal calculation are given in Table 1 (odd parity) and Table 2 (even parity).
628
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 1 Adopted parameters (in cm\) for odd con"gurations in Tm II. The ratios between "tted and ab initio values are also given
Fixed parameter.
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 2 Adopted parameters (in cm\) for even con"gurations in Tm II. The ratios between "tted and ab initio values are also given
Fixed parameter.
629
630
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646
Table 3 Comparison between calculated energy levels and LandeH g-factors obtained in the present work and experimental values from the NBS compilation [24] below 60 000cm\ for the odd parity in Tm II. *E"E !E . All the energies are given in cm\
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 3. Continued.
631
632
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646
Table 4 Comparison between calculated energy levels and LandeH g-factors obtained in the present work and experimental values from the NBS compilation [24] below 45 000cm\ for the even parity in Tm II. *E"E !E . All the energies are given in cm\
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 4. Continued.
633
634
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 4. Continued.
3. Results and discussion The calculated energy levels and LandeH g-factors are compared with experiment, when available, in Tables 3 and 4 for odd and even parities, respectively. In all cases, the agreement between observed and calculated values is very good. The standard deviations, according to the de"nition given by Cowan [18] are equal to 128 cm\ for the odd parity (141 "tted levels with 45 variable parameters) and 118 cm\ for the even parity (157 "tted levels with 45 adjustable parameters). Calculated HFR radiative lifetimes obtained in the present work, q , are compared to the recent laser-induced #uorescence measurements [16], q , in Tables 5 and 6, respectively, for odd parity levels below 55 000 cm\ and even-parity levels below 42 000 cm\. The calculated lifetime values greater than 1.5 ls (resulting from very weak transitions) are not reported in these tables. For the odd-parity levels, the agreement between the theory and experiment is very good for all the levels. In fact, the mean ratio q /q is, for 19 levels, 1.158$0.145 where the uncertainty represents the standard deviation of the mean. In the case of the even-parity levels, the same ratio for 69 levels is 1.070$0.292 but if we exclude from the mean 13 levels for which the ratio is larger than 2 or smaller than 0.5. For 6 of these levels at 26579, 30841, 35773, 38094, 39637 and 40232 cm\ characterized by a large discrepancy, the HFR lifetimes are considerably larger than experiment indicating probable cancellation e!ects in the calculation of the line strength. For the other 7 levels, no clear explanation exist for explaining the discrepancies. One could possibly be
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646
635
Table 5 Calculated and experimental radiative lifetimes for odd-parity levels below 55 000 cm\ in Tm II
From Ref. [24]. Present work. From Ref. [16].
a wrong designation of the levels in Ref. [24] but no evidence has been found for justifying such an explanation. The calculated weighted HFR oscillator strengths, gf , obtained in the present work are reported in Table 7 alongside the lower and upper observed energy levels of the transition and the
636
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 6 Calculated and experimental radiative lifetimes for even-parity levels below 42 000 cm\ in Tm II
From Ref. [24]. Present work. From Ref. [16].
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 7 Calculated and experimental oscillator strengths, gf, for transitions in Tm II. Only transitions with j410 000 As , upper level below 50 000 cm\and calculated gf-values greater than 0.01 are listed
637
638
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 7. Continued.
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 7. Continued.
639
640
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 7. Continued.
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 7. Continued.
641
642
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 7. Continued.
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 7. Continued.
643
644
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646 Table 7. Continued.
Air wavelengths deduced from the experimental levels of Ref. [24]. From Ref. [24]. Present work. From Ref. [17]. (e) Even parity. (o) Odd parity.
air wavelengths in A> . These wavelengths were derived from the Tm II experimental levels compiled by Martin et al. [24]. Only transitions with j410 000 As , the upper level below 50 000 cm\ and calculated gf-values greater than 0.01 are listed in Table 7. The complete table is available upon request to the authors. Experimental weighted oscillator strengths, gf , when available, are also given in Table 7 for comparison. These values were obtained by Wickli!e and Lawler [17] from the combination of branching fractions deduced from Fourier transform spectra with the radiative lifetimes measured by laser-induced #uorescence technique [16]. The comparison between calculated and experimental gf-values is illustrated in Figs. 1 and 2 for transitions with odd and even upper levels, respectively. As seen from these "gures, the general agreement between both sets of results is better in the former case than in the latter one re#ecting the comparison of the lifetimes reported in Tables 5 and 6. In fact, for the transitions involving odd upper levels, the mean di!erence between our oscillator strengths and experimental values is equal to 23% and is even reduced to 20% when excluding from the mean, the lines at 3673.14, 5114.55, 5128.08 and 5342.40 As for which our calculated gf-values are 50}60% smaller than the measurements. For the many transitions for which no experimental data are available, the predictive power is expected to be similar providing for the most intense transitions gf-values with errors reaching typically a few percent. For the strongest transitions with even upper levels, the general agreement between experimental and calculated oscillator strengths is also good. Indeed, for such lines characterized by gf '0.5, the mean di!erence between gf - and gf -values is equal to 19% while, for the lines with 0.14gf 40.5, the mean discrepancy reaches 22% if we exclude from the mean the transitions from the even levels situated at 26 579, 26 838 (J"3), 30 377, 30 509, 30 841 (J"4), 31 927, 33 037 (J"5) and 35 274 (J"2) cm\ for which large di!erences (reaching a factor of two) are observed. These extensive comparisons show that in complex ions like the singly ionized lanthanides, the HFR method is adequate for providing good quality oscillator strengths provided con"guration interaction and polarization e!ects are carefully considered in the calculations. One should also emphasize that such comparisons are themselves dependent upon the uncertainties of the experimental results which are directly related to the di$culty to measure branching fractions for weak transitions. In addition, wrong assignments of some levels cannot be entirely ruled out.
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646
645
Fig. 1. Comparison between the calculated HFR oscillator strengths obtained in the present work, log gf , and the experimental results, log gf , published by Wickli!e and Lawler [17] for Tm II transitions with odd upper levels.
Fig. 2. Comparison between the calculated HFR oscillator strengths obtained in the present work, log gf , and the experimental results, log gf , published by Wickli!e and Lawler [17] for Tm II transitions with even upper levels.
646
P. Quinet et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 625}646
Acknowledgements EB and PQ are respectively Research Director and Research Fellow of the Belgian Fund for Scienti"c Research (FNRS). Financial support from this organisation is acknowledged.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
Grevesse N, Blanquet G. Solar Phys 1969;8:5. Gorshkov VN, Komarovskii VA. Sov. Astron.-A.J. 1986;30:333. Jaschek C, Jaschek M. The behaviour of chemical elements in stars. Cambridge: Cambridge University Press, 1995. Reynolds SE, Hearnshaw JB, Cottrell PL. Mon Not Roy Astron Soc. 1988;235:1423. Poli AA, Bord DJ, Cowley R. PASP 1987;99:623. Cowley CR, Crosswhite HM. PASP 1978;90:108. Jaschek M, Brandi E. Astron Astrophys 1972;20:233. Adelman S. Astrophys J Suppl 1973;26:1. Bidelman WP. Astrophys J 1953;117:377. Waymouth JF. Electric discharge lamps. Cambridge, MA: MIT Press, 1971:206}306. Corliss CH, Bozman WR. Experimental transition probabilities for spectral lines of seventy elements. Nat Bur Stand (US) Monography 1962:53. [12] Curtis LJ, Martinson I, Buchta R. Nucl Inst and Methods 1973;110:391. [13] Andersen T, Sorensen G. Sol Phys 1974;38:343. [14] Blagoev KB, Komarovskii VA, Penkin NP. Opt Spectrosc (USSR) 1978;44:130. [15] Blagoev KB, Komarovskii VA. At Data Nucl Data Tables 1994;56:1. [16] Anderson HM, Den Hartog EA, Lawler JE. J Opt Soc Am 1996;B13:2382. [17] Wickli!e ME, Lawler JE. J Opt Soc Am 1997;B14:737. [18] Cowan RD. The theory of atomic structure and spectra. Berkeley: University of California Press, 1981. [19] BieH mont E, Quinet P, Phys Scripta 1996;54:36. [20] Quinet P, Hansen JE. J Phys 1995;B28:L213. [21] Migdalek J, Baylis WE. J Phys 1978;B11:L497. [22] Cowan RD, Gri$n DC. J Opt Soc Am 1976;66:1010. [23] Fraga S, Karwowski J, Saxena KMS. Handbook of atomic data. Amsterdam: Elsevier, 1976. [24] Martin WC, Zalubas R, Hagan L. Atomic energy levels*the rare-earth elements. Natl Stand Ref Data Series, 1978;NBS-60:261.