A&A 424, 363-369 (2004)
DOI: 10.1051/0004-6361:20040496

Dirac-Fock energy levels and transition probabilities for oxygen-like Fe XIX^,

V. Jonauskas^1, - F. P. Keenan ¹ - M. E. Foord ² - R. F. Heeter ² - S. J. Rose ³ - G. J. Ferland ⁴ - R. Kisielius^1, - P. A. M. van Hoof ¹ - P. H. Norrington ⁵

1 - Department of Pure and Applied Physics, The Queen's University of Belfast, Belfast BT7 1NN, Northern Ireland, UK
2 - University of California, Lawrence Livermore National Laboratory, Livermore, CA 94551, USA
3 - Department of Physics, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK
4 - Department of Physics, University of Kentucky, Lexington, KY 40506, USA
5 - Department of Applied Mathematics and Theoretical Physics, The Queen's University of Belfast, Belfast BT7 1NN, Northern Ireland, UK

Received 22 March 2004 / Accepted 26 May 2004

Abstract
Multiconfigurational Dirac-Fock calculations are reported for 656 energy levels and the 214 840 electric dipole (E1), electric quadrupole (E2) and magnetic dipole (M1) transition probabilities in oxygen-like Fe XIX. The spectroscopic notations as well as the total transition probabilities from each energy level are provided. Good agreement is found with data compiled by NIST.

Key words: atomic data

1 Introduction

Atomic data for O-like Fe XIX have been extensively studied under the framework of the IRON Project (Hummer et al. 1993). Galavis et al. (1997) present radiative rates for forbidden transitions in O-like ions with $Z \leq 28$, for the n=2 complex. Butler & Zeippen (2001) provide energy levels, electron impact collision strengths and rates for transitions among the 92 lowest levels of the n=2 and n=3 complexes of Fe XIX. The atomic structure code SUPERSTRUCTURE (SS) (Eissner et al. 1974) was employed by both sets of authors. On the other hand, McLaughlin et al. (2001) present the 25 lowest energy levels and effective collision strengths, calculated with the atomic structure code CIV3 (Hibbert 1975), and the Breit-Pauli version of the R-matrix codes (Berrington et al. 1995), respectively. Zhang & Sampson (2002) obtain relativistic distorted-wave collision strengths and oscillator strenghts for transitions among 10 lowest levels of the n=2 complex.

High resolution spectra of astrophysical sources require accurate atomic data for reliable plasma modelling. In particular, the Chandra and XMM-Newton satellites provide spectra with particularly large effective areas in the 6 to 18 $\AA$ wavelength range, covering Fe XIX lines arising from the $n \ge 3$ complex (McKernan et al. 2003; Behar et al. 2001; Heyden et al. 2003). To allow the reliable interpretation of these lines, we have studied Fe XIX transitions among the levels of the 25 lowest configurations originating from the n=3,4 and 5 complexes. Previous calculations cover a smaller range of transitions.

In the present paper we report calculations for energy levels, transition probabilities and oscillator strengths for electric dipole (E1), electric quadrupole (E2), and magnetic dipole (M1) transitions among 656 levels of oxygen-like Fe XIX. A comparison between NIST data and our calculations, as well as the energy levels of McLaughlin et al. (2001) and Butler & Zeippen (2001), is made. The five strongest and the sum of all the radiative transition probabilities (useful to obtain decay branching ratios) from the levels are also presented.

2 Calculations

The GRASP code of Parpia et al. (1996); Dyall et al. (1989); Norrington (2003) is used for the calculation of wave functions as well as the matrix elements of the Dirac-Coulomb-Breit Hamiltonian and transition operators. As a result, direct and indirect relativistic effects are included in the calculations. One-electron wave functions, obtained by solving multiconfigurational Dirac-Fock (MCDF) equations, were used to build configuration state wave functions. The intermediate coupling wave functions:

$$\Psi_{\gamma} (J)=\sum\limits_{\alpha}c_{\gamma}(\alpha J)~\phi (\alpha J)$$ (1)

are generated in the basis $\phi (\alpha J)$ of configuration state functions (CSF) by diagonalizing the Dirac-Coulomb-Breit Hamiltonian matrix. The frequency-dependent transverse Breit interaction operator is used for the calculation of Breit matrix elements. QED corrections are considered in the first order of perturbation theory, and the correlation corrections taken into account by the configuration interaction (CI) method. The relativistic line strengths are obtained from matrix elements of frequency-dependent transition operators. Coulomb and Babushkin gauges of the electric transition operators allow us to estimate the accuracy of wave function expansion in the intermediate coupling.

Results are presented for total number of 656 lowest energy levels in Fe XIX. We consider 565 energy levels arising from the 1s²2s²2p⁴, 1s²2s¹2p⁵, 1s²2p⁶, 1s²2s²2p³nl, 1s²2s¹2p⁴3l', 1s²2s¹2p⁴4s¹, 1s²2s¹2p⁴4p¹, and 1s²2p⁵3l' (n=3,4,5, l=0,..,n-1, l'=0,1,2) configurations. Due to the high density of levels between highly excited configurations, and strong configuration interaction between these, there are also included some energy levels of other configurations (38 levels of 1s²2s¹2p⁴4d¹ and 41 levels of 1s²2s¹2p⁴4f¹) with calculated binding energies larger than our chosen cut-off value, which corresponds to the highest level of the 1s²2s²2p³5g¹ configuration.

A large CI basis is used to generate one-electron wave functions solving MCDF equations as well to obtain accurate intermediate coupling wave functions diagonalizing the Dirac-Coulomb-Breit Hamiltonian matrix. We include one-electron excitations from 2s and 2p orbitals of the 1s²2s²2p⁴ configuration up to the 7l orbital, and two-electron excitations from 2s or/and 2p orbitals to all possible combinations of two electrons in the shells with n=3. The level energies and intermediate coupling wave functions obtained after the diagonalization of the Dirac-Coulomb-Breit Hamiltonian matrix are then used to evaluate the transition probabilities, oscillator and line strengths for electric dipole, electric quadrupole and magnetic dipole transitions among the 656 lowest levels in our calculations.

3 Results and discussion

The 656 fine-structure levels which give rise from the above mentioned configurations are listed in Table 1. The indexes for levels presented in the first column of Table 1 are used in all tables. Energy levels are given in cm^-1 relative to the ground state 1s²2s²2p^4 3P₂. Configuration as well as term notations presented in the second and third columns have the primary contribution to the wavefunction of Eq. (1) obtained after Dirac-Coulomb-Breit matrix diagonalization. For brevity, filled 1s² shells are omitted in the notations of configurations, and intermediate many-electron quantum numbers are shown in the parentheses. Leading percentage compositions are equal to squared expansion coefficients $c_{\gamma}(\alpha J)$ for the intermediate coupling function of Eq. (1), and are limited to values larger than 10%. As was mentioned above, diagonalization of the Dirac-Coulomb-Breit matrix, which is obtained in the jj-coupling scheme, provides expansion coefficients for the intermediate coupling functions in the basis of CSF. We present the CSF in the LS-coupling scheme. Therefore, the corresponding percentage compositions are obtained from coefficients calculated in the jj-coupling scheme using term-coupling coefficients. The expansion of the intermediate coupling function in the CSF basis depends on the model used in calculations. Leading percentage compositions obtained after Dirac-Coulomb matrix diagonalization differ from our presented values. The discrepancies between the two sets of coefficients are not large, but in some cases the order of levels identified by the largest contribution of CSF can be different. Calculated energies of levels can also change after inclusion of Breit and QED corrections.

Table 2: Suggested change of spectroscopic identifications of levels to ensure the completeness of spectroscopic dataset. The indexes of levels for which spectroscopic identifications are changed are presented in the first column. The second column contains index of level with the same highest contribution of configuration state function (Table 1) as level from the first column before change. Identification of level with index 492 was changed due to use for identification its LS-configuration state function for level with index 482. Indexes of levels in the first two columns are taken from Table 1.

Some levels identified by the CSF obtained in LS-coupling scheme with the largest contribution to an intermediate coupling function have the same spectroscopic notations. To ensure the completeness of the spectroscopic dataset, we suggest new spectroscopic notations for levels with the same spectroscopic identities. In some cases, the mixing of CSF defined in LS-coupling scheme is so strong that it is difficult to make definite identifications of levels. The mixing of CSF especially increases for highly excited levels where the separation between level energies decreases. Leading percentages of levels show that the LS-coupling scheme is not satisfactory for highly excited states. Proposed spectroscopic notations for levels with the same largest contribution of LS-coupling CSF are presented in Table 2. If two levels have the same many-electron quantum numbers for a given configuration in the second and third columns of Table 1, the identification of the level with the smaller contribution to the intermediate coupling wave function is changed to that of the secondary CSF. If this secondary function is employed for the identification of some other level, then the third CSF is checked. The spectroscopic notation of level 492 has had its LS-configuration state function re-assigned to level 482. However, no other level in the second and third columns of Table 1 has the new spectroscopic notation proposed for level 492. Therefore, no spectroscopic notation has been assigned to level 492 in the second column of Table 2.

In Table 3 we compare our calculated energy levels as well as the data obtained by McLaughlin et al. (2001) and Butler & Zeippen (2001) with values compiled by NIST (National Institute for Standards and Technology: www.physics.nist.gov). Fairly good agreement is obtained for the MCDF calculations, with the energy levels agreeing to better than 2% with the NIST values. Levels 10 (2p^6 1S₀) and 9 (2s¹2p^5 1P₁) show the largest differences of 1.9% and 1.6%, respectively. All highly excited levels agree to better than 1%. Only level 493 has a large deviation from the NIST energy, but even this is only 1.3%. The percentage composition of the LS-coupling CSF for the level shows that there is strong mixing between CSF and no single CSF has a contribution exceeding 50%. Our energy of ground state -230 793 799 cm^-1 is slightly higher than NIST value of $-231~017~600~ \pm~ 142~300$ cm^-1. Due to this some calculated energies of levels relative to the calculated ground energy are lower than NIST values presented in Table 3.

The average deviation between our energy levels and the NIST values is only 0.35%. On the other hand, the average deviation for energies obtained by McLaughlin et al. (2001) is 0.8%. For the energies in common, the average deviation of the Butler & Zeippen (2001) values is 0.8%, while for our data this is only 0.6%. The largest deviation for energies obtained by McLaughlin et al. (2001) is 3.2% for the first excited level (2s²2p^4 3P₀) and 2.7% for the second (2s²2p^4 3P₁). Butler & Zeippen (2001) energies show an average deviation of 0.6% with NIST data, while the average for our values is 0.45% for the same range of levels. The largest deviations for energies obtained by Butler & Zeippen (2001) is 3.3% for level 4 (2s²2p^4 1D₂) and 1.6% for levels 10 (2p^6 1S₀) and 9 (2s¹2p^5 1P₁). Therefore, our energy levels agree better with the NIST values than those calculated by previous authors. The main reason for this is that we use a larger base of CSF, and a fully relativistic approach. Level ordering differs from NIST in both large-scale calculations.

The total number of dipole allowed and intercombination E1-type transitions is 58 390, while the complete set of forbidden transitions totals 136 866. Table 6 includes only E1-type transitions for which oscillator strengths exceed 0.001. All forbidden E2-type transitions with $f \ge 10^{-11}$ are reported in Table 7. Table 8 provides data for M1-type transitions with f-values exceeding 10^-7. Differences between Babushkin and Coulomb gauges (velocity and length forms of the electric transition operators in the nonrelativistic limit), which are an additional indicator of accuracy for the wavefunctions, do not exceed 20% for most strong E1-type transitions ($f \ge 0.1$). The average deviation between the two forms is 5.3% for 812 lines with $f \ge 0.1$ and 9.6% for 4524 lines with $f \ge 0.01$. The electric quadrupole transition has the largest contribution to the lifetime of the first excited level, the difference between the two forms is large, and the oscillator strength is very small. E2-type transition from level 5 (79% 2s²2p^4 1S₀ + 19% 2s²2p^4 3P₀) also shows a large deviation of 80% between the two forms, and a small f-value ( $f=2.5 \times 10^{-11}$). On the other hand, our obtained leading compositions of wavefunctions reports similar values for these three levels as NIST ones; 1: 90% (2s² 2p^4 3P₂) + 10% (2s² 2p^4 1D₂); 2: 80% (2s² 2p^4 3P₀) + 20% (2s² 2p^4 1S₀); 5: 78% (2s² 2p^4 1S₀) + 20% (2s² 2p^4 3P₀). The deviation between calculated (Babushkin gauge) and NIST characteristics of the transition from the first excited level does not exceed 4%.

Table 3: Comparison of some calculated energies of Fe XIX levels with data presented by NIST ( $E^{\rm NIST}$). $E^{\rm McLaughlin}$ - values obtained by McLaughlin et al. (2001), $E^{\rm Butler}$ - energies calculated by Butler & Zeippen (2001), $E^{\rm GRASP}$ - our values. Indexes of levels in the first column are taken from Table 1. Energies are in cm^-1.

Table 4: Comparison of some calculated (GRASP) Fe XIX wavelengths $\lambda$, radiative transition probabilities A^r_ki, oscillator f_ik and line S strengths with values presented by NIST. Indexes of levels in the first two columns are taken from Table 1.

Our calculated wavelengths, transition probabilities, oscillator and line strengths as well as the NIST values are presented in Table 4. The accuracy of the transition wavelengths is well above 1% for short wavelengths ( $\lambda < 16~$Å). The largest deviations of wavelengths (up to 4%) are obtained for transitions which involve levels 4, 9 and 10, as these show the largest differences in energies from NIST data. Our probabilities for most transitions are slightly smaller than the corresponding NIST values, with intercombination E1-type transitions showing the largest deviations. The difference for their radiative probabilities varies from 4% to 20%. Transitions 2s¹2p^5 3P₁ $\rightarrow$ 2s²2p^4 1S₀ and 2s¹2p^5 1P₁ $\rightarrow$ 2s²2p^4 3P₀ have the largest deviation of 20% and 16%, respectively. On the other hand, their contributions to the radiative lifetimes of the levels do not exceed 1%. The deviation for other intercombination E1-type transitions does not exceed 12%, while transition probabilities for other lines agree to better than 10%. This is highly satisfactory for such a large-scale calculation. The average deviation between our radiative transition probabilities and NIST values presented in Table 4 is 5.8%.

The five largest spontaneous radiative transition probabilities from each level are given in Table 5, and the sum of all E1, E2, and M1 radiative transition probabilities from the corresponding level are provided in the last column. Forbidden E2 and M1 transitions have the largest weight for transitions between fine-structure levels of the ground configuration, where E1-type transitions are forbidden. Levels 3, 4, and 5 (2s²2p^4 3P₁, ¹D₂, and ¹S₀) decay mainly due to magnetic dipole transitions. The contribution of electric quadrupole transitions from these levels is negligible. For highly excited levels, the contribution of magnetic dipole transitions is small. On the other hand, the first excited level decays only through E2-type transition. Contribution of E2-type forbidden transitions to the lifetimes of highly excited levels is noticeable for levels of the 2s²2p³3p¹ and 2s¹2p⁴3d¹ configurations. For example, E2-type transitions amount to 40% of the lifetime for level 24 (2s²2p³ (⁴S) 3p^1 3P₀), 28% for level 138 (2s¹2p⁴ (⁴P) 3d^1 3F₄), and 27% for level 33 (2s²2p³ (²D) 3p^1 1F₃). Finally, electric quadrupole transitions have contributions exceeding 10% of the total transition probability for 12 levels (24, 138, 33, 30, 34, 23, 28, 22, 176, 126, 18 and 172).

4 Conclusion

In the present paper we have reported large-scale calculations in the multiconfigurational Dirac-Fock approach of 656 lowest energy levels as well as corresponding transition wavelengths, absorption oscillator strengths and radiative transition probabilities for electric dipole, electric quadrupole as well as magnetic dipole type transitions in oxygen-like Fe XIX. To our knowledge, our work represents the largest calculation to date for Fe XIX. Fairly good agreement is obtained between our calculated and NIST data.

Spectroscopic notations of levels in the LS-coupling scheme have been presented, and checked for their completeness. Contributions of CSF to intermediate coupling wave functions indicate that LS-coupling scheme is not satisfactory for highly excited states.

The five major radiative probabilities from each level and the total transition probability have been provided, taking into account forbidden transitions. The largest contributions of forbidden E2 and M1 transitions have been obtained for the lifetimes of fine-structure levels of the ground configuration.

Agreement between our presented theoretical values and available NIST data as well as large basis of configuration state functions and relativistic Dirac-Fock approach, allows us to conclude that our calculations of energy levels and radiative transition data for Fe XIX are reliable. They may successfully be used in various plasma codes for the interpretation of astronomical and other spectral observations, especially for high resolution spectra provided by the Chandra and XMM-Newton satellites.

Acknowledgements

F.P.K. and S.J.R. are grateful to A. W. E. Aldermaston for the award of William Penney Fellowships. This work was supported by PPARC and EPSRC, and also by NATO Collaborative Linkage Grant CLG.979443. We are also grateful to the Defence Science and Technology Laboratory (dstl) for support under the Joint Grants Scheme.

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Online Material

Table 1: Calculated energy levels of Fe XIX with spectroscopic identification (in cm^-1) relative to the ground energy. The leading percentage compositions of levels which contributions exceed 10% are presented in the last column.

Table 5: Calculated energy levels of Fe XIX (in cm^-1) relative to the ground energy with spectroscopic notations. The five major spontaneous radiative transition probabilities A^r (in s^-1) from each level are given. Arrow marks the final level to which radiative transition happens from the level. The sum of all radiative probabilities from the corresponding level is given in the last column.