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Radiative transitions from singly ionized oxygen
ARTICLE IN PRESS
Journal of Quantitative Spectroscopy & Radiative Transfer 97 (2006) 267–277 www.elsevier.com/locate/jqsrt
Radiative transitions from singly ionized oxygen L. Natarajan Department of Physics, University of Mumbai, Mumbai 400 098, India Received 29 January 2005; accepted 27 April 2005
Abstract Fully relativistic calculations have been performed on the states of 2p3 ; 2p2 3s; 2p2 3p and 2p2 ð3 PÞ3d configurations of singly ionized oxygen. The calculations have been carried out using multi-configuration Dirac–Fock wavefunctions with the inclusion of Breit interaction and quantum electrodynamic contributions. A substantial amount of correlation has also been included in our calculations. The numerical results for more than 150 electric dipole lines in the spectrum of O+ are compared with other theoretical and experimental values. r 2005 Published by Elsevier Ltd. Keywords: Multi-configuration Dirac-Fock wavefunctions; Quantum electrodynamics
1. Introduction The structural properties of singly ionized oxygen are of great importance in identifying and classifying their content in planetary nebulae. Also the radiation and collisional excitation rates of OII ions are used as a diagnostic tool in the laboratory and in astrophysical plasmas, and for the theoretical modeling of supernova remnants (SNR) and OB-type hot stars [1]. The thermal population of plasma can be studied using the spectral properties of OII. The solar wind contains a small amount of singly ionized oxygen. Many workers have earlier investigated the fine-structure transitions from OII theoretically and experimentally. Recently, the transition probabilities of many multiplets have been measured by del Val et al. [2] using a pulsed discharge tube. Veres and Wiese [3] reported their photoelectron E-mail address: [email protected]. 0022-4073/$ - see front matter r 2005 Published by Elsevier Ltd. doi:10.1016/j.jqsrt.2005.04.011
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measurements for many fine structure lines of OII. Theoretical investigations on OII using the Rmatrix method in LS-coupling [4] and configuration interaction formalism in intermediate coupling [5] exist in literature. The compilation of transition probabilities in the NIST atomic spectra database (NIST-ASD) has been reported very recently [6]. The configuration interaction calculations, with the inclusion of relativistic effects in the Breit-Pauli approximation, were carried out by Bell et al. [5] using CIV3 code of Hibbert. The importance of relativistic effects on the radiative lifetimes of singly ionized oxygen has already been detailed by Bell et al. [5] by comparing their values with the R-matrix LS-coupling calculation [4]. In this work, we use a fully relativistic configuration interaction (RCI) model in the multi-configuration Dirac-Fock (MCDF) formalism. This method includes the Breit and quantum electrodynamics (QED) corrections in the Dirac-Coulomb Hamiltonian instead of relativistic contributions from the Breit-Pauli approximation. In addition, the present method differs from the CIV3 code in the orbital optimization procedure. In CIV3, the orbitals are optimized one at a time. In this work all the orbitals that are considered in the computation are optimized simultaneously. The results from both these independent models are complimentary to each other. In addition, a comparison of the present rates with the R-matrix LS-coupling values of Lennon and Burke [4] will indicate the importance of fully relativistic treatment on the dipole rates. Also, the rates with error estimates included in the NIST-ASD compilation and the experimental values will help in judging the accuracy of our results. In the present work, transition rates and oscillator strengths for several multiplets originating from states of 2p23s, 2p23p and 2p23d configurations of OII have been computed in the MCDF formalism with the inclusion of Breit and other QED corrections like self energy, vacuum polarization, specific and normal mass shifts. The calculations have been carried out using the dipole length and the dipole velocity operators. A good amount of 2 s core–valence correlation has been incorporated in the RCI calculations. For convenience, non-relativistic notation is used throughout this paper. The graspVU [8] package used in the present work to generate the MCDF wavefunctions and perform the structure calculations is the modified version of the GRASP92 code [9–10].
2. Numerical procedure The relativistic MCDF basis set in the configuration interaction computational program are described in detail in the literature [9–13]. In the MCDF method the configuration state functions f ðGJ P Þ of a certain J and parity are formed by taking a linear combination of Slater determinants of the Dirac orbitals. A linear combination of these configuration state functions (CSFs) is then used in the construction of atomic state functions (ASFs) with the same J and parity: ncsf X cia fðGa J P Þ, (1) Ci ðJ P Þ ¼ a¼i
where cia are the mixing coefficients for the state i, and ncsf is the number of CSFs included in the evaluation of ASF. Ga represents all the one-electron and intermediate quantum numbers needed to define the CSFs. The ASFs thus constructed were used in solving the Dirac-Fock equation, and
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the Dirac-Coulomb Hamiltonian was used with the atomic nucleus described by a Fermi charge distribution. The contributions from higher-order correction terms like Breit interaction, QED and finite nuclear mass corrections are in general added as a first-order perturbation correction after self-consistency is obtained. The relativistic transition probabilities can be calculated either by using a Coulomb gauge or a Babushkin gauge which, in the non-relativistic limit, correspond respectively, to velocity and length forms. The numerical procedure adopted in this work is mainly the same as in our earlier convergence studies [14–16]. As a starting point, the CSFs were generated from the reference configurations 2p3, 2p23s, 2p23p and 2p23d using the extended optimal level (EOL) scheme. In this method, the radial orbitals and the mixing coefficients are determined by optimizing the energy function, which is the weighted sum of energy values corresponding to a set of eigen states. The zero-order eigenfunctions and energy eigenvalues were first calculated using this code. The generation of CSFs depends on the correlation model and orbital set in the active space. In the valence correlation model, the active set consisted of many virtual shells in addition to the valence shell. In the 2s core–valence correlation model, the 2s core was also included in the active set. In both these models the correlation was affected by considering single and double (SD) excitations of electrons from the reference configurations to the orbitals in the active set. The virtual shells were increased in a systematic way taking into account the convergence criteria. In all our calculations, the 1s core was kept fixed. As the multiplet splitting leads to a number of even and odd parity states, based on the initial and final configurations, we considered the following three expansion sets of orbitals: (i) For 2p23s–2p23p transitions, the CSFs were generated with SD excitation of orbitals in the set {2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d} while, for 2p3–2p23s transitions, we excluded 3d and 4d orbital from the above set. (ii) For 2p3–2p23d and 2p23p–2p23d transitions, we considered single excitation of the set {2p, 3p, 3d, 4p, 4d, 5p, 5d}. As the number of CSFs for double excitation increases many-fold with the increase in the size of the active set, to keep the CSFs at a manageable level we restricted these transitions only to a single excitation. First we optimized the orbitals with np3; where n is the principal quantum number. Then we included the orbitals with n ¼ 4 and optimized the set. This step-by-step increase in the expansion set was carried out in the optimization process. Once the MCDF orbitals were optimized and the energy eigenvalues converged, higher-order correction terms were added to the Dirac-Coulomb Hamiltonian and a further diagonalization in the configuration basis was carried out, so as to recalculate the mixing coefficients. The eigenvectors and eigenvalues from the RCI calculations were then used in the transition rate calculations.
3. Results and discussions The converged average energies of the set of levels considered in this work relative to the ground level are reported in Table 1. Also included in this table are the differences between the
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Table 1 Weighted average energies of different levels of O+ relative to the ground level States
2s22p3 2D 2s22p3 2P 2p23s 4P 2p23s 2P 2p23s 2S 2p23s 2D 2p23p 2S 2p23p 4D 2p23p 4P 2p23p 2D 2p23p 4S 2p23p 2P 2p23p 2F 2p23p 2D 2p23d 4F 2p23d 4P 2p23d 4D 2p23d 2F 2p23d 2D
RCI
0.119785 0.185825 0.844755 0.864603 1.103685 0.953995 0.929975 0.939985 0.947478 0.964418 0.972295 0.977847 1.050525 1.059305 1.052828 1.054990 1.060021 1.156262 1.161621
Energy differences CIV3 (Ref. [5])
Experimental (Ref. [7])
0.009271 0.001442 0.000011 0.004651 0.024971 0.000159 0.002943 0.000036 0.000392 0.000488 0.006706 0.001181 0.001455 0.000816 0.005197 0.003235 0.001740 0.089862 0.094079
0.002401 0.001442 0.000789 0.003417 0.046345 0.010962 0.000746 0.009695 0.002202 0.000132 0.006706 0.001181 0.008323 0.011552 0.001640 0.004406 0.000449 0.092503 0.093443
Also included in this table are differences in the RCI energies and the LS-coupling values of Bell et al. [5] and experimental energies of Wenaker [7].
RCI and CIV3 values of Bell et al. [5], as well as the experimental energies of Wenaker [7]. Though a direct comparison is not possible, it is seen that the present energy values agree very well with the experimental and CIV3 results, except for the states of 2p2(3P)3d configuration, and the agreement justifies the proper choice of orbital sets used in this work. The RCI energies and rates from 2p23s–2p23p, 2p3–2p23s, 2p23d and 2p23p–2p23d groups of transitions are discussed in Subsections 3.1–3.3, respectively. As our RCI energies for the above four groups of transitions are found to have a marginal difference with the earlier results, they are not listed separately in this work. The 2p23s–2p23p and 2p23p–2p23d transitions have been studied both experimentally and theoretically and data are available for many multiplets. For the 2p3–2p23s and 2p23d transitions, earlier data mainly correspond only to the theoretical values. The allowed dipole rates, absorption oscillator strengths and ratio of length to velocity rates for 2p23s–2p23p, 2p3–2p23s, 2p23d and 2p23p–2p23d transitions, respectively, are listed in Tables 2–4. Only those transitions with rates X105 s1 are reported in this work. Also included in these tables are the transition rates and oscillator strengths calculated by Bell et al. [5] in length gauge, the Rmatrix LS-coupling rates of Lennon and Burke [4] and the dipole rates from the NIST-ASD compilation [6]. The available experimental rates obtained by del Val et al. [2] and by Veres and Wiese [3] are also listed in these tables.
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Table 2 Dipole rates (Al) in 108 s1 and absorption oscillator strengths (fl) in length form for 2p23s–2p23p transitions Transition
4
P1/2–4D1/2 P1/2–4D3/2 4 P3/2–4D1/2 4 P3/2–4D3/2 4 P3/2–4D5/2 4 P5/2–4D3/2 4 P5/2–4D5/2 4 P5/2–4D7/2 4 P1/2–4P1/2 4 P1/2–4P3/2 4 P3/2–4P1/2 4 P3/2–4P3/2 4 P3/2–4P5/2 4 P5/2–4P3/2 4 P5/2–4P5/2 4 P1/2–4S3/2 4 P3/2–4S3/2 4 P5/2–4S3/2 2 P1/2–(3P)2D3/2 2 P3/2–(3P)2D5/2 2 P3/2–(3P)2D3/2 2 P3/2–(1D)2D5/2 2 P1/2–(1D)2D3/2 2 P3/2–(1D)2D3/2 2 P3/2–(1D)2D5/2 2 P1/2–(3P)2P1/2 2 P1/2(3P)2P3/2 2 P3/2–(3P)2P1/2 2 P3/2–(3P)2P3/2 2 P1/2–(1D)2P1/2 2 P1/2–(1D)2P3/2 2 P3/2–(1D)2P1/2 2 P3/2–(1D)2P3/2 2 P1/2–(1S)2P1/2 2 P3/2–(1S)2P3/2 2 P3/2–(3P)2S1/2 2 P1/2–(1S)2P3/2 2 P1/2–2S1/2 2 P3/2–2S1/2 2 D3/2–(3P)2P1/2 2 D3/2–(1D)2P1/2 2 D3/2–(1D)2P3/2 2 D5/2–(1D)2P3/2 2 D5/2–(3P)2P3/2 4
Av/Al
Al RCI
CIV3 Ref. [5]
0.718 0.379 0.129 0.435 0.623 0.034 0.226 0.851 0.152 0.403 0.827 0.160 0.269 0.419 0.715 0.327 0.675 1.055 0.805 0.999 0.190 0.498 0.398 0.088 0.540 0.869 0.249 0.453 1.021 0.249 0.106 0.130 0.311 0.331 0.045 0.201 0.008 0.111 0.201 0.001 1.254 0.088 1.176 0.007
0.720 0.382 0.127 0.432 0.626 0.033 0.222 0.849 0.149 0.395 0.826 0.162 0.262 0.420 0.715 0.288 0.595 0.937 0.777 0.926 0.147 0.543 0.458 0.082 0.542 0.857 0.214 0.438 1.075 0.375 0.085 0.193 0.481
Ref. [6]
Ref. [2]
Ref. [3]
Ref. [4]
0.695 0.351
0.142 0.368 0.795 0.153 0.248 0.392 0.675 0.225 0.564 0.897 0.716 0.847 0.137
0.621 0.340 0.122 0.376 0.543 0.030 0.245 0.718 0.135 0.340 0.764 0.144 0.234 0.366 0.635 0.261 0.534 0.937 0.655 0.767 0.126
0.694 0.349 0.137 0.441 0.586 0.040 0.246 0.833 0.164 0.412 0.807 0.130 0.296 0.430 0.676 0.300 0.593 0.874 0.793 0.952 0.155
0.421 0.076 0.498
0.099 0.182 0.002 1.228 0.142 1.093 0.002
1.22 0.141 1.09
0.835 0.411 0.965 0.166 0.260 0.358 0.771
0.856
0.081
0.188 0.393 0.956
0.182
0.589
0.894 0.225 0.437 1.10 0.83 0.80 0.87 0.80 0.91 0.76 1.15 0.71 1.14 1.15 1.90 1.00 1.11 1.00 0.85
0.99 1.00 0.99 1.00 1.00 0.99 0.99 1.00 1.00 1.01 1.00 1.00 1.04 1.00 1.01 0.88 0.85 0.85 1.00 0.97 1.01 1.01 0.77 0.83 0.70 0.99 0.99 1.02 1.00
fl RCI
CIV3 Ref. [5]
0.246 0.258 0.022 0.150 0.319 0.008 0.078 0.368 0.045 0.236 0.123 0.048 0.118 0.084 0.213 0.125 0.130 0.137 0.500 0.047 0.060 0.062 0.065 0.007 0.071 0.215 0.121 0.057 0.252 0.019 0.016 0.005 0.025 0.009 0.001 0.075 0.0005 0.080 0.075 0.004 0.149 0.021 0.173 0.008
0.234 0.247 0.021 0.141 0.306 0.007 0.073 0.367 0.042 0.221 0.117 0.046 0.110 0.080 0.203 0.119 0.124 0.132 0.455 0.041 0.044 0.073 0.081 0.007 0.073 0.201 0.100 0.052 0.260 0.030 0.014 0.008 0.038
0.062 0.066 0.062 0.003 0.142 0.033 0.167 0.004
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Table 2 (continued ) Transition
2
D3/2–(1D)2D3/2 D3/2–(1D)2D5/2 2 D5/2–(1D)2D5/2 2 D5/2–(1D)2D3/2 2 D5/2–(1D)2F5/2 2 D3/2–(1D)2F5/2 2 D5/2–(1D)2F7/2 2 S1/2–(1S)2P1/2 2 S1/2–(1S)2P3/2 2
Av/Al
Al RCI
CIV3 Ref. [5]
Ref. [6]
1.035 0.063 1.048 0.073 0.048 0.826 0.882 0.008 0.005
0.934 0.058 0.99 0.119 0.048 0.827 0.878
0.932 0.058 0.989 0.049 0.834 0.885
Ref. [2]
fl
Ref. [3]
Ref. [4]
RCI
CIV3 Ref. [5]
1.003
0.91 1.01 0.95 0.76 1.00 1.00 1.06 0.51 0.50
0.290 0.022 0.280 0.014 0.016 0.410 0.390 0.003 0.002
0.265 0.025 0.281 0.023 0.015 0.393 0.371
0.761
0.817 0.822
Table 3 Dipole rates (Al) in 108 s1 and absorption oscillator strengths (fl) in length form for 2p3–2p23s and 2p3–2p23d transitions Transition
Al
Av/Al
RCI
CIV3 Ref. [5]
Ref. [6]
P1/2–2P1/2 P1/2–2S1/2 2 P1/2–2P3/2 2 P1/2–2D3/2 2 P3/2–2P1/2 2 P3/2–2P3/2 2 P3/2–2S1/2 2 P3/2–2P3/2 2 P3/2–2D5/2 2 D3/2–2P1/2 2 D3/2–2P3/2 2 D3/2–2D3/2 2 D3/2–2D5/2 2 D5/2–2P3/2 2 D5/2–2D3/2 2 D5/2–2D5/2 4 S3/2–4P1/2 4 S3/2–4P3/2 4 S3/2–4P5/2
4.132 3.087 1.171 3.703 1.116 3.611 10.40 1.015 5.325 17.67 2.707 10.00 0.921 15.79 1.373 9.859 10.46 10.42 10.36
3.677 2.296 0.98 4.184 1.881 4.841 4.400 0.921 4.975 35.04 3.307 9.937 0.789 31.69 0.953 10.25 9.678 9.686 9.703
3.63 3.24 0.968 3.96 1.86 4.78 6.21 0.872 4.71 34.3 3.24 11.01 0.801 31.1 0.967 11.04 9.81 9.81 9.83
2p3–2p2(3P) 3d 4 S3/2–4P1/2 4 S3/2–4D1/2 4 S3/2–4P3/2 4 S3/2–4D3/2
30.67 4.035 29.52 1.035
40.63 5.214 39.44 6.353
2 2
fl RCI
CIV3 Ref. [5]
1.25 0.86 1.21 1.14 1.28 1.35 0.88 1.15 1.15 1.06 1.05 0.97 0.98 1.06 0.97 0.97 0.95 0.95 0.95
0.032 0.017 0.011 0.028 0.008 0.037 0.044 0.005 0.030 0.103 0.011 0.047 0.006 0.092 0.006 0.048 0.045 0.045 0.045
0.025 0.01 0.013 0.045 0.006 0.033 0.01 0.005 0.040 0.100 0.019 0.046 0.006 0.120 0.003 0.047 0.021 0.042 0.063
0.97 0.96 0.99 0.91
0.048 0.006 0.092 0.006
0.056 0.007 0.109 0.017
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Table 3 (continued ) Transition
4
S3/2–4P5/2 S3/2–4D5/2 2 D5/2–2P3/2 2 D5/2–2D3/2 2 D5/2–2F5/2 2 D5/2–2F7/2 2 D5/2–2D5/2 2 D3/2–2P1/2 2 D3/2–2P3/2 2 D3/2–2D3/2 2 D3/2–2F5/2 2 D3/2–2D5/2 2 P3/2–2P1/2 2 P3/2–2P3/2 2 P3/2–2D3/2 2 P3/2–2D5/2 2 P1/2–2P1/2 2 P1/2–2P3/2 2 P1/2–2D3/2 4
fl
Av/Al
Al RCI
CIV3 Ref. [5]
29.17 2.310 15.17 1.579 1.541 21.11 9.265 14.11 4.737 9.601 18.68 10.10 9.241 20.50 4.789 16.47 13.29 5.705 11.57
41.61 3.491 18.15 1.737 1.152 24.92 8.029 20.71 3.251 6.337 18.55
Ref. [6]
10.65 23.55 5.363 19.53 20.76 6.898 14.50
9.35 20.7 5.03 18.31 18.20 6.05 13.61
0.91 1.01 1.04 0.99 0.97 1.02 0.98 1.14 0.95 0.90 1.01 0.96 0.92 0.93 0.91 0.91 0.92 1.01 0.96
RCI
CIV3 Ref. [5]
0.137 0.010 0.033 0.004 0.004 0.121 0.031 0.026 0.017 0.029 0.102 0.047 0.020 0.075 0.020 0.071 0.054 0.044 0.104
0.173 0.015 0.043 0.004 0.004 0.117 0.028 0.036 0.011 0.022 0.098 0.021 0.095 0.021 0.117 0.084 0.056 0.116
Table 4 Dipole rates (Al) in 108 s1 and absorption oscillator strengths (fl) in length form for 2p23p–2p23d transitions Transition
4
D1/2–4P1/2 D1/2–4D1/2 4 D1/2–4F3/2 4 D1/2–4P3/2 4 D1/2–4D3/2 4 D3/2–4P1/2 4 D3/2–4D1/2 4 D3/2–4F3/2 4 D3/2–4P3/2 4 D3/2–4D3/2 4 D3/2–4F5/2 4 D3/2–4D5/2 4 D3/2–4P5/2 4 D5/2–4F3/2 4 D5/2–4P3/2 4 D5/2–4D3/2 4
Av/Al
Al RCI
CIV3 Ref. [5]
Ref. [6]
0.169 0.175 1.620 0.087 0.071 0.007 0.247 0.505 0.079 0.137 1.432 0.069 0.031 0.015 0.009 0.151
0.091 0.195 1.446 0.033 0.094 0.005 0.280 0.535 0.089 0.158 1.553 0.063 0.019 0.034 0.002 0.218
0.091 0.187 1.42 0.033 0.082 0.005 0.245 0.521 0.089 0.122 1.48 0.034 0.039 0.002 0.191
Ref. [2]
Ref. [3]
Ref. [4]
0.179 1.40
0.293 1.40
0.069
0.147
0.210 0.507 0.146
0.441
0.291 0.557
0.006
0.234 1.50 0.014
0.044
0.039
0.165
0.202
1.56 1.14 1.40 0.87 1.15 0.66 1.14 0.83 0.88 1.34 0.89 1.15 0.90 0.75 0.73 1.12
fl RCI
CIV3 Ref. [5]
0.037 0.043 0.621 0.036 0.048 0.008 0.021 0.171 0.006 0.033 0.436 0.025 0.010 0.003 0.002 0.022
0.020 0.043 0.719 0.015 0.042 0.0005 0.031 0.136 0.020 0.035 0.579 0.021 0.006 0.005 0.0003 0.033
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Table 4 (continued ) Transition
4
D5/2–4F5/2 D5/2–4P5/2 4 D5/2–4D5/2 4 D5/2–4F7/2 4 D5/2–4D7/2 4 D7/2–4F5/2 4 D7/2–4P5/2 4 D7/2–4D5/2 4 D7/2–4F7/2 4 D7/2–4D7/2 4 D7/2–4F9/2 4 P1/2–4P1/2 4 P1/2–4D1/2 4 P1/2–4P3/2 4 P1/2–4D3/2 4 P1/2–4F3/2 4 P3/2–4P1/2 4 P3/2–4D1/2 4 P3/2–4F3/2 4 P3/2–4P3/2 4 P3/2–4D3/2 4 P3/2–4F5/2 4 P3/2–4P5/2 4 P3/2–4D5/2 4 P5/2–4P3/2 4 P5/2–4D3/2 4 P5/2–4F3/2 4 P5/2–4P5/2 4 P5/2–4D5/2 4 P5/2–4D7/2 2 D3/2–2P3/2 2 D3/2–2D3/2 2 D3/2–2F5/2 2 D3/2–2D5/2 2 D5/2–2D3/2 2 D5/2–2F3/2 2 D5/2–2D5/2 2 D5/2–2F7/2 2 P3/2–2P3/2 2 P3/2–2D3/2 2 P3/2–2D5/2 2 P1/2–2P1/2 2 P1/2–2P3/2 2 P1/2–2D3/2 2 S1/2–2P1/2 4
Av/Al
Al RCI
CIV3 Ref. [5]
Ref. [6]
Ref. [2]
Ref. [3]
Ref. [4]
0.559 0.136 0.204 1.656 0.089 0.010 0.017 0.062 0.233 0.524 1.881 0.555 0.631 0.771 0.204 0.043 0.486 0.957 0.060 0.004 1.03 0.011 0.637 0.426 0.263 0.234 0.034 0.417 0.253 1.425 0.025 0.332 0.952 0.027 0.060 0.024 0.401 1.012 0.312 0.188 0.809 0.346 0.148 0.594 1.381
0.445 0.086 0.214 1.765 0.061 0.015 0.005 0.110 0.246 0.514 2.010 0.560 0.863 0.737 0.247
0.431 0.086 0.206 1.79 0.060 0.015 0.005 0.107 0.245 0.51 1.98 0.56 0.532 0.84 0.379
0.418
0.465
0.482
0.398 0.603
0.165 0.805
0.398 0.708
0.001 0.935 0.0002 0.664 0.474 0.239 0.260
0.038 0.955 0.0002 0.728 0.329 0.194 0.295
0.376 0.840
0.300 0.625 1.397 0.007 0.345 0.936 0.014 0.042 0.058 0.377 1.241 0.338 0.196 0.861 0.322 0.120 0.667 1.484
0.249 0.225 1.39 0.036 0.357
0.249 0.198 1.22
0.198 1.82 0.060 0.016 0.104 0.244 0.506 1.94
1.677
0.228 0.489 0.686
0.468 0.839 0.333
0.727 0.194
0.876
0.689
0.331 1.71 0.082 0.019 0.100 0.280 1.99 0.169 0.128 0.419 0.642 0.840 0.254 0.133 0.817
0.815 0.289 0.173
0.297 1.07 0.785
1.356
0.686 0.455 1.52
1.32 0.014 0.043 0.043 0.391 1.01 0.201 0.857
0.716 0.9
0.329 0.167 0.715 0.302 0.116 0.540 1.17
1.40 0.540 0.162 0.979 0.444 0.109 0.816 1.25
1.34 0.90 1.16 1.40 1.14 1.28 3.40 1.07 1.36 1.15 0.94 1.34 0.89 0.89 1.31 1.25 0.90 1.31 0.80 1.38 0.92 0.80 1.20 0.89 0.92 1.48 1.40 0.90 1.38 0.95 1.01 1.09 1.41 2.37 0.90 0.81 0.88 0.94 0.83 0.97 0.98 1.10 1.13 1.00 1.10
fl RCI
CIV3 Ref. [5]
0.137 0.030 0.045 0.541 0.020 0.002 0.001 0.011 0.058 0.114 0.002 0.168 0.154 0.383 0.099 0.024 0.060 0.117 0.004 0.001 0.251 0.004 0.237 0.154 0.044 0.037 0.011 0.104 0.230 0.350 0.009 0.081 0.601 0.012 0.012 0.015 0.113 0.546 0.061 0.082 0.470 0.151 0.143 0.432 0.302
0.111 0.020 0.048 0.585 0.018 0.003 0.001 0.019 0.062 0.116 0.006 0.143 0.218 0.378 0.125 0.051 0.076 0.0004 0.236 0.0001 0.258 0.180 0.043 0.044 0.078 0.159 0.474 0.002 0.099 0.465 0.006 0.008 0.020 0.108 0.549 0.137 0.072 0.473 0.129 0.097 0.489 0.254
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Table 4 (continued ) Transition
2
S1/2–2P3/2 S1/2–2D3/2 4 S3/2–4P1/2 4 S3/2–4D1/2 4 S3/2–4P3/2 4 S3/2–4D3/2 4 S3/2–4F5/2 4 S3/2–4P5/2 4 S3/2–4D5/2 2
Av/Al
Al RCI
CIV3 Ref. [5]
Ref. [6]
Ref. [2]
1.633 0.030 0.271 0.136 0.417 0.112 0.559 0.727 0.059
1.501 0.012 0.579 0.081 0.540 0.100
1.31 0.016
1.12
1.24
0.441
0.635
0.417
0.629
0.543 0.056
0.417
Ref. [3]
Ref. [4] 1.01 1.13 1.26 1.29 1.10 1.01 1.01 0.89 0.93
fl RCI
CIV3 Ref. [5]
0.637 0.003 0.090 0.034 0.233 0.038 0.136 0.037 0.017
0.518 0.004 0.104 0.014 0.196 0.036 0.030 0.030
3.1. 2p23p–2p23s transitions The RCI energies for the various multiplets from 2p23s–2p23p transitions (not listed in the tables) differ in general from the earlier experimental and theoretical values by 0.02–0.06 eV, except for the 2P–(1D)2D and 2P–(1D)2P transitions, with an energy difference of 0.1–0.2 eV. It may be noted that, in order to improve the agreement between experimental and theoretical data, Bell et al. used a set of semi-empirical energy corrections for the various LS blocks, and the same correction term was used for all the states in a particular block. The comparison of transition energies shows that the differences between the RCI and CIV3 energies are nearly the same as the semi-empirical correction values used by Bell et al., and we find that the present energies are generally in good agreement with the earlier values. It is seen from Table 2 that the RCI rates in length gauge agree well with the other theoretical and experimental dipole rates for 2p23s–2p23p transitions. The present rates are in excellent agreement with the CIV3 rates and differ only marginally for a few transitions. This marginal discrepancy is mainly due to the full use of relativity. The RCI rates differ by 1–2% from the Rmatrix LS-coupling rates for a number of transitions and deviate by 7–10% for some transitions. Taking into account the error estimates reported in the NIST-ASD compilations, it is found that the present rates are in good agreement with these rates. A comparison of the RCI rates with experimental rates reported by del Val et al. [2] and Veres and Wiese [3] shows that the present rates are slightly larger than the experimental rates for all the transitions considered in Table 1. The RCI rates are in better agreement with the rates obtained by del Val et al. than those of Veres and Wiese. 3.2. 2p3–2p23s and 2p23d The RCI energies (not listed in Table 3) for the fine structure lines from 2p3–2p23s transitions are more than the CIV3 energies for some transitions, and less for some other transitions, and the variation ranges from 0.01–0.12 eV. The RCI energies for 2P–2P and 2P–2D transitions differ
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from the CIV3 values by 0.1 and 0.02 eV, respectively. For the 2D–2D and 2D–2P transitions, the difference is o0.02 eV, whereas for 2P–2S transitions, RCI energies are more than the CIV3 energy values by 0.09 eV. The difference between the present energies and CIV3 values for 4S–4P transitions is 0.03 eV. In general, as seen in Section 3.1, the differences between the RCI energy values and CIV3 values of Bell et al. are nearly the same as the semi-empirical corrections used by them to match their values with the experimental results. A comparison of the RCI and CIV3 rates [5] listed in Table 3 for 2p3–2p23s transitions shows a significant variation in the dipole rates. For some transitions, fairly large deviations are noticed between the RCI and CIV3 rates. Similar variations are noticed between the present rates and NIST-ASD compilations. The RCI rates are larger than the earlier compilations for all dipole lines except for 2D–2P transitions, and differ in general by 9–20%. The present rates are nearly double, and half of the earlier rates, for 2P3/2–2S1/2 and 2D–2P transitions, respectively. The accuracy of the dipole rates is influenced both by the relative positions of the eigenvalues and the composition of the eigenvectors. As the difference in the RCI and CIV3 energies of 2D–2P transitions is very less, the large deviation in the dipole rates is mainly due to orbital functions, whereas, for the 2P–2S transitions, the significant variation in the transition rates is a combined effect of the large difference in energy values and eigenvectors. The present work makes full use of relativity, whereas, CIV3 calculations are known for better use of correlation effects and hence it is not possible to relate the discrepancies specifically to either of them. The RCI rates for a few strong lines arising from 2p3–2p2(3P)3d transitions are also listed in Table 3, along with the CIV3 rates of Bell et al. and values from NIST-ASD compilations. A comparison shows that the RCI rates are in reasonably good agreement with the other theoretical results for most of the transitions and the difference is appreciable for a few transitions. As the RCI average energies for the states of 2p23d configurations are only in fairly good agreement with the experimental and the semi-empirically adjusted CIV3 values, the differences in the RCI and CIV3 rates are due to the combined effect of transition wavelengths and correlation effects. 3.3. 2p23p–2p23d transitions The RCI energies for 2p23p–2p23d transitions also show a systematic variation from CIV3 results. The RCI energies for the 4D–4D, 4D–4P, 4P–4P and 4P–4D transitions differ from CIV3 values by 0.06 eV. For 4D–4F and 4P–4F transitions, the energy difference is 0.02–0.1 eV. The RCI energies for 2D–2D, 2D–2F transitions vary by 0.03 eV. The differences in the RCI and CIV3 transition energies are mainly the same as the semi-empirical scaling factor used to adjust the CIV3 values with the experimental energies. The RCI rates for some selected dipole lines arising from 2p2(3P)3p–2p2(3P)3d transitions of OII are reported in Table 4. The variations in the dipole rates reported in this work and other earlier results are not systematic. For some transitions, the RCI rates are larger than the other theoretical and experimental rates, and, for some other transitions, the present rates are smaller. The present rates for strong transitions agree well with CIV3 and NIST-ASD rates and vary by 2–10%. The RCI rates for most of the weak transitions encounter fairly large deviations from the earlier results. For the rest of the transitions, the differences vary from 2–20%. The RCI rates are generally in good agreement with the experimental rates of del Val et al. and Veres and Weise, except for 4P3/2–4P3/2 and 4D3/2–4D5/2 transitions, for which exceptionally large variations
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between RCI rates and the respective experimental rates of del Val et al. and Veres and Wiese are noticed. The RCI rates in length gauge for the various transitions reported in this work are generally in good agreement with the rates calculated in velocity gauge, except for a few weak lines. 4. Conclusion A comparison of the RCI rates for allowed dipole lines reported in Tables 2–4 and earlier results shows that the present rates are in very good agreement with each other for 2p23s–2p23p and vary on an average by 9–20% for 2p3–2p23s transitions. For 2p3–2p23d and 2p23p–2p23d transitions, the agreement is in general good. Though the present calculation is fully relativistic, the choice of excited orbitals included in this work might probably be responsible for the discrepancies between RCI and CIV3 rates, especially for 2p3–2p23d and 2p23p–2p23d transitions. Also, the differences between the RCI rates and earlier results might partly be due to the different ways in which the relativistic and correlation effects are calculated.
References [1] Decaux V, Beiersdorfer P, Kahn SM, Jacobs VL. High resolution measurement of the Ka spectrum of Fe XXV–XVIII: new spectral diagnostics of non-equilibrium astrophysical plasmas. Astrophys J 1997;482:1076–84. [2] del Val JV, Aparicio JA, Gonzalex VR, Mar S. Transition probability measurment of several OII spectral lines. J Phys B 2001;34:4531–8. [3] Veres G, Wiese WL. Experimental atomic transition probabilities for OII lines. Phys Rev A 1996;54:1999–2003. [4] Lennon DJ, Burke VM. 1991, unpublished. [5] Bell KL, Hibbert A, Stafford RP, MaLaughlin BM. Accurate transition probabilities for some spectral lines of singly ionized oxygen. Phys Scripta 1994;50:343–53. [6] NIST Atomic Spectra database data 2001. [7] Wenaker L. The spectrum of singly ionized OII. Phys Scripta 1990;42:667–84. [8] Jonsson P, He X, Froese Fischer C. 1998. The graspVU relativistic atomic structure package, (private communication). [9] Parpia FA, Froese Fischer C, Grant IP. GRASP92: a package for large-scale relativistic atomic structure calculations. Comput Phys Comm 1996;94:249–71. [10] Dyall KG, Grant IP, Johnson CT, Parpia FA, Plummer EP. GRASP: a general purpose relativistic atomic structure program. Comput Phys Comm 1989;55:425–56. [11] Grant IP. Relativistic atomic structure. In: Drake GWF, editor. Handbook in atomic, molecular and optical physics. New York: AIP; 1996. p. 268–86. [12] Froese Fischer C. Atomic structure: multiconfiguration Hartree-Fock theories. In: Drake GWF, editor. Handbook in atomic, molecular and optical physics. New York: AIP; 1996. p. 243–57. [13] Grant IP. Relativistic atomic structure calculations: methods in computational chemistry. In: Wilson S, editor. Relativistic effects in atoms and molecules, Vol. 2. New York: Phelom; 1988. [14] Natarajan L, Muley YG. Multi Configuration Dirac-Fock and configuration interaction calculations of Ar7+ to Ar17+ ions. J Phys B At Mol Phys 2001;34:1839–47. [15] Natarajan L. Spin-forbidden electric dipole transitions of highly ionized argon. J Phys B 2002;35:3179–87. [16] Anuradha N, Natarajan L. Kb satellites of highly ionized iron. J Phys B 2004;37:4789–801.
Journal of Quantitative Spectroscopy & Radiative Transfer 97 (2006) 267–277 www.elsevier.com/locate/jqsrt
Radiative transitions from singly ionized oxygen L. Natarajan Department of Physics, University of Mumbai, Mumbai 400 098, India Received 29 January 2005; accepted 27 April 2005
Abstract Fully relativistic calculations have been performed on the states of 2p3 ; 2p2 3s; 2p2 3p and 2p2 ð3 PÞ3d configurations of singly ionized oxygen. The calculations have been carried out using multi-configuration Dirac–Fock wavefunctions with the inclusion of Breit interaction and quantum electrodynamic contributions. A substantial amount of correlation has also been included in our calculations. The numerical results for more than 150 electric dipole lines in the spectrum of O+ are compared with other theoretical and experimental values. r 2005 Published by Elsevier Ltd. Keywords: Multi-configuration Dirac-Fock wavefunctions; Quantum electrodynamics
1. Introduction The structural properties of singly ionized oxygen are of great importance in identifying and classifying their content in planetary nebulae. Also the radiation and collisional excitation rates of OII ions are used as a diagnostic tool in the laboratory and in astrophysical plasmas, and for the theoretical modeling of supernova remnants (SNR) and OB-type hot stars [1]. The thermal population of plasma can be studied using the spectral properties of OII. The solar wind contains a small amount of singly ionized oxygen. Many workers have earlier investigated the fine-structure transitions from OII theoretically and experimentally. Recently, the transition probabilities of many multiplets have been measured by del Val et al. [2] using a pulsed discharge tube. Veres and Wiese [3] reported their photoelectron E-mail address: [email protected]. 0022-4073/$ - see front matter r 2005 Published by Elsevier Ltd. doi:10.1016/j.jqsrt.2005.04.011
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measurements for many fine structure lines of OII. Theoretical investigations on OII using the Rmatrix method in LS-coupling [4] and configuration interaction formalism in intermediate coupling [5] exist in literature. The compilation of transition probabilities in the NIST atomic spectra database (NIST-ASD) has been reported very recently [6]. The configuration interaction calculations, with the inclusion of relativistic effects in the Breit-Pauli approximation, were carried out by Bell et al. [5] using CIV3 code of Hibbert. The importance of relativistic effects on the radiative lifetimes of singly ionized oxygen has already been detailed by Bell et al. [5] by comparing their values with the R-matrix LS-coupling calculation [4]. In this work, we use a fully relativistic configuration interaction (RCI) model in the multi-configuration Dirac-Fock (MCDF) formalism. This method includes the Breit and quantum electrodynamics (QED) corrections in the Dirac-Coulomb Hamiltonian instead of relativistic contributions from the Breit-Pauli approximation. In addition, the present method differs from the CIV3 code in the orbital optimization procedure. In CIV3, the orbitals are optimized one at a time. In this work all the orbitals that are considered in the computation are optimized simultaneously. The results from both these independent models are complimentary to each other. In addition, a comparison of the present rates with the R-matrix LS-coupling values of Lennon and Burke [4] will indicate the importance of fully relativistic treatment on the dipole rates. Also, the rates with error estimates included in the NIST-ASD compilation and the experimental values will help in judging the accuracy of our results. In the present work, transition rates and oscillator strengths for several multiplets originating from states of 2p23s, 2p23p and 2p23d configurations of OII have been computed in the MCDF formalism with the inclusion of Breit and other QED corrections like self energy, vacuum polarization, specific and normal mass shifts. The calculations have been carried out using the dipole length and the dipole velocity operators. A good amount of 2 s core–valence correlation has been incorporated in the RCI calculations. For convenience, non-relativistic notation is used throughout this paper. The graspVU [8] package used in the present work to generate the MCDF wavefunctions and perform the structure calculations is the modified version of the GRASP92 code [9–10].
2. Numerical procedure The relativistic MCDF basis set in the configuration interaction computational program are described in detail in the literature [9–13]. In the MCDF method the configuration state functions f ðGJ P Þ of a certain J and parity are formed by taking a linear combination of Slater determinants of the Dirac orbitals. A linear combination of these configuration state functions (CSFs) is then used in the construction of atomic state functions (ASFs) with the same J and parity: ncsf X cia fðGa J P Þ, (1) Ci ðJ P Þ ¼ a¼i
where cia are the mixing coefficients for the state i, and ncsf is the number of CSFs included in the evaluation of ASF. Ga represents all the one-electron and intermediate quantum numbers needed to define the CSFs. The ASFs thus constructed were used in solving the Dirac-Fock equation, and
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the Dirac-Coulomb Hamiltonian was used with the atomic nucleus described by a Fermi charge distribution. The contributions from higher-order correction terms like Breit interaction, QED and finite nuclear mass corrections are in general added as a first-order perturbation correction after self-consistency is obtained. The relativistic transition probabilities can be calculated either by using a Coulomb gauge or a Babushkin gauge which, in the non-relativistic limit, correspond respectively, to velocity and length forms. The numerical procedure adopted in this work is mainly the same as in our earlier convergence studies [14–16]. As a starting point, the CSFs were generated from the reference configurations 2p3, 2p23s, 2p23p and 2p23d using the extended optimal level (EOL) scheme. In this method, the radial orbitals and the mixing coefficients are determined by optimizing the energy function, which is the weighted sum of energy values corresponding to a set of eigen states. The zero-order eigenfunctions and energy eigenvalues were first calculated using this code. The generation of CSFs depends on the correlation model and orbital set in the active space. In the valence correlation model, the active set consisted of many virtual shells in addition to the valence shell. In the 2s core–valence correlation model, the 2s core was also included in the active set. In both these models the correlation was affected by considering single and double (SD) excitations of electrons from the reference configurations to the orbitals in the active set. The virtual shells were increased in a systematic way taking into account the convergence criteria. In all our calculations, the 1s core was kept fixed. As the multiplet splitting leads to a number of even and odd parity states, based on the initial and final configurations, we considered the following three expansion sets of orbitals: (i) For 2p23s–2p23p transitions, the CSFs were generated with SD excitation of orbitals in the set {2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d} while, for 2p3–2p23s transitions, we excluded 3d and 4d orbital from the above set. (ii) For 2p3–2p23d and 2p23p–2p23d transitions, we considered single excitation of the set {2p, 3p, 3d, 4p, 4d, 5p, 5d}. As the number of CSFs for double excitation increases many-fold with the increase in the size of the active set, to keep the CSFs at a manageable level we restricted these transitions only to a single excitation. First we optimized the orbitals with np3; where n is the principal quantum number. Then we included the orbitals with n ¼ 4 and optimized the set. This step-by-step increase in the expansion set was carried out in the optimization process. Once the MCDF orbitals were optimized and the energy eigenvalues converged, higher-order correction terms were added to the Dirac-Coulomb Hamiltonian and a further diagonalization in the configuration basis was carried out, so as to recalculate the mixing coefficients. The eigenvectors and eigenvalues from the RCI calculations were then used in the transition rate calculations.
3. Results and discussions The converged average energies of the set of levels considered in this work relative to the ground level are reported in Table 1. Also included in this table are the differences between the
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Table 1 Weighted average energies of different levels of O+ relative to the ground level States
2s22p3 2D 2s22p3 2P 2p23s 4P 2p23s 2P 2p23s 2S 2p23s 2D 2p23p 2S 2p23p 4D 2p23p 4P 2p23p 2D 2p23p 4S 2p23p 2P 2p23p 2F 2p23p 2D 2p23d 4F 2p23d 4P 2p23d 4D 2p23d 2F 2p23d 2D
RCI
0.119785 0.185825 0.844755 0.864603 1.103685 0.953995 0.929975 0.939985 0.947478 0.964418 0.972295 0.977847 1.050525 1.059305 1.052828 1.054990 1.060021 1.156262 1.161621
Energy differences CIV3 (Ref. [5])
Experimental (Ref. [7])
0.009271 0.001442 0.000011 0.004651 0.024971 0.000159 0.002943 0.000036 0.000392 0.000488 0.006706 0.001181 0.001455 0.000816 0.005197 0.003235 0.001740 0.089862 0.094079
0.002401 0.001442 0.000789 0.003417 0.046345 0.010962 0.000746 0.009695 0.002202 0.000132 0.006706 0.001181 0.008323 0.011552 0.001640 0.004406 0.000449 0.092503 0.093443
Also included in this table are differences in the RCI energies and the LS-coupling values of Bell et al. [5] and experimental energies of Wenaker [7].
RCI and CIV3 values of Bell et al. [5], as well as the experimental energies of Wenaker [7]. Though a direct comparison is not possible, it is seen that the present energy values agree very well with the experimental and CIV3 results, except for the states of 2p2(3P)3d configuration, and the agreement justifies the proper choice of orbital sets used in this work. The RCI energies and rates from 2p23s–2p23p, 2p3–2p23s, 2p23d and 2p23p–2p23d groups of transitions are discussed in Subsections 3.1–3.3, respectively. As our RCI energies for the above four groups of transitions are found to have a marginal difference with the earlier results, they are not listed separately in this work. The 2p23s–2p23p and 2p23p–2p23d transitions have been studied both experimentally and theoretically and data are available for many multiplets. For the 2p3–2p23s and 2p23d transitions, earlier data mainly correspond only to the theoretical values. The allowed dipole rates, absorption oscillator strengths and ratio of length to velocity rates for 2p23s–2p23p, 2p3–2p23s, 2p23d and 2p23p–2p23d transitions, respectively, are listed in Tables 2–4. Only those transitions with rates X105 s1 are reported in this work. Also included in these tables are the transition rates and oscillator strengths calculated by Bell et al. [5] in length gauge, the Rmatrix LS-coupling rates of Lennon and Burke [4] and the dipole rates from the NIST-ASD compilation [6]. The available experimental rates obtained by del Val et al. [2] and by Veres and Wiese [3] are also listed in these tables.
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Table 2 Dipole rates (Al) in 108 s1 and absorption oscillator strengths (fl) in length form for 2p23s–2p23p transitions Transition
4
P1/2–4D1/2 P1/2–4D3/2 4 P3/2–4D1/2 4 P3/2–4D3/2 4 P3/2–4D5/2 4 P5/2–4D3/2 4 P5/2–4D5/2 4 P5/2–4D7/2 4 P1/2–4P1/2 4 P1/2–4P3/2 4 P3/2–4P1/2 4 P3/2–4P3/2 4 P3/2–4P5/2 4 P5/2–4P3/2 4 P5/2–4P5/2 4 P1/2–4S3/2 4 P3/2–4S3/2 4 P5/2–4S3/2 2 P1/2–(3P)2D3/2 2 P3/2–(3P)2D5/2 2 P3/2–(3P)2D3/2 2 P3/2–(1D)2D5/2 2 P1/2–(1D)2D3/2 2 P3/2–(1D)2D3/2 2 P3/2–(1D)2D5/2 2 P1/2–(3P)2P1/2 2 P1/2(3P)2P3/2 2 P3/2–(3P)2P1/2 2 P3/2–(3P)2P3/2 2 P1/2–(1D)2P1/2 2 P1/2–(1D)2P3/2 2 P3/2–(1D)2P1/2 2 P3/2–(1D)2P3/2 2 P1/2–(1S)2P1/2 2 P3/2–(1S)2P3/2 2 P3/2–(3P)2S1/2 2 P1/2–(1S)2P3/2 2 P1/2–2S1/2 2 P3/2–2S1/2 2 D3/2–(3P)2P1/2 2 D3/2–(1D)2P1/2 2 D3/2–(1D)2P3/2 2 D5/2–(1D)2P3/2 2 D5/2–(3P)2P3/2 4
Av/Al
Al RCI
CIV3 Ref. [5]
0.718 0.379 0.129 0.435 0.623 0.034 0.226 0.851 0.152 0.403 0.827 0.160 0.269 0.419 0.715 0.327 0.675 1.055 0.805 0.999 0.190 0.498 0.398 0.088 0.540 0.869 0.249 0.453 1.021 0.249 0.106 0.130 0.311 0.331 0.045 0.201 0.008 0.111 0.201 0.001 1.254 0.088 1.176 0.007
0.720 0.382 0.127 0.432 0.626 0.033 0.222 0.849 0.149 0.395 0.826 0.162 0.262 0.420 0.715 0.288 0.595 0.937 0.777 0.926 0.147 0.543 0.458 0.082 0.542 0.857 0.214 0.438 1.075 0.375 0.085 0.193 0.481
Ref. [6]
Ref. [2]
Ref. [3]
Ref. [4]
0.695 0.351
0.142 0.368 0.795 0.153 0.248 0.392 0.675 0.225 0.564 0.897 0.716 0.847 0.137
0.621 0.340 0.122 0.376 0.543 0.030 0.245 0.718 0.135 0.340 0.764 0.144 0.234 0.366 0.635 0.261 0.534 0.937 0.655 0.767 0.126
0.694 0.349 0.137 0.441 0.586 0.040 0.246 0.833 0.164 0.412 0.807 0.130 0.296 0.430 0.676 0.300 0.593 0.874 0.793 0.952 0.155
0.421 0.076 0.498
0.099 0.182 0.002 1.228 0.142 1.093 0.002
1.22 0.141 1.09
0.835 0.411 0.965 0.166 0.260 0.358 0.771
0.856
0.081
0.188 0.393 0.956
0.182
0.589
0.894 0.225 0.437 1.10 0.83 0.80 0.87 0.80 0.91 0.76 1.15 0.71 1.14 1.15 1.90 1.00 1.11 1.00 0.85
0.99 1.00 0.99 1.00 1.00 0.99 0.99 1.00 1.00 1.01 1.00 1.00 1.04 1.00 1.01 0.88 0.85 0.85 1.00 0.97 1.01 1.01 0.77 0.83 0.70 0.99 0.99 1.02 1.00
fl RCI
CIV3 Ref. [5]
0.246 0.258 0.022 0.150 0.319 0.008 0.078 0.368 0.045 0.236 0.123 0.048 0.118 0.084 0.213 0.125 0.130 0.137 0.500 0.047 0.060 0.062 0.065 0.007 0.071 0.215 0.121 0.057 0.252 0.019 0.016 0.005 0.025 0.009 0.001 0.075 0.0005 0.080 0.075 0.004 0.149 0.021 0.173 0.008
0.234 0.247 0.021 0.141 0.306 0.007 0.073 0.367 0.042 0.221 0.117 0.046 0.110 0.080 0.203 0.119 0.124 0.132 0.455 0.041 0.044 0.073 0.081 0.007 0.073 0.201 0.100 0.052 0.260 0.030 0.014 0.008 0.038
0.062 0.066 0.062 0.003 0.142 0.033 0.167 0.004
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Table 2 (continued ) Transition
2
D3/2–(1D)2D3/2 D3/2–(1D)2D5/2 2 D5/2–(1D)2D5/2 2 D5/2–(1D)2D3/2 2 D5/2–(1D)2F5/2 2 D3/2–(1D)2F5/2 2 D5/2–(1D)2F7/2 2 S1/2–(1S)2P1/2 2 S1/2–(1S)2P3/2 2
Av/Al
Al RCI
CIV3 Ref. [5]
Ref. [6]
1.035 0.063 1.048 0.073 0.048 0.826 0.882 0.008 0.005
0.934 0.058 0.99 0.119 0.048 0.827 0.878
0.932 0.058 0.989 0.049 0.834 0.885
Ref. [2]
fl
Ref. [3]
Ref. [4]
RCI
CIV3 Ref. [5]
1.003
0.91 1.01 0.95 0.76 1.00 1.00 1.06 0.51 0.50
0.290 0.022 0.280 0.014 0.016 0.410 0.390 0.003 0.002
0.265 0.025 0.281 0.023 0.015 0.393 0.371
0.761
0.817 0.822
Table 3 Dipole rates (Al) in 108 s1 and absorption oscillator strengths (fl) in length form for 2p3–2p23s and 2p3–2p23d transitions Transition
Al
Av/Al
RCI
CIV3 Ref. [5]
Ref. [6]
P1/2–2P1/2 P1/2–2S1/2 2 P1/2–2P3/2 2 P1/2–2D3/2 2 P3/2–2P1/2 2 P3/2–2P3/2 2 P3/2–2S1/2 2 P3/2–2P3/2 2 P3/2–2D5/2 2 D3/2–2P1/2 2 D3/2–2P3/2 2 D3/2–2D3/2 2 D3/2–2D5/2 2 D5/2–2P3/2 2 D5/2–2D3/2 2 D5/2–2D5/2 4 S3/2–4P1/2 4 S3/2–4P3/2 4 S3/2–4P5/2
4.132 3.087 1.171 3.703 1.116 3.611 10.40 1.015 5.325 17.67 2.707 10.00 0.921 15.79 1.373 9.859 10.46 10.42 10.36
3.677 2.296 0.98 4.184 1.881 4.841 4.400 0.921 4.975 35.04 3.307 9.937 0.789 31.69 0.953 10.25 9.678 9.686 9.703
3.63 3.24 0.968 3.96 1.86 4.78 6.21 0.872 4.71 34.3 3.24 11.01 0.801 31.1 0.967 11.04 9.81 9.81 9.83
2p3–2p2(3P) 3d 4 S3/2–4P1/2 4 S3/2–4D1/2 4 S3/2–4P3/2 4 S3/2–4D3/2
30.67 4.035 29.52 1.035
40.63 5.214 39.44 6.353
2 2
fl RCI
CIV3 Ref. [5]
1.25 0.86 1.21 1.14 1.28 1.35 0.88 1.15 1.15 1.06 1.05 0.97 0.98 1.06 0.97 0.97 0.95 0.95 0.95
0.032 0.017 0.011 0.028 0.008 0.037 0.044 0.005 0.030 0.103 0.011 0.047 0.006 0.092 0.006 0.048 0.045 0.045 0.045
0.025 0.01 0.013 0.045 0.006 0.033 0.01 0.005 0.040 0.100 0.019 0.046 0.006 0.120 0.003 0.047 0.021 0.042 0.063
0.97 0.96 0.99 0.91
0.048 0.006 0.092 0.006
0.056 0.007 0.109 0.017
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Table 3 (continued ) Transition
4
S3/2–4P5/2 S3/2–4D5/2 2 D5/2–2P3/2 2 D5/2–2D3/2 2 D5/2–2F5/2 2 D5/2–2F7/2 2 D5/2–2D5/2 2 D3/2–2P1/2 2 D3/2–2P3/2 2 D3/2–2D3/2 2 D3/2–2F5/2 2 D3/2–2D5/2 2 P3/2–2P1/2 2 P3/2–2P3/2 2 P3/2–2D3/2 2 P3/2–2D5/2 2 P1/2–2P1/2 2 P1/2–2P3/2 2 P1/2–2D3/2 4
fl
Av/Al
Al RCI
CIV3 Ref. [5]
29.17 2.310 15.17 1.579 1.541 21.11 9.265 14.11 4.737 9.601 18.68 10.10 9.241 20.50 4.789 16.47 13.29 5.705 11.57
41.61 3.491 18.15 1.737 1.152 24.92 8.029 20.71 3.251 6.337 18.55
Ref. [6]
10.65 23.55 5.363 19.53 20.76 6.898 14.50
9.35 20.7 5.03 18.31 18.20 6.05 13.61
0.91 1.01 1.04 0.99 0.97 1.02 0.98 1.14 0.95 0.90 1.01 0.96 0.92 0.93 0.91 0.91 0.92 1.01 0.96
RCI
CIV3 Ref. [5]
0.137 0.010 0.033 0.004 0.004 0.121 0.031 0.026 0.017 0.029 0.102 0.047 0.020 0.075 0.020 0.071 0.054 0.044 0.104
0.173 0.015 0.043 0.004 0.004 0.117 0.028 0.036 0.011 0.022 0.098 0.021 0.095 0.021 0.117 0.084 0.056 0.116
Table 4 Dipole rates (Al) in 108 s1 and absorption oscillator strengths (fl) in length form for 2p23p–2p23d transitions Transition
4
D1/2–4P1/2 D1/2–4D1/2 4 D1/2–4F3/2 4 D1/2–4P3/2 4 D1/2–4D3/2 4 D3/2–4P1/2 4 D3/2–4D1/2 4 D3/2–4F3/2 4 D3/2–4P3/2 4 D3/2–4D3/2 4 D3/2–4F5/2 4 D3/2–4D5/2 4 D3/2–4P5/2 4 D5/2–4F3/2 4 D5/2–4P3/2 4 D5/2–4D3/2 4
Av/Al
Al RCI
CIV3 Ref. [5]
Ref. [6]
0.169 0.175 1.620 0.087 0.071 0.007 0.247 0.505 0.079 0.137 1.432 0.069 0.031 0.015 0.009 0.151
0.091 0.195 1.446 0.033 0.094 0.005 0.280 0.535 0.089 0.158 1.553 0.063 0.019 0.034 0.002 0.218
0.091 0.187 1.42 0.033 0.082 0.005 0.245 0.521 0.089 0.122 1.48 0.034 0.039 0.002 0.191
Ref. [2]
Ref. [3]
Ref. [4]
0.179 1.40
0.293 1.40
0.069
0.147
0.210 0.507 0.146
0.441
0.291 0.557
0.006
0.234 1.50 0.014
0.044
0.039
0.165
0.202
1.56 1.14 1.40 0.87 1.15 0.66 1.14 0.83 0.88 1.34 0.89 1.15 0.90 0.75 0.73 1.12
fl RCI
CIV3 Ref. [5]
0.037 0.043 0.621 0.036 0.048 0.008 0.021 0.171 0.006 0.033 0.436 0.025 0.010 0.003 0.002 0.022
0.020 0.043 0.719 0.015 0.042 0.0005 0.031 0.136 0.020 0.035 0.579 0.021 0.006 0.005 0.0003 0.033
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Table 4 (continued ) Transition
4
D5/2–4F5/2 D5/2–4P5/2 4 D5/2–4D5/2 4 D5/2–4F7/2 4 D5/2–4D7/2 4 D7/2–4F5/2 4 D7/2–4P5/2 4 D7/2–4D5/2 4 D7/2–4F7/2 4 D7/2–4D7/2 4 D7/2–4F9/2 4 P1/2–4P1/2 4 P1/2–4D1/2 4 P1/2–4P3/2 4 P1/2–4D3/2 4 P1/2–4F3/2 4 P3/2–4P1/2 4 P3/2–4D1/2 4 P3/2–4F3/2 4 P3/2–4P3/2 4 P3/2–4D3/2 4 P3/2–4F5/2 4 P3/2–4P5/2 4 P3/2–4D5/2 4 P5/2–4P3/2 4 P5/2–4D3/2 4 P5/2–4F3/2 4 P5/2–4P5/2 4 P5/2–4D5/2 4 P5/2–4D7/2 2 D3/2–2P3/2 2 D3/2–2D3/2 2 D3/2–2F5/2 2 D3/2–2D5/2 2 D5/2–2D3/2 2 D5/2–2F3/2 2 D5/2–2D5/2 2 D5/2–2F7/2 2 P3/2–2P3/2 2 P3/2–2D3/2 2 P3/2–2D5/2 2 P1/2–2P1/2 2 P1/2–2P3/2 2 P1/2–2D3/2 2 S1/2–2P1/2 4
Av/Al
Al RCI
CIV3 Ref. [5]
Ref. [6]
Ref. [2]
Ref. [3]
Ref. [4]
0.559 0.136 0.204 1.656 0.089 0.010 0.017 0.062 0.233 0.524 1.881 0.555 0.631 0.771 0.204 0.043 0.486 0.957 0.060 0.004 1.03 0.011 0.637 0.426 0.263 0.234 0.034 0.417 0.253 1.425 0.025 0.332 0.952 0.027 0.060 0.024 0.401 1.012 0.312 0.188 0.809 0.346 0.148 0.594 1.381
0.445 0.086 0.214 1.765 0.061 0.015 0.005 0.110 0.246 0.514 2.010 0.560 0.863 0.737 0.247
0.431 0.086 0.206 1.79 0.060 0.015 0.005 0.107 0.245 0.51 1.98 0.56 0.532 0.84 0.379
0.418
0.465
0.482
0.398 0.603
0.165 0.805
0.398 0.708
0.001 0.935 0.0002 0.664 0.474 0.239 0.260
0.038 0.955 0.0002 0.728 0.329 0.194 0.295
0.376 0.840
0.300 0.625 1.397 0.007 0.345 0.936 0.014 0.042 0.058 0.377 1.241 0.338 0.196 0.861 0.322 0.120 0.667 1.484
0.249 0.225 1.39 0.036 0.357
0.249 0.198 1.22
0.198 1.82 0.060 0.016 0.104 0.244 0.506 1.94
1.677
0.228 0.489 0.686
0.468 0.839 0.333
0.727 0.194
0.876
0.689
0.331 1.71 0.082 0.019 0.100 0.280 1.99 0.169 0.128 0.419 0.642 0.840 0.254 0.133 0.817
0.815 0.289 0.173
0.297 1.07 0.785
1.356
0.686 0.455 1.52
1.32 0.014 0.043 0.043 0.391 1.01 0.201 0.857
0.716 0.9
0.329 0.167 0.715 0.302 0.116 0.540 1.17
1.40 0.540 0.162 0.979 0.444 0.109 0.816 1.25
1.34 0.90 1.16 1.40 1.14 1.28 3.40 1.07 1.36 1.15 0.94 1.34 0.89 0.89 1.31 1.25 0.90 1.31 0.80 1.38 0.92 0.80 1.20 0.89 0.92 1.48 1.40 0.90 1.38 0.95 1.01 1.09 1.41 2.37 0.90 0.81 0.88 0.94 0.83 0.97 0.98 1.10 1.13 1.00 1.10
fl RCI
CIV3 Ref. [5]
0.137 0.030 0.045 0.541 0.020 0.002 0.001 0.011 0.058 0.114 0.002 0.168 0.154 0.383 0.099 0.024 0.060 0.117 0.004 0.001 0.251 0.004 0.237 0.154 0.044 0.037 0.011 0.104 0.230 0.350 0.009 0.081 0.601 0.012 0.012 0.015 0.113 0.546 0.061 0.082 0.470 0.151 0.143 0.432 0.302
0.111 0.020 0.048 0.585 0.018 0.003 0.001 0.019 0.062 0.116 0.006 0.143 0.218 0.378 0.125 0.051 0.076 0.0004 0.236 0.0001 0.258 0.180 0.043 0.044 0.078 0.159 0.474 0.002 0.099 0.465 0.006 0.008 0.020 0.108 0.549 0.137 0.072 0.473 0.129 0.097 0.489 0.254
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Table 4 (continued ) Transition
2
S1/2–2P3/2 S1/2–2D3/2 4 S3/2–4P1/2 4 S3/2–4D1/2 4 S3/2–4P3/2 4 S3/2–4D3/2 4 S3/2–4F5/2 4 S3/2–4P5/2 4 S3/2–4D5/2 2
Av/Al
Al RCI
CIV3 Ref. [5]
Ref. [6]
Ref. [2]
1.633 0.030 0.271 0.136 0.417 0.112 0.559 0.727 0.059
1.501 0.012 0.579 0.081 0.540 0.100
1.31 0.016
1.12
1.24
0.441
0.635
0.417
0.629
0.543 0.056
0.417
Ref. [3]
Ref. [4] 1.01 1.13 1.26 1.29 1.10 1.01 1.01 0.89 0.93
fl RCI
CIV3 Ref. [5]
0.637 0.003 0.090 0.034 0.233 0.038 0.136 0.037 0.017
0.518 0.004 0.104 0.014 0.196 0.036 0.030 0.030
3.1. 2p23p–2p23s transitions The RCI energies for the various multiplets from 2p23s–2p23p transitions (not listed in the tables) differ in general from the earlier experimental and theoretical values by 0.02–0.06 eV, except for the 2P–(1D)2D and 2P–(1D)2P transitions, with an energy difference of 0.1–0.2 eV. It may be noted that, in order to improve the agreement between experimental and theoretical data, Bell et al. used a set of semi-empirical energy corrections for the various LS blocks, and the same correction term was used for all the states in a particular block. The comparison of transition energies shows that the differences between the RCI and CIV3 energies are nearly the same as the semi-empirical correction values used by Bell et al., and we find that the present energies are generally in good agreement with the earlier values. It is seen from Table 2 that the RCI rates in length gauge agree well with the other theoretical and experimental dipole rates for 2p23s–2p23p transitions. The present rates are in excellent agreement with the CIV3 rates and differ only marginally for a few transitions. This marginal discrepancy is mainly due to the full use of relativity. The RCI rates differ by 1–2% from the Rmatrix LS-coupling rates for a number of transitions and deviate by 7–10% for some transitions. Taking into account the error estimates reported in the NIST-ASD compilations, it is found that the present rates are in good agreement with these rates. A comparison of the RCI rates with experimental rates reported by del Val et al. [2] and Veres and Wiese [3] shows that the present rates are slightly larger than the experimental rates for all the transitions considered in Table 1. The RCI rates are in better agreement with the rates obtained by del Val et al. than those of Veres and Wiese. 3.2. 2p3–2p23s and 2p23d The RCI energies (not listed in Table 3) for the fine structure lines from 2p3–2p23s transitions are more than the CIV3 energies for some transitions, and less for some other transitions, and the variation ranges from 0.01–0.12 eV. The RCI energies for 2P–2P and 2P–2D transitions differ
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from the CIV3 values by 0.1 and 0.02 eV, respectively. For the 2D–2D and 2D–2P transitions, the difference is o0.02 eV, whereas for 2P–2S transitions, RCI energies are more than the CIV3 energy values by 0.09 eV. The difference between the present energies and CIV3 values for 4S–4P transitions is 0.03 eV. In general, as seen in Section 3.1, the differences between the RCI energy values and CIV3 values of Bell et al. are nearly the same as the semi-empirical corrections used by them to match their values with the experimental results. A comparison of the RCI and CIV3 rates [5] listed in Table 3 for 2p3–2p23s transitions shows a significant variation in the dipole rates. For some transitions, fairly large deviations are noticed between the RCI and CIV3 rates. Similar variations are noticed between the present rates and NIST-ASD compilations. The RCI rates are larger than the earlier compilations for all dipole lines except for 2D–2P transitions, and differ in general by 9–20%. The present rates are nearly double, and half of the earlier rates, for 2P3/2–2S1/2 and 2D–2P transitions, respectively. The accuracy of the dipole rates is influenced both by the relative positions of the eigenvalues and the composition of the eigenvectors. As the difference in the RCI and CIV3 energies of 2D–2P transitions is very less, the large deviation in the dipole rates is mainly due to orbital functions, whereas, for the 2P–2S transitions, the significant variation in the transition rates is a combined effect of the large difference in energy values and eigenvectors. The present work makes full use of relativity, whereas, CIV3 calculations are known for better use of correlation effects and hence it is not possible to relate the discrepancies specifically to either of them. The RCI rates for a few strong lines arising from 2p3–2p2(3P)3d transitions are also listed in Table 3, along with the CIV3 rates of Bell et al. and values from NIST-ASD compilations. A comparison shows that the RCI rates are in reasonably good agreement with the other theoretical results for most of the transitions and the difference is appreciable for a few transitions. As the RCI average energies for the states of 2p23d configurations are only in fairly good agreement with the experimental and the semi-empirically adjusted CIV3 values, the differences in the RCI and CIV3 rates are due to the combined effect of transition wavelengths and correlation effects. 3.3. 2p23p–2p23d transitions The RCI energies for 2p23p–2p23d transitions also show a systematic variation from CIV3 results. The RCI energies for the 4D–4D, 4D–4P, 4P–4P and 4P–4D transitions differ from CIV3 values by 0.06 eV. For 4D–4F and 4P–4F transitions, the energy difference is 0.02–0.1 eV. The RCI energies for 2D–2D, 2D–2F transitions vary by 0.03 eV. The differences in the RCI and CIV3 transition energies are mainly the same as the semi-empirical scaling factor used to adjust the CIV3 values with the experimental energies. The RCI rates for some selected dipole lines arising from 2p2(3P)3p–2p2(3P)3d transitions of OII are reported in Table 4. The variations in the dipole rates reported in this work and other earlier results are not systematic. For some transitions, the RCI rates are larger than the other theoretical and experimental rates, and, for some other transitions, the present rates are smaller. The present rates for strong transitions agree well with CIV3 and NIST-ASD rates and vary by 2–10%. The RCI rates for most of the weak transitions encounter fairly large deviations from the earlier results. For the rest of the transitions, the differences vary from 2–20%. The RCI rates are generally in good agreement with the experimental rates of del Val et al. and Veres and Weise, except for 4P3/2–4P3/2 and 4D3/2–4D5/2 transitions, for which exceptionally large variations
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between RCI rates and the respective experimental rates of del Val et al. and Veres and Wiese are noticed. The RCI rates in length gauge for the various transitions reported in this work are generally in good agreement with the rates calculated in velocity gauge, except for a few weak lines. 4. Conclusion A comparison of the RCI rates for allowed dipole lines reported in Tables 2–4 and earlier results shows that the present rates are in very good agreement with each other for 2p23s–2p23p and vary on an average by 9–20% for 2p3–2p23s transitions. For 2p3–2p23d and 2p23p–2p23d transitions, the agreement is in general good. Though the present calculation is fully relativistic, the choice of excited orbitals included in this work might probably be responsible for the discrepancies between RCI and CIV3 rates, especially for 2p3–2p23d and 2p23p–2p23d transitions. Also, the differences between the RCI rates and earlier results might partly be due to the different ways in which the relativistic and correlation effects are calculated.
References [1] Decaux V, Beiersdorfer P, Kahn SM, Jacobs VL. High resolution measurement of the Ka spectrum of Fe XXV–XVIII: new spectral diagnostics of non-equilibrium astrophysical plasmas. Astrophys J 1997;482:1076–84. [2] del Val JV, Aparicio JA, Gonzalex VR, Mar S. Transition probability measurment of several OII spectral lines. J Phys B 2001;34:4531–8. [3] Veres G, Wiese WL. Experimental atomic transition probabilities for OII lines. Phys Rev A 1996;54:1999–2003. [4] Lennon DJ, Burke VM. 1991, unpublished. [5] Bell KL, Hibbert A, Stafford RP, MaLaughlin BM. Accurate transition probabilities for some spectral lines of singly ionized oxygen. Phys Scripta 1994;50:343–53. [6] NIST Atomic Spectra database data 2001. [7] Wenaker L. The spectrum of singly ionized OII. Phys Scripta 1990;42:667–84. [8] Jonsson P, He X, Froese Fischer C. 1998. The graspVU relativistic atomic structure package, (private communication). [9] Parpia FA, Froese Fischer C, Grant IP. GRASP92: a package for large-scale relativistic atomic structure calculations. Comput Phys Comm 1996;94:249–71. [10] Dyall KG, Grant IP, Johnson CT, Parpia FA, Plummer EP. GRASP: a general purpose relativistic atomic structure program. Comput Phys Comm 1989;55:425–56. [11] Grant IP. Relativistic atomic structure. In: Drake GWF, editor. Handbook in atomic, molecular and optical physics. New York: AIP; 1996. p. 268–86. [12] Froese Fischer C. Atomic structure: multiconfiguration Hartree-Fock theories. In: Drake GWF, editor. Handbook in atomic, molecular and optical physics. New York: AIP; 1996. p. 243–57. [13] Grant IP. Relativistic atomic structure calculations: methods in computational chemistry. In: Wilson S, editor. Relativistic effects in atoms and molecules, Vol. 2. New York: Phelom; 1988. [14] Natarajan L, Muley YG. Multi Configuration Dirac-Fock and configuration interaction calculations of Ar7+ to Ar17+ ions. J Phys B At Mol Phys 2001;34:1839–47. [15] Natarajan L. Spin-forbidden electric dipole transitions of highly ionized argon. J Phys B 2002;35:3179–87. [16] Anuradha N, Natarajan L. Kb satellites of highly ionized iron. J Phys B 2004;37:4789–801.