Branching ratios and lifetimes for the low-lying levels of Fe xx

Abstract

Transition energies, probabilities and branching ratios for electric dipole allowed (E1) and forbidden (M1, E2, M2) lines have been calculated for the 3s²3p⁵, 3s3p⁶ and 3s²3p⁴3d configurations of Fe x. From the transition probabilities, lifetimes of all 31 levels of these low-lying configurations are also derived, and compared with experiment. By applying systematically enlarged multiconfiguration Dirac-Fock wavefunctions, most important effects of relativity, correlation and the rearrangement of the electron density are treated within the same (computational) model.

atomic data

Introduction

Iron is one of the most abundant heavy elements in the Universe. It is created during the life cycle of stars and distributed by supernova explosions, so that it is already abundant by the second generation of stars. The emission and absorption spectra of multiply charged Fe ions have therefore been observed in a number of different astronomical objects and, in particular, in the extreme-ultraviolet spectrum of the Sun (Gabriel, Fawcett & Jordan 1965; Feldman, Doschek & Seely 1988; Shirai et al. 1990; Feldman et al. 1997). For Fe x, an important spectrum for astrophysical diagnostics, transitions arise from the low-lying excited levels of the 3s3p⁶ and 3s²3p⁴3d configurations. However, remarkable discrepancies between theoretical and experimental intensities have been found for this spectrum over the last three decades. For example, large deviations between theory and observations were obtained for the intensity ratios between the 3s²3p⁵²P_J-3s²3p⁴3d²S_1/2 doublet and the 3s²3p⁵²P_J-3s²3p⁴3d ²P, ²D transitions, as well as for a few other lines (Malinovsky & Heroux 1973; Nussbaumer 1976). Since that earlier time, therefore, several theoretical studies have attempted to improve the data base on transition probabilities and intensity ratios for Fe x (see Fawcett 1991 for a detailed bibliography). Over the years, these calculations have included both relativistic effects (often in the Breit-Pauli approximation) and configuration interactions to an increasingly large extent. By applying a few semi-empirical adjustments, in addition, Fawcett (1991) arrived at an improved data set for the strong E1 transitions of the 3s²3p⁵-3s3p⁶ and 3s²3p⁵−3s²3p⁴3d configurations. However, his work reports neither transition probabilities for the weak and forbidden transitions of these configurations, nor the lifetimes of these levels. Moreover, the discrepancy between the theoretical lifetimes for the (lowest) 3s3p⁶²S_1/2 even-parity level obtained by Fawcett and others, when compared with beam-foil measurements (Träbert 1996), could be resolved only very recently by using an extensive wavefunction expansion which takes into account the effects of relativity, correlation and the rearrangement of the bound electrons within the same computational model (Kohstall et al. 1999).

In this contribution, we present results of a large-scale computation on the electric-dipole allowed (E1) and forbidden (M1, E2, M2) transitions among the levels of the 3s²3p⁵ and the 3s3p⁶, 3s²3p⁴3d configurations for Fe⁹⁺ ions. We have applied relativistic multiconfiguration Dirac-Fock (MCDF) wavefunctions of elaborate size and incorporated relaxation effects by a level-dependent optimization of the wavefunctions for different groups of levels. This computational approach enables us to present a consistent and clearly improved data set of all important transitions of the Fe x spectrum, which might be useful in further astrophysical investigations.

Computational procedure

In this study, wavefunctions have been generated by the widely used atomic structure package grasp92 (Parpia, Froese Fischer & Grant 1996), which is an implementation of the MCDF method. In its revised 1996 version, this program enables large-scale relativistic computations as needed for most multiply charged ions with more than just a few electrons. However, only a brief description of the MCDF method and the generation of wavefunctions will be given here, since these aspects have been explained before in several publications (see Grant 1988; Fritzsche, Froese Fischer & Fricke 1998). In the MCDF method, an atomic state is approximated by a linear combination of configuration state functions (CSFs) of the same symmetry:  

(1)

where n_c is the number of CSFs and {c_r(α)} denotes the representation of the atomic state in this basis. In standard calculations, the CSFs are antisymmetrized products of a common set of orthonormal orbitals which are optimized on the basis of the Dirac-Coulomb Hamiltonian. Further relativistic contributions to the representation {c_r(α)} of the atomic states owing to (transverse) Breit interactions were also added by diagonalizing the Dirac-Coulomb-Breit Hamiltonian matrix. Moreover, we also estimated the dominant quantum electrodynamics (QED) contributions to the transition energies, but these corrections were found to be small and have not been included in the computations.

For chlorine-like ions, there are two odd-parity levels of the 3s²3p⁵ ground configuration with total angular momenta J=1/2 and 3/2, and 29 excited levels which belong to the 3s3p⁶ and 3s²3p⁴3d configurations. These excited levels have even parity and angular momenta in the range J=1/2, …hile all levels with J≤5/2 may decay by allowed E1 transitions to the ground configuration, the five J=7/2 levels are coupled only via magnetic quadrupole (M2) lines to the J=3/2 ground state, and the two J=9/2 levels are practically stable against a direct decay to the ground configuration. Both the J=7/2 and 9/2 levels may, however, decay by forbidden M1 and E2 transitions to lower levels of the same parity. Compared with the E1 allowed transitions, these M1, E2 and M2 forbidden transitions are suppressed by several orders of magnitude and, thus, finally result in lifetimes for the J=7/2 and 9/2 levels of a few (tens of) milliseconds.

To exploit the symmetry of the atomic levels in the generation of wavefunctions, we divided these levels into six groups, one for the ground-state levels and five other groups for each total angular momentum J=1/2, …, 9/2 of the excited levels. In relativistic computations, the different J symmetries are ‘block diagonal’ and, consequently, wavefunctions can most easily be obtained by an independent optimization procedure for each of these groups. In addition, this procedure has the advantage in studying transition probabilities that electron relaxation effects arising from the transitions are automatically included to a large extent. An independent variation of wavefunctions yields, however, orbital functions for each group of levels that are not quite orthogonal to the orbitals of any other group, and hence require some additional effort as we discuss below.

The calculation of accurate transition probabilities and lifetimes enforces, in particular for weak and forbidden lines, the inclusion of a large number of those correlation contributions even though they have only a negligible effect on the transition energies. These contributions can most easily be incorporated by configuration interaction, i.e. by allowing for virtual excitations of the electrons into (spectroscopically) unoccupied orbitals. In the past, various authors have applied and discussed different classes of excitations for studying the transition probabilities of Fe x (Glass 1983; Cowan, Bromage & Fawcett 1984; Fawcett 1991). A more systematic approach to the inclusion of electron-electron correlations applies the active space method whereby (large) lists of CSFs [cf. equation (1)] are generated by exciting electrons from the spectroscopically occupied configurations within an active set of orbital functions. Here, we will not discuss in detail such CSF expansions for different groups of levels, since it has become a standard procedure in both non-relativistic and relativistic structure calculations (Godefroid, Froese Fischer & Jünsson 1996; Fritzsche et al. 1998). In the present computations we included up to quadruple excitations (SDTQ) of all valence electrons outside the Ne-like core into the 3l subshells, as well as single (S) and double (D) excitations into the 4l and 5l shells. A very similar scheme of excitations was shown to be sufficient to incorporate all important valence-valence correlations for transitions along the silicon isoelectronic sequence (Kohstall et al. 1998). By including these classes of excitations for 3s²3p⁵ with respect to the 3s3p⁶ and 3s²3p⁴3d electron configurations, we finally arrived for the six groups of levels at wavefunction expansions of 4754 CSFs for the two ground-state levels and of 11-113 (J=1/2), 19-049 (J=3/2), 22-499 (J=5/2), 21-277 (J=7/2) and 16-973 (J=9/2) CSFs for the corresponding groups of excited levels.

Results and discussion

Results of our computations for the electric dipole allowed and forbidden transitions among the low-lying levels of Fe x are compiled below within five tables. Table 1 displays the level energies and lifetimes for all 31 levels of the 3s²3p⁵, 3s3p⁶ and 3s²3p⁴3d configurations. Excitation energies are given relative to the 3s²3p⁵J=3/2 ground state and are compared with experimental data from the National Institute of Science and Technology (NIST) Atomic Spectroscopic Database (Fuhr et al. 1998). In this table, we assign a level number in ascending order of energy to each level, which is used in the other tables to denote individual transitions. For the lower lying levels we find very good agreement with experiment, significantly better than obtained before in any ab initio investigation. As the excitation energies increase, then (as usual) the levels become slightly less well represented in the given configuration basis. For the highest levels, this results in a ‘systematic’ shift of the excitation energies of up to about 8000 cm⁻¹ which corresponds to an accuracy of about 1 per cent. As far as available, Table 1 also compares our theoretical lifetimes with experimental data by Träbert et al. (1993; see also Träbert 1996). It is clear that five of these lifetimes from beam-foil measurements (mostly for the lower levels) are in good agreement with the present calculation, whereas, for some of the longer lived levels, deviations of up to a factor of 2 or more occur. So far, however, experimental lifetimes are indeed scarce, and further investigations are needed to understand these deviations.

Table 1.

Excitation energies and lifetimes for the low-lying levels of the 3s²3p⁵, 3s3p⁶ and 3s²3p⁴3d configurations for Fe x.

The separate optimization of the wavefunctions for each group of levels (see above) allowed us to incorporate the main relaxation effects into our (theoretical) transition probabilities and lifetimes. Although the importance of the rearrangement of the electron density owing to an absorption or emission of photons is now well-established (Fritzsche & Grant 1994; Fritzsche et al. 1995; Kohstall et al. 1998), these effects have so far been included only in a few selected case studies of transition probabilities (and often only for a few individual lines) since, in practice, they require the inclusion of a large number of non-vanishing overlaps of orbital functions of the same symmetry. To facilitate the incorporation of these effects for larger arrays also, we have recently developed two new program modules (Fritzsche & Grant 1997; Fritzsche & Froese Fischer 1997) which exploit the relativistic wavefunctions from grasp92 (Parpia et al. 1996). Such a relaxed orbital approach is computationally expansive compared with an orthogonal orbital basis; however, it often leads to a considerably better agreement of transition probabilities and lifetimes for different gauges of the radiation field and also with experiment. In relativistic calculations, two gauges that are frequently applied to the transition operators are the Babushkin and Coulomb gauges which, in the non-relativistic limit, correspond to the length and velocity forms of these operators. Even though an agreement of transition probabilities and lifetimes from different gauges cannot prove any individual result, their averaged deviation, if taken for a whole transition array of several tens of hundreds of lines, certainly provides insight into the quality of the approximation. For this reason, we display the E1 and E2 transition probabilities in both gauges, even though (at least for E1 transitions) the results in length gauge are usually considered to be more reliable.

Table 2 lists the transition energies, probabilities and oscillator strengths of the (allowed) E1 transitions of the 3s²3p⁵−3s3p⁶ and 3s²3p⁵−3s²3p⁴3d configurations. While transition probabilities are presented in length and velocity gauge, the weighted oscillator strengths are shown only in length gauge. To facilitate the identification of these lines and of lifetime measurements, we also show the branching ratio of each line for the upper level in the last column (‘br’). As seen from this table, remarkably good agreement for the probabilities in length and velocity gauge is obtained for (almost) all medium and strong lines.

Table 2.

E1 transition energies, emission rates and oscillator strengths for the 3s²3p⁵−3s3p⁶ and 3s²3p⁵−3s²3p⁴3d lines of Fe x. The level numbers of the lower and upper levels refer to Table 1. The weighted oscillator strength (gf) is displayed in length gauge only; br denotes the branching fraction of this line for the corresponding upper level.

A brief comparison of our calculation of transition probabilities with previous computations by Huang et al. (1983); Fawcett (1991); Bhatia & Doschek (1995) is shown in Table 3 for the E1 decay of the six excited levels with J=1/2. Generally, our results agree well with the semi-empirical calculations by Fawcett (1991) as far as data are available from this reference. Recently, Bhatia & Doschek (1995) presented a revised set for the E1 transition data of Fe x which, however, differs considerably from Fawcett's and ours, and thus seems to us to be less reliable. For most weaker lines (not included by Fawcett), we consider neither the calculations by Huang et al. nor those by Bhatia & Doschek as accurate, because of the small configuration expansions that were used by those authors. In contrast, the systematically enlarged wavefunction in the present case should be flexible enough to include the important correlations and to provide clearly improved data for weak E1 lines and, in fact, even for the electric dipole forbidden transitions (see below). In a number of test calculations, moreover, we analysed the influence of the different classes of excitations into the 3l and 4l orbital layers, and found that triple (T) and quadruple (Q) excitations of the valence electrons into the 3p and/or 3d shells are not negligible, particularly for weak transitions. From the comparison in Table 3 and by considering the agreement of our results from different gauges, we estimate the transition probabilities in Table 2 for the strong E1 lines to be accurate to within about 5-15 per cent, despite the larger discrepancies with some of the lifetime measurements, which correspond, however, to weaker lines.

Table 3.

Calculated E1 transition probabilities (in length gauge) for a few selected 3s²3p⁵−3s3p⁶ and 3s²3p⁵−3s²3p⁴3d transitions, compared with previous computations.

The 3s²3p⁴3d excited levels with J=7/2 and 9/2 may decay by magnetic dipole (M1) and electric quadrupole (E2) transitions to lower lying levels of the same symmetry. The corresponding transition energies and probabilities are displayed in Table 4 along with the oscillator strengths and branching ratios. Only transitions with probabilities A>10⁻¹ s⁻¹ have been included in this table. As seen from the table, larger deviations than for the E1 lines of up to a factor of 3 occur for the E2 transition probabilities calculated in different gauges. For the J=7/2 levels, furthermore, forbidden M2 transitions to the 3s²3p⁵J=3/2 ground state also appear (Table 5), which are typically of the same order of magnitude as the competitive decay to levels of the same parity. Although there are no experimental data available so far with which to compare our calculations for these forbidden lines, Tables 4 and 5 may support the future identification of those lines, for instance from the solar spectrum. For the M2 transitions of the J=7/2 levels to the ground state, our probabilities agree to within 30 per cent with earlier calculations by Mason & Nussbaumer (1977).

Table 4.

M1 and E2 transition energies, emission rates and oscillator strengths for lines of the 3s²3p⁵ and 3s²3p⁴3d configurations of Fe x. The notation is the same as in Table 2.

Table 5.

M2 transition energies, emission rates and oscillator strengths for the 3s²3p⁵-3s²3p⁴3d lines of Fe x. The notation is the same as in Table 2.

In conclusion, we have calculated the electric dipole allowed and forbidden transitions among the 31 low-lying levels of Fe x ions. A detailed and consistent data set is presented which we expect to be accurate enough to support further line identifications for these ions. Apart from the E1 resonance and intercombination lines of the 3s3p⁶ and 3s²3p⁴3d configurations, the decay branches of the metastable J=7/2 and 9/2 levels have been analysed. Our study demonstrates the capability of relativistic ab initio calculations for rather complex, open-shell ions if electron correlations can be included sufficiently. It also confirms that typically excitations into (at least) two additional layers of orbitals are needed to arrive at satisfactorily converged transition probabilities and lifetimes. For many ions, in particular those having more than two electrons in an open d- or f-shell, this requirement is on the limit or even beyond of present-day computer technology.

Acknowledgments

We thank Dr E. Träbert for helpful discussions and suggestions regarding the manuscript. CZD is grateful for the support of the National Nature Science Foundation of China under contract No. 19874051. This work has been supported by the Deutsche Forschungsgemeinschaft (DFG) in the framework of the Schwerpunkt ‘Wechselwirkung intensiver Laserfelder mit Materie’.