Energy levels, radiative rates and electron impact excitation rates for transitions in Si ii

Abstract

1 INTRODUCTION

Emission lines of Si ii have been observed at optical and ultraviolet (UV) wavelengths in a variety of plasmas, such as planetary nebulae, quasars and the interstellar medium (see e.g. Judge, Carpenter & Harper 1991). Particularly useful for diagnostic purposes are the multiplets 3s² 3p ²P^o–3s 3p^2 4P and 3s² 3p ²P^o–3s 3p^2 2D at around 2340 and 1810 Å, respectively. Similarly, Hubrig & González (2007) have detected two emission lines (∼7849 Å) in the magnetic Bp star a Centauri (HD 125823). Recently, Shaltout et al. (2013) have determined the silicon abundance in the solar atmosphere, based on lines of Si ions, including those of Si ii. Many lines of Si ii in the 750–6680 Å wavelength range are listed in the CHIANTI data base at http://www.chiantidatabase.org/ and the Atomic Line List (v2.04) of Peter van Hoof at http://www.pa.uky.edu/peter/atomic/. Silicon ions, including Si ii, are also important for the studies of fusion plasmas, particularly because amorphous silicon is used for coating the first wall of the devices, such as Tokamak Experiment for Technology Oriented Research (TEXTOR). Huber et al. (2003) have measured intensities of several Si ii lines in the TEXTOR tokamak in the 290–640 Å wavelength range, belonging to the n ≤ 5 levels. The importance of data for Si ions has further increased with the developing International Thermonuclear Experimental Reactor (ITER) project.

For plasma diagnostics and modelling, atomic data are required for a range of parameters, particularly energy levels, radiative rates (A values) and excitation rates (or equivalently the effective collision strengths ϒ). Measured values of energy levels have been compiled by the National Institute of Standards and Technology (NIST) team (Kramida et al. 2013) and are available at their website http://www.nist.gov/pml/data/asd.cfm. Theoretical energy levels have been determined by several authors, and the most notable results are those of Tayal (2007) and Bautista et al. (2009). Both these workers have also listed the A values.

Results for collision strengths (Ω) and effective collision strengths (ϒ) are also available. Dufton & Kingston (1991) have reported ϒ data for transitions among the lowest seven levels of the 3s² 3p and 3s 3p² configurations and from the 3s² 3p ²P|$^o_{1/2,3/2}$| ground state levels to higher excited ones, up to 15 (see Table 1). These limited results are based on the R-matrix method and are primarily in LS coupling (Russell–Saunders or spin–orbit coupling), but corresponding data for fine-structure transitions were determined through algebraic transformation. Although calculations for Ω were performed up to an energy of 10 Ryd, which is fully sufficient to determine ϒ up to T_e = 10^4.6 K, the range of partial waves adopted by them (J ≤ 6) was too limited to obtain convergence of Ω, not only for the allowed but also the forbidden transitions. Furthermore, Judge et al. (1991) analysed several observations using their data and noted discrepancies with theory by up to a factor of 2 for several lines. On the other hand, Baldwin et al. (1996) studied broad emission lines of high-luminosity quasi-stellar objects (QSOs), including some of Si ii, particularly λ ∼ 1263 Å (3s² 3p ²P^o–3s² 3d ²D), 1307 Å (3s² 3p ²P^o–3s 3p²²S) and 1814 Å (3s² 3p ²P^o–3s 3p²²D). For the 1814 Å line there was a satisfactory agreement between prediction and observation, but the discrepancies for the other two lines were over an order of magnitude. Therefore, there was a clear need to re-examine the theoretical atomic data.

Subsequently, Tayal (2008) made significant improvements over the atomic data of Dufton & Kingston (1991), mainly by extending the range of partial waves up to angular momentum J = 36. Furthermore, he performed calculations in the Breit–Pauli B-spline R-matrix (BSRM) approach and reported ϒ not only over a wider temperature range (up to log T_e = 5.4 K) but also for all transitions among 31 levels of the 3s² 3p, 3s 3p², 3s 3p 3d, 3s² 4ℓ, 3s² 5s/p/d/f and 3s² 6s/p configurations. As a result of these improvements the differences between his values of ϒ and those of Dufton & Kingston (1991) were significant for some transitions, in both magnitude and behaviour, particularly the allowed ones, such as 1–12 (3s² 3p ²P|$^o_{1/2}$|–3s² 3d ²D_3/2) and 2–13 (3s² 3p ²P|$^o_{3/2}$|–3s² 3d ²D_5/2), as shown in his fig. 6. However, Tayal included only two levels (²D|$^o_{3/2,5/2}$|⁠) of the 3s 3p 3d configuration, whereas it generates 23 in total (see Table 1), and some of these lie in between those of the 3s² 4ℓ and 3s² 5ℓ configurations, included in his calculations. The omission of these levels affects the resonance structure of Ω and subsequently calculations of ϒ. Therefore, there is a scope for improvement as well as confirmation of accuracy of the Tayal results, so that data can be confidently applied to plasma studies.

Bautista et al. (2009) performed another calculation adopting the Breit–Pauli R-matrix code of Berrington, Eissner & Norrington (1995). Although a large number of (43) LS terms were included in the collisional calculations, they reported values of ϒ only at three temperatures and for eight transitions from the ground level. More importantly, discrepancies with the corresponding results of Dufton & Kingston (1991) as well as Tayal (2008) are up to 50 per cent for several transitions. In general, there is poor agreement among the three R-matrix calculations. Bautista et al. (2009) speculated that differences with the calculations of Tayal (2008) could be because of the non-orthogonal orbitals adopted by him. However, Dufton & Kingston (1991) adopted orthogonal orbitals, as did Bautista et al. (2009), but differences between the two sets of ϒ are still up to 50 per cent at all temperatures between 5000 and 20 000 K. A closer examination of the collision strengths shown by Tayal (2008) and Bautista et al. (2009) for three transitions, namely 1–2 (3s² 3p ²P|$^o_{1/2}$|–3s² 3p ²P|$^o_{3/2}$|⁠), 2–4 (3s² 3p ²P|$^o_{3/2}$|–3s 3p²⁴P_1/2) and 2–7 (3s² 3p ²P|$^o_{3/2}$|–3s 3p²²D_5/2), reveals that background values of Ω_B are lower in the latter's calculations, particularly at energies below 0.8 Ryd (equivalent to ∼1.25 × 10⁵ K). These differences clearly lead to the lowest values of ϒ reported by Bautista et al. (2009) for all transitions and at all temperatures.

Bautista et al. (2009) included a larger range of partial waves (L ≤ 16) in comparison to those of Dufton & Kingston (1991), who included only L ≤ 8, and yet their values of ϒ are the lowest, not only for the allowed (such as 1–3, 4 and 6) but also the forbidden (1–2 and 1–5) transitions. Therefore, neither the range of partial waves nor the use of (non)-orthogonal orbitals appears to be a convincing cause of the discrepancies in ϒ. For this reason we have performed another calculation for the important Si ii ion and report a complete set of results for energy levels, A values and ϒ for all transitions among 56 levels of the 3s² 3p, 3s 3p², 3p³, 3s² 3d, 3s 3p 3d, 3s² 4ℓ and 3s² 5ℓ configurations.

2 ENERGY LEVELS

To calculate energy levels and A values we have employed the multiconfiguration Dirac–Fock (mcdf) code, developed by Grant et al. (1980). It is a fully relativistic code, based on the jj coupling scheme, and includes higher order relativistic corrections arising from the Breit (magnetic) interaction and quantum electrodynamics (QED) effects (vacuum polarization and Lamb shift). This code has undergone several revisions by the names General-purpose Relativistic Atomic Structure Package (grasp), grasp92 and grasp2K, revised by Dyall et al. (1989), Parpia et al. (1989) and Jönsson et al. (2007, 2013), respectively. However, the version adopted here has been revised by one of its authors (Dr P. H. Norrington) is known as grasp0 and is freely available at the website http://web.am.qub.ac.uk/DARC/. This provides comparable results with other revised versions and has been extensively applied by ourselves and other workers to a wide range of ions. Furthermore, as in our earlier work, the option of extended average level (EAL) has been adopted in which a weighted (proportional to 2j+1) trace of the Hamiltonian matrix is minimized. We note that results obtained for energy levels and A values are comparable to other options, such as average level (AL), as demonstrated by Aggarwal, Keenan & Lawson (2008) for Kr ions.

For the determination of atomic structure we have included 14 configurations, namely (1s² 2s² 2p⁶) 3s² 3p, 3s 3p², 3p³, 3s² 3d, 3s 3p 3d, 3s² 4ℓ and 3s² 5ℓ, which generate 56 levels listed in Table 1. Our calculated level energies, obtained with the inclusion of Breit and QED effects, are given in Table 1 along with the NIST compilations of experimental energies and those of Tayal (2008) who adopted the multiconfiguration Hartree–Fock (mchf) code of Zatsarinny & Fischer (1999). Since Si ii is only moderately heavy, the contributions of Breit and QED effects are negligible. However, differences with the NIST listings are up to ∼0.1 Ryd for some of the levels, particularly 3s 3p²²P_{1/2, 3/2} (14 and 15). Additionally, the orderings are also slightly different in a few instances – see for example levels 17–22. We note here that for some species, such as Al-like Ti X (Aggarwal & Keenan 2013), it is not possible to unambiguously identify the levels, because of strong mixing. However, that is not the case for Si ii. The energies obtained by Tayal (2008) are in closer agreement with those of NIST for all levels, because he has used non-orthogonal orbitals. However, his energy level ordering differs with that of NIST in a few instances, such as for 6–7 and 21–22. Furthermore, using the similar non-orthogonal orbitals, in his earlier calculations (Tayal 2007) the energy obtained for the 3s² 3p ²P|$^o_{3/2}$| level is lower by 13 per cent. Nevertheless, a clear advantage of this approach is that orbitals can be independently optimized on individual states/levels and thus a higher accuracy can be achieved by minimizing the differences with the measured results. However, this approach can only be successfully applied if experimental energies are already available, which is the case for Si ii, but not for all levels, because several are missing from the NIST compilations as seen in Table 1. Since Tayal (2008) performed collisional calculations among experimentally determined levels only, he could apply the approach of non-orthogonal orbitals. Nevertheless, most of the scattering codes, including the one adopted here and discussed in Section 5, do not have a provision of this option.

A most commonly and widely used methodology for an accurate determination of energy levels (and hence subsequent other parameters) is the inclusion of configuration interaction (CI). This approach has been extensively applied to a wide range of ions by many workers, including ourselves for other Si ions (Aggarwal & Keenan 2010). Therefore, to assess the effect of additional CI we have performed further calculations with the Flexible Atomic Code (fac) of Gu (2008). This code, apart from being highly efficient, provides results for energy levels and A values of comparable accuracy with other atomic structure codes, such as configuration interaction version 3 (civ3) of Hibbert (1975) and grasp, as shown by Aggarwal et al. (2007) for three Mg-like ions.

We have performed three calculations with the fac code with increasing amount of CI, namely (i) fac1, which includes the same configurations/levels as in grasp; (ii) fac2, which includes all possible combinations of the 3ℓ orbitals (3*3) plus 3s² 4ℓ and 3s² 5ℓ, generating 164 levels in total and finally (iii) fac3, which includes a further 11 levels of 3s² 6ℓ, because some of these intermix with those of fac2. Energies obtained from these three calculations are listed in Table 1 for comparison with other results.

Our fac1 energies agree closely with those from grasp (within 0.03 Ryd) and the ordering is also similar, although a few differ slightly, such as levels 17–22. We also note that small discrepancies in the grasp and fac energies mainly arise due to the different ways the calculations of central potential for radial orbitals and recoupling schemes of angular parts are performed – see the detailed discussion in the fac manual (http://sprg.ssl.berkeley.edu/∼mfgu/fac/). Such small differences have also been noted for several other ions. Inclusion of larger CI in fac2 lowers the energies, by up to 0.07 Ryd, for some of the levels, such as 12–15, but differences with the NIST compilations remain of up to ∼0.1 Ryd. Finally, energies obtained in fac3 are comparable (within 0.01 Ryd) to those from fac2 indicating that the inclusion of the 3s² 6ℓ configuration has little effect on the energy levels of Table 1. Furthermore, the energy for level 2 (3s² 3p ²P|$^o_{3/2}$|⁠) has become worse than in fac1 or grasp, and hence there in no overall advantage of including more CI than that already considered in grasp. We discuss this further below.

Bautista et al. (2009) have performed a series of calculations with the atomic structure (as) code of Badnell (1997). Apart from adopting different optimization procedures (including the minimization with observed energies) they have included extensive CI with up to n = 5 orbitals in 44 configurations. They have also opened the n = 2 shell (frozen in our work with grasp and fac) to account for the core–valence correlation. However, none of their nine calculations yielded accurate energies for the lowest 15 levels. Differences with the measurements for all sets of energies are over 15 per cent for some levels or others, as shown in their table 2. It is clear that inclusion of a large amount of CI is not helpful for the accurate determination of Si ii energy levels.

Apart from the inclusion of CI, another possibility of improving the accuracy of wave functions is to add correlation effects through pseudo-orbitals, i.e. all orbitals need not be spectroscopic as has been the case in the calculations described above with the grasp, fac and as codes. This is a normal practice in standard R-matrix calculations for Ω, for which input wave functions are generated through the civ3 code, but all orbitals are orthogonal. Therefore, Dufton et al. (1983) adopted this approach but their energy levels still differ by up to 6 per cent with the measurements. Additionally, a disadvantage of this approach is the presence of unphysical pseudo-resonances, because the corresponding eigenstates are not included in the calculations of Ω (see e.g. fig. 1 of Aggarwal & Hibbert 1991). If these pseudo-resonances are not properly removed then the subsequent results for ϒ may be significantly overestimated (see e.g. figs 1–6 of Aggarwal et al. 2000). Therefore, scope remains for improvement over our energy levels listed in Table 1, but keeping in mind our further calculations for more important parameters (Ω and ϒ), the accuracy achieved should be satisfactory for a majority of the levels.

Figure 1.

Partial collision strengths at three energies of 2 Ryd (circles), 6 Ryd (triangles) and 12 Ryd (stars) for three transitions of Si ii, namely (a) 1–12 (3s² 3p ²P|$^o_{1/2}$|–3s² 3d ²D_3/2), (b) 2–13 (3s² 3p ²P|$^o_{3/2}$|–3s² 3d ²D_5/2) and (c) 2–15 (3s² 3p ²P|$^o_{3/2}$|–3s 3p²²P_3/2).

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3 RADIATIVE RATES

In Table 2 we list transition (energies) wavelengths (λ, in Å), radiative rates (A_ji, in s⁻¹), oscillator strengths (f_ij, dimensionless) and line strengths (S, in a.u.) for all 460 electric dipole (E1) transitions among the 56 levels of Si ii. The A, f and S values have been calculated in both Babushkin and Coulomb gauges, i.e. the length and velocity forms in the widely used non-relativistic nomenclature. However, in Table 2 results are listed in the length form alone, because the velocity form is generally considered to be comparatively less accurate. Nevertheless, we will discuss later the velocity/length form ratio, as this provides some assessment of the accuracy of the results. Also note that the indices used to represent the lower and upper levels of a transition correspond to those in Table 1. Apart from the above E1 transitions, there are 733 electric quadrupole (E2), 567 magnetic dipole (M1) and 574 magnetic quadrupole (M2) transitions among the same 56 levels. However, for these only the A values are listed in Table 2, because these are the ones required for plasma modelling. The corresponding results for f values can be easily obtained through equation (1). Similarly, if required, corresponding S values can be obtained through equations (2)–(5) of Aggarwal & Keenan (2012).

In Table 3 we compare our f values from grasp with those of Tayal (2007) and Bautista et al. (2009) for transitions among the lowest 30 levels of Table 1. Bautista et al. (2009) have listed several sets of f values, but those included in Table 3 correspond to their ‘recommended’ results, which are based on the averages of a variety of theoretical and experimental (to be discussed later) values. These authors reported f values only for transitions among the lowest 15 levels, but many are missing from the work of Tayal (2007). Most of the missing transitions are weak (i.e. f ∼ 10⁻⁵ or even less), but some are rather strong, such as 6–22 (f = 0.125), 7–21 (f = 0.120) and 12–22 (f = 0.734). This is because he included the 3s² 4f (21–22) levels in his calculations for collisional data (Tayal 2008), but not for the radiative rates. Nevertheless, the absence of f (or A) values for these transitions may affect the modelling of plasmas.