Single photoionization of the Kr-like Rb ii ion in the photon energy range 22–46.0 eV

Abstract

Single photoionization cross-sections for Kr-like Rb⁺ ions are reported in the energy (wavelength) range from 22 (564 Å) to 46 eV (270 Å). Theoretical cross-section calculations for this trans-Fe element are compared with measurements from the ASTRID radiation facility in Aarhus, Denmark and from the dual laser plasma technique, at respectively 40 and 35 meV FWHM energy resolution. In the photon energy region 22–32 eV the spectrum is dominated by excitation auto-ionizing resonance states. Above 32 eV the cross-section exhibits classic Fano window resonances features, which are analysed and discussed. Large-scale theoretical photoionization cross-section calculations, performed using a Dirac Coulomb R-matrix approximation are benchmarked against these high-resolution experimental results. Comparison of the theoretical work with the experimental studies allowed the identification of resonance features and their parameters in the spectra in addition to contributions from excited metastable states of the Rb⁺ ions.

1 INTRODUCTION

Rb i optical pumping lines have been observed in AGB stars, and both isotopes of Rb i (⁸⁵Rb i and ⁸⁷Rb i) are present based on the presence of s-process elements such as Zr (Darling 2018). A major source of discrepancy is the quality of the atomic data used in the modelling (Sneden, Gratton & Crocker 1991; Mishenina et al. 2002; Roederer et al. 2010; Roederer, Marino & Sneden 2011; Frebel, Simon & Kirby 2014).

The motivation for this study of the trans-Fe element, Rb ii, is to provide benchmark photoionization (PI) cross-section data for applications in astrophysics. High-resolution measurements of the photoionization cross-sections of Rb⁺ were recently performed at the Advanced Light Source (ALS) synchrotron radiation facility in Berkeley, California (Mueller et al. 2013), over the photon energy range 22–46 eV at a resolution of 18 meV FWHM. Many excited Rydberg states have been identified in the energy (wavelength) range from 22 (564 Å) to 46 eV (270 Å). Large-scale darc PI cross-section calculations when compared with previous synchrotron radiation (SR) and dual laser plasma (DLP) experimental studies (Kilbane et al. 2007) indicate excellent agreement. Such comparisons give confidence in our theoretical data for use in astrophysical applications.

To the authors knowledge, we are aware only of experimental and theoretical work on PI for this trans-Fe element, Rb ii, performed recently at the ALS (Mueller et al. 2013), the joint study by Kilbane et al. (2007), using a merged-beams technique by combining SR with a beam of Rb⁺ (Rb ii) ions at the ASTRID synchrotron radiation facility, the DLP technique, at Dublin City University (DCU), and the theoretical study of Demekhin et al. (2007). This updated earlier work carried out using the DLP technique by Neogi et al. (2003).

2 THEORY

2.1 Atomic structure

The grasp code (Dyall et al. 1989; Parpia, Froese-Fisher & Grant 2006; Grant 2007) generated the target wave functions employed in this work. All orbitals were physical up to n = 3, 4s, 4p, and 4d. We initially used an extended averaged level (EAL) calculation for the n  = 3 orbitals. The EAL calculations were performed on the lowest 13 fine-structure levels of the residual Rb iii ion. In our work we retained all the 456 – levels originating from one, two and three – electron promotions from the n = 4 levels into the orbital space of this ion. All 456 levels arising from the six configurations were included in the darc close-coupling calculation, namely: 3s²3p⁶3d¹⁰4s²4p⁵, 3s²3p⁶3d¹⁰4s4p⁶, 3s²3p⁶3d¹⁰4s²4p⁴4d, 3s²3p⁶3d¹⁰4s4p⁵4d, 3s²3p⁶3d¹⁰4s²4p³4d², and 3s²3p⁶3d¹⁰4s²4p²4d³.

Table 1 shows the theoretical energy levels from the 456-level grasp calculations for the lowest 13 levels of the residual Rb²⁺ ion, compared to the values available from the NIST tabulations (Kramida et al. 2018). The average percentage difference of our theoretical energy levels compared with the NIST values is approximately 3 per cent higher.

Photoionization cross-sections calculations were carried out on the Rb ii ion for the 3d¹⁰4s²4p^6 1S₀ ground state, 3d⁹4s²4p⁵5s ³P_2,1,0, 3d⁹4s²4p⁵5s ¹P₁, and the 3d⁹4s²4p⁵4d ³P_2,1,0 metastable levels using the darc codes.

2.2 Photoionization calculations

The scattering calculations were performed for photoionization cross-sections using the above large-scale configuration interaction (CI) target wavefunctions as input to the parallel darc suite of R-matrix codes. The latest examples of the darcR-matrix method, implemented in our parallel suite of codes to predict accurate photoionization cross-sections are the recent experimental and theoretical studies on the Zn ii trans-Fe ion by Hinojosa et al. (2017) and the Ca ii ion by Müller et al. (2017).

We used 16 continuum orbitals in our scattering calculations. A boundary radius of 9.82 a₀ was necessary to accommodate the diffuse n  = 4 bound state orbitals of the residual Rb iii ion. To resolve all the fine resonance features in the spectra, we used an energy grid of 2 × 10⁻⁷|${\cal {Z}}^2\,\,Ry$| (13.6 |$\mu$|eV), where |$\cal {Z}$|  = 2, in our calculations.

Photoionization cross-section calculations with this 456-level model were carried out using the above energy mesh for the 3d¹⁰4s²4p^6 1S₀ ground state, the 3d⁹4s²4p⁵5s ³P^o_{2, 1, 0}, 3d⁹4s²4p⁵5s ¹P^o₁, and 3d⁹4s²4p⁵4d ³P^o_2,1,0 metastable levels of this ion, over the photon energy range similar to experimental studies.

For the ¹S₀ ground state level we require only the bound-free dipole matrices, J^π  = 0^e, → J|$^{{\prime }\pi ^{\prime }}$|  = 1^o. In the case of the metastable levels we required the following, J^π  = 2^o, → J|$^{{\prime }{\rm \pi }^{\prime }}$|  = 1^e, 2^e, 3^e, for J^π  = 1⁰, → J|$^{{\prime }{\rm \pi }^{\prime }}$|  = 0^e, 1^e, 2^e and J^π  = 0^o, → J|$^{{\prime }{\rm \pi }^{\prime }}$|  = 1^e, bound-free dipole matrices. The Hamiltonian diagonal matrices were shifted to the recommended experimental NIST tabulated (Kramida et al. 2018) values. Such an energy adjustment provides better positioning of resonances energies relative to all thresholds.

2.3 Resonances

The NIST tabulations (Kramida et al. 2018) for the Rb ii energy levels were a helpful guidance for the present resonance assignments. The time delay of the S-matrix method (Smith 1960), applicable to atomic and molecular complexes, for locating narrow resonances, as developed by Berrington and coworkers (Quigley & Berrington 1996; Quigley, Berrington & Pelan 1998; Ballance, Berrington & McLaughlin 1999) was used to locate and determine the resonance positions. This procedure was used in our recent study on the trans-Fe element Zn ii (Hinojosa et al. 2017) to locate all the resonances in the spectra.

3 RESULTS AND DISCUSSION

The 456-level darc PI calculations are shown in Fig. 1. to illustrate the single photoionization cross-sections contributions from both the ground and the metastable states in the region from threshold to 45 eV, for the Rb⁺ ions in their 4s²4p^6 1S₀ ground state, the 4s²4p⁵5s ³P_2,1,0, 4s²4p⁵5s ¹P₁, and the 4s²4p⁵4d ³P_0,1,2 metastable states. We statistically averaged the contribution from the metastable levels and then use a best fit with the experimental measurements to identify the metastable content in the beam.

In Fig. 2 we illustrate our theoretical results over the photon energy range 22–46 eV. We make the assumption that 98 per cent of the 4s²4p^6 1S₀ ground state and 2 per cent of the statistically averaged excited metastable states : 4s²4p⁵5s ³P_2,1,0, 4s²4p⁵5s ¹P₁, and 4s²4p⁵4d ³P_0,1,2 will give the best match with the ALS experimental data taken at 18 meV FWHM (Mueller et al. 2013).

Prior DLP measurements at DCU made at 35 meV and synchrotron measurements performed on ASTRID at 40 meV (Kilbane et al. 2007) are compared with our darc calculations. These comparisons are illustrated in Figs 3 and 4, respectively. Fig. 3 shows the comparison of our present darc calculations with the DLP measurements taken at 35 meV FWHM in the photon energy region 27–28.6 where excellent agreement between theory and experiment is seen. In Fig. 4, a comparison with our present darc PI calculations and the measurements from the ASTRID radiation facility in Aarhus (taken at a photon resolution of 40 meV) is made in the photon region where the prominent Fano window resonances are located. Here again excellent agreement between theory and experiment is observed. The good agreement with the available experimental measurements provides further confidence in our theoretical cross-section data for astrophysical applications.

From these investigations we see that the PI cross-sections below 32 eV are dominated by strong Feshbach resonances, while above 32 eV there are strong dips in the PI cross-section due to Fano window style resonances. In the energy range from the ground state threshold to 28 eV, Feshbach Rb⁺(3d¹⁰4s²4p⁵ns, md ¹S₀) resonances dominate the cross-section. The Fano Rydberg window resonance series originates from photoexcitation of the inner shell 4 s-electron to the outer np orbital, n ≥ 5, when the Rb⁺(3d¹⁰4s²4p^6 1S₀) ion is in its ground state. The narrow Feshbach 4s²4p⁵ns resonances (∼0.2 meV or less), which are tabulated in Table 2, are not detectable in the experimental studies due to the limited experimental resolution, at best 18 meV, from the ALS work. The results for the broader 4s²4p⁵nd resonances are presented in Table 3, where they are compared with previous experimental studies (Neogi et al. 2003; Kilbane et al. 2007). Finally, in Table 4, the first few members of the Fano window resonances, 4s → np, n ≥ 5, are tabulated and their values compared with the previous DLP results and those from the ASTRID SR facility experimental studies (Neogi et al. 2003; Kilbane et al. 2007).

The darc theoretical R-matrix cross-sections gave a value of 3.60 for 98 per cent of the 4s²4p^6 1S₀ ground state and 2 per cent of the statistical average of the 4s²4p⁵5s ³P^o_{2, 1, 0}, 4s²4p⁵5s ¹P^o₁, and 4s²4p⁵4d ³P^o_{2, 1, 0} metastable states.

4 CONCLUSIONS

Theoretical results from large-scale darc photoionization cross-section calculations were used to interpret the experimental data from the DLP and ASTRID facilities  . From our darc results, a resonance analysis of both the Feshbach and the Fano window resonances illustrate excellent agreement with the available experimental data. The present theoretical work may be incorporated into astrophysical modelling codes like cloudy (Ferland et al. 1998; Ferland 2003), xstar (Kallman 2001), and atomdb (Foster et al. 2012) used to numerically simulate the thermal and ionization structure of ionized astrophysical nebulae.

ACKNOWLEDGEMENTS

BMMcL acknowledges support by the US National Science Foundation through a grant to itamp at the Harvard-Smithsonian Center for Astrophysics under the visitors program, the University of Georgia at Athens for the award of an adjunct professorship, and Queen’s University Belfast for the award of a visiting research fellowship (vrf). itamp is supported in part by NSF Grant No. PHY-1607396. The hospitality of Professor Thomas J Morgan and Wesleyan University, Middletown, CT, USA are gratefully acknowledged, where this research was completed. We thank Captain Thomas J Lavery USN Ret. for his constructive comments which enhanced the quality of this manuscript. Professor John Costello and Dr Deirdre Kilbane are thanked for the provision of the astrid and dlp data in numerical format. The authors acknowledge this research used grants of computing time at the National Energy Research Scientific Computing Centre (nersc), which is supported by the Office of Science of the U.S. Department of Energy (doe) under Contract No. DE-AC02-05CH11231. The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for funding this project by providing computing time on the GCS Supercomputer hazel hen at Höchstleistungsrechenzentrum Stuttgart (http://www.hlrs.de). itamp is supported in part by NSF Grant No. PHY-1607396.

REFERENCES