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

TL;DR

This study measures and analyzes the photoionization cross sections of Kr-like Rb$^+$ ions across 22-46 eV, comparing experimental data with advanced theoretical calculations to identify resonance features and metastable state contributions.

Contribution

It provides high-resolution experimental data and benchmarks large-scale Dirac Coulomb R-matrix theoretical calculations for Rb$^+$ photoionization, including resonance identification.

Findings

Resonance features dominate 22-32 eV range.

Fano window resonances observed above 32 eV.

Theoretical calculations agree well with experimental data.

Abstract

Single photoionization cross sections for Kr-like Rb$^+$ ions are reported in the energy (wavelength) range 22 eV (564 \AA) to 46 eV (270 \AA). Theoretical cross section calculations for this {\it trans}-Fe element are compared with measurements from the ASTRID radiation facility in Aarhus, Denmark and the dual laser plasma (DLP) technique, at respectively 40 meV and 35 meV FWHM energy resolution. In the photon energy region 22 - 32 eV the spectrum is dominated by excitation autoionizing resonance states. Above 32 eV the cross section exhibit 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 bench marked 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.

Tables

Table 1. Table 1: Rb 2+ energy levels in Rydbergs (Ry) from the large-scale grasp calculations compared with the available tabulations from the NIST database (Kramida et al., 2018 ) . A sample of the lowest 13 levels for the residual Rb 2+ ion from the 456-level grasp calculations are shown compared to experiment. The percentage difference Δ ( % ) \Delta(\%) compared to experimentf individual energy levels is given for completeness.

Level	STATE	TERM	NIST	GRASP	$\Delta {(\%)}^a$
			(Ry)	(Ry)
1	$4s^{2}4p^5$	${}_{}{}^{2}P_{3/2}^o$	0.00000	0.00000	0.0
2	$4s^{2}4p^5$	${}_{}{}^{2}P_{1/2}^o$	0.06720	0.06678	0.6
3	$4s4p^6$	${}_{}{}^{2}S_{1/2}^{}$	1.18494	1.22287	3.2
4	$4s^{2}4p^4\left({}_{}{}^{3}P\right)4d$	${}_{}{}^{4}D_{7/2}^{}$	1.40992	1.44446	2.5
5	$4s^{2}4p^4\left({}_{}{}^{3}P\right)4d$	${}_{}{}^{4}D_{5/2}^{}$	1.41135	1.44680	2.5
6	$4s^{2}4p^4\left({}_{}{}^{3}P\right)4d$	${}_{}{}^{4}D_{3/2}^{}$	1.41761	1.45368	2.5
7	$4s^{2}4p^4\left({}_{}{}^{3}P\right)4d$	${}_{}{}^{4}D_{1/2}^{}$	1.42516	1.46140	2.5
8	$4s^{2}4p^4\left({}_{}{}^{3}P\right)4d$	${}_{}{}^{4}F_{9/2}^{}$	1.48462	1.53650	3.5
9	$4s^{2}4p^4\left({}_{}{}^{3}P\right)4d$	${}_{}{}^{4}F_{7/2}^{}$	1.50853	1.56335	3.6
10	$4s^{2}4p^4\left({}_{}{}^{3}P\right)4d$	${}_{}{}^{4}F_{5/2}^{}$	1.52929	1.58192	3.4
11	$4s^{2}4p^4\left({}_{}{}^{3}P\right)4d$	${}_{}{}^{4}F_{3/2}^{}$	1.53777	1.59050	3.4
12	$4s^{2}4p^4\left({}_{}{}^{1}P\right)4d$	${}_{}{}^{2}P_{1/2}^{}$	1.51351	1.59218	5.2
13	$4s^{2}4p^4\left({}_{}{}^{1}P\right)4d$	${}_{}{}^{2}D_{3/2}^{}$	1.56776	1.62083	3.4

Table 2. Table 2: Resonance energies E n subscript 𝐸 𝑛 E_{n} of the 4 ​ s 2 ​ 4 ​ p 5 ​ ( P o 1 / 2 2 ) ​ n ​ s ​ P 1 o 1 4 superscript 𝑠 2 4 superscript 𝑝 5 superscript subscript superscript P o 1 2 2 𝑛 𝑠 superscript subscript superscript P o 1 1 4s^{2}4p^{5}({\rm{}^{2}P^{o}}_{1/2})\ ns\;{\rm{}^{1}P^{o}_{1}} Rydberg series from the present 456-level darc calculations, converging to the Rb ( 3 d 10 4 s 2 4 p 5 P 1 / 2 o 2 ) 2 + {}^{2+}(3d^{10}4s^{2}4p^{5}\;\rm{}^{2}P^{o}_{1/2}) threshold, originating from the Rb ( 3 d 10 4 s 2 4 p 6 S 0 1 ) + {}^{+}(3d^{10}4s^{2}4p^{6}\;{\rm{}^{1}S_{0}}) ground state. The quantum defect μ 𝜇 \mu for the Rydberg series and linewidths Γ Γ \Gamma ( μ 𝜇 \mu eV) are included for completeness. This series is not detectable in the experimental studies due to the extremely narrow resonance linewidths.

Label	Theory	Theory	Theory
$ns$	$E_n$ (eV)	${\mu }_n$	$\Gamma$ ($\mu$eV)
$8s$	–	–	–
$9s$	27.4233	0.6502	214
$10s$	27.5815	0.6493	151
$11s$	27.6960	0.6487	110
$12s$	27.7816	0.6482	83
$13s$	27.8472	0.6479	64
$14s$	27.8986	0.6476	51
$15s$	27.9397	0.6474	41
$16s$	27.9730	0.6474	33
⋮
Limit	28.204	–	–

Table 3. Table 3: Resonance energies E n subscript 𝐸 𝑛 E_{n} of the 4 ​ s 2 ​ 4 ​ p 5 ​ ( P o 1 / 2 2 ) ​ n ​ d ​ P 1 o 1 4 superscript 𝑠 2 4 superscript 𝑝 5 superscript subscript superscript P o 1 2 2 𝑛 𝑑 superscript subscript superscript P o 1 1 4s^{2}4p^{5}({\rm{}^{2}P^{o}}_{1/2})\ nd\;{\rm{}^{1}P^{o}_{1}} Rydberg series from the experimental measurements and the theoretical calculations (HXR approximation, Cowan code) of Kilbane et al. ( 2007 ) with the present 456-level darc calculations, converging to the Rb ( 3 d 10 4 s 2 4 p 5 P 1 / 2 o 2 ) 2 + {}^{2+}(3d^{10}4s^{2}4p^{5}\;\rm{}^{2}P^{o}_{1/2}) threshold. The Feshbach resonances in the photoionization cross section originate out of the 4 ​ p 4 𝑝 4p -subshell from the Rb ( 3 d 10 4 s 2 4 p 6 S 0 1 ) + {}^{+}(3d^{10}4s^{2}4p^{6}\;{\rm{}^{1}S_{0}}) ion in its ground state. The quantum defect μ 𝜇 \mu for the Rydberg series and the autoionization linewidths Γ Γ \Gamma (meV) are included for completeness.

Experimental and theoretical autoionizing Rb II resonance energies, quantum defects and linewidths
Label	Theorya	Exptb	Theoryc	Theorya	Exptb	Theoryc	Theorya
$nd$	$E_n$ (eV)	$E_n$ (eV)	$E_n$(eV)	${\mu }_n$	${\mu }_n$	${\mu }_n$	$\Gamma$ (meV)
$7d$	–	–	–	–	–	–	–
$8d$	27.3008	27.30	27.3099	0.2376	0.1977	0.1982	32.4
$9d$	27.4805	27.50	27.4595	0.3265	0.2077	0.3922	17.2
$10d$	27.6223	27.63	27.5900	0.3257	0.2628	0.5853	12.3
$11d$	27.7263	27.74	27.6814	0.3253	0.1699	0.7952	9.1
$12d$	27.8046	27.82	27.7519	0.3246	0.0951	1.0283	6.9
$13d$	27.8652	27.89	27.8072	0.3243	-0.1651	1.2887	5.4
$14d$	27.9129	27.95	27.8515	0.3241	-0.6377	1.5746	4.3
$15d$	27.9512	–	–	0.3239	–	–	3.4
$16d$	27.9824	–	–	0.3239	–	–	2.8
⋮
Limit	28.204	28.204	28.204	–	–	–	–

Table 4. Table 4: Rb  4 + s → n p {}^{+}\;4s\rightarrow np Fano window Rydberg resonance parameters for the first few members of the 4 ​ s ​ 4 ​ p 6 ​ ( S 1 / 2 2 ) ​ n ​ p ​ P 1 o 1 4 𝑠 4 superscript 𝑝 6 superscript subscript S 1 2 2 𝑛 𝑝 superscript subscript superscript P o 1 1 4s4p^{6}({\rm{}^{2}S}_{1/2})\ np\;{\rm{}^{1}P^{o}_{1}} series from the present 456-level darc calculations, and experiment, converging to the Rb ( 3 d 10 4 s 4 p 6 S 1 / 2 2 ) 2 + {}^{2+}(3d^{10}4s4p^{6}\;\rm{}^{2}S_{1/2}) threshold. Theoretical results obtained from the darc 456-level approximation are compared with previous experimental values from the ASTRID synchrotron radiation (SR) facility and the dual laser plasma (DLP) technique. The Rydberg resonance energy positions E r subscript 𝐸 𝑟 E_{r} are in eV and the autoionization resonances linewidths Γ Γ \Gamma are in meV.

Label	Experiment	Theory	Experiment	Theory
$np$	$E_r$ (eV)	$E_r$ (eV)	$\Gamma$ (meV)	$\Gamma$ (meV)
$5p$	35.710 $\pm$ 0.02a	35.720d	90 $\pm$ 30a	137d
	35.708b		117b
	35.714c		143c
$6p$	39.442b	39.448d	–	38d
	39.436c
$7p$	–	40.942d	–	17d

Equations

$$f$$

$${\overline{\sigma }}_{\mathrm{PI}}={\int }_{E_1}^{E_2}\sigma (h\nu )dh\nu$$

Full text

††pubyear: 2017††pagerange: Single photoionization of the Kr-like Rb II ion in the photon energy range 22 - 46.0 eV–References

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 et al., 1991; Mishenina et al., 2002; Roederer et al., 2011; Roederer et al., 2010; Frebel et al., 2014).

The motivation for the current study of this trans-Fe element, Rb II, is to provide benchmark PI cross section data for applications in astrophysics. High-resolution measurements of the photoionization cross section of Rb*+* were recently performed at the 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 22 eV (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 Advanced Light Source (ALS) (Mueller et al., 2013), the joint study by Kilbane et al. (2007), using a merged-beams technique by combining synchrotron radiation (SR) with a beam of Rb*+* (Rb II) ions at the ASTRID synchrotron radiation facility, the dual laser plasma (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 et al., 2006; Grant, 2007) generated the target wave functions employed in the present work. All orbitals were physical up to $n$=3, 4$s$, 4$p$ 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: $3s^{2}3p^{6}3d^{10}4s^{2}4p^{5}$, $3s^{2}3p^{6}3d^{10}4s4p^{6}$, $3s^{2}3p^{6}3d^{10}4s^{2}4p^{4}4d$, $3s^{2}3p^{6}3d^{10}4s4p^{5}4d$, $3s^{2}3p^{6}3d^{10}4s^{2}4p^{3}4d^{2}$, and $3s^{2}3p^{6}3d^{10}4s^{2}4p^{2}4d^{3}$.

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% higher.

Photoionization cross sections calculations were carried out on the Rb II ion for the $3d^{10}4s^{2}4p^{6}\;{\rm{}^{1}S}_{0}$ ground state, $3d^{9}4s^{2}4p^{5}5s\;{\rm{}^{3}P}_{2,1,0}$, $3d^{9}4s^{2}4p^{5}5s\;{\rm{}^{1}P}_{1}$ and the $3d^{9}4s^{2}4p^{5}4d\;{\rm{}^{3}P}_{2,1,0}$ 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 darc R-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).

Sixteen continuum orbitals were used 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 $\times$ 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 $3d^{10}4s^{2}4p^{6}\;{\rm{}^{1}S}_{0}$ ground state the $3d^{9}4s^{2}4p^{5}5s\;{\rm{}^{3}P^{o}}_{2,1,0}$, $3d^{9}4s^{2}4p^{5}5s\;{\rm{}^{1}P^{o}}_{1,}$ and $3d^{9}4s^{2}4p^{5}4d\;{\rm{}^{3}P^{o}}_{2,1,0}$ metastable levels of this ion, over the photon energy range similar to experimental studies.

For the ${\rm{}^{1}S}_{0}$ ground state level we require only the bound-free dipole matrices, Jπ = 0e,  $\rightarrow$  J${}^{{\prime}\pi^{\prime}}$ = 1o. In the case of the metastable levels we required the following, Jπ = 2o,  $\rightarrow$  J${}^{{\prime}{\rm\pi}^{\prime}}$ = 1e , 2e, 3e, for Jπ = 10,  $\rightarrow$  J${}^{{\prime}{\rm\pi}^{\prime}}$ = 0e , 1e, 2e and Jπ = 0o,  $\rightarrow$  J${}^{{\prime}{\rm\pi}^{\prime}}$ =1e, 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 co-workers (Quigley & Berrington, 1996; Quigley et al., 1998; Ballance et al., 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 Figure 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 $4s^{2}4p^{6}\;{\rm{}^{1}S_{0}}$ ground state, the $4s^{2}4p^{5}5s\;{\rm{}^{3}P_{2,1,0}}$, $4s^{2}4p^{5}5s\;{\rm{}^{1}P_{1}}$ and the $4s^{2}4p^{5}4d\;{\rm{}^{3}P_{0,1,2}}$ 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 Figure 2 we illustrate our DARC 456-level calculations over the photon energy range 22 – 46 eV. We make the assumption that 98% of the $4s^{2}4p^{6}\;{\rm{}^{1}S_{0}}$ ground state and 2% of the statistically averaged excited metastable states : $4s^{2}4p^{5}5s\;{\rm{}^{3}P_{2,1,0}}$, $4s^{2}4p^{5}5s\;{\rm{}^{1}P_{1}}$ and $4s^{2}4p^{5}4d\;{\rm{}^{3}P_{0,1,2}}$ will give the best match with the ALS experimental data taken at 18 meV FWHM (Mueller et al., 2013).

Prior dual laser plasma (DLP) measurements at Dublin City University (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 figure 3 and figure 4 respectively. Figure 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 figure 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${}^{+}(3d^{10}4s^{2}4p^{5}ns,md\;{\rm{}^{1}S_{0}})$ dominate the cross section. The Fano Rydberg window resonance series originates from photo-excitation of the inner shell $4s$-electron to the outer $np$ orbital, $n\geq 5$, when the Rb${}^{+}(3d^{10}4s^{2}4p^{6}\;{\rm{}^{1}S_{0}})$ ion is in its ground state. The narrow Feshbach $4s^{2}4p^{5}ns$ resonances ($\sim$ 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 $4s^{2}4p^{5}nd$ 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, $4s\rightarrow np$, $n\geq 5$, are tabulated and their values compared with the previous dual laser plasma (DLP) and those from the ASTRID synchrotron radiation (SR) facility experimental studies (Neogi et al., 2003; Kilbane et al., 2007).

The theoretical integrated continuum oscillator strength $f$ may also be compared with experiment when available. The integrated continuum oscillator strength $f$ of the experimental spectra may be calculated over the energy grid [$E_{1}$, $E_{2}$], where $E_{1}$ is the minimum experimental energy and $E_{2}$ is the maximum experimental energy measured, using (Shore, 1967; Fano & Cooper, 1968; Berkowitz, 1979),

[TABLE]

where

[TABLE]

is the resonance strength. Evaluating the continuum oscillator strength from the darc theoretical R-matrix cross sections gave a value of 3.60 for 98% of the $4s^{2}4p^{6}\;{\rm{}^{1}S}_{0}$ ground state and 2% of the statistical average of the $4s^{2}4p^{5}5s\;{\rm{}^{3}P^{o}}_{2,1,0}$, $4s^{2}4p^{5}5s\;{\rm{}^{1}P^{o}}_{1,}$ and $4s^{2}4p^{5}4d\;{\rm{}^{3}P^{o}}_{2,1,0}$ 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 (www.hlrs.de). itamp is supported in part by NSF Grant No. PHY-1607396.