L-Shell Photoionization of Magnesium-like Ions with New Results for Cl⁵⁺

Abstract

This study reports on the absolute photoionization cross sections for the magnesium-like Cl⁵⁺ ion over the 190–370 eV photon energy range, corresponding to the L-shell (2s and 2p subshells) excitation regime. The experiments were performed using the Multi-Analysis Ion Apparatus (MAIA) on the PLéIADES beamline at the SOLEIL synchrotron radiation storage ring facility. Single and double ionization ion yields, produced by photoionization of the 2p subshell of the Cl⁵⁺ ion from the 2p⁶3s^2 1S₀ ground state and the 2p⁶3s3p ³P_0,1,2 metastable levels, were observed, as well as 2s excitations. Theoretical calculations of the photoionization cross sections using the Multi-Configuration Dirac-Fock and R-matrix approaches were carried out, and the results were compared with the experimental data. The Cl⁵⁺ results were examined within the overall evolution of L-shell excitation for the early members of the Mg-like isoelectronic sequence (Mg, Al⁺, Si²⁺, S⁴⁺, Cl⁵⁺). Characteristic photon energies for P³⁺ were estimated by interpolation.

1. Introduction

Photon-driven excitation and ionization along the magnesium isoelectronic sequence have long been of interest in experimental and theoretical investigations e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]. This is not surprising, given the nature of magnesium-like systems with their simple closed-subshell ground state configuration 2p⁶3s². The ¹S₀ ground state tends to make it a relatively stable ion configuration in many plasma environments; therefore, from a theoretical point of view, photoionization calculations should focus mainly on the effects of excited state configurations. However, this ideal picture largely fails in practice, as electron correlations, e.g., two-electron excitations, turn out to be extremely important in this system, leading to a significant departure from the simple one-electron excitation picture. Many early investigations concentrating on the first ionization threshold energy region highlighted the complexity of such electron interactions, along with consequent demands on appropriate theories, see e.g., [17] and the references therein. As short wavelength photon interactions lead to inner-shell excitations, which couple with multiple excitations of the valence electrons, photoionization calculations become even more challenging due to the increased number of unfilled (sub)shells. Reliable photoionization laboratory investigations of magnesium-like ions in the inner-shell excitation photon energy regime are therefore of continuing interest as they provide robust benchmarking opportunities for different theoretical approaches.

Experimental and theoretical investigations of the photoionization of free atomic ions are becoming ever more important, stimulated by the abundant data obtained from X-ray satellite observatories such as Chandra and XMM-Newton. Future space missions such as XRISM and Athena promise even greater sensitivity and higher spectral resolution. Optimizing the value of astrophysical spectral observations, and the concomitant provision of greater physical insights, requires improved theoretical modelling and simulation of astrophysical plasma environments. The latter rely on fundamental atomic data, such as those provided by ongoing systematic and complementary experimental and theoretical investigations of the interaction of ionizing photons with free atomic ions. This is evidenced by the recent growth in laboratory astrophysics initiatives and associated data banks [19,20,21,22,23,24].

In the absence of experimental results, much of the atomic photoionization data exploited in modelling ionized environments has to rely on atomic physics calculations. Investigations of isoelectronic magnesium-like ions (twelve bound electrons) provide one important avenue for the calibration of different theoretical approaches. Along the isoelectronic sequence from neutral magnesium through Al⁺, Si²⁺, P³⁺, S⁴⁺ to Cl⁵⁺ ions, the increasing core charge leads to significant changes in the relative effects of the nuclear charge versus electron–electron interactions on the photoionization properties. The sequence provides an ideal case study to examine the ability of theoretical models to predict and interpret these changes, which involve level crossing and plunging configurations [17].

Plasma-based light source experiments were successful in recording resonances and ionization thresholds for early members of the magnesium sequence for both ground and excited states, see, for example [1,5,6,7,25].

The biggest step, however, in the systematic investigation of the inner-shell photoionization of positive ions in general has resulted from dedicated merged photon-ion beam facilities at synchrotron radiation storage rings, notably, Daresbury (UK), SuperACO and later SOLEIL (France), ASTRID (Denmark), ALS (USA), and PETRA III (Germany), see e.g., [26,27,28,29,30] and the references therein. The great advantage of such merged beam experiments is that they isolate specific ion stages and allow quantitative results to be obtained at high spectral resolution. This often results in absolute cross sections for different ionization channels, thereby providing the highest quality fundamental data for atomic and plasma modelling, as well as for benchmarking the most advanced atomic theories.

Absolute cross sections for L-shell ($2\mathrm{s}$ and $2\mathrm{p}$) photoionization along the magnesium series have been previously obtained through synchrotron radiation-based merged-beam experiments for Al⁺ [9,31], Si²⁺ [10], S⁴⁺ [18], and Fe¹⁴⁺ [32], with the latter produced in an electron beam ion trap. These experiments have been complemented by various theoretical investigations. These investigations have interpreted the evolution of the observed resonances in the early part of the sequence, showing that considerable changes in the relative intensities and positions of resonances take place as the core charge increases along the sequence. The L-shell single and double photoionization cross sections of magnesium itself have been studied extensively using synchrotron radiation in high resolution experiments, but only on a relative scale, see [16] and the references therein.

The next member of the sequence is Cl⁵⁺. Chlorine is an extremely reactive element, appearing in many gaseous, liquid, and solid molecular forms of considerable scientific and industrial significance. Similarly to all halogens, it is strongly oxidizing as it is one electron short of the neighboring inert gas configuration. Chlorine ions play key roles in many terrestrial and astrophysical environments; therefore, understanding their interaction with radiation is important [33,34,35,36,37,38].

In this article, we report the first study of L-shell photoionization of magnesium-like Cl⁵⁺ for photon energies between 190 and 370 eV, straddling the 2s and 2p inner-shell excitation regimes. It is interesting to compare the success of the Multi-Configuration Dirac Fock (MCDF) and the R-matrix theoretical treatments of Cl⁵⁺ with the success of the previous analogous treatments of S⁴⁺ [18]. Our results provide experimental absolute photoionization cross sections for Cl⁵⁺, a comparison with the theoretical predictions of both the MCDF and the R-matrix approaches, as well as insights into the evolution along the isoelectronic sequence from Al⁺ to Cl⁵⁺. To the best of our knowledge, no such photoionization data for P³⁺ is available in the open literature. From the observed trends along the sequence, we estimated the values of some relevant characteristic energies for P³⁺.

2. Experimental Details

The experimental results for Cl⁵⁺ reported in this study were obtained using the dedicated merged photon-ion apparatus, MAIA (Multi-Analysis Ion Apparatus), on the ultra-high resolution soft X-ray (10 eV to 1 keV photons) PLÉIADES beamline at SOLEIL. Well-characterized synchrotron photon and ion beams overlap, and the resulting photoionization ions are selectively measured, providing cross section information for the different ionization channels. An advantage of working with ions over neutral species is that the number density in the overlap region can be determined. This allows the absolute cross sections to be measured. A comprehensive description of MAIA, and how absolute photoionization cross section results may be determined, is provided in [26]. Here we provide a shorter description to include the specific experimental parameters used for the Cl⁵⁺ investigations.

HCl gas was introduced into an Electron Cyclotron Resonance (ECR) ion source in order to generate chlorine ions. The Cl⁵⁺ ions were extracted and accelerated by an applied potential difference of −4 kV. They were then guided by a 90⁰ bending magnet into the overlap region to meet the counter-propagating synchrotron radiation beam. An input power of 32 W at 12.36 GHz was used to optimize the production of Cl⁵⁺ ions. The magnetic filter enabled the selection of the most abundant ³⁵Cl⁵⁺ isotope. The ion current, measured in a Faraday cup placed after the magnet exit, was of the order of 10 μA. This beam was then focused and shaped to match the size of the counter-propagating photon beam. The remaining current of ions interacting with the photons was typically of the order of 300 nA. The length of the interaction region was determined by a 57 cm long tube placed in the path of the two beams and polarized at a voltage of −2kV. Three transverse profilers, located at the center and at both ends of this pipe, respectively, allowed measurement of the overlap of the photon beam and the ion beam; a Form Factor [26] of 32,000 m⁻¹ was reached. The primary ion beam current was measured after the interaction region using a Faraday cup. The photoionized ions, either singly ionized Cl⁶⁺ or doubly ionized Cl⁷⁺, were separated from the primary Cl⁵⁺ beam by a second dipole magnet, selected in speed by an electrostatic analyzer, and measured with a microchannel-plate detector coupled to a counter. A photon chopper was used to subtract the contribution of ions produced by collisions in the residual gas (background pressure of 1.5 × 10⁻⁹ mbar) to the Cl⁶⁺ or Cl⁷⁺ signal. The photon beam flux was monitored by a calibrated photodiode, and a typical current of ~100 μA at 195 eV photon energy and 150 meV bandwidth was measured. Knowing the photodiode current, the Cl⁵⁺ ion current, the form factor characterizing the overlap of the two beams, the length of the interaction region, the ion speed, and the efficiencies of the photodiode and channel-plate detectors, and recording the Cl⁶⁺ and the Cl⁷⁺ signals as functions of the photon energy, allowed the determination of the single and double absolute photoionization cross sections of the Cl⁵⁺ ions. During the Cl⁵⁺ experiments, SOLEIL was operated with a current of 450 mA. Circular left polarization, which delivers the highest flux in the photon range of interest, was used. The photon energy, corrected for the Doppler shift due to ion velocity, was calibrated using a gas cell containing argon, where the 2p_3/2-> 4s transition in argon at 244.39 eV [39] was measured. The energy uncertainties were of the order of 20 meV, but varied depending on the resonance (see tables). The total uncertainty of the measured cross sections was estimated to be not greater than 15%, and was mostly due to the combined effects of the inaccuracy of the determination of the beam overlaps (the form factor), the efficiency of the detector, and the photon flux.

The heating microwaves interacted directly with the electrons at the ion source through the ECR mechanism, and a complicated range of processes involving energetic electrons was responsible for producing the ions. The detailed electron energy distribution was not known and the ECR ion source plasma was not in equilibrium [40]. A well-recognized result is that in the merged beam technique, the ions in the interaction region can often exist in excited states as well as the ground state [26,27,28,29,30]. The population of excited levels depends on their lifetimes and whether they survive the journey from the source into the interaction region. In general, this can be the case for metastable levels, some of which can have relatively long lifetimes. We will see below how the issue of excited states for the specific case of magnesium-like ions can be satisfactorily resolved.

3. Theoretical Details

For the recent analogous investigations of photoionization of S⁴⁺, we used two quite disparate theoretical approaches––MCDF and R-matrix theory [18]. We used the same methods to predict and interpret the new Cl⁵⁺ results presented here. While long-established and used to calculate many photoionization cross sections for a wide variety of ion species, both methods are being continuously reviewed and improved. In the MCDF approach, the twelve-electron magnesium-like ion is tackled directly. The quality of the results critically depends on including the most relevant electron configurations, although this is subject to calculational limitations. The R-matrix approach treats the magnesium-like challenge as a problem of an incident electron scattering off the eleven-electron sodium-like ion. Therefore, the quality of the final overall results depends on the accuracy of the intermediate target description. In many cases, some of the thresholds of the target system are already known, and this information can be used to help in the R-matrix approach. This was indeed the case for S⁴⁺ [18], but not for Cl⁵⁺. It is therefore interesting to see how the R-matrix code works for the latter when compared with the former. Both length and velocity gauge calculations were carried out using both approaches, and very satisfactory agreement was shown each time. Consequently, for the sake of brevity, only the length gauge results are presented in this paper.

Because of the similarity of the computations for the Cl⁵⁺ system with those previously carried out for the S⁴⁺ ion [18], we provide only brief descriptions of the R-matrix and MCDF calculations. In order to focus on the problem at hand, viz., the L-shell photoionization of Cl⁵⁺ and its behavior along the beginning of the magnesium sequence, we do not provide lengthy mathematical descriptions to support the fundamental aspects of the theoretical descriptions. Only the most important details are provided, i.e., those specific to the case of Cl⁵⁺. For additional information on the more fundamental and mathematical aspects of the MCDF and R-matrix theories, see the following recent references that will provide didactic descriptors as well as extensive lists of anterior references, namely, [41,42,43] for MCDF and [44,45,46,47] for R-Matrix.

3.1. R-Matrix Calculations

The calculational convenience of transitioning from Mg-like S⁴⁺ to Cl⁵⁺ is that the electronic orbital and configurational descriptions are the same once proper Z-scaling is accounted for. Previous R-matrix calculations for S⁴⁺ have been described in detail [18,48,49]; the changes for the new calculations of Cl⁵⁺ simply require adding one more proton to the nucleus, while keeping the same number of electrons. Indeed, from simple Z-scaling of the radial coordinate $\rho =Zr$, the radial orbital $P\left(\rho \right)\to Z^{1/2}P\left(r\right)$, and the cross section $\sigma \left(Z\right)\sim \sigma \left(1\right)/Z^2$, the plots of the corresponding radial orbitals $P\left(r\right)$ (shown in Figure 1) and the resultant photoabsorption cross sections are both remarkably similar. The Z-scaling also naturally leads to a reduction in the electron correlation effect, manifested via the electron–electron repulsion $1/\left|\overrightarrow{r_1}-\overrightarrow{r_2}\right|$ that scales as $1/Z$ and becomes less significant at higher $Z$. Therefore, Cl⁵⁺ is expected to show a more hydrogenic structure than S⁴⁺. This behavior is seen in Figure 1, where both sets of orbitals exhibit very similar patterns, with a slightly greater localization of the orbitals near the nuclear core for Cl⁵⁺.

For the present Cl⁵⁺ work, the main series considered within the R-matrix formulation are the 2s²2p⁶3l channels of Cl⁶⁺ and the 2s²2p⁵3s3l and 2s²2p⁵3s² inner-shell channels. Note that again, as for S⁴⁺, the 2s2p⁶3s3p channels are omitted to keep the computation tractable. This means that certain weak resonances, which are seen in the MCDF results, will be absent. Nevertheless, the main 2p → nd Rydberg resonance series are well featured in the computed end results, and the weaker interspersed 2p → (n + 1)s series and 2s → np series at higher photon energy are also included.

It can never be overstated that the primary source of uncertainty in any R-matrix calculation is due to a lack of convergence in the initial and/or final state energies, given that the calculation is based on a variational principle. This leads, in turn, to a corresponding non-convergence of the photon transition energies. In order to simplify the comparison, we present ab initio results for Cl⁵⁺ rather than using certain empirical energy information [50], knowing in advance that certain energy shifts must be aligned with the experimental results. Additionally, the R-matrix results are preconvolved with a resonance width of ${\Gamma }_{spectator}=2.5\times {10}^{-3}$ Ryd for the spectator Auger decay process. This is to ensure the full resolution of the various infinite Rydberg series of otherwise narrowing resonances $({\Gamma }_{participator}\sim 1/n^3$). ${\Gamma }_{participator}$ refers to the resonance width of the participator Auger decay in which the initially photon-excited electron engages in the Auger process. While in contrast, this does not occur in spectator Auger decay.

As a final note, the R-matrix method does include the Fano interference [51] between direct photoionization and indirect photoexcitation–autoionization. Therefore, asymmetric profiles can be revealed in general. This is in contrast to perturbative methods, such as the MCDF method, where the resonance is treated as a single final state, giving a Breit–Wigner (symmetric Lorentzian) resonance profile added incoherently to the (uncoupled) smooth background photoionization cross section. Hence, the asymmetrical profile of the experimental results of the 2s → np Rydberg series, in particular, can only be reproduced by the R-matrix calculations here.

3.2. MCDF Calculations

An updated version of the MCDF code developed by Bruneau [42] has been used to compute the relevant photoexcitation and photoionization cross sections. Calculations were performed using a full intermediate coupling scheme in a jj basis set. The calculations were restricted to electric dipole transitions using both the Coulomb and the Babushkin gauges, which correspond to the velocity and the length form of the electric dipole operator in the non-relativistic limit, respectively [42]. Preliminary calculations were performed to compute a set of one-electron wave functions resulting from the energy minimization of the Slater transition state [52] using the following configuration set: [Ne]3snl, [Ne]3s, and [F]3s², and [He]2s2p⁶3s², where n = 3,…7 and l = s, p, d, and where [He], [F], and [Ne] mean 1s², 1s²2s²2p⁵, and 1s²2s²2p⁶, respectively. This one-electron wave function set (OWFS) was used to calculate the 2s and 2p photoexcitation cross sections from the ¹S₀ level of the [Ne]3s² ground configuration and from the ³P_0,1,2 metastable levels of the [Ne]3s3p configuration. With regard to the photoexcitation cross sections from the ground level, the following photo-excited configurations were retained: [F]3s²3d, [F]3s3p², [F]3s²n’l’, and [He]2s2p⁶3s²nd, where n’ = 4,…, 7 and l’ = s, d. For the photoexcitation cross sections from the [Ne]3s3p ³P₀, ₁, ₂ metastable levels, the photo-excited configurations retained were: [F]3s²3p, [F]3s3p3d, [F]3s3pn’l’, [He]2s2p⁶3s3p², and [He]2s2p⁶3s3pn’p. The initial OWFS was also used to compute the autoionization rates for [F]3s²3d and [F]3s²3p/3s3p3d levels photoexcited from the [Ne]3s^{2 1}S₀ ground level and the [Ne]3s3p ³P₀, ₁, ₂ metastable levels, respectively. The largest calculated autoionization rates were found for the [F]3s²3d ¹P₁ and ³P₁ excited levels, with corresponding Auger widths equal to 86 and 92 meV, respectively. The Auger widths for the [F]3s²3p excited levels were all lower than 3.6 meV. The direct 2p and 3s photoionization cross sections were also calculated using the initial OWFS for the [Ne]3s^{2 1}S₀ and [Ne]3s3p ³P_0,1,2 initial levels. The 3p photoionization cross sections were also computed for the [Ne]3s3p ³P_0,1,2 levels.

4. Results

4.1. L-Shell Photoionization of Cl⁵⁺

This section presents the results for the single and double ionization of Cl⁵⁺ ions over the photon energy range corresponding to the L-shell excitations (both 2s and 2p). Figure 2 shows a schematic energy level diagram indicating some of the main ionization pathways due to photon absorption in the L-shell excitation regime. In particular, it shows excitation from the ground 2p⁶3s^{2 1}S, and the metastable 2p⁶3s3p ³P levels which lie just over 12 eV above the ground state (the vertical upward lines indicate photoabsorption). The final inner-shell excited states shown lying above ~211 eV then give rise to Auger decay resonances in the photoionization cross section. These can interact with the underlying direct photoionization process, and the downward dotted arrows illustrate the ensuing production of either Cl⁶⁺ or Cl⁷⁺ ions due such non-radiative decay process(es) of the original 2p vacancy.

Figure 3 shows the measured photoionization cross section over the photon energy region from 190 to 370 eV. The 285–320 eV photon region was not scanned as resonances were not expected in this interval. All cross sections between 190 and 285 eV were measured in the very dominant single ionization (SI) channel, while all the cross sections above 326 eV were measured in the double ionization (DI) channel (which becomes the dominant channel above the 2p⁻¹ ionization limit). We also show the results of the ab initio MCDF and R-matrix calculations where the theoretically predicted resonances are folded with the experimentally determined bandpasses for the different spectral regions. Because of the weakness of the resonances near 200 eV and those above 320 eV, their cross sections (both experimental and theoretical) were multiplied by a factor of ten, which accounts for the apparent vertical displacements of the cross sections on the right-hand side of Figure 3.

We first discuss how we estimated the contribution of metastable levels to the overall photoionization cross section. We then analyse the experimental cross section in comparison with the MCDF and R-matrix predictions in separate contiguous energy regions corresponding to changing photon interaction processes and taking into account the contribution due to metastable levels.

4.1.1. Metastable Resonance Region

The important contribution of metastable levels to photoabsorption of plasmas containing magnesium-like ions has been recognized in some early dual laser produced plasma experiments [6]. These experiments showed that for Mg, Al⁺, and Si²⁺, the resonances arising from the 2p⁶3s3p ³P levels appear in a separate photon energy region to the resonances arising from the ground state [5,7]. This feature allows the estimation of the metastable states population fractions by comparing the theoretically predicted cross sections from these states with the measured ones at those photon energies. Similar advantageous behavior was observed for S⁴⁺ [18], and was again the case for the current Cl⁵⁺ results.

In Figure 3, the photoionization resonances arising from the excitation from the ³P metastable levels (2p⁶3s3p ³P → 2p⁵3s²3p ³S, ³P, ³D) can be seen in the 195–201 eV photon energy region, well separated from the ground state resonances lying above 220 eV photon energy. We can estimate the relative populations of the metastable levels by comparing the experimental data with the MCDF and R-matrix predicted cross sections. Figure 4 shows details of the measured cross section due to the metastable levels and the comparative cross sections from both the MCDF (blue line) and the R-matrix (red line) calculations. The detailed comparison of the relative strengths of the individual resonances is determined by the relative populations of the individual ³P_0,1,2 J-levels. The summation of the cross section over all the resonances depends on the relative metastable to ground state ratio. The best fit of the theoretical with experimental data implies relative populations of 77% ¹S₀ + 4% ³P₀ + 1% ³P₁ + 18% ³P₂. As for S⁴⁺, the very low contribution of the 2p⁶3s3p ³P₁ state is readily explained by its E1 radiative decay lifetime of ~2.6 μs [52], which contributes to significantly repopulating the ground state by the time the fast (3.23 × 10⁵ ms⁻¹) Cl⁵⁺ sample ions reach the interaction zone. These relative population factors are used in all future figures where we compare the experimental data with the theoretical predictions.

It should be noted that in comparing the experimental data of Figure 4 with theory, the ab initio MCDF predictions were shifted by −0.54 eV, while the ab initio R-matrix results were shifted by −0.64 eV. The integrated cross sections over the metastable resonances, taking into account the relative populations, were: experimental data (2.9 ± 0.4 Mb.eV), MCDF (2.36 Mb.eV), and R-matrix (2.91 Mb.eV). In

Table 1. Experimental energies, theoretical energies, line strengths, and assignments in LSJ notation (based on MCDF results) for Cl⁵⁺ resonances in the 195 eV–200 eV photon energy range, arising from 2p → 3s inner-shell excitations from the valence-excited metastable 2p⁶3s3p ³P_0,1,2 states.

	Energy (eV)/Strength (Mb.eV)
Experimental *	R-Matrix	MCDF	Assignment
195.66(4)/0.09(2)	196.61/0.083	196.84/0.064	3P2-3S1
195.95(3)/0.15(3)	196.83/0.074	197.05/0.064	3P0-3S1
197.29(2)/0.85(13)	198.20/0.864	198.13/0.700	3P2-3D3
197.54(4)/0.09(2)	198.47/0.118	198.38/0.036	3P1-3D2
198.12(3)/0.27(4)	199.04/0.268	198.97/0.211	3P0-3D1
198.31(4)/0.28(5)	199.24/0.076	199.10/0.273	3P2-3P2
199.09(4)/0.07(3)	200.04/0.054	199.94/0.048	3P2-3D1
199.33(4)/0.09(2)	200.18/0.013	200.08/0.012	3P1-3D1
199.49(4)/0.11(2)	200.25/0.028	200.15/0.024	3P0-3D1
199.59(2)/0.47(8)	200.51/0.478	200.36/0.306	3P2-3D2
	200.57/0.252	200.47/0.156	3P2-3P1
199.97(5)/0.05(2)	200.79/0.067	200.68/0.50	3P0-3P1

* The number in brackets is the experimental uncertainty on the last digit, i.e., 195.66(4) eV is the same as (195.66 ± 0.04) eV. The same convention is used for all the data presented in this column.

, we show experimental and theoretical (both R-matrix and MCDF) resonance energies, together with resonance line strengths (cross section integrated over a single resonance profile) and assignments for the structures in the 195 eV to 200 eV photon energy region arising from inner-shell excitations of the valence-excited metastable 3P levels. The assignments are based on best intensity and energy position matchings of the observed resonances and the MCDF predictions.

4.1.2. 2p → 3d Excitation Region

The dominant resonance structures observed between 235 and 240 eV in Figure 3 are due to the expected 2p–3d excitations, accompanied by considerably weaker 2p–ns resonances. These are shown in detail in Figure 5. The relative strengths of the individual resonances within this group vary considerably as one progresses from neutral magnesium toward Cl⁵⁺. The detailed spectroscopic assignments of the corresponding resonances for the early members of the sequence have given rise to considerable early discussions [1,2,5,16], with the labelling to some extent depending on the particular theoretical approach. This is not surprising because, as previously noted in Section 3., considerable challenge arise in the theoretical calculations due to the complexity associated with multiply excited outer electron configurations combining with the inner-shell excited 2p (or 2s) hole. Further complexity arises from the nominally closed shell ground state of 3s^{2 1}S₀ mixing with configurations such as 3p² [5,9,10]. The theoretical calculations indicate that some of the weaker resonances in Figure 5 can be attributed to such electron correlation effects between energetically close-lying, multiply excited states.

As can be seen in Figure 5, the R-matrix results mimic the experimental data in this region somewhat better than the corresponding MCDF predictions, however, only by a small amount. Both theoretical approaches require systematic energy shifts in order to bring the strongest resonances into approximate agreement with the experimental data (MCDF and R-matrix shifts of −0.46 eV and −1.86 eV, respectively). When compared with the experimental data, the R-matrix relative strengths of the main resonances are somewhat underestimated, while the energy separation of the strongest resonances is slightly overestimated (typically by an amount of less than 0.5 eV) in the MCDF calculations. For the latter approach, the use of an even more extended set of ground state and photoexcited configurations would likely improve the value of this energy separation, via both improved OWFS and description of ground and excited state correlations. This relatively minor improvement would require significant additional computational efforts, which were not available to the authors. The cross sections integrated over the photon energies of the spectral region of Figure 5 are: experimental data (129 ± 19 Mb.eV), MCDF (111.2 Mb.eV), and R-matrix (118.9 Mb.eV), respectively.

4.1.3. Region of 2p → nd Excitations

Figure 6 shows the corresponding comparisons for the experimental and ab initio theoretical predictions for the resonances in the 2p → nd (n > 3) photon energy region. The systematic energy shifts required to bring the theoretical predictions into approximate agreement with the experimental data were +0.97 eV for the MCDF and a larger shift of −4.36 eV for the R-matrix calculations. The integrated cross sections over the spectral region covered by Figure 6 were in good agreement, and were: experimental data (87 ± 13 Mb.eV), MCDF (71.8 Mb.eV), and R-matrix (85.4 Mb.eV), respectively. Both the R-matrix and MCDF results suggest that the strong resonance feature observed at 282.66 eV likely corresponds with the first member (n = 3) of the 2s → np ¹P₁ series. Numerical fitting of the Rydberg series for the 2p → nd series using the basic $E_n=I_p-\frac{13.6\times 6^2}{{\left(n-\delta \right)}^2}$ hydrogenic quantum defect formula (in eV) led to values of $299.6\left(2\right)\mathrm{eV}$ and $0.22\left(1\right)$ for the (2s²2p⁵3s²) ²P ionization limit $I_p$ and quantum defect $\delta$, respectively. In the experimental conditions, it was not possible to reliably access the energy values of the two fine-structure limits ²P_1/2,3/2. The comments in Section 4.1.2 about the use of more extended OFWS and configuration sets are also applicable here.

4.1.4. Region of 2s → np Excitations

Above the 2p⁵3s^{2 2}P limit, we observed a regular series of relatively weak resonances, analogous in shape to those previously observed [18] for S⁴⁺. As experimental absolute cross sections were not available here, the data were instead normalized to the theoretical cross sections, as shown in Figure 7. The strongest resonance lay at ~330 eV, identified as the 2s → 4p member of the series. The most striking feature of the resonances in this region was the strongly asymmetrical Fano profiles due to the strong autoionization interaction with the underlying continua. As noted in Section 3.1, the R-matrix calculation includes the Fano interference between the resonance and the underlying continua, and mimics rather well the observed asymmetric profiles. Hydrogenic extrapolation (see previous paragraph) of the experimental 2s → np series led to the best fit values of 373.2(6) eV and 0.70(1) for the $2\mathrm{s}{}_{}{}^{2}S{}_{1/2}$ ionization threshold energy and quantum defect of the $2snp{}_{}{}^{1}P{}_1$ levels, respectively.

In

Table 2. Experimental and theoretical (MCDF and R-matrix) energies, as measured line strengths and, where available, Auger widths for the Cl⁵⁺ resonances due to 2p → nd and 2s → np inner-shell excitations from the 2p⁶3s^{2 1}S₀ ground state. Where identifiable from the output of the MCDF theory, the dominant electron configuration and leading LSJ character of the final state of the resonance is given in the outermost right column.

Energy (eV)/Strength (Mb eV)/Auger Width (meV) * Experimental **	Energy (eV)/Strength (Mb eV)/Auger Width (meV) * R-matrix	Energy (eV)/Strength (Mb eV)/Auger Width (meV) * MCDF	Assignment (MCDF)
225.28(3)/4.0(7)/x	226.77/3.94/x	224.78/5.06/x	$2{\mathrm{p}}^{5}3\mathrm{s}3{\mathrm{p}}^2{}_{}{}^{3}D{}_1$
226.56(7)/0.7(4)/x	228.08/0.63/x	225.93/6.57/x	$2{\mathrm{p}}^{5}3\mathrm{s}3{\mathrm{p}}^2{}_{}{}^{3}P{}_1$
230.03(7)/0.9(4)/x	231.91/1.07/x
231.01(3)/2.9(6)/x	232.88/2.75/x	231.25/1.32/x	$2{\mathrm{p}}^{5}3\mathrm{s}3{\mathrm{p}}^2{}_{}{}^{3}P{}_1$
231.93(4)/2.2(5)/x	233.80/2.12/x
232.49(7)/0.8(4)/x
233.05(5)/1.2(4)/x	235.18/0.19/x
233.64(10)/0.5(4)/x
234.26(4)/2.4(5)/x		234.48/2.83/x	$2{\mathrm{p}}^{5}3\mathrm{s}3\mathrm{p}3\mathrm{d}{}_{}{}^{3}D{}_3$
234.73(6)/1.0(4) /x
235.98(2)/26(4)/51(4)	237.81/29/37	236.01/39.8/45	$2{\mathrm{p}}^{5}3{\mathrm{s}}^{2}3\mathrm{d}{}_{}{}^{3}D{}_1$
236.35(3)/1.1(4)/x
236.54(3)/1.5(4)/x
236.71(3)/9(2)/158(65)	238.69/20.3/134	236.97/28/92	$2{\mathrm{p}}^{5}3{\mathrm{s}}^{2}3\mathrm{d}{}_{}{}^{3}P{}_1$
237.27(3)/2.2(3)/x
237.34(3)/4.7(8)/x		237.73/5.5/x	$2{\mathrm{p}}^{5}3\mathrm{s}3\mathrm{p}3\mathrm{d}{}_{}{}^{3}D{}_2$***
237.49(4)/0.5(4)/x
237.66(3)/4.8(8)/x		238.16/2.7/x	$2{\mathrm{p}}^{5}3\mathrm{s}3\mathrm{p}3\mathrm{d}{}_{}{}^{3}D{}_3$***
238.01(3)/2.3(5)/x		238.45/1.72/ x	$2{\mathrm{p}}^{5}3\mathrm{s}3\mathrm{p}3\mathrm{d}{}_{}{}^{3}D{}_1$***
238.61(2)/78(12)/46(2)	240.58/58.1/69	239.46/42.6/86	$2{\mathrm{p}}^{5}3{\mathrm{s}}^{2}3\mathrm{d}{}_{}{}^{1}P{}_1$
240.26(4)/2.0(5)/x		240/2.24/x	$2{\mathrm{p}}^{5}3\mathrm{s}3\mathrm{p}3\mathrm{d}{}_{}{}^{3}S{}_1$***
240.47(7)/9(4)/x
251.98(2)/1.5(2)/x	255.97/1.9/ 87	251.10/1.0/x	$2{\mathrm{p}}^{5}3{\mathrm{s}}^{2}4\mathrm{s}{}_{}{}^{3}P{}_1$
253.55(2)/1.4(2)/x	257.51/1.8/87	252.71/0.8/x	$2{\mathrm{p}}^{5}3{\mathrm{s}}^{2}4\mathrm{s}{}_{}{}^{3}P{}_1$
265.33(3)/13(2)/x	269.73/12.2/59	264.36/14.0/x	$2{\mathrm{p}}^{5}3{\mathrm{s}}^{2}4\mathrm{d}{}_{}{}^{3}D{}_1$
266.88(3)/21(3)/x	271.18/17.9/53	265.96/21.8/x	$2{\mathrm{p}}^{5}3{\mathrm{s}}^{2}4\mathrm{d}{}_{}{}^{3}P{}_1$
278.01(3)/4.1(7)/x
278.75/8.5(1.3)/x		276.98/9.09/x	$2{\mathrm{p}}^{5}3{\mathrm{s}}^{2}5\mathrm{d}{}_{}{}^{3}P{}_1$
280.05(3)/8.4(1.3)/x		278.56/7.57/x	$2{\mathrm{p}}^{5}3{\mathrm{s}}^{2}5\mathrm{d}{}_{}{}^{3}P{}_1$
282.66(3)/6.5(1.0)/x		282.60/21.9/x	$2\mathrm{s}2{\mathrm{p}}^{6}3{\mathrm{s}}^{2}3\mathrm{p}{}_{}{}^{1}P{}_1$
329.6(1)/x/x	336.88/x/x	328.85/4.27/x	$2\mathrm{s}2{\mathrm{p}}^{6}3{\mathrm{s}}^{2}4\mathrm{p}{}_{}{}^{1}P{}_1$
346.7(1)/x/x	353.38/x/x	345.47/1.77/x	$2\mathrm{s}2{\mathrm{p}}^{6}3{\mathrm{s}}^{2}5\mathrm{p}{}_{}{}^{1}P{}_1$
355.3(1)/x/x	362.03/x/x	353.69/0.88/x	$2\mathrm{s}2{\mathrm{p}}^{6}3{\mathrm{s}}^{2}6\mathrm{p}{}_{}{}^{1}P{}_1$
360.4(1)/x/x	366.67/x/x	358.38/0.49/x	$2\mathrm{s}2{\mathrm{p}}^{6}3{\mathrm{s}}^{2}7\mathrm{p}{}_{}{}^{1}P{}_1$

* The symbol x indicates partial absence of data for that entry. No entry at all means that there are no data available. ** The number in brackets is the experimental uncertainty on the last digit of the data, i.e., 235.98(2)/26(4)/51(4) is the same as (235.98 ± 0.02) eV/(26 ± 4) Mb.eV/(51 ± 4) meV. The same notation convention is used for all the data presented in this column. *** These resonances are due to 2p → 3d excitations from the initial $2p^{6}3s3p{}_{}{}^{3}P{}_{0,2}$ metastable states.

, we gather together the experimental measurements and the corresponding theoretical predictions for all resonances observed between 225 and 370 eV, most of which have been shown in detail in Figure 5, Figure 6 and Figure 7. Table 2 shows that the vast majority of the observed resonances are reasonably accounted for by both theories in terms of resonance energy, strength, and width. In addition, the supernumerary low-intensity peaks seen in Figure 5 are readily interpreted as transitions from the $2{\mathrm{p}}^{6}3\mathrm{s}3\mathrm{p}{}_{}{}^{3}P{}_{0,2}$ metastable states to ${}^{3}S{,}^{3}D$ states of the multiply excited $2{\mathrm{p}}^{5}3\mathrm{s}3\mathrm{p}3\mathrm{d}$ configuration. This is consistent with our interpretation of the 195 eV–201 eV low-energy photon region (see Section 4.1.1).

4.2. Evolution along the Mg-like Sequence

Studies of isoelectronic sequences of atomic energy levels [53] and absorption oscillator strengths (f-values) [54], line strengths or cross sections are of fundamental interest. New atomic data can be inferred from isoelectronic sequences via the interpolation/extrapolation to neighboring atoms and ions of sets of $f$-values/energies exhibiting smooth and systematic variations. Further, trends along isoelectronic sequences are also of great general interest because a smooth trend can be modeled using $Z$-dependent perturbation theory. Irregular, non-monotonic trends are also known to occur, and are commonly related to the crossing of levels of correlation–mixed states from interacting electronic configurations or spin–orbit mixed states [55]. The magnesium spectrum is a case in point for correlation mixing due to the strongly perturbing effects of the $3p^2$ configuration on the $3s3d$ configuration as well as over the entire $3dns$ series indeed. As pointed out in the introduction, these effects are found to persist in the presence of a $2p$ vacancy. Therefore, it is interesting to view the Cl⁵⁺ results within the overall context of the magnesium isoelectronic sequence.

Figure 8 provides a full view of both the single and double experimental photoionization cross sections, measured in synchrotron radiation-based, merged-beam experiments, including Al⁺ [9] (ASTRID), Si²⁺ [10] (SuperACO), S⁴⁺ [18] (SOLEIL), and Cl⁵⁺ (SOLEIL, this study). Although it is not possible to make detailed and/or quantitative intercomparisons between the four traces of Figure 8 due to their very different photon energy ranges with associated different spectral dispersions (determined by the beamline instrumentation), as well as the choice of experimental monochromator bandpass(es), the following qualitative remarks can be made. As expected, the resonance structures shift markedly to higher energies as the nuclear charge increases. The corresponding trends are explored in greater detail below for selected transitions. Progressing along the series, the progressive discretization of the SI resonance structure (i.e., mostly the $2{\mathrm{p}}^{5}3{\mathrm{s}}^2\mathrm{nd}$ resonances) into a seemingly hydrogenic Rydberg series for Cl⁵⁺, as well as the concomitant redistribution of the line strengths into the first $2{\mathrm{p}}^{5}3{\mathrm{s}}^{2}3\mathrm{d}$ series member, are apparent.

In order to better visualize and characterize the main features of isoelectronic evolution, Figure 9 shows the total cross sections for the ions Al⁺ to Cl⁵⁺; however, these are displayed on modified photon energy scales so that the resulting spectra line up for better inter-comparison. To accomplish this, the energy scales are multiplied by appropriate numerical factors and suitably shifted so that the main $2\mathrm{p}\to 3\mathrm{d}$ and $2\mathrm{p}\to 4\mathrm{d}$ resonance structures line up (i.e., the same $2\mathrm{p}\to 3\mathrm{d}$, $2\mathrm{p}\to 4\mathrm{d}$ energy difference is maintained for the four ions) below each other. It is clear from Figure 9 that for S⁴⁺ and Cl⁵⁺, the overall resonance structures are very similar. This confirms that by S⁴⁺, the increased core charge has become the dominating factor, simplifying the spectra as they become more hydrogenic in character. For the early members of the sequence, the $2\mathrm{p}\to 3\mathrm{d}$ structures show considerable changes as we move from Al⁺ to S⁴⁺: the supernumerary resonances to the lower energy side of the $2\mathrm{p}\to 3\mathrm{d}$ group, in particular, are strong for Al⁺, weaker for Si²⁺, and much weaker for S⁴⁺ and Cl⁵⁺. This shows the rapidly diminishing effects of the main $2{\mathrm{p}}^{5}3\mathrm{s}\left(3{\mathrm{p}}^2\right)$ series perturber (see above).

Table 3. Top line: Experimental energy (eV), middle line: Non radiative (NR) width (meV), and bottom line: Strength (Mb eV) for the strongest $2\mathrm{p}\to 3{\mathrm{d}}^1{P}_1,{}^3{D}_1{,}^3{P}_1$ resonances, energy of the 2p limit (eV), and experimental energy of the $2\mathrm{s}\to 3\mathrm{p}$ ${}^1{P}_1$ resonance along the magnesium-like sequence. Experimental data and their references are provided for all members of the sequence up to Cl⁵⁺, apart from P³⁺, for which interpolated values are deduced. The symbol x indicates that data are not available. The number in brackets is the experimental uncertainty on the last digit of the data, i.e., 55.492(1) is the same as (55.492 ± 0.001) etc. The same convention is used for all the data presented in this table where error values are available.

Atom/Ion	$2\mathit{p}\to 3\mathit{d}$ ${}^1{\mathit{P}}_1$	$2\mathit{p}\to 3\mathit{d}$ ${}^3{\mathit{D}}_1$	$2\mathit{p}\to 3\mathit{d}$ ${}^3{\mathit{P}}_1$	2p Limit	$2\mathit{s}\to 3\mathit{p}$ ${}^1{\mathit{P}}_1$
Mg I	55.492(1)	55.677(1)	55.838(1)	57.658(2) e	x
	4.6(9)	5.0(1)	8.2(4)
	x	x	x
Al II	84.99(2)	85.36(2)	85.57(2)	91.75(15) c	121.5(5)
	x	x	x
	5.6 b	0.54 b	1.5 b
Si III	117.6(1)	118.1(1)	118.8(1)	133.5(1) c	155.7(4)
	x	x	x
	37(7) b	17(3) b	5(1) b
P IV (present work) a	154.4	153.9	154.7	180.5	193.6
	x x	x x	x x
S V	194.88 51 46(7) b	192.74 47 32(5) b	193.42 149 13(2) b	237.35 c	235.4
Cl VI (present work)	238.61(2) 46(2) 68(10) b	235.98(2) 51(4) 26(4) b	236.71(3) 158(65) 9(2) b	299.6 d	282.66

^a For P IV, the entries are obtained by interpolation of the isoelectronic plots of Figure 10 and italicized for clarity. ^b Not corrected for possible contributions of metastable states. ^c Values estimated from the experimental photon energy of the onset of double ionization. ^d From hydrogenic fit (see text). ^e Statistically averaged over the ${}^2{P}_{1/2}$ and ${}^2{P}_{3/2}$ components.

shows the main energy level and strength data (resonance energies and inner ionization thresholds) for all the early members of the sequence including magnesium. Where necessary, explanations regarding the origin of some data entries are provided in the footnote of Table 3. We plotted this data in two different ways in order to establish isoelectronic trends and interpolate with the missing P³⁺ ion.

Firstly, as shown in Figure 10, the cross section data (represented to scale for all the ions where electronic data are available) are displayed as a function of photon energy, and parametrically, as a function of the net core charge $\mathsf{\zeta }$ seen by a Rydberg electron ($\mathsf{\zeta }=1$ for Mg, $\mathsf{\zeta }=2$ for Al⁺, etc…). The spectra are shifted vertically from one to the next by an arbitrary amount of 100 Mb for clarity. The three curves in Figure 10 show the movement of the strongest $2\mathrm{p}\to 3\mathrm{d}$ (the energy differences between the three possible final states of Table 3 are too small to be seen in Figure 9) and $2\mathrm{s}\to 3\mathrm{p}$ resonances, and the 2p ionization threshold as the net core charge increases from one to six. Noting the very smooth evolution of the curves along the sequence, the data are very accurately represented by 3rd-order polynomial fits (red-, blue- and green-colored broken lines, respectively), it seems justified to derive results for the missing member P³⁺ by interpolation, we show the corresponding data in Table 3.

We note that for the first four members, the $2\mathrm{s}\to 3\mathrm{p}$ resonance is found at energies above the $2\mathrm{p}$ threshold. For a value of the net core charge close to five (vicinity of S⁴⁺), this situation is reversed. This is discussed in greater detail below.

Secondly, we examine the isoelectronic trends of characteristic (resonance and threshold) energies and absorption oscillator strength, as shown in Figure 11a,b, respectively. Figure 11a plots the $E_{2\mathrm{p}\to 3\mathrm{d}}$ and $E_{2\mathrm{s}\to 3\mathrm{p}}$ transition energies divided by the net core charge, referred to the energy value of the first inner limit $2{\mathrm{p}}^{5}3{\mathrm{s}}^2{}_{}{}^{2}P$ $\left(E_{2\mathrm{p}}\right),$ divided by the net core charge ($\zeta$), i.e., $\left(E_{2\mathrm{p}\to 3\mathrm{d}}-E_{2\mathrm{p}}\right)/\zeta$ and $\left(E_{2\mathrm{s}\to 3\mathrm{p}}-E_{2\mathrm{p}}\right)/\zeta$, respectively. The energy differences between the three possible final states ${}^1{P}_1,{ }^3{D}_1{,}^3{P}_1$ are too small to be seen in Figure 11a. Further relevant data in the form of the theoretical 2s⁻¹ threshold energies taken from [56], referred to the 2p⁻¹ threshold in the same manner, are also included in Figure 11a.

From Figure 11a, we observe a regular (linear) series behavior for the $2{\mathrm{p}}^{5}3{\mathrm{s}}^{2}3\mathrm{d}$ average term energy, while for both the $2\mathrm{s}2{\mathrm{p}}^{6}3{\mathrm{s}}^{2}3\mathrm{p}{}_{}{}^{1}P{}_1$ and the 2s⁻¹ threshold energies, the series behaviors are clearly non-linear, although smooth. Notably, the energy position of the first member of the asymmetric profile $2\mathrm{s}\to 3\mathrm{p}{}_{}{}^{1}P{}_1$ excitation moves closer to the 2p⁻¹ threshold as the core charge increases, and drops just below and well below the 2p⁻¹ threshold for S⁴⁺ [18] and Cl⁵⁺, respectively. This crossing of the 2p⁻¹ threshold by the $2\mathrm{s}2{\mathrm{p}}^{6}3{\mathrm{s}}^{2}3\mathrm{p}{}_{}{}^{1}P{}_1$ state is accompanied by a noticeable change from a Fano to a Lorentzian profile and a prominence of the resonance in the single ionization channel only. As one moves further up the sequence, this downward movement toward the 2p⁻¹ threshold (and finally below) is expected to be replicated for the higher members of the series (n > 3). We note the similarity of this behavior with the previously observed behavior for the $2\mathrm{s}\to \mathrm{np}$ transitions in the neon-like sequence [57]. This suggests a lesser role played by the aforementioned ($3{\mathrm{s}}^2+3{\mathrm{p}}^2+3{\mathrm{d}}^2)$ correlations on the $2\mathrm{s}\to 3\mathrm{p}$ transition compared with the $2\mathrm{p}\to 3\mathrm{d}$ for the early members of the Mg-like sequence. If we interpolate the data for P³⁺ from Figure 11a, we obtain the values 154 eV (2p-3d) and 195 eV (2s-3p), which compare favorably with the corresponding predictions from Figure 10, as shown in Table 3. The value of the 2p⁻¹ threshold for P³⁺, which is needed to obtain the data just quoted, was estimated from [58] to be (183 ± 1) eV. This is in reasonable agreement with the interpolated value shown in Table 3.

In Figure 11b, we plot the oscillator strengths $f_{i\to j}$ for the $2\mathrm{p}\to 3\mathrm{d}{ }^1{P}_1,{ }^3{D}_1{,}^3{P}_1$ transitions obtained from the integrated cross sections (in Mb eV) shown in Table 3, using the standard equation ${\int }_{line}\sigma \left(E\right)dE=109.8f_{i\to j}{\int }_{line}^{}{\Phi }_{ij}\left(E\right)dE$, where ${\Phi }_{ij}$ is the normalized line profile as a function of $1/Z$, where $Z$ is the atomic number. Correction factors of 0.95, 0.975, 0.78, and 0.77 were applied to the Al⁺, Si²⁺, S⁴⁺, and Cl⁵⁺ strength values, respectively, in Table 3 to take into account the different initial ground state populations. The curves for the $2\mathrm{p}\to 3\mathrm{d}{ }^3{D}_1{,}^3{P}_1$ transitions are seen to present a maximum (near S⁴⁺), which is characteristic of series where configuration interaction effects are known to be important [55]. This is fully compatible with our previous discussions regarding the effects of the ($3{\mathrm{s}}^{2}3\mathrm{d}+3\mathrm{s}3{\mathrm{p}}^2)$ mixing along the magnesium sequence. As the nuclear charge increases, the oscillator strength transfers into the fully LS allowed $2\mathrm{p}{}_{}{}^{1}S{}_0\to 3\mathrm{d}{ }^1{P}_1$ transition, in tandem with the weakening of the configuration interaction effects discussed above.

5. Conclusions

The development of the storage ring-based, merged-beam technique has allowed the systematic study of inner-shell photoionization for extended isoelectronic sequences. For magnesium-like ions, the current results extend the sequence up to Cl⁵⁺. The similarity of the S⁴⁺ and Cl⁵⁺ ion yields implies that the photoionization behavior settles down when compared with earlier members of the sequence. The new experimental results for Cl⁵⁺ are compared with ab initio MCDF and R-matrix calculations. Differences between the experimental data and the theoretical predictions underscore the importance of the ongoing benchmarking of theory. While the relative energies and strengths of resonances are reasonably predicted by theory, it is clear that significant systematic energy shifts are required to bring the theoretically predicted resonance structures into reasonable coincidence with experimental data. Interpolation of the results along the sequence provides estimates for the energies of the 2p–3d and 2s–3p resonances and the 2p⁻¹ and 2s⁻¹ thresholds for the P³⁺ ion, which have not yet been experimentally studied.