Observation of strong two-electron–one-photon transitions in few-electron ions

We resonantly excite the K series of O 5 + and O 6 + up to principal quantum number n = 11 with monochromatic x rays, producing K -shell holes, and observe their relaxation by soft-x-ray emission. Some photoabsorption resonances of O 5 + reveal strong two-electron–one-photon (TEOP) transitions. We ﬁnd that for the [ ( 1 s 2 s ) 1 5 p 3 / 2 ] 3 / 2;1 / 2 states, TEOP relaxation is by far stronger than the radiative decay and competes with the usually much faster Auger decay path. This enhanced TEOP decay arises from a strong correlation with the near-degenerate upper states [ ( 1 s 2 p 3 / 2 ) 1 4 s ] 3 / 2;1 / 2 of a Li-like satellite blend of the He-like K α transition. Even in three-electron systems, TEOP transitions can play a dominant role, and the present results should guide further research on the ubiquitous and abundant many-electron ions where electronic energy degeneracies are far more common and conﬁguration mixing is stronger.


I. INTRODUCTION
In hot astrophysical plasmas, the most common elements, hydrogen and helium, are fully ionized, and only those with higher nuclear charge can keep some bound electrons, appearing as highly charged ions (HCIs) [1].The widths, Doppler shifts and relative intensities of their characteristic lines are recorded by x-ray observatories and analyzed for plasma diagnostics, relying not only on tabulated calculations but also on more scarce laboratory data.To fully exploit the data of current and upcoming high-resolution x-ray missions such as XRISM [2] and Athena [3], more accurate laboratory tests of the atomic models used in astrophysics are needed [1,4].Light elements such as carbon, nitrogen, and the oxygen studied here abundantly appear as HCIs over a broad range of temperatures and can thus serve as unique spectroscopic probes of, e.g., the warm-hot intergalactic medium (WHIM), which is critical to a complete census of baryonic matter in the universe [5][6][7].It is important to have knowledge of both the photoabsorption cross sections and the various decay channels that govern the fluorescence yield and the ionization balance in plasmas.After x-ray absorption takes place, the most common relaxation processes are direct radiative decay and autoionization.However, even in few-electron systems, more complex processes and multi-electron transitions also compete with them.Including such mechanisms in models is computationally intensive, and hence laboratory data are needed to guide those efforts [8].
Here, we report on resonant excitation of the K series of He-like and Li-like oxygen ions between 570 and 750 eV using monoenergetic soft x rays.We detect their fluorescencephoton yield and energy as a function of the incident photon energy, and observe surprisingly strong and sometimes dominating two-electron-one-photon (TEOP) transitions in Li-like oxygen.

II. EXPERIMENTAL SETUP
Our experiment was conducted at the variable-polarization XUV beamline P04 [45] of the PETRA III synchrotron facility with a portable electron beam ion trap (EBIT), PolarX [46] (see Fig. 1).Molecular oxygen was injected into the EBIT, dissociated and successively ionized yielding a large He-like O 6+ population and a small Li-like O 5+ fraction.These HCI are radially trapped by the electron beam (here 3 mA, to reduce ion heating), and axially confined within a potential well formed by making the central drift tube slightly more negative than the adjacent ones.With an electron-beam energy of ∼200 eV just above the Li-like ionization threshold, we produce He-like O 6+ , but stay below the excitation threshold of Kα or higher Kn series transitions.This ensures a lowbackground measurement of the K series fluorescence by a silicon-drift detector (SDD) mounted side-on above the central drift tube where the ions are confined.
The P04 beamline is equipped with an APPLE-II undulator covering the photon energy range 250 −-3000 eV, and a grating monochromator (1200 lines/mm) providing circularly polarized light at a resolving power of more than 10 4 [45].Expected long-time drifts of the monochromator recommended for this overview measurement a fast-scan mode lasting less than one hour, forcing the use of a wide slit (200 µm) for better statistics.This gives us a photon flux on the order of 10 12 photons/s at a moderate resolving power.Nonetheless, we could determine excitation energies with a relative uncertainty of ∆E/E ≈ 10 −5 .For this, we digitize the SDD-energy signal (y axis) for each photon-detection event while continuously scanning the monochromator, i. e., the incident photon energy (x axis), obtaining a two-dimensional fluorescence histogram (Fig. 3a).To remove background events due to electron recombination, we subtract at each resonance the offresonance mean count rate from the on-resonance signal (see an example in Fig. 2).Then, we project the region of interest containing the resonance onto both axes, and fit Gaussians with a full widths at half maximum of order of 350 meV (x axis) and order of 100 eV (y axis) to those projections.

III. MEASUREMENT OF PHOTOEXCITATION ENERGIES
A monochromator scan from 570 eV to 740 eV at 500 meV s −1 resolved the core excited K series of He-like oxygen up to Kκ, as well as several other weaker Li-like resonances (Fig. 3).After determining their centroid positions (see Fig. 3a), we assign to the six transitions Kα up to Kζ (identified on the approximately calibrated x axis for the incident photon-energy in Fig. 3) energies taken from accurate calculations by Yerokhin and Surzhykov [48] with uncertainties on the order of 0.5 meV and determine the final monochromator-dispersion curve with a linear fit.Its confidence interval is basically dominated by the order of 20-meV statistical uncertainties of the individual transitions in our fast overview scan.By extrapolating the dispersion curve we obtain the excitation energies of the Kη, Kθ, Kι and Kκ transitions (see Table I).Using these data points, we are able to determine the ionization potential of O 6+ .We use a quantumdefect model based on the Rydberg formula, with the Rydberg energy E R , the effective nuclear charge Z eff and the quantum defect δ n,l for principal n and orbital l quantum numbers, respectively: Fitting this model (Fig. 4) yields an ionization potential E IP ) of O 6+ of E IP = 739.336(16) eV, which agrees very well with the 739.326 82 (6) eV predicted by Drake [49] and 739.326 262 eV by Tupitsyn et al. [50] (Table II).
For the fluorescence-photon energy calibration of the SDD we also use the K series transitions of He-like oxygen (Fig. 3a) up to Kζ.Each component of this series shows a wellresolved, elastic single photon decay to the ground state, thus allowing us to assign to the centroids of their y projections the same energies as the respective exciting photon of the x projection.

A. Measurement of fluorescence-photon energies
We measure the TEOP-transition energies in fluorescence to distinguish them from other channels.For both Li-like Kβ 1 at 640 eV and Kγ at 666 eV, we observed the OEOP radiative decay channel into the ground state: 1s 2s 3p → 1s 2 2s and 1s 2s 4p → 1s 2 2s.
Hereafter, we refer the radiative decay channel towards the ground state as direct decay (DD), to distinguish from sequential two-photon decays (TPD), such as 1s 2s 3p → 1s 2s 2 → 1s 2 2p.Direct decay is the time-inverse process of photoexcitation (PE), and the overall process of PE plus DD is equivalent to elastic fluorescence emission, as apparent for Kβ 1 and Kγ in the decay spectrum of Fig. 3a. Figure 5 shows the decay spectra for Kβ 1 and Kγ, confirming these DD channels.However, as also displayed in Fig. 5 calculations of the main decay channels of the lines presented in Fig. 5 with the Flexible Atomic Code (FAC) [83], which provide us with transition rates missing in the high-accuracy calculations of Yerokhin and Surzhykov [48].
While doubly-excited states commonly relax by AD, our FAC [83] calculations show that this channel is only relevant for Kβ 1 and Kγ (see Fig. 6), and also confirm a main DD for Kβ 1 and Kγ.After PE of Kβ 2 , DD competes with the TEOP transition feeding into the 1s 2 3s and 1s 2 3d states (roughly 5 eV apart) that can radiatively decay through various cascades.This results from configuration mixing with the 1s 2p 3s and 1s 2p 3d states.For Kδ, no significant DD could be observed.Here, the upper state dominantly relaxes through a TEOP transition to the 1s 2 4s state.
The question is, what makes the usual DD to the ground state so weak?As shown in Fig. 3c, the excited 1s 2s 5p state has a near degeneracy (0.8 eV) with a state of the same total angular momentum and parity, 1s 2p 4s, which is also the upper state of a Li-like satellite of the He-like Kα line.Thus, the excited states strongly mix with these, which have much higher decay rates towards 1s 2 4s (on the order of 10 12 s −1 ).This suppresses the one-photon DD to the ground state, as can be seen in Fig. 5.Here DD denotes the direct one-photon decay to the ground state, TEOP the two-electron-one photon decay, TPD the two-photon decay cascade, and AD Auger decay.

B. Role of configuration mixing: Calculations
We investigate the underlying quantum processes by calculating electronic energies, transition rates, and Auger rates with the relativistic configuration-interaction package FAC [83].The convergence of the configuration mixing was studied by varying the size of the configuration-interaction (CI) basis set, as listed in Table III.This allowed us to identify the key configurations leading to a strong TEOP rate.
Figure 7 displays the effect of the CI basis size on the transition rates from the upper state [(1s2s) 1 5p 3/2 ] 1/2 towards the final states [1s 2 2s] 1/2 (the OEOP transition) and [1s 2 4s] 1/2 (the TEOP transition), and on the Auger rate to the 1s 2 state (the only possible Auger channel).All the decay rates have an allowed electric dipole contribution that dominates higher-order multipoles.The energy degeneracy, defined as the smallest energy difference between this initial state and the nearest one having the same total momentum and parity symmetry in a different configuration, is also represented.While the OEOP rate is nearly independent of the CI-basis set, a sudden increase of three orders of magnitude in the TEOP transition appears when the 1s2p4s configuration is included, which leads to a mixed state with contributions of order of 75% from [(1s2s) 1 5p 1/2,3/2 ] 1/2 and of order of 25% from [(1s2p 1/2,3/2 ) 1 4s] 1/2 .Adding further configurations lets the TEOP rate converge towards a value that is approximately twice that of the Auger process, and eight times that of OEOP decay.The above-mentioned energy degeneracy becomes more pronounced with the inclusion of further configurations, as the energy separation decreases from tens of eV to order of 0.8 eV.This causes the mixing coefficient to grow by order of 28% for [(1s2p 1/2,3/2 ) 1 4s] 1/2 , which combined with the high decay rate of the [1s2p4s] 1/2 → [1s 2 4s] 1/2 transition (the satellite of the He-like Kα) makes the TEOP rate for [1s2s5p] predominant.The initial [(1s2s) 1 5p 3/2 ] 3/2 state also follows a similar behavior.The Auger rates drop due to mixing with configurations of higher orbital momen-  III in text).Both transition rates to final states [1s 2 2s] 1/2 (the OEOP transition) and [1s 2 4s] 1/2 (the TEOP transition) are included, along with the Auger rate to 1s 2 state (left y-scale).Open circles represent the energy difference (degeneracy) in eV (right y-scale).TABLE IV.Core-hole relaxation following Kβ1,2, Kγ, and Kδ resonant photon excitation in O 5+ ions.The first column lists the excitation transitions.Subsequent columns indicate the decay channels' respective initial and final configurations, peak photon energies (from the silicon-drift detector), final-state energies (calculated with FAC [83]), and transition type (DD denotes direct photon decay, and TEOP the two-electron-one-photon transition).The DD-to-TEOP intensity ratio is also given.All energies are in eV.tum having lower Auger rates.The TEOP rate for the initial state [(1s2s) 0 5p 1/2 ] 1/2 does not show a significant increase with inclusion of [1s2p4s] 1/2 in the basis set.This is due to the energy difference with [1s 2 4s] 1/2 being of order of a few eV, which reduces the mixing coefficient of the [1s2p4s] 1/2 configuration to 1.3%.Summarizing our theoretical analysis, Fig. 6 shows that including the near degenerate 1s 2p 4s state in the calculation of the Kδ decay rates increases the TEOP rate by more than three orders of magnitude.Besides this, we also checked another plausible photoexcitation channel 1s 2 2s → 1s 2p 4s that can also decay to 1s 2 4s ground state.This alternative path can possibly further enhance the observed TEOP channel.How-ever, a comparison of photoexcitation rates between 1s 2 2s → 1s 2p 4s (approximately equal to 2 × 10 10 s −1 ) and 1s 2 2s → 1s 2s 5p (approximately equal to 1 × 10 11 s −1 ) shows an order of magnitude difference between them.Therefore, we emphasize that the 1s 2p 4s state can only be populated via Kδ photoexcitation and decay via the TEOP channel, as observed in the present experiment.

C. Determination of the ratio DD/TEOP
Our measured Kβ 2 decay energies are listed in Table IV.Because the respective fluorescence-decay channels can be barely resolved by the SDD (cf.Fig. 5), their centroids were only determined with an uncertainty in the 0.5-1.3%range; this can be improved in the future, e. g., by using a highresolution x-ray microcalorimeter [84].Since measuring line ratios is an essential plasma diagnostic tool, and stringently tests theory, we extract the ratio of the DD rate to the TEOP transition rate from the measured spectra (Fig. 5).While the DD-to-TEOP ratio of Kδ could not be accurately determined due to low statistics in the DD channel, it was nonetheless possible to quantify the DD-to-TEOP ratio for the Li-like Kβ 2 emission.For this we characterized the x-ray detector taking into account the transmission of the 500-nm Al filter placed in front of it to block visible and UV radiation, and also the low-energy Compton tail of the detector response line shape.This tail was obtained from a fit to the Kβ 1 transition, which should only have the elastic channel, as configuration mixing with other states is small, and used as fit function for the DD channel of Kβ 2 .We estimate the uncertainty of the Al-filter transmission from its thickness (500 ± 100 nm), adding a contribution of ±15% to the ratio-error budget.The soft x-ray spectral sensitivity of the SDD strongly depends on the thickness of native silicon dioxide layer on its surface, which is not well known (20 to 50 nm), on the top-electrode materials, and on the slow condensation of water on its cold surface during the experiment.Since one of the decay channels (the TEOP transition) of Kβ 2 is close to the oxygen K edge, these layers can significantly change the ratio of the transmission coefficients for the 1s2s3p → 1s 2 transition at 644 eV versus the one of the 1s2s3p → 1s 2 3d at 565 eV.With these caveats, and assuming a filter transmission ratio (644 eV-to-565 eV) approximately equal to 1.25 ± 0.2 and a SDD sensitivity ratio (644 eV-to-565 eV) of approximately 1.15 ± 0.5, the observed intensity ratio of 2.93 yields a ratio of approximately 2.0 ± 0.8, which is compatible with the FAC prediction.In a second approach, we determine the DD-to-TEOP ratio by comparing the intensity ratio Kα/Kβ = 2.06 ± 0.05 of the simultaneously observed He-like transitions with the ratio of their theoretical Einstein coefficients, A ik /A ik = 3.53 according to the NIST database [54].We normalize the observed intensities to the excitation-photon flux, and obtain a sensitivity ratio (665.61eV-to-573.94eV) approximately equal to 1.70 ± 0.17, whereby the decay channels very closely match the energies of the Li-like transitions under investigation.This takes into account all previously mentioned effects of the filter and detector efficiency.When we interpolate this sensitivity ratio to the close-by Li-like transitions, the observed intensity ratio (approximately equal to 2.93) for the Li-like Kβ 2 decay channels results in a ratio of approximately 1.73±0.19, in fair agreement with our FAC calculation.
Emission of Kβ 2 follows PE of the states [(1s2s) 0 3p 1/2 ] 1/2 and [(1s2s) 0 3p 3/2 ] 3/2 feeding the radiative decay channels listed in Table V.Their respective strengths yield a DD-to-TEOP ratio of 1.39 for an observation angle of 90 • (see the Appendix) and 1.43 for the solid-angle integrated total emission.Note that the decay rates from these states to the ground state are similar and likewise the corresponding PE cross sections.This cancels the effect of state population on the ratio.

V. CONCLUSION
We have demonstrated how a complex process that is difficult to disentangle in astrophysical plasmas can be isolated and studied in detail by high-resolution photon excitation.Unexpectedly strong TEOP transitions were found in an essential species, the relatively simple Li-like O 5+ , showing, among other observations, evidence that the upper state of the Kδ line in Li-like O 5+ mainly decays as a satellite of He-like O 6+ Kα.This produces a problematic blend in a key feature for the diagnostics of photoionized plasmas (e.g.[85]).Although a strong suppression of the direct photo decay by TEOP transitions was observed in just one of several lines in Li-like oxygen, it is not far-fetched to assume that TEOP-dominated relaxation also happens in other multiply excited, multi-electron systems, and thus its contribution should not be neglected in accurate astrophysical plasma models.Systems with more than three electrons have richer overlapping excitations with manifold decay channels not only cause similar blends in emission and absorption spectra, but also affect the ionization balance of plasmas.The three-electron system studied here is more tractable by current theory and has allowed us to stringently test the underlying electronic correlations.This is of great importance for the diagnostics of hot gas in astrophysics.The upcoming launches of XRISM [2] and Athena [3] urgently call for studying the position and strength of TEOP transitions that can cause shifts, or broaden the strong diagnostically important O-K and Fe-L lines in the 15-23 Å range, which are crucial for determining gas-outflow velocities of warm absorbers and density diagnostics of photoionized plasmas [39,[85][86][87][88], and needed for accurately modeling the x-ray continuum flux.

FIG. 2
FIG.2.a) Fluorescence yield (recorded by the SDD) of He-like O 6+ K β under excitation with 663 −668 eV photons (x axis) versus fluorescence-photon energy (y axis), and (black curve) projection of the photon yield onto x. b) Projections onto the y axis of on/off resonance slices (red/blue histograms).The off-projection is used for subtraction of electron-induced background and SDD-noise.

FIG. 3 .
FIG.3.a) Shown on top is a histogram of events registered with the SDD: fluorescence photon energy (y axis) versus excitation photon energy (x axis).The dashed diagonal line indicates elastic channels, with excitation and the fluorescence photon at equal energies.Shown on the bottom is fluorescence from resonantly photoexcited He-like O 6+ and Li-like O 5+ , labeled in black and in color, respectively.b) He-like and c) Li-like level diagrams.Configuration mixing of the upper levels of Kδ causes a predominance of the TEOP process over both the elastic direct decay to the ground state and the Auger channels.Rates are in units of s −1 .

FIG. 4 . 1 FIG. 5 .
FIG.4.Fit of the Rydberg formula to the measured He-like K series with nuclear effective charge Ze, quantum defect δ and ionization potential (EIP) as a free parameter.The blue shaded area indicates the confidence interval.

FIG. 6
FIG.6.a) FAC transition-rate calculations for the strongest decay channels following resonant photoexcitation of the Kβ1,2, Kγ, and Kδ transitions in O 5+ ions, and b) their convergence depending on the configuration basis size (see the text).For Kδ, two stages of convergence are highlighted: red indicates incomplete and blue the final result.The respective final states and transition energies are labeled accordingly.Here DD denotes the direct one-photon decay to the ground state, TEOP the two-electron-one photon decay, TPD the two-photon decay cascade, and AD Auger decay.

TABLE I .
Experimental excitation energies of He-like and Li-like oxygen absorption resonances (in eV).Results are compared to calculations using the relativistic configuration-interaction method (FAC: O VI, this work; RCI-QED: O VII

TABLE II .
Fit results in comparison with available theoretical and experimental values.

TABLE III .
Configurations added to the CI basis set for each calculation having the CI label of Fig.7; each set keeps all the configurations of the sets above it.

TABLE VI .
Angular coefficients βJ d →J f for angular momenta J d and Jf .