High-accuracy Measurements of Core-excited Transitions in Light Li-like Ions

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High-resolution grating spectrometers onboard the Earthorbiting X-ray telescopes Chandra and XMM-Newton have enabled the detection of inner-shell X-ray absorption lines in ionized outflows of active galactic nuclei (AGN) and the neutral interstellar medium (ISM) and warm-hot intergalactic medium (WHIM) [1][2][3][4].They reveal the physical conditions of the ionized absorbing medium, among others the velocities of outflows, plasma densities and temperatures [5][6][7].The strongest 1s − 2p inner-shell absorption lines of light Lilike ions (referred to as q, r, following notation of Gabriel) are among the most important lines observed in such environments.However, inaccurate transition energies introduced systematic uncertainties, e. g., discrepancies of up to 1.3 eV have been seen in predictions of q and r of oxygen at ∼ 560 eV.This caused a velocity uncertainty of 700 km/s, as large as the outflow velocities in nearby active galaxies, which hampered the understanding of multi-component outflows [8,9].Uncertain transition energies also hindered disentanglement of absorption from different charge states of oxygen in the ISM [10].Experiments at the LLNL EBIT [11] reduced the uncertainty to 20-40 km/s, but individual core-excited finestructure levels were not resolved.The q and r lines of Ne 7+ have also been proposed as electron-density diagnostics in flares of stellar coronae, but suffer both from their overlap with L-shell lines from iron and theoretical uncertainties [12].Similar problems occur with the X-ray lines q and r emitted following dielectronic recombination of helium-like ions needed for determining electron densities and temperatures in magnetically confined fusion plasmas [12][13][14].The utility the aforementioned cases depend on the quality of the utilized atomic data.
Unfortunately, Li-like ions still challenge contemporary calculations, which have uncertainties at the few-meV level for core-excited levels [15][16][17][18], far greater than in H-like and He-like systems [19,20].With few exceptions [21,22], earlier X-ray measurements could not benchmark predictions [23] of the core-excited states (1s( 2 S)2s2p( 3 P) 2 P 3/2 and 2 P 1/2 ) due to limited resolution [11,[24][25][26].Machado et al. [21] and Schlesser et al. [22] used for q and r in Ar and S a crystal spectrometer with a resolving power of ∼12000, achieving a wavelength accuracy better than 2-5 ppm, but at energies below 1 keV available crystals and gratings offer neither enough reflectivity nor resolving power.Other experiments using merged beams of ions and photons [27] were limited by insufficiently accurate calibrations with molecular X-ray absorption spectra [28,29].Systematic uncertainties were recently substantially reduced down to 15 ppm by using 1s − np transitions in He-like ions to independently recalibrate molecular transitions [30,31].State-of-the-art soft X-ray monochromators with resolution exceeding 20 000 [32] still suffer from calibration problems resulting from periodic errors of angular encoders in use [33].
In this Letter, we present measurements at the P04 beamline of the PETRA-III storage ring in Hamburg, Germany [34] resolving the q and r lines in Li-like oxygen, fluorine, and neon, and yielding transition energies with an accuracy of approximately 5 ppm.A more than 10-fold improvement made possible by mitigating systematic fluctuations of angu- lar encoders by simultaneous high-resolution photoelectron spectroscopy (XPS).Our accurate measurements benchmark predictions accounting for electron-electron correlation, QED and nuclear size effects, as well as general and specific relativistic mass shifts.Moreover, we provide accompanying large-scale calculations, which exhibit theoretical uncertainties on par with state-of-the-art predictions while taking the effect of autoionization shifts into account [18].We thereby provide, both experiment and theory to test and benchmark the state-of-the-art calculations of Li-like theory for few selected elements.
We acquire absorption spectra by scanning the incident photonenergy while counting the number of fluorescence photons after resonant excitation, which are detected by silicondrift detectors (SDD).At P04 [34], a photon beam of 10 14 γ/s at 0.1% bandwidth is generated by an undulator, which after monochromatization and transport losses results in an approximately 10 11 γ/s flux at the focus, which is placed at center of our compact electron beam ion trap, PolarX-EBIT [35].The ions are produced by PolarX-EBIT, by injecting a tenuous atomic or molecular beam containing the element of interest.It crosses the electron beam, which is focused by a magnetic field and set to an energy sufficient to generate and trap the respective He-like and Li-like ions, but below their K-shell excitation threshold.This ensures a good signal-tonoise in the silicon-drift detectors, which are equipped with 500 nm aluminium filters for blocking most of the low energy stray light.Both detectors are mounted side-on for registering soft fluorescence X-rays produced by electron impact and, crucially, upon resonant photoexcitation.This method was demonstrated at the free-electron laser FLASH as soft X-ray laser spectroscopy [36] and later also applied at synchrotronradiation facilities [30,33,[37][38][39][40].
After optimizing the monochromator using the strong, narrow w (1s2p 1 P 1 to 1s 2 1 S 0 ) line of oxygen [32] to achieve a resolving power of more than 30 000, we can resolve q and r.We then scan several times in discrete monochromator steps of nominal energies a range containing the q and r transitions of the Li-like ion, and calibration lines including the w line of the He-like ion of the same element, as well as a short series of 1s 2 to 1snp transitions of the next lower element in atomic number.
Knowing the actual photon energy depends on accurate readings of grating and mirror angles (θ 1 , θ 2 in Fig. 1) in the monochromator.Angular encoders measure them by the transmission of light between several glass disks patterned with opaque marks that overlap only at certain rotation-angle steps.While the narrow line width of the recorded transitions calls for steps of roughly 4 × 10 −5 degree, the spacing between the 36 000 reference marks of the Heidenhain RON 905 encoders used in the P04 monochromator delivers only 10 −2 -degree dark-bright cycles.Between those marks, encoders interpolate the angle changes thousand-fold using the quadrature signals of several diodes measuring the light modulated by minor disk rotations.This procedure is very sensitive to imperfections of those analog signals, and thus empirical look-up tables have to be regularly generated and stored in the hardware.However, residual errors remain.Previous observations at P04 and beamlines elsewhere showed that nominal photon energies calculated from such interpolated readouts had periodic sub-divisional errors [41,42], leading to fluctuations in the photon-energy readout.With two encoders needed to measure the diffraction angle, a double interpolation uncertainty should affect most X-ray monochromators worldwide [41,42].As we will discuss in the following, periodic changes with peak-to-peak amplitudes of up to 70 meV have been found in our case.
Since the off-axis electron gun of PolarX-EBIT lets the photon beam exit downstream unimpeded, we perform XPS [43] measurements for monitoring fluctuations of the actual photon-beam energy (See Fig. 1).The hemispherical photoelectron spectrometer (PES), ASPHERE [44] is permanently installed at P04 several meters downstream of the open port where PolarX-EBIT is mounted.After passing through it, the photon beam illuminates a gold target mounted in PES that is electrostatically biased, where photoelectrons are emitted from the Au 4 f 5/2,7/2 states known for their large cross sections [43,45,46] with a kinetic energy given by E kin = E γ − E 4 f + V bias .While V bias is scanned in par with the nominal photon energy in order to keep E kin nominally constant, PES selects 4 f 5/2,7/2 photoelectrons within a narrow (≈15 eV) range encompassing both states, and guides them to a microchannel-plate-amplified phosphor screen imaged on a camera.This high selectivity together with the short term stability of ASPHERE allows us to monitor periodic energy fluctuations from the nominal, linearly growing energy of each   The trace covers the energy range of the Li-like neon q and r measurement.(B,C,D) Example scans of q and r lines for neon, fluorine and oxygen and respective calibration lines are superimposed with their, simultaneously acquired, photoelectron trace, which monitors the deviations of the demanded photon energy from the actual energy.Each scan contains the resolved q and transitions of the Li-like ion, as well as various He-like calibration lines.
scan.While the 4 f 5/2,7/2 photoelectron peaks should have fixed positions at the detector, actual energy fluctuations induce small centroid shifts of both peaks.After projection of the detector image onto its dispersive axis, we continuously monitor them with a few meV statistical uncertainty by fitting two Voigt peaks and a linear background.Alternative fit models did not significantly improve the fits.During photon energy scans, the 4 f 5/2,7/2 peaks' oscillations (see Fig. 2 A) reflect the interpolation inaccuracies of the two angular encoders.Two distinct oscillation periods arise from the different distances of mirror and grating to undulator X-ray source and exit slit.This recording yields the photoelectron traces used for correction.To calibrate the kinetic-energy range covered by these traces on the photoelectron detector image, we scan the bias voltage at a constant photon energy, shifting the 4 f 5/2,7/2 peaks across the detector.Subsequently, traces are locally modeled at resonances using low-degree polynomials within narrow energy windows (See Fig. 2), and added as corrections to the nominal, yet uncalibrated photon energy scale derived from the monochromator-angle readout.Using this modified scale, the centroids of the HCI fluorescence resonances are determined by fitting Voigt functions.Under present experimental conditions, we see several sources comprising Gaussian and Lorentzian components, respectively.We associate the Gaussian contribution with the inherent limitations in resolution of the monochromator and the thermal motion of the ions.The Lorentzian width, as shown in [33], stems from the finite lifetime of the excited levels and the pseudo-Lorentz instrumental component due to X-ray diffraction at beamline components [42].For absolute calibration of the photon energy in each scan, we assign to the measured positions of the He-like transitions predicted energy values from Ref. [19], and fit the corresponding dispersion curves using linear functions, except for q and r of Li-like oxygen, where a 3. Comparison of our results with theory from Refs.[15,16].Theory-experiment energy difference of q and r (A) and their fine-structure splitting (B).Dashed lines mark experimental 1-sigma uncertainties, excluding those of our XPS data.Area shaded in green includes all uncertainties.Predictions and uncertainties of Refs.[15,16] as well as of our calculations are shaded in red and purple, respectively.The bold right-side axes in sub-figure (A) show the accuracy in units of km/s, corresponding to the uncertainty of the AGN-outflow velocity.
second-order polynomial was needed.We found a systematic shift in the energies of q and r depending on whether the correction was derived from the Au 4 f 7/2 or the 4 f 5/2 peak (blue and red data points in Fig. 3).By taking a weighted average of these individual results, we find this shift being largest for neon with approximately 1.8 meV, for fluorine 1 meV and negligible for oxygen measurements, and take it into account with an accordingly enlarged systematic uncertainty.These uncertainty bands are depicted as dashed lines in Fig. 3. Repeated XPS measurements also revealed a broader distribution of 4 f 7/2 , 4 f 5/2 centroids than statistically expected, which we attribute to instabilities in the voltage sources of PES.We estimate this systematic error from the distribution widths found respectively as 5.3 meV, 2.4 meV and 2.3 meV for Ne, F, and O and add it in quadrature to the total.See Supplemental Material [47] for details on error estimation.
We then compare the measured energies of q and r with high-precision calculations of both the ground state and the excited 1s2s2p states, including contributions from electronic correlations, quantum electrodynamics (QED), and nuclear recoil.Since the excited states can decay via electron emission, a so-called Auger-Meitner channel (see Fig. 1), they do not have square integrable wave functions.For this reason, the energies of autoionizing states can exhibit a strong dependence on the basis set parameters, which limits the accuracy of the standard high precision approaches such as CI or Coupled Cluster.To properly account for the energy shift resulting from the Auger-Meitner channel [48], we have used the complex scaling method [18,[49][50][51] to evaluate the energies of 1s2l2l ′ levels of Li-like oxygen, fluorine, and neon with extended configuration space.
In Table I and Fig. 3, we compare the experimental data and our new calculations with the existing predictions of Yerokhin et al. [15] based on the basis-balancing method for the treat-ment of the autoionization channel.For all elements, our theoretical values for the q and r transitions show a shift of ∼5 meV and ∼2 meV, respectively, from those of Ref. [15,16].Although our complex rotation method is better suited for the complete treatment of states with autoionization channels, our experiment shows overall better agreement with Ref. [15].It is interesting to note that both predictions of fine structure splitting, i.e. the differences between q and r, agree well with our experiment.This suggests that the likely cause of the discrepancy observed in the absolute energy comparison may be due to electron correlation effects rather than the QED corrections.
Fig. 4 compares our results with predictions and measurements of the unresolved oxygen qr-blend of other works.We also include earlier predictions from Vainshtein and Safronova [52] showing one significant digit more than other theoretical works [8,53,54].Interestingly, the center-of-gravity value of 562.9419 eV by Vainshtein and Safronova [52] agrees better with astrophysical observations ( [55][56][57]) than other laboratory measurements.However, both our theoretical and experimental results align more favorably with Refs.[15,16].
strong fields using heavier ions, where correlation effects become smaller.Furthermore, our results re-calibrate earlier works, and immediately benefit XRISM [58], a recently launched X-ray observatory furnished with a high-resolution X-ray microcalorimeter.Additionally, our data provides accurate reference lines, which allow full utilization of upcoming X-ray observatories such as Athena [59] and Arcus [60], which have targeted resolving power of 1000-3500 and uncertainty of below 10 km/s, call for high accuracy rest wavelength standards of essential soft X-ray transitions.Financial support was provided by the Max-Planck-Gesellschaft (MPG) and Bundesministerium für Bildung und Forschung (BMBF) through project 05K13SJ2.C.S. acknowledges support from NASA under award number 80GSFC21M0002 and Max-Planck-Gesellschaft (MPG).M.A.L. acknowledges support from NASA's Astrophysics Program.We acknowledge DESY (Hamburg, Germany), a member of the Helmholtz Association HGF, for the provision of experimental facilities.Parts of this research were carried out at PETRA III.We thank Jens Viefhaus and Rolf Follath for valuable discussions on X-ray monochromator resolution and performance, and the synchrotron-operation team at PE-TRA III for their skillful and reliable work.

4 FIG. 1 .
FIG. 1. Scheme of our experiment.Passing a monochromator (1) photons are focused onto the trapped HCI (2), with the relevant transitions and energy levels indicated.Fluorescence photons following resonant excitation due to the incident photon beam are recorded by a SDD (3).The outgoing beam continuously generates photoelectrons from a gold target (4).A PES resolves the two prominent 4 f lines on a detector (5) to monitor energy fluctuations.

F 7 FIG. 2 .
FIG. 2. (A)Trace of energy deviations recorded by means of XPS of Au 4f 5/2 photoelectrons.The trace covers the energy range of the Li-like neon q and r measurement.(B,C,D) Example scans of q and r lines for neon, fluorine and oxygen and respective calibration lines are superimposed with their, simultaneously acquired, photoelectron trace, which monitors the deviations of the demanded photon energy from the actual energy.Each scan contains the resolved q and transitions of the Li-like ion, as well as various He-like calibration lines.

TABLE I .
Measured energies of the 1s2s2p 2 P 3/2 (q) and 1s2s2p 2 P 1/2 (r) transitions, derived center-of-gravity (c.g.) and differences from predictions.All values are given in eV FIG. 4. Experimental values for the center-of-gravity energy of the blended qr line of Li-like oxygen ([