High-precision measurements of n=2\to n=1 transition energies and level widths in He- and Be-like Argon Ions

High-precision measurements of transition energies and level widths in He- and Be-like Argon Ions

J. Machado Laboratório de Instrumentação, Engenharia Biomédica e Física da Radiação (LIBPhys-UNL), Departamento de Física, Faculdade de Ciências e Tecnologia,
FCT, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal
Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-PSL Research University, Collège de France, Case 74; 4, place Jussieu, F-75005 Paris, France
   C. I. Szabo National Institute of Standards and Technology, Gaithersburg, MD20899, USA Theiss Research, La Jolla, CA 92037, USA    J. P. Santos Laboratório de Instrumentação, Engenharia Biomédica e Física da Radiação (LIBPhys-UNL), Departamento de Física, Faculdade de Ciências e Tecnologia,
FCT, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal
jps@fct.unl.pt
   P. Amaro Laboratório de Instrumentação, Engenharia Biomédica e Física da Radiação (LIBPhys-UNL), Departamento de Física, Faculdade de Ciências e Tecnologia,
FCT, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal
   M. Guerra Laboratório de Instrumentação, Engenharia Biomédica e Física da Radiação (LIBPhys-UNL), Departamento de Física, Faculdade de Ciências e Tecnologia,
FCT, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal
   A. Gumberidze ExtreMe Matter Institute EMMI and Research Division, GSI Helmholtzzentrum für Schwerionenforschung, D-64291 Darmstadt, Germany    Guojie Bian Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-PSL Research University, Collège de France, Case 74; 4, place Jussieu, F-75005 Paris, France Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, P.R. China    J. M. Isac Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-PSL Research University, Collège de France, Case 74; 4, place Jussieu, F-75005 Paris, France    P. Indelicato Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-PSL Research University, Collège de France, Case 74; 4, place Jussieu, F-75005 Paris, France paul.indelicato@lkb.upmc.fr
July 12, 2019
Abstract

We performed a reference-free measurement of the transition energies of the line in He-like argon, and of the line in Be-like argon ions. The highly-charged ions were produced in the plasma of an Electron-Cyclotron Resonance Ion Source. Both energy measurements were performed with an accuracy better than 3 parts in , using a double flat-crystal spectrometer, without reference to any theoretical or experimental energy. The transition measurement is the first reference-free measurement for this core-excited transition. The transition measurement confirms recent measurement performed at the Heidelberg Electron-Beam Ion Trap (EBIT). The width measurement in the He-like transition provides test of a purely radiative decay calculation. In the case of the Be-like argon transition, the width results from the sum of a radiative channel and three main Auger channels. We also performed Multiconfiguration Dirac-Fock (MCDF) calculations of transition energies and rates and have done an extensive comparison with theory and other experimental data. For both measurements reported here, we find agreement with the most recent theoretical calculations within the combined theoretical and experimental uncertainties.

pacs:
34.80.Kw. 32.30 RJ
pacs:
34.80.Dp, 34.50.Fa, 34.10.+x

I Introduction

Bound-states quantum electrodynamics (BSQED) and the relativistic many-body problem have been undergoing important progress in the past few years. Yet there are several issues that require increasing the number of high-precision tests. High-precision measurements of transition energies on medium to high- elements Bruhns et al. (2007); Amaro et al. (2012); Chantler et al. (2012); Rudolph et al. (2013); Chantler et al. (2014); Payne et al. (2014); Kubiçek et al. (2014); Beiersdorfer and Brown (2015); Epp et al. (2015), Landé -Factors Sturm et al. (2011); Wagner et al. (2013); Sturm et al. (2014); Köhler et al. (2015); Ullmann et al. (2015); Marques et al. (2016); Yerokhin et al. (2016) and hyperfine structureBeiersdorfer et al. (1998); Seelig et al. (1998); Boucard and Indelicato (2000); Beiersdorfer et al. (2001a); Yerokhin and Shabaev (2001); Shabaev et al. (2001); Volotka et al. (2012); Andreev et al. (2012); Nörtershäuser et al. (2013); Lochmann et al. (2014); Beiersdorfer et al. (2014); Vollbrecht et al. (2015); Ullmann et al. (2017), just to name a few, are needed either to improve our understanding or to provide tests of higher-order QED-corrections, the calculations of which are very demanding.

Recent measurement of the proton size in muonic hydrogen Pohl et al. (2010); Antognini et al. (2013) and of the deuteron in muonic deuterium Pohl et al. (2016), which disagree by 7 and 3.5 standard deviations respectively from measurements in their electronic counterparts triggered experimental and theoretical research regarding not only the specific issue of the proton and deuteron size, but also the possible anomalies in BSQED. A discrepancy of this magnitude corresponds to a difference in the muonic hydrogen energy of , which is far outside the calculations uncertainty of about and is much larger than what can be expected from any omitted QED contribution. Another large discrepancy of 7 standard deviations between theory and experiment has also been observed recently in a specific difference between the hyperfine structures of hydrogenlike and lithiumlike bismuth measured at the Experimental Storage Ring (ESR) at GSI in Darmstadt Ullmann et al. (2017), designed to eliminate the effect of the nuclear magnetization distribution (the Bohr-Weisskopf correction) Shabaev et al. (2001).

Medium and high- few-electron ions with a hole are the object of the present work. They have been studied first in laser-produced plasmas Aglitskii et al. (1974) and beam-foil spectroscopy (see, e.g., Dohmann and Mann (1979); Briand et al. (1983)), low-inductance vacuum spark Aglitsky et al. (1988), or by using the interaction of fast ion beams with gas targets in heavy-ion accelerators. Ion storage rings have also been used (see, e.g., Stöhlker et al. (1994, 2000); Gumberidze et al. (2005)). The limitation in precision of those measurements is mostly due to the large Doppler effect, which affects energy measurements, and the Doppler broadening, which affects any possible width measurement.

Recoil ion spectroscopy Deslattes et al. (1984), which has also been used, is not affected by the Doppler effect, and provides an interesting check. Plasma machines, such as tokamaks, have also provided spectra Bitter et al. (1985); Hsuan et al. (1987), leading to relative measurements, without Doppler shift, usually using He-like lines as a reference. Solar measurements Neupert (1971) have also been reported.

Accurate transition energy measurements in medium and high-, few-electron ions have been reported using either Electron Beam Ion Trap (EBIT) or Electron-Cyclotron Ion Sources (ECRIS) to produce ions at rest in the laboratory. Such measurements, using an EBIT, have been performed by the Livermore group (see, e.g., MacLaren et al. (1992); Widmann et al. (1996); Beiersdorfer et al. (2001b); Thorn et al. (2008); Beiersdorfer and Brown (2015) and reference there in), Heidelberg group Bruhns et al. (2007); Kubiçek et al. (2012); Rudolph et al. (2013); Epp et al. (2015) and the Melbourne and National Institute of Standards and Technology (NIST) collaboration  Chantler et al. (2012, 2014); Payne et al. (2014). The present collaboration has reported values using an ECRIS  Amaro et al. (2012).

The Heidelberg group reported the measurement of the He-like argon line with a relative accuracy of without the use of a reference line Kubiçek et al. (2012). In that work, the spectrometer used is made of a single flat Bragg crystal coupled to a charge-coupled device (CCD) camera, which can be positioned very accurately with a laser beam reflected by the same crystal as the x rays Kubiçek et al. (2012). The Melbourne-NIST collaboration reported the measurement of all the transitions in He-like titanium with a relative accuracy of , using a calibration based on neutral x-ray lines emitted from an electron fluorescence x-ray source Chantler et al. (2012, 2014); Payne et al. (2014). The Livermore group reported a measurement of all lines in heliumlike copper Beiersdorfer and Brown (2015), using hydrogenlike lines in argon as calibration. It also reported measurement of all 4 lines in He-like xenon, using a micro-calorimeter and calibration with x-ray standards Thorn et al. (2009). It should be emphasized that measurements in both type of ion sources do not require Doppler shift correction to transition energy measurements, because the ions have only thermal motion.

Measurements of the line in Be-like ions are scarce. Some measurements are relative measurements using tokamaks, where the Be-like line appears as a satellite line for the He-like transitions. The line is often used as a calibration. Measurements of that type for Be-like Ar have been performed at the Tokamak de Fontenay aux Roses (TFR) TFR Group et al. (1985a), and for Ni at the Tokamak Fusion Test Reactor (TFTR) Bitter et al. (1985); Hsuan et al. (1987). Such relative measurements, which use theoretical results on the He-like line, must be re-calibrated using the most recent theoretical values. Several other observations have been made on different elements, but no experimental energy reported (see, e.g., Ref. Rice et al. (2015) for Cl, Ar and Ca), or the experimental accuracy is not completely documented (see e.g., TFR Group et al. (1985b); Rice et al. (1995, 2014)). Measurements in EBIT are also known, as in vanadium Beiersdorfer et al. (1991) and iron Beiersdorfer et al. (1993), for terrestrial and astrophysics plasma applications. There have also been relative measurements in ECRIS for sulfur, chlorine and argon Schlesser et al. (2013), using the relativistic M1 transition as a reference.

Chantler et al.  Chantler and Kimpton (2009); Chantler et al. (2012, 2014), have claimed that existing data show the evidence of a discrepancy between the most advanced BSQED calculation Artemyev et al. (2005) and measurements in the He-like isoelectronic sequence, leading to a deviation that scales as . They speculated  Chantler et al. (2014) that this supposed systematic effect could provide insight into the proton size puzzle, the Rydberg and fine-structure constants, or missing three-body BSQED terms. Here we make a detailed analysis, including all available experimental results, to check this claim.

We emphasize the advantage of studying highly-charged, medium- systems, such as argon ions, to test QED. The BSQED contributions have a strong -dependence: the retardation correction to the electron-electron interaction contribution scales as , and the one-electron corrections, self-energy and vacuum polarization, scale as . Yet, at high-, the strong enhancement of the nuclear size contribution and associated uncertainty limits the degree to which available experimental measurements can be used to test QED Plunien et al. (1991); Plunien and Soff (1995, 1996); Beier et al. (1998); Chantler and Kimpton (2009). At very low-, experiments can be much more accurate, but tests of QED can be limited as well, even for very accurate measurements of transitions to the ground state of He Eikema et al. (1996, 1997); Kandula et al. (2011). For few-electron atoms and ions, they are limited by the large size of electron-electron correlation and by the evaluation of the needed higher-order QED screening corrections, in the non-relativistic QED formalism (NRQED) Pachucki (1998a, b, 2006a, 2006b); Yerokhin and Pachucki (2010). It can also be limited by the slow convergence of all-order QED contributions at low-, which may be required for comparison, and because of the insufficient knowledge of some nuclear parameters, namely the form factors and polarizability  Pohl et al. (2010); Antognini et al. (2013); Pohl et al. (2016); Artemyev et al. (2005). In medium- elements like argon or iron, the nuclear mean spherical radii are sufficiently well known (see, e.g., Angeli and Marinova (2013)) and nuclear polarization contribution to the ion level energies is very small. So uncertainties related to the nucleus are small compared to experimental and theoretical accuracy. This can be seen in the theoretical uncertainties claimed in Ref. Artemyev et al. (2005).

Besides the fundamental aspect, knowledge of transition energies and wavelengths of highly-charged ions is very important for many sectors of research, such as astrophysics or plasma physics. For example, an unidentified line was recently detected in the energy range to in an X-ray Multi-Mirror (XMM-Newton) space x-ray telescope spectrum of 73 galaxy clustersBulbul et al. (2014) and at for another XMM spectrum in the Andromeda galaxy and the Perseus galaxy cluster Boyarsky et al. (2014). The next year a line at was observed in the deep exposure dataset of the Galactic center region with the same instrument. A possible connection with a dark matter decay line has been put forward, yet measurements performed with an EBIT seem to show that it could be a set of lines in highly charged sulfur ions, induced by charge exchange Shah et al. (2016), while a recently published search with the high-resolution x-ray spectrometer of the HITOMI satellite does not find evidence for such lines in the Perseus cluster Aharonian et al. (2017).

In the present work, we apply the method we have developed to measure the energy and line-width of the M1 transition reported in Ref. Amaro et al. (2012), to the transition in He-like argon and to the transition in Be-like argon ions. We also present a multi-configuration Dirac-Fock (MCDF) calculation for the two transition energies and widths. These calculations are performed with a new version of the mcdfgme code that uses the effective operators developed by the St Petersburg group to evaluate the self-energy screening Shabaev et al. (2013).

The article is organized as follows. In the next section we briefly describe the experimental setup used in this work. A detailed description of the analysis method that provides the energy, width and uncertainties is given in Sec. III. A brief description of the calculations of transition energy and widths is given in Sec. IV. We present our experimental result for the transition in Sec. V. In the same section we present all available experimental results for and transitions in He-like ions. We do a very detailed comparison between theory from Ref. Artemyev et al. (2005), which covers and the available measurements in this -range. Our results and comparison with theory for the line in Be-like argon ions are presented in Sec. VI. The conclusions are provided in Sec. VII.

Ii Experimental Method

ECRIS plasmas have been shown to be very intense sources of x rays, and have diameters of a few cm. Therefore, they are better adapted to spectrometers that can use an extended source. At low energies one can thus use cylindrically or spherically bent crystal spectrometers as well as double-crystal spectrometers (DCSs).

A single flat-crystal spectrometer, combined with an accurate positioning of the detector, and alternate measurements, symmetrical with respect to the optical axis of the instrument, as used in Heidelberg Kubiçek et al. (2012), and the double-crystal spectrometers Deslattes (1967); Amaro et al. (2014) are the only two methods that can provide high-accuracy, reference-free measurements in the x-ray domain. We use here reference-free with the same meaning as in Ref. Kessler Jr. et al. (1979), i.e., the measured wavelengths are directly connected to the meter as defined in the International System of Units, through the lattice spacing of the crystals Amaro et al. (2014). Our group reported in 2012 such a measurement of the transition energy in He-like argon with an uncertainty of without the use of an external reference Amaro et al. (2012), using the same experimental device as in the present work: a DCS connected to an ECRIS, the “Source d’ Ions Multichargés de Paris” (SIMPA)Gumberidze et al. (2010), jointly operated by the Laboratoire Kastler Brossel and the Institute des Nanosciences de Paris on the Université Pierre and Marie Curie campus.

A detailed description of the experimental setup of the DCS at the SIMPA ECRIS used in this work is given in Ref. Amaro et al. (2014). A neutral gas (Ar in the present study) is injected into the plasma chamber inside a magnetic system with minimum fields at the very center of the vacuum chamber. Microwaves at a frequency of heat the electrons that are trapped by the magnetic field. The energetic electrons ionize the gas through repeated collisions reaching up to heliumlike charge states Guerra et al. (2013). The ions are, in turn, trapped by the space charge of the electrons, which have a density around . This corresponds to a trapping potential of a fraction of , leading to an ion-speed distribution of per charge, and thus to a small Doppler broadening of all the observed lines. In contrast, EBITs have a trapping potential of several hundred , and the Doppler broadening is then much larger.

The state is mostly created by electron ionization of the ground state of Li-like argon, and therefore the line is the most intense line we observed in He-like argon. The line observed here results from the excitation of the He-like argon ground state, which is much less abundant, leading to a weaker line. The Be-like excited level, , is mostly produced by ionization of the ground state of boronlike argon, which is a well-populated charge-state (see Fig. 21, Ref. Amaro et al. (2014)). The line is thus the most intense we observed.

The spectra are recorded by a specially-designed, reflection vacuum double-crystal spectrometer described in detail in Ref. Amaro et al. (2014). The two , -thick Si(111) crystals were made at the National Institute for Standards and Technology (NIST). Their lattice spacing in vacuum was measured and found to be (relative uncertainty of ) at a temperature of  Amaro et al. (2014), relative to the standard value Fujimoto et al. (2011); Mohr et al. (2012). More details will be found in Ref. Kessler Jr et al. (2017). Using this lattice spacing, our measurement provides wavelengths directly tied to the definition of the meter Mohr et al. (2012). The DCS is connected to the ion source in such a way that the axis of the spectrometer is aligned with the ECRIS axis and is located at from the plasma (a sphere of in diameter).

To analyze the experimental spectra, we developed a simulation code Amaro et al. (2014), which uses the geometry of the instrument and of the x-ray source, the shape of the crystal reflectivity profile, as well as the natural Lorentzian shape of the atomic line and its Gaussian Doppler broadening to perform high-precision ray-tracing. The reflectivity profile is calculated using XOP (X-ray Oriented Programs)  Sanchez del Rio and Dejus (2004), which uses dynamical diffraction theory from Ref. Zachariasen (1967), and the result is checked with the X0H program, which calculates crystal susceptibilities and  Stepanov (); Lugovskaya and Stepanov (1991).

The first crystal is maintained at a fixed angle. A spectrum is obtained by a series of scans of the second crystal. A stepping motor, driven by a micro-stepper, runs continuously, between two predetermined angles that define the angular range of one spectrum. X rays are recorded continuously and stored in a histogram, together with both crystals temperatures. Successive spectra are recorded in opposite directions. Both crystal angles are measured with Heidenhain111Certain commercial equipment, instruments, or materials are identified in this paper to foster understanding. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose. high-precision angular encoders. The experiment is performed in the following way: a nondispersive-mode (NDM) spectrum is recorded first. Then a dispersive-mode (DM) spectrum is recorded. The sequence is completed with the recording of a second NDM spectrum. Due to the low counting rate, such a sequence of three spectra takes a full day to record. In order to obtain enough statistics, the one-day sequence is repeated typically 7 to 15 times.

Iii Data analysis

The data analysis is performed in three steps. First we derive a value for the experimental natural width of the line. For this, each experimental dispersive-mode spectrum is fitted with simulated spectra, using an approximate energy (e.g., the theoretical value) and a set of Lorentzian widths. A weighted one-parameter fit is performed on all the results for all recorded dispersive-mode spectra providing a width value and its uncertainty. This experimental width is then used to generate a new set of simulations, using several different energies and crystal temperatures. These simulations are used to fit each dispersive-mode and nondispersive-mode experimental spectrum in order to obtain the line energy. For each day of data recording this leads to two Bragg angle values, obtained by taking the angular difference between a nondispersive-mode spectrum and a dispersive-mode spectrum:

  • one Bragg angle value is obtained by comparing the first nondispersive-mode spectrum of the day and the dispersive-mode spectrum obtained immediately after;

  • a second Bragg angle value is obtained by comparing the same dispersive-mode spectrum with the nondispersive-mode spectrum obtained immediately after.

In that way a number of possible time-dependent drifts in the experiment are compensated. We now describe these processes in more detail.

iii.1 Evaluation of the widths

The ion temperature, which is necessary to calculate the Gaussian broadening was obtained by measuring first a line with a completely negligible natural width, the M1 , transition. The width of this transition is , which is totally negligible when compared to our spectrometer inherent energy resolution. From this analysis we obtained the Gaussian broadening Amaro et al. (2012). This value also provides the depth of the trapping potential due to the electron space charge. Knowing the experimental Gaussian broadening value , we can perform all the needed simulations. For each line under study we then proceed as follows:

  • Perform simulations for the dispersive-mode spectra for a set of natural width values and the theoretical transition energy , using the already known , and crystal temperature ;

  • Interpolate each simulation result with a piece-wise spline function to obtain a set of continuous, parametrized functions , where correspond to the angle at which the simulation reaches its maximum value, and ;

  • Normalize all the functions above to have the same maximum value (we chose the one with as reference);

  • Fit each experimental spectrum with the functions obtained above

    (1)

    where is the line intensity, the crystal angle, the background intensity and the background slope. The parameters , , and are adjusted to minimize the reduced . We perform a series of fits of each experimental spectrum, with 27 simulated spectra, each evaluated with a different width , to obtain a set of values. The width values go from to by steps of , completed by a point at . A typical experimental spectrum and the fitted simulated functions, for 5 of the 27 values of used to make the analysis, are shown in Fig. 1 ;

  • Fit a third degree polynomial to the set of points ;

  • Find the minimum of the third degree polynomial to get the corresponding optimal , being the experiment run number (see Fig. 2 for an example);

  • Get the error bar for experiment run by finding the values of the width for which Press et al. (2007)

    (2)
  • Finally a weighted average of the values in the set of all the obtained for all measured spectra is performed to obtain the experimental value and its error bar:

    (3)

    The sets of for both lines studied here are plotted in Fig. 3.

The two first steps are performed by two different methods, one based on the CERN (Centre Européen de Recherche Nucléaire) program ROOT, version 6.08  Brun and Rademakers (1997); Antcheva et al. (2009, 2011) and one based on MATHEMATICA, version 11  Wolfram Research (2016).

Figure 1: (Color online) Example of a dispersive-mode experimental spectrum for the He-like Ar transition (black dots), together with a few plots of the function in Eq. (1), for different values of the natural line width . The four parameters have been adjusted to minimize the reduced (see text for more explanations).
Figure 2: (Color online) Third degree polynomial fitted to the set of points (black dots), for the He-like Ar 1s2p 1s transition. The values were obtained from the fits, a few of which are represented in Fig. 1, with 27 different values of . The blue dashed-doted line corresponds to Eq. 2.
(a) He-like argon transition.
(b) Be-like argon transition.
Figure 3: (Color online) Natural width values of all the spectra recorded during the experiment, with weighted average and uncertainties evaluated with Eq. (III.1).

iii.2 Transition energy values

Once we obtained the experimental width value of a measured line (cf. Sec. III.1), the determination of the correspondent experimental transition energy value is achieved using the following scheme:

  • Perform simulations in the nondispersive and dispersive modes for a set of transition energy values , where is the theoretical energy value, an energy increment and an integer that can take positive or negative values. The simulations are done with the experimental natural width and Gaussian broadening . The simulations are performed at various crystal temperature values for each energy.

  • As in Sec. III.1, interpolate each simulation result with a spline function for both the nondispersive and dispersive modes, to obtain a set of functions depending on all the pairs;

  • Fit each experimental spectrum, using Eq. (1) with and , to obtain the angle difference between the simulation and the experimental spectrum, both in dispersive and nondispersive mode;

  • For each pair of dispersive and nondispersive modes experimental spectra, calculate the offsets between the simulated spectra and the experimental value obtained in the step above. This offset should be 0 if the energy and temperature used in the simulation were identical to the experimental values;

  • Fit the bidimensional function

    (4)

    where and are adjustable parameters, to the set of points obtained in the previous step (see Fig. 4 as an example);

  • The experimental line energy for spectrum pair number , is the energy such that where , stands for the average measured temperature on the second crystal;

  • As a check, we also used the line energy such that (). This leads to a temperature-dependent energy. We then fitted a straight line to the line energy, as a function of the second crystal temperature, and extrapolated to . Both methods lead to very close values, well within the uncertainties.

  • As in Sec. III.1, we calculate the weighted average of all the pairs to obtain the final experimental energy. The error bar on each point is the quadratic combination of the instrumental uncertainty, as given in Table 1 and of the statistical error.

  • To check the result, we also fit the set of pairs with the function to check that there is no residual temperature dependence.

Contribution: Value (eV)
Crystal tilts ( for each crystal)
Vertical misalignment of collimators ()
X-ray source size ( to )
Form factors
X-ray polarization
Angle encoder error
Lattice spacing error
Index of refraction
Coefficient of thermal expansion
X-ray polarization
Energy-wavelength correction
Temperature ()
Table 1: Instrumental contributions to the uncertainties in the analysis of the daily experiments (see Refs. Amaro et al. (2012, 2014)).
Figure 4: (Color online) Fitted two-dimensional function from Eq. (4), and experimental results (white spheres), for the He-like Ar transition. The fit is performed taking into account the statistical error bars in each point.

Iv Theoretical calculation

The core-excited transition in Be-like ions has been calculated with the most recent methods, only very recently and only for iron Yerokhin et al. (2014), and argon Yerokhin et al. (2015). Previous calculations Safronova and Urnov (1979); Safronova and Lisina (1979); Chen (1985); Chen and Crasemann (1987) did not take into account QED and relativistic effects to the extent possible today.

For the preparation of this experiment, we performed a calculation of the energy value for the transition in Be-like argon, using the multiconfiguration Dirac-Fock (MCDF) approach as implemented in the 2017.2 version of the relativistic MCDF code (MCDFGME), developed by Desclaux and Indelicato Desclaux (1975); Indelicato and Desclaux (1990); Indelicato et al. (1987); Indelicato and Desclaux (2005). The full description of the method and the code can be obtained from Refs. Grant (1970); Desclaux (1975); Grant and Quiney (1988); Indelicato (1995). The present version also takes into account the normal and specific mass shifts, evaluated following the method of Shabaev Shabaev (1985); Shabaev and Artemyev (1994); Shabaev (1998), as described in Li et al. (2012); Sampaio et al. (2013).

The main advantage of the MCDF approach is the ability to include a large amount of electronic correlation by taking into account a limited number of configurations Santos et al. (1999, 2006); Martins et al. (2009). All calculations were done for a finite nucleus using a uniformly charged sphere. The atomic masses and the nuclear radii were taken from the tables by Audi et al.  Audi et al. (2003) and Angeli and Marinova Angeli (2004); Angeli and Marinova (2013), respectively.

Radiative corrections are introduced from a full QED treatment. The one-electron self-energy is evaluated using the one-electron values of Mohr and co-workers  Mohr (1974, 1992); Mohr and Kim (1992); Indelicato and Mohr (1992); Le Bigot et al. (2001), and corrected for finite nuclear size Mohr and Soff (1993). The self-energy screening and vacuum polarization were included using the methods developed by Indelicato and co-workers Indelicato et al. (1987); Indelicato and Desclaux (1990); Indelicato and Lindroth (1992); Indelicato et al. (1998); Indelicato and Mohr (1998). In previous work, the self-energy screening in this code was based on the Welton approximation Indelicato et al. (1987); Indelicato and Desclaux (1990). Here we also evaluate the self-energy screening following the model operator approach recently developed by Shabaev et al. Shabaev et al. (2013, 2015), which has been added to MCDFGME. A detailed description of this new code will be given elsewhere.

In order to assess the quality of this new method for calculating the self-energy screening we can compare the different values for the He-like transition measured here. The QED value of Indelicato and Mohr Indelicato and Mohr (2001) is , the one from Ref. Artemyev et al. (2005) (Table IV) is The Welton method provides , while the implementation of the Saint-Petersburg effective operator method gives , closer to the ab initio methods. We can thus assume an uncertainty of and for the effective operator and Welton operator methods respectively. The same procedure applied to the Be-like transitions provides using Ref. Indelicato and Mohr (2001), for the effective operator method and for the Welton method. We can conclude that at intermediate , both the Welton and effective operator methods provide very similar results, the effective operator method being in slightly better agreement with ab initio calculation. This is consistent with earlier comparisons for fine-structure transitions, (see, e.g., , Ref. Blundell (1993)).

Lifetime evaluations are done using the method described in Ref. Indelicato et al. (1989). The orbitals contributing to the wave function were fully relaxed, and the resulting non-orthogonality between initial and final wave functions fully taken into account, following Löwdin (1955); Indelicato (1997).

The full Breit interaction and the Uehling potential are included in the self-consistent field process. Projection operators have been included Indelicato (1995) to avoid coupling with the negative energy continuum.

As a check, we also performed a calculation of the He-like argon lines measured in the present work and in Ref. Amaro et al. (2012). Following Refs. Froese Fischer (1977); Gorceix et al. (1987); Indelicato (1995, 1996), we use for the excited state the following configurations:

(5)

where the indicates an orbital with identical angular function as the one, but with another radial wave function, for which the orthogonality with orbitals of the same symmetry in other configuration is not enforced. The ground state wave function is taken as usual as . We also evaluated

(6)

in order to calculate the M1 transition energies measured Ref. Amaro et al. (2012), which allowed to compare also energy differences.

For Be-like argon, the correlation contributions result from the inclusion of all single, double and triple electron excitations of the and 2 electrons in the unperturbed configuration up to . For the ground state it corresponds to configurations and for the excited state to configurations. We performed an estimation of the full correlation energy by doing a fit with the function , and extrapolation to for each level, for both the Welton and the Model operator values. The results are presented in Table 2. By comparing the extrapolated value and the changes in QED due to the use of either the Welton or effective operator method we estimated the theoretical uncertainty provided in the table. There is however a contribution that is not included, the Auger shift. This shift is due to the fact that the being core-excited is degenerate with a continuum. To our knowledge, such shifts have been evaluated only in the case of neutral atoms x-ray spectra Indelicato and Lindroth (1992); Indelicato et al. (1998); D. et al. (2003). For argon with a hole, the shift is , while for a hole it is . Here we have a 4-electron system, with only 3 possible Auger channels, and the shell is closed, so the effect is expected to be small. We assume an extra theoretical uncertainty of for this uncalculated term.

Welton QED Model operator QEDShabaev et al. (2013, 2015)
Initial Final Transition Initial Final Transition
DF
2
3
4
5
Table 2: Total energy and transition energies ( in ) for the transition in Be-like argon, as a function of the maximum principal quantum number of the correlation orbitals. All correlation from the Coulomb, retardation and QED parts is included. Extrapolation for is done by fitting the function to the correlation energy (Difference with the energy for and the Dirac-Fock (DF) value) of each level and retaining only the constant term . The uncertainty combines the difference between the extrapolated and best directly calculated value, the missing Auger shift and the self-energy screening model.

The Auger width of the level is calculated with the MCDFGME code, following the method described in Ref. Howat et al. (1978) with full relaxation and final-state channel mixing, again taking into account the non-orthogonality between the initial and final state. For the first time, we combine this method with fully correlated wave functions, up to . The convergence of the transition energy and width are presented in Table 3. This table shows that the Auger width values vary rather strongly when increasing the maximum of correlation orbitals, when non-orthogonality and full relaxation are included. This behavior is due to the fact that the free electron wave functions have to be orthogonal to all the occupied and correlation orbitals of the same symmetry, which provides a lot of constraints.

We have also performed calculations of the transition energies and rates with the “flexible atomic code” (FAC), widely used in plasma physics Gu (2008). This code is based on the relativistic configuration interaction (RCI), with independent particle basis wave functions that are derived from a local central potential. This local potential is derived self-consistently to include the screening of the nuclear potential by the electrons.

The final results are compared to other calculations from Refs. Chen (1985); Costa et al. (2001); Natarajan (2003) in Table 4. The relatively large difference between our present MCDF calculation and the Dirac-Fock calculation from Ref. Costa et al. (2001) , made with an earlier version of our code, is due to correlation and to the evaluation of Auger rates using fully relaxed initial and final states.

The contributions of all the other possible transitions to the levels, , was evaluated by computing all Auger widths up to . We then fitted a function to the total Auger width for each principal quantum number , summing all values of and for each value of , to evaluate the contribution from up to infinity. We find and . The total value for the contribution of all levels with is and is thus negligible.

Radiative Auger
Max. Ener. Width Ener. Width Ener. width Ener. Width Total width
DF
2
3
4
5
Table 3: Convergence of theoretical partial radiative widths, Auger widths and energies for transitions originating from the Be-like level. Transition energies are in and widths in .
MCDF, Chen (1985) Chen (1985) MCDF, Costa et al. 2001 Costa et al. (2001) RCI, Natarajan (2003) Natarajan (2003)
Initial Level final level energy rate energy rate energy rate
Total Auger
Level width
MCDF (this work) FAC (this work)
energy rate energy rate
Total Auger
Level width
Table 4: Comparison between theoretical partial radiative widths, Auger widths and energies for transitions originating from the Be-like level. Transition energies are in and widths in .

V Results and comparison with theory for the He-like transition

v.1 Line widths

Our experimental values for the line widths, obtained as explained in Sec. III.1 and Fig. 2(a), are presented in Table 5, together with several theoretical results. There are several possible E1 radiative transitions originating from the level. Because of the large energy difference, the contribution of the transition to the level width is strongly dominant. The next largest contribution, due to the transition, contributes only to the width. The width of the transitions has been calculated using Drake’s unified method Drake (1979), relativistic random phase approximation, MCDF, relativistic configuration interaction (RCI) and QED Johnson et al. (1995). The effect of the negative energy continuum has been discussed in Refs. Indelicato (1996); Derevianko et al. (1998). Radiative corrections to the photon emission have also been evaluated Indelicato et al. (2004). The differences between all theoretical values and our measurement are well within the experimental error bar.

Transition Experiment Theory Reference
75 (17) 70.4778 (25) MCDF (this work)
70.40 MBPT, Si et al. (2016) Si et al. (2016)
70.43 MCDHF, Si et al. (2016) Si et al. (2016)
70.43 Johnson et al. (1995) Johnson et al. (1995)
70.49 (14) Drake (1979) Drake (1979)
Table 5: Measured and computed natural line width values for the transitions in He-like Ar. All values are given in , and estimated uncertainties are shown in parentheses.

v.2 Transition energies

We present in Fig. 5 the transition energy values obtained from the successive pairs of dispersive and nondispersive-modes spectra, recorded during the experiment for the He-like argon following the method presented in Sec. III. The weighted average and bands are plotted as well.

Figure 5: (Color online) He-like argon transition energy values of the different spectra recorded during the experiment. Error bars in each point correspond to the quadratic sum of the peak fitting uncertainty with the uncertainties from Table 1, which have random fluctuations only, i.e., the angle measurement and the temperature correction. The (pink) shaded area correspond to the weighted average of the peak position statistical uncertainty obtained from the fit. The lines combine this statistical uncertainties with all systematic errors from Table 1. Every pair of points correspond to one-day data taking (see text for explanations).

Table 6 presents the measured He-like argon transition energy, together with all known experimental and theoretical results. The final experimental accuracy, combining the instrumental contributions from Table 1 is . The value is in agreement with a preliminary result, obtained with the same set-up, but using fit with Voigt profiles of both the experimental spectra and the simulations Szabo et al. (2013a, b). The agreement with the most precise experiments, i.e., the two reference-free experiments Bruhns et al. (2007); Kubiçek et al. (2012) and the recoil ion experiment of Deslattes et al. Deslattes et al. (1984) is well within combined error bars. The agreement with the calculation of Artemyev et al. Artemyev et al. (2005) is also within the linearly combined error bars.

Energy Reference Exp. method
Experiment
3139.5927 (50)(63)(80) This Work (stat.)(syst.)(tot.) ECRIS
3139.567 (11) Schlesser et al. (2013)  Schlesser et al. (2013) ECRIS
3139.581 (5) Kubiček et al. (2012)  Kubiçek et al. (2012) EBIT
3139.583 (63) Bruhns et al. (2007)  Bruhns et al. (2007) EBIT
3139.552 ( 37 ) Deslattes et al. (1984)  Deslattes et al. (1984) Recoil ions
3139.60 ( 25 ) Briand et al. (1983)  Briand et al. (1983) Beam-foil
3140.1 ( 7 ) Dohmann et al. (1979)  Dohmann and Mann (1979) Beam-foil
3138.9 ( 9 ) Neupert et al. (1971)  Neupert (1971) Solar emission
Theory
3139.559 (10) (13) This work using model operators Shabaev et al. (2013, 2015)
(correlation)(SE screening)
3139.553 (10) (18) This work using Welton model (correlation)(SE screening)
3139.538 MBPT, Si et al. (2016)  Si et al. (2016)
3139.449 MCDHF, Si et al. (2016)  Si et al. (2016)
3139.5821 (4) Artemyev et al. (2005)  Artemyev et al. (2005)
3139.582 Plante et al. (1994)  Plante et al. (1994)
3139.617 Cheng et al. (1994)  Cheng et al. (1994)
3139.576 Drake (1988)  Drake (1988)
3139.649 Indelicato et al. (1987)  Indelicato et al. (1987)
3139.56 Safronova (1981)  Safronova (1981)
3140.15 Johnson et al. (1976)  Johnson and Sapirstein (1992)
3140.46 Gabriel (1972)  Gabriel (1972)
Table 6: Comparison of our He-like argon experimental transition energy with previous experimental and theoretical values. All energies are given in eV, and estimated uncertainties are shown in parentheses.
(a) -range.
(b) (Color online) Zoom on the -range, and small energy differences.
Figure 6: (Color online) Comparison between the theoretical values by Artemyev et al.  Artemyev et al. (2005) and experimental data for transition in He-like ions presented in Tables 7 and 8 for all . The continuous lines represent the weighted fits with , , and functions, and the shaded area the bands, representing the confidence interval from the fit. The experimental values for are not plotted as they have very large error bars, but were included in the fit. Values of different experiments for a given are slightly shifted horizontally to make the figure easier to read.

v.3 Comparison between measurements and calculations for

There have been many measurements of transition energies in He-like ions. The reference-free measurements, of the kind reported in the present work, and the measurements calibrated against x-ray standards or transitions in H-like ions are summarized in Tables 7 and 8 for . Relative measurements, using the theoretical value for one of the He-like lines in the spectrum, originating from ECRIS or Tokamak experiments are summarized in Table 9. When older calculations were used as a reference, we used the energies of Ref. Artemyev et al. (2005) to obtain an updated value for this table.

A detailed analysis of the difference between theory Artemyev et al. (2005) and experiment has been performed in previous work Chantler et al. (2012, 2014); Beiersdorfer and Brown (2015). Here we provide an updated analysis, which include our new result and the data from Tables 7 and 8 .

The differences between these experimental values and Artemyev et al. Artemyev et al. (2005) theoretical values are plotted in Fig. 6 together with weighted fits by several functions of the shape , to 3. The error bands for the fits are also plotted. These error bands show that there is no significant deviation between theory and experiment.

In order to reinforce this conclusion, we have performed a systematic significance analysis. This analysis has been performed fitting functions of the form , , on three datasets build using the data presented in Tables 7 and 8. One dataset contains only the transition, one contains all , , , and transitions, and the last one is the same, from which the experimental values of this work, of Kubiçek et al. Kubiçek et al. (2014) and of Amaro et al. Amaro et al. (2012) have been removed. The values of the reduced are plotted as a function of in Fig. 7 for the three subsets. It should be noted that the reduced increases as a function of , although in two of the subsets there is a weak local minimum near . We present in Fig. 8 the uncertainty of the fit coefficient in standard-error units as a function of for all three datasets. The figure shows that the maximum deviation from zero is obtained for . The deviation of the fit coefficient tends to zero with increasing value of while the reduced increases. For the other two datasets considered, i.e., all experimental values presented in Tables 7 and 8 or the subset consisting only of the -lines, there is a local maximum for each dataset around . For all experimental data the local maximum happens at with a coefficient significance of standard errors, while for the -lines the local maximum is at with a deviation of standard errors from zero. In spite of the presence of this local maximum for different monomial orders of , the maximum deviation from zero of the fit parameter is at as well as the minimum reduced value. This leads to the conclusion that is the most probable model to describe the data when considering a power law dependence with .

To sustain this conclusion, a goodness of a fit test was performed. Fig. 9 shows the result probability (p-value) of the observed cumulative distribution function (upper tail) as a function of , for the given number of degrees of freedom and the minimum value of each performed fit. This probability, that the observed for degrees of freedom is larger than , is given by Press et al. (2007)

where is the incomplete function. When all data from Tables 7 and 8 are included, . It can be noticed that the highest p-value for the three considered datasets is for , and, as before, one can see a local maximum when considering all experimental results from Tables 7 and 8 or just the -lines for the same value as from Fig. 8. Considering the standard significance level of to evaluate the acceptance or rejection of the null hypotheses (i.e., the fact that the data can be described by the function), and since the highest p-value is for the three considered datasets, the null hypotheses has a very small probability to be true, with the caveats noted in Ref. Wasserstein and Lazar (2016). We also performed a t-student test, which shows that is the most probable value for all . Therefore, we conclude that it is highly unlikely that the experiment–theory difference has a dependence in of the form for any given with .

Figure 7: (Color online) Values of the reduced function as a function of , when fitting , to 12, to the experiment-theory differences from Tables 7 and 8. Solid line: reduced fitting only the () values. Dotted line: reduced fitting all 4 , , and transition energies differences with theory. Dashed line: same data as dotted line, but removing the reference-free values from this work and from Refs. Amaro et al. (2012); Kubiçek et al. (2014).
Figure 8: (Color online) Values of the significance of the fit coefficient in standard-error units as a function of when fitting to the experiment-theory differences from Tables 7 and 8.
Figure 9: (Color online) -value as a function of when fitting to the experiment-theory differences from Tables 7 and 8. See legend of Fig. 8 for explanations of the data included in each curve.
(w) (x) (y) (z)
Exp. (eV) Err. Theory Exp. (eV) Err. Theory Exp. (eV) Err. Theory Exp. (eV) Err. Theory Method Ref.
SR Engstrom and Litzen (1995)
SR Engstrom and Litzen (1995)
SR Aglitskii et al. (1974)
SR Aglitskii et al. (1974)
SR Aglitskii et al. (1974)
SR Aglitskii et al. (1974)
SR Aglitskii et al. (1974)
SR Aglitskii et al. (1974)
SR Aglitsky et al. (1988)
RF Kubiçek et al. (2014)
SR Schleinkofer et al. (1982)
RF Amaro et al. (2012)
SR Dohmann et al. (1978)
RF this work
RF Kubiçek et al. (2014)
SR Deslattes et al. (1984)
SR Briand et al. (1983)
SR Beiersdorfer et al. (1989)
SR Aglitsky et al. (1988)
SR Rice et al. (2014)
SR Beiersdorfer et al. (1989)
SR Rice et al. (1995)
SR Beiersdorfer et al. (1989)
SR Payne et al. (2014)
SR Aglitsky et al. (1988)
SR Beiersdorfer et al. (1989)
SR Chantler et al. (2000)
Table 7: Summary of all measured transition energies in He-like ions . The theoretical values are from Ref. Artemyev et al. (2005), which are available for . The experimental values are either reference-free measurements (RF) or measurements calibrated against standard reference x-ray transitions, or hydrogen-like transitions (SR).
(w) (x) (y) (z)
Exp. (eV) Err. Theory Exp. (eV) Err. Theory Exp. (eV) Err. Theory Exp. (eV) Err. Theory Method Ref.
SR Aglitsky et al. (1988)
SR Beiersdorfer et al. (1989)
SR Aglitsky et al. (1988)
SR Beiersdorfer et al. (1989)
RF Kubiçek et al. (2014)
SR Briand et al. (1984)
RF Rudolph et al. (2013)
SR Aglitsky et al. (1988)
SR Aglitsky et al. (1988)
SR Aglitsky et al. (1988)
SR Beiersdorfer and Brown (2015)
SR Aglitsky et al. (1988)
SR Aglitsky et al. (1988)
SR MacLaren et al. (1992)
SR Indelicato et al. (1986)
SR Widmann et al. (1996)
RF Epp et al. (2015)
SR Aglitsky et al. (1988)
SR Aglitsky et al. (1988)
SR Briand et al. (1989)
SR Widmann et al. (2000)
SR Thorn et al. (2009)
SR Thorn et al. (2008)
SR Briand et al. (1990)
SR Lupton et al. (1994)
Table 8: Summary of all measured transition energies in He-like ions . The theoretical values are from Ref. Artemyev et al. (2005). The experimental values are either reference-free measurements (RF) or measurements calibrated against standard reference x-ray transitions, or hydrogenlike transitions (SR).
(w) (x) (y) (z)
Exp. (eV) Err. Theory Exp. (eV) Err. Theory Exp. (eV) Err. Theory Exp. (eV) Err. Theory Ref.
Ref. Schlesser et al. (2013)
Ref. Schlesser et al. (2013)
Ref. TFR Group et al. (1985a)
Ref.