# Critical Assessment of Theoretical Calculations of Atomic Structure and Transition Probabilities: An Experimenter’s View

^{1}

^{2}

## Abstract

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**PACS**32.30.-r; 32.70.Cs; 32.10.Fn

## 1. Introduction

#### 1.1. Codes

## 2. Historical Remarks

#### 2.1. Be-Like Ions

#### 2.1.1. Resonance Transition Rate

**Figure 1.**Lifetime data on the 2s2p ${}^{1}$P${}_{1}^{\mathrm{o}}$ level, the lowest excited singlet level in Be-like ions. The data are shown by their deviation from the lifetime predictions in the Multi-Configuration Hartree-Fock/Multi-Configuration Dirac-Hartree-Fock (MCHF/MCDHF) collection data base provided by C. Froese Fischer and G. Tachiev (the horizontal line) [28]. Also shown (broken line) is the result of a fit to the experimental data [29]. For some elements there have been several measurements, of which only the result of best quality (judged by the author) has been selected. For Be there are three data entries (the most accurate ones), and open questions remain (see text).

#### 2.1.2. Intercombination Transition Rate

**Figure 2.**Intercombination transition rate in C III (adopted from Figure 4 in [6]). The timeline combines calculated (T) and measured (E) results and is discussed in the text. The publications in this incomplete listing are identified by authors and year. The uncertainty of the 1997 measurement by Doerfert et al. is as small as the center of the cross mark. Note that a number of recent computations also bear error estimates.

#### 2.2. General Problems

## 3. Isoelectronic Sequences

#### 3.1. H-Like Ions

#### 3.2. He-Like Ions

**Figure 3.**Deviation of the measured M1 decay rate from the lowest triplet level in He-like ions ($Z=3$ through 54) from the results of computations by Johnson, Plante, and Sapirstein (JPS) [79]. The experimental data points result from work at the Heidelberg heavy-ion storage ring TSR, the Livermore electron beam ion traps (EBIT), a recoil ion beam experiment at GSI Darmstadt, and from beam-foil measurements in various places (Berkeley, Argonne, Darmstadt). Some older results that have been technically superseded are not shown.

#### 3.3. Li-Like Ions

#### 3.3.1. Levels

#### 3.3.2. Lifetimes

#### 3.4. Be-Like Ions

**Figure 4.**Measurements (full black symbols) and Relativistic Configuration Interaction (RCI) computational results (open blue squares and broken line [129]) of the 2s2p ${}^{3}$P${}_{1}$ level in Be-like ions and their deviation from Many-Body Perturbation Theory (MBPT) computations by Safronova et al. [126]. Note that at low Z, the computation by Chen and Cheng [129] is close to experiment, while the other one is not, and vice versa at higher Z.

#### 3.4.1. 2s2p Level Lifetimes

**Figure 5.**Lifetime of the 2s2p ${}^{3}$P${}_{1}^{\mathrm{o}}$ level in Be-like ions (intercombination decay) displayed as the relative deviation from the computational web resources (MCHF/MCDHF collection) made available by Froese Fischer and Tachiev [28]. A problem lies in the only partial coverage by any given computation, and therefore one cannot refer all of the available measurements and computations to a single reference data set. Because of this, the trends shown all suffer a step function (by 7%) above Ne ($Z=10$), because this is where one of the references ends and the comparison changes to another one. The only continuous reference data set would have been the one derived by a semiempirical averaging procedure applied by Curtis and Ellis [29] (broken red line) to computations and measurements, which thus represents a secondary trend. The broken blue line refers to computations by Jönsson and Froese Fischer [27], the short-dash broken black line connects the computational results reported by Brage et al. [131] (open circles, the study has to be lauded for point-by-point estimates of uncertainty). Not all of the experimental data (filled circles) are shown, because for several of them the size of the error bars exceeds the plot boundaries.

#### 3.4.2. Displaced Level Lifetimes

#### 3.4.3. Hyperfine-Induced Decays

#### 3.5. B-Like Ions

**Figure 6.**The 2s${}^{2}$2p ${}^{2}$P${}^{\mathrm{o}}$ $J=1/2$ - $J=3/2$ ground state fine structure splitting in B-like ions, shown as the deviation from the computations by Safronova et al. [138]. Up to about $Z=30$, experiment (filled symbols) and other computations (open symbols) stay close to the trend predicted by Safronova et al., but then they diverge. Unfortunately, in the latter range the accurate computations are extremely sparse. The EBIT data point for Xe ($Z=54$) represents the sum of two transition energies, because the lowest quartet level drops below the upper fine structure level of the doublet ground term and thus opens an alternate (two-step) decay path to ground that is more easily observed and resolved spectroscopically than the single-step decay branch.

#### 3.5.1. E1-Forbidden Transition Rates

**Figure 7.**Transition rate of the 2s${}^{2}$2p ${}^{2}$P${}^{\mathrm{o}}$ $J=1/2$ - $J=3/2$ ground state fine structure transition in B-like ions, depicted as the deviation from the predictions by Rynkun et al. [144]. Measurements have been done at an electrostatic ion trap [153], electron beam ion traps [154,155,156,157], and at a heavy-ion storage ring [158]. The EBIT measurements have eventually reached high accuracy, as is seen in the data point for Ar that has an uncertainty smaller than the data marker. Three representatives of recent computations are shown (identified by the lead authors), and the best experiment [156,157] agrees with only one of them, the computations by Rynkun et al. [144] (full line). It would be good to know whether this computation has taken the anomalous magnetic moment of the electron (EAMM) contribution to the M1 transition moment into account, an effect that amounts to 0.45% and thus to much more than the experimental uncertainty (smaller than the symbol size), but I have not seen it being mentioned.

#### 3.6. CNO-Like Ions

#### 3.7. F-Like Ions

**Figure 8.**Lifetime of the 2s${}^{2}$2p${}^{2,4}$ ${}^{1}$D${}_{2}$ level in C- and O-like ions of Si, P, and S. All measurements (full markers with error bars) have been done at the TSR heavy-ion storage ring. The Si measurements could nowadays be redone with much higher statistical reliability, if the storage ring was still available. The data are shown relative to two computations. The computations shown scatter by a few percent around the experimental data, whereas some earlier computations differed by more than 10% (see [168]).

**Figure 9.**Lifetime of the 2s${}^{2}$2p${}^{5}$ ${}^{2}$P${}_{1/2}^{\mathrm{o}}$ level in F-like ions from Cl through Ti. Measurements (full markers with error bars) have been done at an electrostatic ion trap (EST) (Ar), at the Livermore EBIT (Ar, K), and at the TSR heavy-ion storage ring (Sc, Ti). Two of the more recent computations are shown; the experimental errors are still too large to discriminate among these two computations.

#### 3.8. Ne-Like Ions

#### 3.8.1. Levels

#### 3.8.2. Lifetimes

#### 3.8.3. Line Ratio

#### 3.9. Na-Like Ions

^{th}century; half a century later the corresponding emission lines in the spectrum of a flame were identified first with table salt and then with sodium, thus helping to establish spectral analysis as a laboratory tool and with finding out what the stars consist of. The strong Na I resonance line doublet has intrigued theorists and experimenters, and nowadays observations and computations [100,101] of the spectra of Na-like ions reach all the way up to U [188,189]. For high values of Z, the fine structure interval between the two D lines (the 3p ${}^{2}$P “fine structure” splitting) is much larger than the excitation energy of the 3p${}_{1/2}$ level. Recent measurements at the NIST electron beam ion trap have tended to a wide range of elements in the upper half of the periodic table [190,191]; in those papers, the new results along with the many earlier data points compare well to the 20 years old Blundell computations.

**Figure 10.**Level scheme of the lowest 37 levels in the Ne-like ion Fe${}^{16+}$. The two arrows point to the 2p${}^{6}$ ${}^{1}$S${}_{0}$ – 2p${}^{5}$3d ${}^{1}$P${}_{1}^{\mathrm{o}}$, ${}^{3}$D${}_{1}^{\mathrm{o}}$ transitions discussed in the text and illustrate the resonant photo excitation experiment performed at the LCLS X-ray light source [187].

#### 3.10. Mg-Like Ions

#### 3.11. Al-Like Ions

**Figure 11.**Timeline of the transition rate of the ground state fine structure transition in the Al-like ion (Fe${}^{13+}$ of iron. The four experimental data points result from two experiments with an electrostatic ion trap and from two that used EBITs. In the latest one [205] the error bar is smaller than the symbol size. The vertical red line (left one of the pair) shows the result of naive theory combining the experimental transition energy and the single-configuration picture line strength. The accompanying dark blue line indicates the same expectation, but corrected for the EAMM contribution from quantum electrodynamics (QED). Note that the best experimental data point does not agree with what presently is seen as a correct theoretical prediction. All of the non-ab initio computations fall close to these two lines, because they have been adjusted to the experimental transition energy. Of the four ab initio computations only one (MR-MP) [199] coincides with the range of successful experiments by merit (and still misses the EAMM correction). The computational result provided by Huang [67] coincides with experiment as well, but by comparison with other atomic systems one has to conclude that these computations are not accurate enough to call this more than a chance agreement. (Figure adapted from Figure 1 of [198], which includes a list of all the references.)

#### 3.12. Si-Like Ions

#### 3.13. P-Like Ions

#### 3.14. S-Like Ions

#### 3.15. Cl-Like Ions

**Figure 12.**The lifetime $\tau $ (scaled by the ion core charge ζ) of the 3s3p${}^{6}$ ${}^{2}$S${}_{1/2}$ level lifetime in Cl-like ions. Only a selection of experimental data (full squares) and computational results (open symbols for Ar, Ca and Cu, and eye-guiding lines; most of the computations in the literature do not cover low-Z systems) are shown. Note how at low Z the lifetime predictions diverge by up to an order of magnitude. The computation by Berrington et al. comes closest overall to the low-Z measurements—except for the measurement on Ar II by Lauer et al. [244] (the error bar is smaller than the symbol size) that Berrington did not know of at the time. Saha and Fritzsche, however, matched that one data point well by their computation, but they calculated only this one. The computations by Wilson seemed better than most others at the time—but covered too few elements. Mohan did well in 1992—but why were only two elements computed?

#### 3.16. Ar-Like Ions

#### 3.17. Cu-Like Ions

#### 3.18. Zn-Like Ions

#### 3.19. Ga-Like and Ge-Like Ions

## 4. Pm-Like, Sm-Like, Eu-Like Ions

## 5. W

## 6. Discussion

## Acknowledgments

## Conflicts of Interest

## References

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Träbert, E. Critical Assessment of Theoretical Calculations of Atomic Structure and Transition Probabilities: An Experimenter’s View. *Atoms* **2014**, *2*, 15-85.
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Träbert E. Critical Assessment of Theoretical Calculations of Atomic Structure and Transition Probabilities: An Experimenter’s View. *Atoms*. 2014; 2(1):15-85.
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Träbert, Elmar. 2014. "Critical Assessment of Theoretical Calculations of Atomic Structure and Transition Probabilities: An Experimenter’s View" *Atoms* 2, no. 1: 15-85.
https://doi.org/10.3390/atoms2010015