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REVIEW 3 major objections 6 minor 204 references

Calibrated Lanthanide Atomic Data for Kilonova Radiative Transfer. I. Atomic Structure and Opacities

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper builds a wavelength-calibrated lanthanide atomic dataset—66,591 transitions pinned to experiment out of 28.7 million—so kilonova spectra can be matched line by line.

desk verdict A useful lanthanide dataset with a real but addressable calibration-validation gap; the opacity benchmarks are solid, but the 66k calibrated wavelengths need a direct test before line-identification claims carry weight. read the letter →

arxiv 2507.07785 v1 pith:IKZG2ZGM submitted 2025-07-10 astro-ph.HE physics.atom-ph

Andreas Flörs , Ricardo Ferreira da Silva , José P. Marques , Jorge M. Sampaio , Gabriel Martínez-Pinedo This is my paper
classification astro-ph.HEphysics.atom-ph
keywords lanthanideskilonovaatomicopacitiesenergylevelsoscillatorstrengthslevelcalibrationradiativetransferr-processnucleosynthesis
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

After a neutron-star merger, the ejected heavy elements imprint their absorption lines on the optical and near-infrared light we call a kilonova, and the lanthanide elements (atomic numbers 57–70) are the main source of that blanketing. This paper tries to give radiative-transfer models atomic data accurate enough for line-by-line spectral matching: it computes the energy levels and electric-dipole transitions of all 28 singly and doubly ionized lanthanides with a relativistic configuration-interaction code, then calibrates low-lying levels against measured laboratory energies. The payoff claimed is 146,856 energy levels and 28,690,443 E1 transitions, of which 66,591 have experimentally anchored wavelengths—enough, the authors argue, to make lanthanide line identifications in kilonova spectra feasible. They also report that strong lines agree with experiment and with the best prior calculations, and that the calibration barely changes the derived opacities, which matters because opacity governs the brightness and color of the transient.

What carries the argument

The load-bearing machinery is a two-stage pipeline. First, the Flexible Atomic Code diagonalizes the Dirac–Coulomb Hamiltonian for a carefully chosen set of configurations, with the local central potential optimized by a Bayesian sequential-model optimization (SMBO) that minimizes a Boltzmann-weighted root-mean-square deviation between computed and reference energies—this weighting is what makes low-lying levels, the ones thermally populated in kilonova ejecta, come out right. Second, a manual calibration step converts the code's $jj$-coupled levels into LS labels via a transformation routine, matches them to experimental energies within parity and total-angular-momentum groups, and applies corrections that range from a direct replacement (when the dominant LS component exceeds 0.7) through a scaled partial correction (0.5–0.7) to a symmetry-group average (below 0.5). The output carries per-level and per-transition metadata stating whether both, one, or neither endpoint was matched to experiment, which is what lets a modeler trust the 66,591 calibrated wavelengths and treat the rest as statistically placed lines.

What would settle it

Take a random sample of the 66,591 calibrated transitions for ions with independent high-resolution laboratory data, such as Fourier-transform spectra recorded after this dataset was assembled, and check whether each predicted wavelength coincides with an observed line within the paper's stated calibration uncertainty. A mismatch rate well above the few-percent level for transitions flagged as having both levels matched to experiment would show the manual level-assignment step is biased, whereas near-complete agreement would confirm the method and the opacities derived from it.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is that the bottleneck for identifying heavy elements in kilonova spectra is not a lack of lines but a lack of lines with trustworthy wavelengths, and the paper removes that bottleneck for the lanthanides. Using the Flexible Atomic Code with an optimized local central potential, the authors computed relativistic configuration-interaction structures for all 28 singly and doubly ionized lanthanide ions (La II/III through Yb II/III), yielding 146,856 bound energy levels below the ionization threshold and 28,690,443 E1 transitions between them, with the correct ground-state configuration recovered for every ion except Gd II. Low-lying levels were then matched by hand, within parity and total-angular-momentum groups, to experimental levels from standard databases, shifting or partially correcting each theoretical level so that 66,591 transitions have both endpoints anchored to measurement; the average calibration corrections are about 2950 $\mathrm{cm}^{-1}$ for singly and 2571 $\mathrm{cm}^{-1}$ for doubly ionized species. The authors further claim that strong transitions ($\log(gf) > -1$) agree with experimental and semi-empirical values to a scatter of roughly 0.5 dex, while theoretical weak lines are systematically weaker than measured ones; that the resulting LTE opacities agree with the updated HULLAC and GRASP datasets to within about 20 percent but disagree with one recent HFR dataset whose near-ground level density is judged inconsistent with experiment; and that calibrating the energies leaves the opacities almost unchanged, because the bulk of the opacity comes from the many uncalibrated lines that fill in between the measured ones.

Load-bearing premise

The load-bearing premise is that the manual matching of computed levels to measured ones—which the authors note is ambiguous because multiple computed levels share the same dominant LS label—is essentially correct; a modest rate of mismatch would silently corrupt the 66,591 calibrated wavelengths and everything built on them.

Editorial extensions

If this is right

  • Radiative-transfer models fitted to AT2017gfo can now be run with wavelength-calibrated lanthanide line lists, and any synthetic feature that survives this test becomes a candidate real line rather than a numerical artifact.
  • The 66,591 calibrated lines constitute a concrete search list of optical and near-infrared wavelengths where lanthanide absorption should appear, giving observers a direct target for confirming or excluding third-peak r-process elements.
  • Because strong lines are reliable to about 0.5 dex while theoretical weak lines are systematically too weak, opacity and ejecta-mass inferences from synthetic spectra should be re-checked for sensitivity to the weak-line population.
  • The comparison discriminates between existing opacity datasets: calibrated FAC opacities sit within about 20% of the updated HULLAC and GRASP results, while the HFR dataset's factor-of-several higher opacities for some ions are explained by level densities that conflict with measured level counts.
  • Per-transition calibration flags let future spectral synthesis report, for every predicted absorption feature, whether its wavelengths are experimentally anchored or purely theoretical.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the manual matching bottleneck could be automated, the same calibration pipeline would transfer to actinides and to higher ionization stages, where experimental anchors are sparser and the payoff for kilonova and supernova modeling would be comparable.
  • The systematic weakness of theoretical gf values for faint lines, which the authors note appears across independent codes and experiments, implies that light-curve models built purely on strong-line data may understate line blanketing; testing this would mean recomputing synthetic spectra with the weak-line population boosted to match experiment.
  • Since calibration moves opacities little but moves wavelengths a lot, the dataset's decisive test is spectral: synthetic spectra computed with calibrated lines should match narrow observed absorption features in AT2017gfo markedly better than the same model with uncalibrated lines.
  • The authors' statement that Pm III alone lacks calibration anchors suggests promethium will remain the hardest lanthanide to identify; a targeted laboratory measurement campaign on Pm III would be the most direct way to extend the calibrated list.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. This manuscript presents FAC-based relativistic configuration-interaction calculations of atomic structure and E1 transitions for all 28 singly and doubly ionized lanthanide ions (La II through Yb III). The authors optimize the fractional-mean-configuration potential with an SMBO algorithm (Eq. 3) and then manually calibrate computed levels to NIST and DREAM reference data (§II C). The headline deliverable is a public dataset of 146,856 levels, 28,690,443 E1 transitions, and 66,591 transitions whose upper and lower levels are both flagged as calibrated, which the abstract claims carry 'experimentally calibrated wavelength information.' The paper benchmarks level energies, oscillator strengths against experimental and semi-empirical data (§IV A), and LTE expansion and Planck-mean opacities against HULLAC, GRASP, and HFR datasets (§IV B). The authors report roughly 0.5 dex scatter for strong transitions (log(gf) > -1), agreement of Planck-mean opacities with HULLAC V2 and GRASP to within about 20%, and factors-of-several-to-ten differences with newer HFR opacities, which they attribute to level-density differences at intermediate excitation energies.

Significance. The dataset addresses a recognized bottleneck in kilonova spectroscopy: the scarcity of wavelength-calibrated atomic data for lanthanides. If the calibration pipeline is trustworthy, the 66,591 anchored transitions would materially advance line-identification efforts for AT2017gfo-like spectra, and the level/transition tables plus the Zenodo data release provide a reproducible resource for radiative-transfer modeling. Strengths of the paper include systematic coverage of 28 ions, explicit benchmarks against several independent codes (HULLAC V2, GRASP, HFR) and against recent laboratory gf measurements, a transparent statement of the non-bijective JJ2LSJ labeling problem, and machine-readable public data. The oscillator-strength and opacity benchmarks are genuinely independent and generally supportive of the calculations. The main risk is concentrated in the unvalidated manual level-matching step that underpins the calibrated-wavelength count; this is fixable within the manuscript's scope because the authors already cite the measured line lists needed for a direct wavelength validation.

major comments (3)
  1. [II C, IV A, Table V] The paper's headline deliverable, the 66,591 transitions claimed to have 'experimentally calibrated wavelength information' (abstract; Table V), is not validated against independent measured wavelengths. The calibration pipeline in §II C depends entirely on manual LS-label matching between FAC levels and NIST/DREAM levels, with the JJ2LSJ transformation explicitly acknowledged to be non-bijective and with all calibrations performed by hand. The reported benchmarks do not test the correctness of these matches: Figures 1-2 show residuals for the very levels used in the calibration, and Figures 6-9 compare oscillator strengths, not line positions, for transitions whose levels were both identified. Because a wrong match still receives the 'xmatch' flag and produces a Ritz wavelength that can be off by thousands of cm-1, the abstract's feasibility claim for line identifications is not yet supported. I request a direct validation of the calibrated wavelengths against measured line positions, for example the NIST line lists and the Ferrara et al. (2024), Den Hartog et al. (2024), and Voith et al. (2025) line lists already used in §IV A, or a hold-out test in which a subset of known levels is excluded from calibration and used only for verification.
  2. [II C, Table V] The 'experimentally calibrated' label overstates the anchoring for a substantial, unquantified fraction of the 66,591 transitions. Section II C states that calibration aligns computed levels with 'experimental energy levels from the NIST ASD and, where applicable, theoretical data from DREAM'; DREAM is a semi-empirical HFR database, not a set of measurements. Table V shows DREAM reference levels for many ions (e.g., Ce II, Pr II, Tm II), so some of the 66,591 transitions are anchored on both ends to theoretical levels. The abstract and §III B should either qualify the claim (for example, 'experimentally or semi-empirically anchored') or report the breakdown of NIST-anchored versus DREAM-anchored transitions in Table V. This is not a purely semantic point: a user relying on the calibrated wavelengths for line identifications needs to know which transitions carry which anchoring status.
  3. [II B, II C, Figures 1-2] The level-energy 'agreement' reported in Figures 1-2 and discussed throughout §III is partly by construction and should be framed as fit residuals rather than predictive validation. The FMC occupation weights are optimized against experimental reference levels via the loss function in Eq. (3), and the calibration shifts of §II C are applied to the same experimental levels that are then used as benchmarks. The genuinely independent accuracy evidence comes from the gf comparisons against experiments (§IV A) and the opacity comparisons against other codes (§IV B), and that evidence is supportive; however, the manuscript should state this distinction explicitly and, ideally, report residuals on levels lying above the calibration range or on a hold-out subset of levels. In addition, because 'all calibrations were performed manually,' the Zenodo release should include the mapping between each matched FAC level and the specific NIST or DREAM energy level; without that mapping, the 66,591-transition claim is not auditable by other groups.
minor comments (6)
  1. [Figures 1 and 2 captions] The captions of Figures 1 and 2 both read 'doubly ionized lanthanide ions,' but the ion labels and the text of §II C indicate that Figure 1 shows singly ionized species and Figure 2 doubly ionized species; please correct the Figure 1 caption.
  2. [II B] The sentence 'These constraints ensure that the optimization explores physically meaningful configurations while allowing sufficient flexibility to improve the accuracy of the calculations.' appears twice verbatim; please delete the duplicate.
  3. [III A 8 and Table IV] The text in the gadolinium subsection states that the FAC calculation 'recover[s] a ground state configuration of 4f7 5d1 6s1, which differs from the experimental ground state 4f7 5d2,' whereas Table IV lists the FAC ground state as 4f7 5d2 and the NIST ground state as 4f7 5d1 6s1; these statements contradict each other and must be reconciled.
  4. [Table VII excerpt] The transition table excerpt contains formatting artifacts in which wavelength and log(gf) values run together (e.g., '7068.15-1.0213' and '4437.09.-1.4071'); the machine-readable version may be unaffected, but the printed excerpt should be cleaned up.
  5. [Tables II and III headers] The continuation tables of Table I are headed 'TABLE II. *' and 'TABLE III. *'; the stray asterisk placeholder should be removed.
  6. [Figure 36 caption] The caption of Figure 36 says the black horizontal lines show NIST experimental data, but the text of §III A 10 states that Dy III calibration uses level information from Spector et al. (1997) because the NIST ASD only contains the ground state for Dy III; the caption should be updated to reflect the actual calibration source.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the calibration and FMC optimization are explicit fits, and the central quantitative claims are benchmarked against independent experiments and independent atomic-structure calculations.

full rationale

The paper's calibration procedure and FMC-weight optimization are openly fitting procedures, not disguised predictions. The loss function (Eq. 3) and the calibration description (Sec. II C) use experimental NIST/DREAM levels to adjust computed energies, and the reported deviations in Figs. 1-2 are pre-calibration residuals; the paper does not claim that the calibrated level energies are independent predictions. The central quantitative claims that could be construed as predictions—the oscillator-strength agreement for strong lines (Figs. 6-9) and the opacity comparisons (Figs. 11-14)—are benchmarked against external experimental data (Ferrara et al., Den Hartog et al., Voith et al.) and independent calculations (HULLAC, GRASP, HFR, DREAM) that were not used to set the FMC weights or calibration shifts. The 66,591 experimentally calibrated transitions are explicitly presented as calibrated data products rather than as derived predictions. The self-citations to refs. [42] and [61] provide methodological background and are not invoked as uniqueness theorems or as the sole support for any load-bearing claim. The remaining caveat concerning manual LS-label matching is a correctness/validation risk, not a circularity, because no quoted step reduces a prediction to its own input by construction.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central calculation rests on a standard relativistic CI code, but the FMC weights and the calibration shifts are fitted to experimental levels, and the SMBO objective temperature is not stated. No new physical entities are introduced.

free parameters (3)
  • FMC occupation weights per ion = Listed in Table I; e.g., La II: 4f 1.62, 5d 0.38; Ce II: 4f 1.86, 5d 0.96, 6s 0.10, 6p 0.08
    Optimized with SMBO to minimize a weighted RMSD against experimental energy levels (Eq. 3). These weights set the radial potential and therefore all level energies and transitions.
  • Calibration corrections for energy levels = Per ion median corrections 617-9047 cm^-1; individual level shifts in Zenodo release
    Applied manually to match experimental levels; unmatched levels receive group and configuration average shifts. These corrections directly define the calibrated wavelengths.
  • Excitation temperature T in SMBO loss function = not stated
    The Boltzmann weighting in Eq. 3 depends on T, but the value is not given in the text, leaving the optimization objective incompletely specified.
assumptions (6)
  • domain assumption FAC's Dirac-Coulomb Hamiltonian with local central potential and screened hydrogenic QED corrections is adequate for lanthanide ions.
    This is the standard physical model behind the Flexible Atomic Code, invoked throughout Section II A.
  • domain assumption Experimental energy levels from NIST ASD and DREAM have correct energies, configurations, and term assignments.
    These levels are the calibration reference; if they are wrong, the calibrated wavelengths inherit the error.
  • ad hoc to paper The LS label purity threshold of 0.7 reliably identifies a theoretical level with its experimental counterpart.
    Section II C treats squared LS composition above 0.7 as reliable, 0.5-0.7 as partially reliable, and below 0.5 as unreliable; no independent validation of this threshold is given.
  • ad hoc to paper The manual matching of theoretical to experimental levels is correct for the matched levels.
    Section II C states that no automated calibration procedure exists and that all calibrations were performed manually.
  • domain assumption LTE and Saha/Boltzmann statistics apply in kilonova ejecta at 2500-8500 K.
    Used for opacities in Section IV B; the paper states the analysis is valid only where singly and doubly ionized species dominate.
  • domain assumption The Sobolev approximation is valid for the expanding kilonova ejecta.
    Adopted for expansion opacities in Section IV B with reference to prior kilonova studies.

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Cite this review

Pith. "Pith review of Calibrated Lanthanide Atomic Data for Kilonova Radiative Transfer. I. Atomic Structure and Opacities." pith.science (2026). https://pith.science/paper/IKZG2ZGM

@misc{pith2026250707785,
  author       = {Pith},
  title        = {Pith review of: Calibrated Lanthanide Atomic Data for Kilonova Radiative Transfer. I. Atomic Structure and Opacities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IKZG2ZGM}},
  note         = {Machine review of arXiv:2507.07785}
}
abstract

The early spectra of the kilonova (KN) AT2017gfo following the binary neutron star merger GW170817 exhibit numerous features shaped by r-process nucleosynthesis products. Although a few species were tentatively detected, no third-peak elements were unambiguously identified, as the amount of atomic data required for radiative transfer modeling is immense. Although comprehensive atomic data, including atomic opacities, is now available for many elements, wavelength-calibrated data remains limited to a select few ions. To examine the atomic opacities of all singly and doubly ionized lanthanides, from La (Z = 57) to Yb (Z = 70), we perform atomic structure calculations using the FAC code. Our calculations incorporate an innovative optimization of the local central potential and the number of configurations considered, alongside a calibration technique aimed at enhancing agreement between theoretical and experimental atomic energy levels. We assess the accuracy of the computed data, including energy levels and electric dipole (E1) transition strengths, as well as their impact on KN opacities. We find that strong transitions ($\log(gf)>-1$) are in good agreement with both experiments and semi-empirical calculations. For ions with substantial experimental data, the computed opacities exhibit good agreement with prior calculations. By calibrating low-lying energy levels with experimental data, we have identified 66,591 transitions with experimentally calibrated wavelength information, rendering future lanthanide line identifications through radiative transfer modeling feasible. In total, our calculations encompass 28 ions, yielding 146,856 energy levels below the ionization threshold and 28,690,443 transitions among these levels.

Figures

Figures reproduced from arXiv: 2507.07785 by the authors.

Figure 1
Figure 1. FIG. 1. Deviations between theoretical and experimental energy levels for doubly ionized lanthanide ions. The figure includes [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Deviations between theoretical and experimental energy levels for doubly ionized lanthanide ions. The figure includes [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Comparison of the energy levels for Nd [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (40 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Distributions of energy levels for singly ionized lanthanide ions. Different colors represent datasets computed with [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Distributions of energy levels for doubly ionized lanthanide ions. Different colors represent datasets computed with [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Oscillator strengths derived from experimental measurements by Ferrara [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Oscillator strengths obtained from experimental measurements by Voith [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Same as figure [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Same as figure [PITH_FULL_IMAGE:figures/full_fig_p023_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Ion fractions of neutral to triply ionized lanthanides [PITH_FULL_IMAGE:figures/full_fig_p024_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Expansion opacity results for ions from Z=57 (La) to Z=64 (Gd) are presented at [PITH_FULL_IMAGE:figures/full_fig_p025_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Expansion opacity results for ions from Z=65 (Tb) to Z=70 (Yb) are presented at [PITH_FULL_IMAGE:figures/full_fig_p026_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Planck mean opacities are shown for mono-atomic [PITH_FULL_IMAGE:figures/full_fig_p027_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Planck mean opacities are shown for mono-atomic [PITH_FULL_IMAGE:figures/full_fig_p028_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Planck mean opacities are presented for mono [PITH_FULL_IMAGE:figures/full_fig_p028_15.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Energy levels for La [PITH_FULL_IMAGE:figures/full_fig_p037_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Energy levels for La [PITH_FULL_IMAGE:figures/full_fig_p037_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19. Energy levels for Ce [PITH_FULL_IMAGE:figures/full_fig_p038_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20. Energy levels for Ce [PITH_FULL_IMAGE:figures/full_fig_p038_20.png]
Figure 21
Figure 21. Figure 21: FIG. 21. Energy levels for Pr [PITH_FULL_IMAGE:figures/full_fig_p039_21.png]
Figure 22
Figure 22. Figure 22: FIG. 22. Energy levels for Pr [PITH_FULL_IMAGE:figures/full_fig_p039_22.png]
Figure 23
Figure 23. Figure 23: FIG. 23. Energy levels for Nd [PITH_FULL_IMAGE:figures/full_fig_p040_23.png]
Figure 24
Figure 24. Figure 24: FIG. 24. Energy levels for Nd [PITH_FULL_IMAGE:figures/full_fig_p040_24.png]
Figure 25
Figure 25. Figure 25: FIG. 25. Energy levels for Pm [PITH_FULL_IMAGE:figures/full_fig_p041_25.png]
Figure 26
Figure 26. Figure 26: FIG. 26. Energy levels for Pm [PITH_FULL_IMAGE:figures/full_fig_p041_26.png]
Figure 27
Figure 27. Figure 27: FIG. 27. Energy levels for Sm [PITH_FULL_IMAGE:figures/full_fig_p042_27.png]
Figure 28
Figure 28. Figure 28: FIG. 28. Energy levels for Sm [PITH_FULL_IMAGE:figures/full_fig_p042_28.png]
Figure 29
Figure 29. Figure 29: FIG. 29. Energy levels for Eu [PITH_FULL_IMAGE:figures/full_fig_p043_29.png]
Figure 30
Figure 30. Figure 30: FIG. 30. Energy levels for Eu [PITH_FULL_IMAGE:figures/full_fig_p043_30.png]
Figure 31
Figure 31. Figure 31: FIG. 31. Energy levels for Gd [PITH_FULL_IMAGE:figures/full_fig_p044_31.png]
Figure 32
Figure 32. Figure 32: FIG. 32. Energy levels for Gd [PITH_FULL_IMAGE:figures/full_fig_p044_32.png]
Figure 33
Figure 33. Figure 33: FIG. 33. Energy levels for Tb [PITH_FULL_IMAGE:figures/full_fig_p045_33.png]
Figure 34
Figure 34. Figure 34: FIG. 34. Energy levels for Tb [PITH_FULL_IMAGE:figures/full_fig_p045_34.png]
Figure 35
Figure 35. Figure 35: FIG. 35. Energy levels for Dy [PITH_FULL_IMAGE:figures/full_fig_p046_35.png]
Figure 36
Figure 36. Figure 36: FIG. 36. Energy levels for Dy [PITH_FULL_IMAGE:figures/full_fig_p046_36.png]
Figure 37
Figure 37. Figure 37: FIG. 37. Energy levels for Ho [PITH_FULL_IMAGE:figures/full_fig_p047_37.png]
Figure 38
Figure 38. Figure 38: FIG. 38. Energy levels for Ho [PITH_FULL_IMAGE:figures/full_fig_p047_38.png]
Figure 39
Figure 39. Figure 39: FIG. 39. Energy levels for Er [PITH_FULL_IMAGE:figures/full_fig_p048_39.png]
Figure 40
Figure 40. Figure 40: FIG. 40. Energy levels for Er [PITH_FULL_IMAGE:figures/full_fig_p048_40.png]
Figure 41
Figure 41. Figure 41: FIG. 41. Energy levels for Tm [PITH_FULL_IMAGE:figures/full_fig_p049_41.png]
Figure 42
Figure 42. Figure 42: FIG. 42. Energy levels for Tm [PITH_FULL_IMAGE:figures/full_fig_p049_42.png]
Figure 43
Figure 43. Figure 43: FIG. 43. Energy levels for Yb [PITH_FULL_IMAGE:figures/full_fig_p050_43.png]
Figure 44
Figure 44. Figure 44: FIG. 44. Energy levels for Yb [PITH_FULL_IMAGE:figures/full_fig_p050_44.png]

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