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Study of the elusive $5s-4f$ level crossing in highly charged osmium with optical transitions suitable for physics beyond the Standard Model searches

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Pith's one-line read This paper measures the nearly degenerate 5s–4f configurations in Os16+ and identifies two ultra-narrow electric-quadrupole transitions as optical clock candidates for beyond-Standard-Model searches.

desk verdict Solid M1 line list and two E2 clock transitions in Os16+, but the claimed E1 non-detection is weaker than the abstract suggests because the search wavelengths carry a ±3490 cm^-1 uncertainty. read the letter →

arxiv 2509.06710 v1 pith:SPYKHB6U submitted 2025-09-08 physics.atom-ph

Nils-Holger Rehbehn , Lakshmi Priya Kozhiparambil Sajith , Michael K. Rosner , Charles Cheung , Sergey G. Porsev , Marianna S. Safronova , Steven Worm , Dmitry Budker
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Thomas Pfeifer José R. Crespo López-Urrutia Hendrik Bekker
This is my paper · ORCID
classification physics.atom-ph
keywords highlychargedionsosmium5s-4flevelcrossingopticalclocktransitionselectricquadrupoleconfigurationinteractionbeyond-Standard-Modelsearcheselectronbeamiontrap
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

This paper aims to fix the energy-level structure of the osmium ion Os16+ at the 5s–4f orbital crossing, where outer-electron shells are nearly degenerate and transitions are exceptionally sensitive to physics beyond the Standard Model. Combining large-scale relativistic configuration-interaction theory with emission spectroscopy in an electron beam ion trap, the authors identify fifteen magnetic-dipole lines and reconstruct nearly the entire fine structure of the two crossing configurations. From those measured levels they derive two electric-quadrupole transitions with a natural linewidth of 44 microhertz—one at 1139.2103(78) nm and one at 502.3391(12) nm—and propose them as ultra-narrow clock transitions for frequency metrology and local Lorentz-invariance searches. They also find that the long-sought interconfiguration electric-dipole lines are predicted to be far weaker than earlier calculations indicated, because inner-shell correlation suppresses their amplitudes; the paper concludes these lines are too weak to detect with the present setup. If correct, the result is a measured wavelength catalog that shortens the route to an osmium-based optical clock and to isotope-shift searches for hypothetical fifth forces.

What carries the argument

The mechanism is the near degeneracy of the [Pd]4f12 5s2 and [Pd]4f13 5s configurations at the 5s–4f orbital crossing in Os16+. The measured M1 transitions provide an experimental backbone: they fix the energies of both configurations, so the two E2 clock-line wavelengths are derived from data, not from theory alone. The theory component—relativistic configuration interaction with inner shells opened successively (4d, then 4p, 4s, n=3)—is what exposes the suppression of the E1 amplitudes. The 44-microhertz linewidth of the E2 lines, set by the absence of lower-energy decay channels, is what makes them suitable for a clock.

What would settle it

Attempt to drive the predicted 3F2–3F4 E2 clock transition at 502.3391(12) nm in a single trapped Os16+ ion using quantum logic spectroscopy. Finding it at the predicted frequency with the expected narrow linewidth confirms the central clock claim; failing to find it within the stated search window would falsify the level identification and the derived clock-line assignments.

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Extended reading notes

Core claim

The central discovery is experimental. Fifteen forbidden M1 lines measured in Os16+ fix the fine structure of the 4f12 5s2 and 4f13 5s configurations, and their Ritz-Rydberg combinations yield two electric-quadrupole transitions—3H6–3F4 at 1139.2103(78) nm and 3F2–3F4 at 502.3391(12) nm—with a computed natural linewidth of 44 microhertz. On the theory side, the paper shows that the E1 interconfiguration rates are strongly suppressed once inner-shell correlations are included: the leading 3F4–3Fo4 rate falls from 12.6 s-1 to below 1 s-1, so previous rate estimates were overestimates. The predicted interconfiguration lines were not detected, and the paper attributes this to the suppressed rate

Load-bearing premise

The conclusion that the interconfiguration E1 transitions are too weak rests on the CI prediction of the 4f13 5s configuration offset x: if the true offset falls outside the quoted ~3490 cm-1 uncertainty—which the authors say may be underestimated because computing-cluster limits prevented full inclusion of some correlations—the sought lines could lie outside the observed spectral window and the non-detection would be a wavelength miss, not a rate suppression.

Editorial extensions

If this is right

  • Os16+ acquires two E2 transitions with measured wavelengths, allowing quantum logic spectroscopy to narrow its search range for laser excitation to sub-GHz windows.
  • The reconstructed fine-structure levels, combined with the seven stable osmium isotopes, provide a basis for generalized King-plot searches for a hypothetical fifth force.
  • The suppressed E1 rates mean direct optical detection of the 5s–4f crossing lines is out of reach for current electron-beam ion-trap spectroscopy; longer exposures or alternative population schemes would be required.
  • Inner-shell excitations must be included when predicting interconfiguration transition rates in complex highly charged ions; neglecting them overestimates E1 rates by orders of magnitude.
  • The roughly 0.6% agreement between the most complete calculations and the measured levels adds confidence to predictions for neighbouring ions near the same orbital crossing.

Reading between the lines

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

  • If the E2 clock lines perform as predicted, the 5s–4f crossing mechanism in osmium could be tuned by moving to neighbouring atomic numbers, potentially yielding even more sensitive crossings than Os16+.
  • The non-detection of the E1 interconfiguration lines is consistent with rate suppression, but it is also consistent with a wavelength miss within the ~3500 cm-1 theoretical uncertainty; a scan of the full quoted uncertainty window at optimized wavelengths would separate the two explanations.
  • The measured M1 and E2 energies could serve as anchors to refine the configuration-interaction treatment, specifically reducing the uncertainty of the configuration offset x, and thereby sharpen predictions for the same crossing in other Nd-like ions.
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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

1 major / 6 minor

Summary. The paper reports a combined experimental and theoretical study of the 5s–4f level crossing region in highly charged osmium. Using the Heidelberg EBIT with a grating spectrometer, the authors measure M1 transitions in Os15+, Os16+, and Os17+ with fractional wavelength uncertainties down to a few times 10^-7. For Os16+, the M1 identifications are supported by Zeeman line-shape fits, measured g-factors, and Ritz–Rydberg combinations, and the authors reconstruct the fine-structure level energies of the 4f^12 5s^2 configuration and the relative level energies of the 4f^13 5s configuration (the absolute offset x remains theory-dependent). From these levels they derive two E2 clock-transition wavelengths: the ground-state 3H6–3F4 transition at 1139.2103(78) nm and the 3F2–3F4 transition at 502.3391(12) nm, with natural linewidths in the μHz–mHz range. Large-scale pCI calculations are used to predict E1 interconfiguration transition rates, and it is found that inner-shell correlations suppress these rates by up to two orders of magnitude compared with earlier calculations. The predicted interconfiguration E1 lines were not detected in the observed spectral range, and the authors conclude they are too weak to be observed with the current setup.

Significance. If the identifications hold, the two E2 transitions in Os16+ are promising candidates for optical clock operation and searches for local Lorentz invariance, with the measured level energies providing the required laser-addressable frequencies. The paper also provides a catalog of M1 transitions in three charge states of osmium, useful for King-plot analyses of fifth-force searches. The theoretical result that inner-shell excitations drastically reduce E1 rates is an important cautionary lesson for CI calculations in complex ions. The strengths of the paper include the use of multiple independent atomic codes (pCI, AMBiT, FAC), explicit uncertainty budgets for the CI calculations, high-precision wavelength measurements, and internal consistency checks (Ritz combinations, CRM intensity ratios). The main weakness is that the conclusion about the interconfiguration E1 transitions being too weak to detect is not independently confirmed by the experiment because the search wavelengths are theory-dependent; this is partially mitigated by the wide spectral coverage.

major comments (1)
  1. [Identifications, Conclusions; Tables I, III, IV] The non-detection of the interconfiguration E1 lines is presented as evidence for the small computed rates, but the search wavelengths depend on the unmeasured 4f^13 5s offset x with uncertainty 3490 cm^-1 (Table IV). This uncertainty is propagated into Table I; e.g., the 3H4−3Fo3 line at 743+261/−153 nm has an upper bound (1004 nm) outside the observed 227–810 nm range, so a wavelength miss cannot be excluded for this line. For the strongest candidates (374 nm, 451 nm, 543 nm), the 1σ ranges are within the covered region, so the non-detection does constrain the old-theory rates (e.g., 42.95 s^-1 for 3F4−3Fo3). Nevertheless, the conclusion in the abstract that the predicted transitions are 'too weak to be detected' should be softened to state that non-detection is consistent with, but does not independently confirm, the low rates; please add the caveat and, if possible, an SNR estimate f
minor comments (6)
  1. [Abstract; Table II] The '44 μHz' linewidth appears to be the Einstein A coefficient (4.36×10^-5 s^-1) rather than the FWHM natural linewidth, which would be A/(2π) ≈ 6.9 μHz. Please clarify the convention or correct the value. This affects the quantitative statement but not the qualitative claim of ultra-narrowness.
  2. [After Table III] The text quotes frequency uncertainties of 9.3 GHz and 584 MHz for the two E2 transitions; the wavelength uncertainties in Table II (7.8 pm and 1.2 pm) correspond to about 1.8 GHz and 1.4 GHz, respectively. Please verify these numbers and explain the derivation, or correct the text.
  3. [Table I] For 3F4−3Fo4, the uncertainty on the reduced matrix element D is larger than D itself; the stated rate uncertainty (0.16 s^-1) seems small compared with the D uncertainty. Please detail the error propagation used.
  4. [Measurements] The total observed spectral coverage could be stated more clearly: the 150 grooves/mm grating covers 316–810 nm and the 1800 grooves/mm grating covers 227–323 nm, so the combined continuous coverage is 227–810 nm. This is relevant for the wavelength-miss discussion.
  5. [Table II] The E2 transitions are marked 'R' (Ritz-Rydberg combination); the text should explicitly remind the reader that these frequencies were not directly observed but inferred from M1 level energies, to avoid confusion with 'found' in the abstract.
  6. [Conclusions] Minor typo: '4f 1252' should be '4f 125s2'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the experimental E2 clock frequencies are Ritz-Rydberg combinations of measured M1 lines, and the E1 non-detection is a genuine prediction subject to wavelength uncertainty, not a fitted tautology.

full rationale

The central measured results—the 15 M1 line wavelengths and the two inferred E2 clock transitions—are derived from EBIT spectroscopy, not from theory fitted to the data. The E2 lines in Table II are explicitly marked as Ritz-Rydberg combinations (e.g., 3F2-3F4 at 502.3391(12) nm and 3F4-3H6 at 1139.2103(78) nm) built from measured M1 energies. The pCI calculations provide g-factors and level energies used for line identification, but identifications are independently supported by multi-line Ritz-Rydberg consistency and wavelength scaling laws, so the assignment is not forced by the theory. The predicted E1 interconfiguration wavelengths and rates in Table I are genuine predictions from a large-scale CI calculation with assigned uncertainties; no E1 line was fitted to data, and the quoted off-diagonal matrix-element uncertainty for the strongest line exceeds its value, making it an order-of-magnitude estimate. The conclusion that the interconfiguration E1 transitions are too weak to observe combines these predicted rates with CRM population modeling and detector-efficiency estimates. The CRM model is validated against a simultaneously measured 437-nm line pair, not fitted to the E1 non-detection. Citations to prior work by overlapping authors (e.g., Refs. [19] and [36]) are methodological or comparative, not load-bearing proofs of the new claims. The skeptic's concern that the search may have missed the E1 lines because the unmeasured configuration offset x carries a ~3490 cm^-1 uncertainty is a legitimate scientific caveat about the E1 null result, but it does not make the argument circular: a prediction can be wrong without being tautological. No step in the derivation reduces to its own input by construction.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central measurements rest on the experimental identification of M1 lines through Zeeman modeling, while the theoretical E1-rate predictions rest on a large-scale CI treatment with extrapolated partial waves and an unmeasured interconfiguration offset x. These are the load-bearing inputs that the paper does not fully deliver as data or code.

free parameters (1)
  • x (relative offset of 4f135s vs 4f125s2 configurations)
    Table III lists 4f135s levels as x + ...; x is not directly measured because no interconfiguration transitions were detected. It is taken from the CI calculations whose uncertainty is about 3490 cm-1, and it underpins the predicted E1 wavelengths used for the search.
assumptions (4)
  • domain assumption The Zeeman line-shape model with calculated g-factors correctly identifies M1 lines and rules out alternative identifications.
    Used in 'Identifications' to assign lines to Os16+ levels; the measured levels and E2 frequencies depend on these assignments.
  • ad hoc to paper The CI treatment with the pCI package and the basis set up to 13spdfg12h11i10k, with extrapolation of l>5 partial waves, is converged enough for the quoted uncertainties.
    Table IV shows the extrapolation procedure; the authors assign uncertainties by adding all correlation corrections in quadrature, which is a heuristic rather than a rigorous bound.
  • domain assumption The collisional-radiative model (FAC-CRM) predicts steady-state populations accurately enough to conclude the E1 lines are too weak to detect.
    Used in the section on non-detection of interconfiguration lines; the model is validated only via a single line-intensity ratio at 437 nm.
  • domain assumption The upper-state lifetimes and branching ratios are dominated by the E2 and M1 rates used to compute the 44 microHz linewidth.
    Required to quote the clock-transition linewidth; alternative decay channels could alter the lifetime.

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Cite this review

Pith. "Pith review of Study of the elusive $5s-4f$ level crossing in highly charged osmium with optical transitions suitable for physics beyond the Standard Model searches." pith.science (2026). https://pith.science/paper/SPYKHB6U

@misc{pith2026250906710,
  author       = {Pith},
  title        = {Pith review of: Study of the elusive $5s-4f$ level crossing in highly charged osmium with optical transitions suitable for physics beyond the Standard Model searches},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SPYKHB6U}},
  note         = {Machine review of arXiv:2509.06710}
}
abstract

Optical transitions of highly charged ions can be very sensitive to hypothetical beyond-the-Standard-Model phenomena. Those near the $5s-4f$ level crossing, where the $5s$ and $4f$ are degenerate are especially promising. We present predictions from atomic theory and measurements of Os$^{15,16,17+}$ at an electron beam ion trap for identification of several transitions suitable for searches for a hypothetical fifth force and possible violations of local Lorentz invariance. The electric quadrupole (E2) transitions of Os$^{16+}$ that were found are especially suitable for frequency metrology due to their small linewidth of 44 ${\mu}$Hz. Our calculations show the need for including enough inner-shell excitations to predict transition rates between configurations, which can otherwise be overestimated. Ultimately, the predicted interconfiguration transitions were too weak to be detected.

Figures

Figures reproduced from arXiv: 2509.06710 by the authors.

Figure 1
Figure 1. FIG. 1. Examples of recorded spectra. a) Overview spec [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Level scheme of Os [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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    +4f 14” of Table IV. Most levels show small shifts (under 20 cm−1), except for the 4f 125s2 3P0 state, which decreases by 499 cm−1. In the “extras

    P. Bilous, C. Cheung, and M. S. Safronova, A neural- network-based python package for performing large-scale atomic CI using pCI and other high-performance atomic codes, Computer Physics Communications315, 109731 (2025). 8 END MA TTER Details of computations–We treat Os 16+ as...

    2025

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.