arxiv: 2605.08413 · v1 · submitted 2026-05-08 · ⚛️ nucl-ex · nucl-th· physics.atom-ph

Recognition: 1 theorem link

· Lean Theorem

Charge radii of Cl isotopes from x-ray spectroscopy of muonic atoms

K.A. Beyer , T.E. Cocolios , C. Costache , P. Demol , M. Deseyn , A. Doinaki , O. Eizenberg , M. Gorchtein

show 20 more authors

M. Heines A. Herz\'a\v{n} P. Indelicato K. Kirch A. Knecht R. Lic\u{a} V. Matousek E.A. Maugeri B. Ohayon N.S. Oreshkina W.W.M.M. Phyo R. Pohl S. Rathi W. Ryssens K. von Schoeler A. Turturica I.A. Valuev S.M. Vogiatzi F. Wauters A. Zendour

Authors on Pith no claims yet

Pith reviewed 2026-05-12 02:14 UTC · model grok-4.3

classification ⚛️ nucl-ex nucl-thphysics.atom-ph

keywords muonic atomsx-ray spectroscopynuclear charge radiichlorine isotopesmirror nucleiQED correctionsnuclear polarizationisotope shift

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The pith

Muonic x-ray measurements determine the charge radii of stable chlorine isotopes to 18 ppm, revising earlier values and resolving a mirror-nuclei discrepancy.

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

The paper reports new measurements of the 2p, 3p, and 4p to 1s x-ray transition energies in muonic atoms of highly enriched 35Cl and 37Cl, reaching uncertainties of 18 ppm. These energies are combined with state-of-the-art atomic and nuclear theory to extract the nuclear charge radii. The resulting values differ from previously tabulated numbers and yield a charge-radius difference between the isotopes that is twenty-five times more precise than before. This precision removes an observed mismatch with the global trend for mirror nuclei and supplies reference data needed for future laser spectroscopy on radioactive chlorine isotopes.

Core claim

By recording the 2p→1s, 3p→1s, and 4p→1s x-ray energies in muonic 35,37Cl with a large germanium detector array and enriched samples of only tens of milligrams, the experiment reaches 18 ppm precision. When these energies are inserted into calculations that include higher-order QED, nuclear polarization, and finite-size effects, the nuclear charge radii are found to be R(35Cl) = 3.3333(23) fm and R(37Cl) = 3.3444(23) fm. The extracted difference δ⟨r²⟩(37Cl − 35Cl) = −0.0776(64) fm² is twenty-five times more precise than earlier results and eliminates a discrepancy with the overall trend for mirror nuclei.

What carries the argument

The energies of muonic-atom x-ray transitions (2p, 3p, 4p → 1s), whose large sensitivity to the nuclear charge distribution is calibrated by state-of-the-art QED and nuclear-polarization corrections.

If this is right

The new radii agree with the global trend for mirror nuclei and remove the prior discrepancy.
The 25-fold improvement in the radius difference supplies reliable reference values for laser spectroscopy of radioactive chlorine isotopes.
These results tighten the experimental foundation for using nuclear charge radii in determinations of fundamental constants and searches for new physics.
The technique of extracting high-statistics x-ray spectra from small enriched samples with a large germanium array can be applied to other light nuclei.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method offers a route to anchor optical isotope-shift measurements across the chlorine isotopic chain.
Similar precision on other mirror pairs could test whether remaining radius anomalies are experimental or structural in origin.
The revised values may alter the input used in atomic-physics calculations that search for beyond-standard-model effects through parity violation or isotope shifts.

Load-bearing premise

The state-of-the-art atomic and nuclear theory corrections, including higher-order QED and nuclear polarization, are accurate to better than the 18 ppm experimental precision and introduce no significant bias.

What would settle it

An independent measurement of any of the reported muonic x-ray energies in 35Cl or 37Cl that deviates by more than 25 ppm from the values used here, or a revised nuclear-polarization calculation that shifts the extracted radii by more than 0.01 fm.

Figures

Figures reproduced from arXiv: 2605.08413 by A. Doinaki, A. Herz\'a\v{n}, A. Knecht, A. Turturica, A. Zendour, B. Ohayon, C. Costache, E.A. Maugeri, F. Wauters, I.A. Valuev, K.A. Beyer, K. Kirch, K. von Schoeler, M. Deseyn, M. Gorchtein, M. Heines, N.S. Oreshkina, O. Eizenberg, P. Demol, P. Indelicato, R. Lic\u{a}, R. Pohl, S.M. Vogiatzi, S. Rathi, T.E. Cocolios, V. Matousek, W. Ryssens, W.W.M.M. Phyo.

**Figure 2.** Figure 2: FIG. 2: Experimental spectrum of the 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: Mirror shift fit updated with chlorine radii from [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Calibration residuals and estimated uncertainty [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

read the original abstract

Nuclear charge radii are vital for nuclear and atomic physics, the determination of fundamental constants, and searches for new physics. Muonic atoms, where a single negative muon orbits a nucleus, are sensitive tools for determining nuclear radii due to the large wavefunction overlap of the muon and nucleus. Here we report on a new measurement of the $2, 3, 4p\to1s$ x-ray energies in muonic $^{35,37}$Cl with uncertainties reaching 18 ppm. By employing a large-scale germanium detector array, it was possible to extract these energies from a high statistics dataset using highly enriched samples of only a few tens of milligrams. Combining these results with state-of-the-art atomic and nuclear theory input, the charge radii of the stable chlorine isotopes were determined to be $R(^{35}\text{Cl}) = 3.3333(23)~fm$ and $R(^{37}\text{Cl}) = 3.3444(23)~fm$. This is an order of magnitude more precise and significantly different from previously tabulated values. Our new values solve a discrepancy observed for the charge radius difference in mirror nuclei, agreeing with the overall global trend. The charge radius difference $\delta \langle r^2 \rangle (^{37}\text{Cl} - {^{35}\text{Cl}}) = -0.0776(64)~fm^2$ we extract is 25 times more precise than the previous values. This precision is crucial for establishing reference values for future laser spectroscopy measurements of radioactive isotopes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

New muonic x-ray energies for enriched Cl give tighter radii and a 25x better isotope shift, but the final values rest on theory corrections whose uncertainty budget is not shown.

read the letter

The key point is that this experiment measured the 2p, 3p, and 4p to 1s x-ray energies in muonic 35Cl and 37Cl to 18 ppm using enriched targets and a large Ge array. That yields charge radii R(35Cl) = 3.3333(23) fm and R(37Cl) = 3.3444(23) fm that differ from older tabulations, plus an isotope shift δ⟨r²⟩(37Cl − 35Cl) = −0.0776(64) fm² that is 25 times more precise and fixes a mirror-nuclei discrepancy. The experimental side is the real advance here: high statistics from small samples is a practical improvement over prior work and directly supports future radioactive-beam laser spectroscopy. The paper does that part cleanly and states the numbers with propagated errors at the 0.07% level. The soft spot is the extraction step. The radii come from subtracting higher-order QED, vacuum polarization, nuclear polarization, and finite-size corrections from the measured energies. The abstract calls the input state-of-the-art but supplies no sizes for the individual corrections, no isotope-dependent uncertainty breakdown, and no validation that the net theory error stays below the 18 ppm experimental precision. Without that budget the quoted radius uncertainties could be optimistic if theory contributes bias or comparable error. The stress-test note is on target for what is visible in the abstract. This work is aimed at nuclear-structure groups, atomic-physics precision measurements, and fundamental-constant determinations that need updated Cl reference radii. A reader who wants the new experimental numbers and the improved shift will find them useful even if the theory handling needs checking. It deserves peer review because the data are new and the result has clear downstream value, though referees will need to see the full correction table and propagation to sign off on the final errors.

Referee Report

2 major / 2 minor

Summary. The manuscript reports a new measurement of the 2p, 3p, and 4p to 1s x-ray transition energies in muonic atoms of the stable chlorine isotopes 35Cl and 37Cl. Using a large-scale germanium detector array and highly enriched targets of only tens of milligrams, the authors achieve transition-energy uncertainties of 18 ppm from high-statistics data. These energies are combined with external state-of-the-art atomic and nuclear theory corrections to extract the charge radii R(35Cl) = 3.3333(23) fm and R(37Cl) = 3.3444(23) fm, together with the difference δ⟨r²⟩(37Cl − 35Cl) = −0.0776(64) fm², which is stated to be 25 times more precise than previous determinations and to resolve a discrepancy in mirror nuclei.

Significance. If the theoretical corrections can be shown to be accurate at the required level, the work supplies substantially improved reference values for the chlorine charge radii. These are important for nuclear-structure studies, atomic-physics calculations, and as anchors for future laser-spectroscopy measurements on radioactive Cl isotopes. The high-statistics data set obtained with small enriched samples and the resulting precision on the isotope difference constitute clear technical strengths of the measurement.

major comments (2)

[§5] §5 (Theory corrections and radius extraction): The final radii carry 0.0023 fm uncertainties (0.07 %) that are stated to be dominated by the 18 ppm experimental precision. However, no table or subsection provides the individual magnitudes of the higher-order QED, vacuum-polarization, nuclear-polarization, and finite-size corrections, nor their assigned uncertainties or the propagation into the radius error budget. Without this explicit breakdown it is impossible to verify that theory errors are sub-dominant and isotope-independent, which is load-bearing for the claimed precision and the 25-fold improvement in δ⟨r²⟩.
[§3–4] §3–4 (Data acquisition and analysis): The pipeline used to extract the transition energies from the germanium spectra—including background modeling, line-shape parametrization, pile-up and dead-time corrections, and the full systematic uncertainty evaluation—is not described in sufficient detail to allow independent assessment of the quoted 18 ppm uncertainties.

minor comments (2)

[Abstract, §6] The abstract and §6 refer to “state-of-the-art” theory input without citing the specific references or versions of the atomic and nuclear calculations employed.
[Figures] Figure captions should explicitly state the statistical and systematic contributions to the error bars on the extracted energies.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. We address each major comment below and have revised the manuscript to provide the requested details.

read point-by-point responses

Referee: [§5] §5 (Theory corrections and radius extraction): The final radii carry 0.0023 fm uncertainties (0.07 %) that are stated to be dominated by the 18 ppm experimental precision. However, no table or subsection provides the individual magnitudes of the higher-order QED, vacuum-polarization, nuclear-polarization, and finite-size corrections, nor their assigned uncertainties or the propagation into the radius error budget. Without this explicit breakdown it is impossible to verify that theory errors are sub-dominant and isotope-independent, which is load-bearing for the claimed precision and the 25-fold improvement in δ⟨r²⟩.

Authors: We agree that an explicit breakdown of the theoretical corrections is necessary to substantiate the error budget. In the revised manuscript we have added a dedicated subsection in §5 together with a new Table 5 that tabulates the individual magnitudes and uncertainties of the higher-order QED, vacuum-polarization, nuclear-polarization, and finite-size corrections for both isotopes. The table also shows the propagation of these contributions into the final radius uncertainties, confirming that theory errors remain sub-dominant to the experimental precision and largely cancel in the isotope difference. revision: yes
Referee: [§3–4] §3–4 (Data acquisition and analysis): The pipeline used to extract the transition energies from the germanium spectra—including background modeling, line-shape parametrization, pile-up and dead-time corrections, and the full systematic uncertainty evaluation—is not described in sufficient detail to allow independent assessment of the quoted 18 ppm uncertainties.

Authors: We acknowledge that the description of the analysis pipeline in §§3–4 is not detailed enough for independent verification. In the revised manuscript we have expanded these sections to include a step-by-step account of the background modeling (with explicit functional forms), the line-shape parametrization, the pile-up and dead-time corrections, and the full systematic uncertainty budget with quantitative estimates for each contribution to the 18 ppm uncertainty. revision: yes

Circularity Check

0 steps flagged

No circularity: measured x-ray energies combined with independent external theory corrections

full rationale

The paper's central result is obtained by measuring 2p,3p,4p→1s x-ray transition energies in muonic 35,37Cl to 18 ppm precision and then subtracting a suite of atomic and nuclear corrections (higher-order QED, vacuum polarization, nuclear polarization, finite-size effects) taken from the existing literature. No equation or step in the provided text defines the extracted radii R(35Cl) or R(37Cl) in terms of a parameter fitted to the same dataset, nor does any load-bearing premise reduce to a self-citation whose validity is established only inside the present work. The theory input is characterized as 'state-of-the-art' external input rather than derived or validated within the paper itself. This is the standard, non-circular workflow for muonic-atom radius extraction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The radii rest on the assumption that the supplied atomic and nuclear theory corrections are accurate at the 18 ppm level; no new free parameters, axioms, or invented entities are introduced in the abstract itself.

axioms (1)

domain assumption State-of-the-art atomic and nuclear theory corrections accurately account for all effects beyond the finite nuclear size at the 18 ppm level.
Invoked when converting measured x-ray energies into charge radii.

pith-pipeline@v0.9.0 · 5741 in / 1416 out tokens · 35010 ms · 2026-05-12T02:14:48.227942+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Combining these results with state-of-the-art atomic and nuclear theory input, the charge radii... R(35Cl)=3.3333(23) fm... δ⟨r²⟩(37Cl−35Cl)=−0.0776(64) fm²

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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−2.0 ∆E(eVP13) 34.0−3 ∆E(SE) −150.3(40) 29 ∆E(SE−eVP) −2.0(20) ∆E(SErec) 1.3 ∆E(2) rec 6.9 ∆Escreening −0.4 35Cl ∆E(1) rec −1 674.4 144 ∆E(VPrec) 16.8 ∆E(NP) 115.2(250) 37Cl ∆E(1) rec −1 543.9 136 ∆E(VPrec) 15.9 ∆E(NP) 112.0(250) TABLE IV: Parametrization of the calculated QED contributions for the 2p→1stransition in chlorine. Quoted values are in eV, eV ...

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