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The nuclear charge radius of $^{13}\mathrm{C}$

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Pith's one-line read Collinear laser spectroscopy on helium-like 13C4+ ions, referenced to a frequency comb, determines the 13C charge radius to 2.4464(45) fm — six times sharper than the electron-scattering value and in roughly 3-sigma tension with the…

desk verdict A clean, well-documented laser-spectroscopy measurement of 13C that sharpens the electronic-sector radius sixfold and sharpens a real electron-muon puzzle; the main caveat is the external QED mass shift, a standard reliance rather than a demonstrated flaw. read the letter →

arxiv 2507.05680 v1 pith:MZEODEUN submitted 2025-07-08 physics.atom-ph nucl-th

Patrick Müller , Matthias Heinz , Phillip Imgram , Kristian König , Bernhard Maass , Takayuki Miyagi , Wilfried Nörtershäuser , Robert Roth
show 1 more author
Achim Schwenk
This is my paper · ORCID
classification physics.atom-phnucl-th PACS 21.10.Ft32.30.Jc
keywords nuclearchargeradiuscarbon-13collinearlaserspectroscopyisotopeshifthelium-likeionshyperfinestructureNRQEDmassmuonicatoms
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

The paper claims that the nuclear charge radius of 13C can be pinned down to 2.4464(45) fm purely from the electronic sector, using collinear laser spectroscopy on helium-like 13C4+ ions referenced to a frequency comb. The value is derived from a measured center-of-gravity isotope shift of 51745.6(1.4) MHz between 13C and 12C, minus a calculated NRQED mass shift of 51719.29(25) MHz, divided by a field-shift factor of -211.5(1) MHz/fm2. The result agrees with the electron-scattering radius while cutting its uncertainty by a factor of six, and it disagrees with the comparable-accuracy muonic-atom value by about 3 sigma, the same pattern already seen in 12C. This matters because carbon is one of the few light elements where electron, muon, and laser-spectroscopic radii can be compared at all, so the disagreement is a live test of whether the two electromagnetic probes of nuclear size are consistent. The measurement also required overcoming a technical obstacle: hyperfine-induced mixing shifts individual spectral lines by several GHz, and only a full accounting of all nine hyperfine components recovers a stable center of gravity.

What carries the argument

The central object is the isotope-shift relation $\delta\nu^{A,A'} = \delta\nu_M + F\,\delta\langle r^2\rangle^{A,A'}$, which turns a measured frequency difference between isotopes into a model-independent change of mean-square charge radius once the mass shift $\delta\nu_M$ and the field-shift constant $F$ are supplied by QED theory; here the finite-size signal is just $26.3(1.4)$ MHz out of a transition frequency near $1.3\times10^9$ MHz. The load-bearing experimental innovation is measuring all nine hyperfine components of the $2\,{}^3\mathrm{S}_1 \to 2\,{}^3\mathrm{P}_{0,1,2}$ fine-structure triplet in ${}^{13}\mathrm{C}^{4+}$ with frequency-comb-referenced collinear and anticollinear excitation: second-order hyperfine-induced mixing displaces individual lines by several GHz, yet the weighted center of gravity computed with $6j$-symbol weights, and corrected with the magnetic-dipole matrix elements of Ref. [52], agrees between two independent analysis routes to 500 kHz.

What would settle it

Measure the isotope shift on a second He-like carbon transition with a different field-shift factor, or obtain an independent recalculation of $\delta\nu_M$; a shift of the extracted $\delta\langle r^2\rangle^{12,13}$ by more than about 0.007 fm${}^2$ (a few MHz) would break the claimed value, as would a new muonic-atom radius for 13C that disagrees with 2.4464(45) fm at the 0.001 fm level.

Watch

Extended reading notes

Core claim

The paper establishes an improved electronic-sector value for the 13C charge radius: $R_\mathrm{c}({}^{13}\mathrm{C}) = 2.4464(45)$ fm, reached by combining the measured center-of-gravity isotope shift $\delta\nu^{12,13} = 51\,745.6(1.4)$ MHz with the NRQED mass shift $\delta\nu_M = 51\,719.29(25)$ MHz and the field-shift factor $F = -211.5(1)$ MHz/fm${}^2$. In nuclear-model-independent terms, the shift between the isotopes is $\delta\langle r^2\rangle^{12,13} = -0.1245(66)$ fm${}^2$, a difference in rms radii of $\delta R_c = -0.0253(14)$ fm, so 13C is the smaller nucleus. The electronic value agrees with the 13C electron-scattering result while reducing its uncertainty by a factor of six, and it reaches about the same accuracy as the muonic-atom measurement, with which it disagrees by roughly 3 $\sigma$. The paper interprets this as a systematic electron-versus-muon offset in absolute radii, since the differential radius obtained from the isotope shift agrees with the muonic difference within its larger uncertainty.

Load-bearing premise

The entire extraction rests on the assumption that the quoted NRQED mass shift of $51\,719.29(25)$ MHz carries no hidden error at the few-MHz level: the finite-size signal is only $26.3(1.4)$ MHz, so any unquantified error in the atomic theory translates almost one-for-one into the extracted radius.

Editorial extensions

If this is right

  • If the central claim holds, 12C and 13C become the pair of light nuclei outside hydrogen and helium with the best-known charge radii, and the 13C electronic-sector accuracy now matches the muonic value.
  • The electron-versus-muon offset, about 2.4 sigma in 12C and 2.8-3 sigma in 13C, would become a systematic pattern in absolute radii rather than a single-isotope anomaly, since the differential radius from the new measurement agrees with the muonic difference within uncertainties.
  • The ab initio nuclear-structure calculations reported here (VS-IMSRG and IM-NCSM across seven chiral Hamiltonians) all predict that 13C is smaller than 12C, but overestimate the size reduction by up to a factor of two, signalling missing many-body correlations in a small, precisely measured difference.
  • The full-multiplet measurement strategy that defeats hyperfine-induced mixing is directly applicable to the upcoming measurements in B3+ and to 14C, where the same second-order shifts would otherwise bias the extracted radius at the GHz level.

Reading between the lines

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

  • An experimental cross-check the authors do not pursue: measuring the isotope shift on a second He-like transition with a different field-shift factor would separate the mass-shift and field-shift terms experimentally, testing the NRQED input without requiring new theory.
  • The electron-muon offset now visible at different strengths in 12C, 13C, and the proton could share a common cause inside the electron-scattering analyses of light nuclei, such as an unaccounted radiative or normalization correction; a reanalysis of the 12C and 13C form-factor data would test this directly.
  • Because the finite-size contribution is only 26.3(1.4) MHz, the extracted radius scales almost linearly with any error in F; pinning F empirically, for instance from a second transition in the same ion, would convert the main theory dependence into a checked quantity.
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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 / 4 minor

Summary. The manuscript reports a collinear/anticollinear laser-spectroscopic measurement of all nine hyperfine components of the 2^3S1 → 2^3P_J multiplet in 13C4+ at COALA, with frequencies referenced to a frequency comb. From the centre-of-gravity isotope shift of 51745.6(1.4) MHz relative to 12C4+, and using the NRQED mass shift δν_M = 51719.29(25) MHz and field-shift factor F = -211.5(1) MHz/fm^2 from Ref. [54], the authors extract δ<r^2>^{12,13} = -0.1245(66) fm^2 and R_c(13C) = 2.4464(45) fm. They compare this with electron-scattering and muonic-atom values and with VS-IMSRG, IM-NCSM, and NLEFT ab initio calculations. The paper includes a full systematic budget (dominant 1.72 MHz beam-alignment term; total 1.8 MHz), an internal check of second-order hyperfine mixing, and a data-availability DOI.

Significance. If correct, the result is significant: it provides the most precise electronic-sector radius of 13C, a six-fold improvement over electron scattering, and a benchmark for ab initio nuclear structure calculations of carbon isotopes. The experimental analysis is careful: all nine hyperfine transitions were measured; second-order hyperfine-induced mixing between the 3P_J states is handled explicitly with theoretical matrix elements and cross-checked by two independent centre-of-gravity constructions that agree at 500 kHz; systematic uncertainties are itemized in Table III. The main limitation is the dependence of the radius extraction on external atomic-structure calculations, which is clearly acknowledged but not independently verified.

major comments (1)
  1. [Section II, Eq. (2)] The finite-size signal is the difference between the measured 51745.6(1.4) MHz isotope shift and the external NRQED mass shift 51719.29(25) MHz, leaving only 26.3(1.4) MHz out of a roughly 50 GHz transition. Because F = -211.5(1) MHz/fm^2, a 1 MHz systematic error in δν_M changes δR_c^{12,13} by about 0.0010 fm, i.e., 0.7σ of the quoted 0.0014 fm differential uncertainty (and 0.22σ of the 0.0045 fm absolute uncertainty); a 5 MHz error would move δR_c by 3.5σ. The quoted 0.25 MHz uncertainty of Ref. [54] is adopted without a statement about omitted mα^7, higher-order recoil, or nuclear-size corrections. I recommend adding a short sensitivity paragraph that explains why the 0.25 MHz uncertainty is complete, or explicitly enlarging the theory error; without this, the precision claim is conditional on an unquantified external assumption. The same qualification applies in weaker form to the field-shift factor F, whose 0.1 MHz/fm^2 uncertainty is negligible but which is also not independently verified.
minor comments (4)
  1. [Fig. 3 caption] The abbreviation 'IM-NSCM' appears twice in the caption; the correct abbreviation used elsewhere in the manuscript is 'IM-NCSM'.
  2. [Abstract and final paragraph] The '3σ discrepancy' with the muonic atom result is inherited largely from the 2.4σ offset between the 12C electron-scattering anchor and the muonic 12C radius, since the differential δR_c agrees with the muonic value at about 1.1σ; I suggest stating this inheritance explicitly in the abstract or conclusion.
  3. [Section II, Table I] The three J-component isotope shifts after the mixing correction still differ by up to 30 MHz (51727.6, 51732.6, and 51757.8 MHz). A sentence explaining why this 'splitting isotope shift' does not enter the centre-of-gravity extraction or its uncertainty would help readers assess the 500 kHz agreement between the two analysis methods.
  4. [Section IV, Eq. (8)] The symbol ̃J and the two-step construction of the centre of gravity are introduced only in Methods; a one-sentence cross-reference in Section II where Table I is first discussed would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the 13C radius is extracted from an independent isotope-shift measurement combined with external QED and electron-scattering inputs; no fitted parameter or load-bearing self-citation carries the derivation.

full rationale

The derivation chain is Eq. (1)-(2): δ⟨r²⟩ = (δν - δν_M)/F. The measured δν = 51745.6(1.4) MHz comes from new frequency-comb-referenced collinear-anticollinear laser spectroscopy of 13C4+ referenced to prior 12C4+ frequencies [47]; δν_M = 51719.29(25) MHz and F = -211.5(1) MHz/fm² are taken from Yerokhin et al. [54], an external NRQED calculation, with no parameter adjusted to the target 13C radius. The anchor Rc(12C) = 2.4717(42) fm is a weighted average of independent electron-scattering results [25,69-71]. The ab initio VS-IMSRG and IM-NCSM calculations solve the many-body Schrödinger equation with chiral Hamiltonians and are compared, not fitted, to the measured differential radius. Self-citations [47,48,50,82] supply the experimental method and the 12C reference frequencies; they are methodological and not the load-bearing evidence for the radius value. The skeptic's concern that a few-MHz unquantified error in the external mass shift would shift Rc(13C) is a statement about external systematic uncertainty and model dependence, not about the derivation reducing to its own inputs. Therefore no circular step is present.

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

The paper introduces no new particles, forces, or entities. The free-parameter list is empty because the experiment measures frequencies directly and the mass shift and field shift are borrowed from external QED calculations. The main burden is on the reliability of those external calculations, the hyperfine matrix elements, and the 12C anchor.

assumptions (4)
  • domain assumption The QED mass-shift and field-shift factor for the 3S-3P transition in C4+ are correct to the quoted accuracy (delta_nu_M = 51719.29(25) MHz, F = -211.5(1) MHz/fm^2), without independent verification in this paper.
    The central radius extraction in Section II uses these external values from Ref. [54] directly. The finite-size contribution is only 26.3(1.4) MHz, so a few-MHz error in the mass shift would change the radius at the quoted uncertainty level.
  • domain assumption The magnetic dipole matrix elements of Ref. [52] used in the hyperfine-mixing correction are accurate to about 1e-4, and the 1P1 mixing is adequately treated.
    The second-order hyperfine reduction in Methods, Eqs. (8)-(9), relies on off-diagonal matrix elements from Ref. [52] with no quoted uncertainty. The authors estimate a 1e-4 uncertainty gives shifts smaller than 0.5 MHz, which they state is below the experimental uncertainty.
  • domain assumption The Gaussian lineshape model and the Doppler-tuning assumption that collinear and anticollinear beams address the same velocity class are adequate.
    The fitting in Methods and the systematic budget in Table III rest on these assumptions; the dominant systematic uncertainty of 1.72 MHz is a measured and simulated spread, not a fitted parameter.
  • domain assumption The electron-scattering value for Rc(12C) used as the anchor is correct to its quoted uncertainty.
    The final Rc(13C) value is obtained by combining the measured isotope shift with the 12C anchor. The quoted e- scattering value is itself in 2.4-sigma tension with the muonic 12C value, which is part of the discrepancy pattern discussed.

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Pith. "Pith review of The nuclear charge radius of $^{13}\mathrm{C}$." pith.science (2026). https://pith.science/paper/MZEODEUN

@misc{pith2026250705680,
  author       = {Pith},
  title        = {Pith review of: The nuclear charge radius of $^13\mathrmC$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MZEODEUN}},
  note         = {Machine review of arXiv:2507.05680}
}
abstract

The size is a key property of a nucleus. Accurate nuclear radii are extracted from elastic electron scattering, laser spectroscopy, and muonic atom spectroscopy. The results are not always compatible, as the proton-radius puzzle has shown most dramatically. Beyond helium, precision data from muonic and electronic sources are scarce in the light-mass region. The stable isotopes of carbon are an exception. We present a laser spectroscopic measurement of the root-mean-square (rms) charge radius of $^{13}\mathrm{C}$ and compare this with ab initio nuclear structure calculations. Measuring all hyperfine components of the $2\,^3\mathrm{S} \rightarrow 2\,^3\mathrm{P}$ fine-structure triplet in $^{13}\mathrm{C}^{4+}$ ions referenced to a frequency comb allows us to determine its center-of-gravity with accuracy better than $2\,\mathrm{MHz}$ although second-order hyperfine-structure effects shift individual lines by several $\mathrm{GHz}$. We improved the uncertainty of $R_\mathrm{c}(^{13}\mathrm{C})$ determined with electrons by a factor of $6$ and found a $3\sigma$ discrepancy with the muonic atom result of similar accuracy.

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