Pith. sign in

REVIEW 1 major objections 5 minor 1 cited by

A microcalorimeter measurement of muonic beryllium-9 yields a nuclear charge radius 2.4 times more precise than the electron-scattering value, and 2.3σ larger.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Muonic X-ray spectroscopy with a microcalorimeter yields r_c(9Be)=2.5506(51) fm, 2.4× more precise than and 2.3σ above the electron-scattering value.

T0 review reviewed 2026-08-02 challenge →

load-bearing objection A real, carefully executed muonic measurement with a 30x precision gain; the extracted radius is plausible but rests on nuclear-polarization theory that is not yet public. the 1 major comments →

arxiv 2607.13690 v1 pith:7BGM4BES submitted 2026-07-15 nucl-ex nucl-thphysics.atom-ph

Nuclear Charge Radius of $^9$Be from Muonic Atom Spectroscopy Using a Microcalorimeter

Ofir Eizenberg , Shikha Rathi , Andreas Abeln , Sonia Bacca , Gon\c{c}alo Baptista , Nir Barnea , Noam Burger , Thomas Elias Cocolios
show 29 more authors
Marie Deseyn Tim Egert Christian Enss Andreas Fleischmann Loredana Gastaldo C\'esar Godinho Nitzan Goldberg Michael Heines Daniel Hengstler Paul Indelicato Weiguang Jiang Klaus Kirch Andreas Knecht Daniel Kreuzberger Jorge Machado Ulf-G. Mei{\ss}ner Ben Ohayon Nancy Paul Randolf Pohl Tim Redelbach Michael Roosa Katharina von Schoeler Quentin Senetaire Shihang Shen Daniel Unger Stergiani Marina Vogiatzi Johanna Walch Frederik Wauters Aziza Zendour
This is my paper
classification nucl-ex nucl-thphysics.atom-ph PACS 21.10.Ft36.10.-k07.85.Nc
keywords muonic atomnuclear charge radiusberyllium-9metallic magnetic calorimetermicrocalorimeterx-ray spectroscopynuclear polarization2p-1s transition
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

Metallic magnetic calorimeters, cryogenic detectors that record the heat deposited by each incoming x-ray, have reached the precision needed to measure muonic-atom transition energies to sub-eV accuracy in a beam environment. This paper applies them to muonic beryllium-9 and measures the 2p→1s transition at 33 391.48(34) eV, thirty times more precisely than the best earlier value. Plugging that energy into a theoretical energy-radius relation yields a nuclear charge radius r_c = 2.5506(51) fm, 2.4 times more precise than the accepted electron-scattering value and larger by 2.3 combined standard errors. A sympathetic reader would see this as the first demonstration of a new metrological path for charge radii of light nuclei, one that does not rely on the model-dependent momentum-transfer extrapolation of electron scattering.

Core claim

The paper claims that the 2p→1s centroid energy of muonic 9Be is 33 391.48(34) eV, obtained from the spectrum of a metallic magnetic calorimeter array operated with an in-situ lanthanum Kα1 x-ray fluorescence calibration line. Combining this with a theoretical parameterization that includes QED, finite nuclear size, and nuclear-structure corrections (shape, nucleon polarization, and nuclear polarization), the authors extract r_c(9Be) = 2.5506(51) fm. This is the first nuclear charge radius determined by muonic x-ray spectroscopy with microcalorimeters. The result is 2.4 times more precise than the commonly cited electron-scattering radius of 2.519(12) fm and disagrees with it by 2.3 combined

What carries the argument

The measuring instrument is a metallic magnetic calorimeter (MMC): a cryogenic microcalorimeter in which each absorbed photon raises the temperature of a magnetized absorber, altering its magnetization detected by a SQUID. The array achieved about 20 eV FWHM at 33 keV, roughly 15 times better than conventional semiconductor detectors. The analysis chain includes a per-pixel temperature-drift correction, a residual-nonlinearity calibration using reference lines on both sides of the region of interest, and a simultaneous maximum-likelihood fit of muon-coincident and anti-coincident spectra, from which the centroid separation between the La Kα1 line and the muonic beryllium line is obtained as

Load-bearing premise

The entire radius extraction hangs on the theoretical energy-radius relation, specifically the nuclear-polarization correction ΔE_NP = 1.00(15) eV, which is computed with an artificial neural network that is not yet published; if this correction is off by more than about 0.2 eV, the 2.3σ discrepancy with electron scattering could disappear.

What would settle it

Re-measure the muonic 9Be 2p→1s transition using a calibration line with an independently verified absolute energy (traceable to a primary standard) and check whether 33 391.48(34) eV reproduces; or, theoretically, recompute ΔE_NP using a fully documented, published neural network and an explicit treatment of the >5 MeV dipole strength, and test whether the correction remains 1.00(15) eV rather than shifting by 0.2 eV or more.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • An improved 9Be anchor shifts the radii of the whole beryllium isotope chain (7Be to 12Be) by about 0.03 fm, bringing most of them into better agreement with modern ab initio lattice calculations.
  • The 30-fold precision gain demonstrates that microcalorimeters can close the gap between muonic x-ray spectroscopy and laser spectroscopy for light nuclei beyond helium.
  • The 2.3σ tension with electron scattering implies that electron-scattering radii for light nuclei, where only limited momentum-transfer ranges are explored, may carry underestimated systematic uncertainties.
  • The updated 9Be radius, combined with mirror-nucleus relations, yields an improved prediction for the 10C charge radius, relevant to superallowed beta-decay precision tests.

Where Pith is reading between the lines

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

  • If the central claim holds, the same detector technology could resolve the long-standing discrepancies between muonic and electron-scattering radii in other light nuclei such as boron, carbon, and nitrogen; a systematic survey would reveal whether the 9Be offset is a single-isotope anomaly or a general trend.
  • The uncertainty budget is now dominated by the calibration-line energy; a more accurate measurement of the lanthanum Kα1 reference energy, or replacement with a line traceable to a primary standard, would immediately reduce the radius uncertainty without additional beam time.
  • The nuclear-polarization correction currently rests on an unpublished neural-network model; publishing that model with open validation against the photodisintegration data below 5 MeV would allow an independent check of the 0.15 eV uncertainty, which is comparable to the experimental statistical uncertainty.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The paper reports a measurement of the 2p→1s transition energy in muonic 9Be using a metallic magnetic calorimeter (MMC) array at PSI. The measured centroid energy is E = 33,391.48(34) eV, a 30-fold improvement over the previous muonic measurement. Combining this with a theoretical energy-radius relation that includes QED and nuclear-structure corrections (shape, nucleon polarization, nuclear polarization), the authors extract r_c(9Be) = 2.5506(52) fm, 2.4 times more precise than and 2.3 combined standard deviations above the electron-scattering value. The paper also updates the radii of the Be isotopic chain using known isotope shifts and compares with ab initio calculations.

Significance. If the result holds, this is the first nuclear charge radius determined with muonic x-ray spectroscopy using microcalorimeters, demonstrating a new metrology path for light nuclei. The experimental chain is careful: temperature-drift correction, per-pixel nonlinearity, coincidence/anti-coincidence separation, hierarchical fits with a 50 meV line-shape systematic, and a clear error budget. However, the extracted radius relies on the theoretical nuclear-polarization correction whose central value and uncertainty are partly based on unpublished work. The claimed 2.3σ discrepancy with electron scattering depends on this correction; therefore the central claim is conditional on the validity of the nuclear-structure calculation.

major comments (1)
  1. [End Matter, Eqs. (10)–(16)] The nuclear-polarization correction ΔE_NP = 1.00(15) eV is the largest nuclear-structure term in the energy-radius relation used to extract r_c. Its leading order ΔE_NP^(0) rests on an artificial neural network strength function from Ref. [59] (in preparation) and a dedicated fit to photodisintegration data [60–62]; its next-to-leading order ΔE_NP^(1) is deferred to 'a separate publication.' A 0.2 eV error in this correction shifts r_c by ≈0.003 fm, about half the quoted total uncertainty (0.0052 fm). As written, the manuscript does not allow the reader to assess the reliability of this correction, which is load-bearing for the central radius claim and for the 2.3σ discrepancy with electron scattering. The authors should provide the ANN and fit details in the paper or supplement, or at minimum give a sensitivity analysis of r_c to the assumed E1 strength model.
minor comments (5)
  1. [Abstract and Eq. (3)] The abstract quotes E = 33,391.48(34) eV, while Eq. (3) gives separate fit and systematic uncertainties of (22) and (27) eV, which combine to 0.35 eV. Please report the combined uncertainty consistently.
  2. [Main text, radius extraction paragraph] The sentence 'we subtract the 66 meV taken by the recoiling atom' is misleading. Since E_transition = E_photon + E_recoil for photon emission, the recoil energy must be added to the photon energy. The numerical result indicates the correct addition was made; please fix the wording.
  3. [End Matter, Eq. (10)] Equation (10) lists '±η²ΔE_NP^(0)' as a term in the sum for ΔE_NP, but the text describes it as an uncertainty. Clarify whether this term enters the central value or only the uncertainty budget, and how the final 0.15 eV uncertainty in Eq. (14) is obtained.
  4. [Table II] The GFMC column is empty for 7Be, but the text states that 'Both the literature value and our updated radius are consistent with Green's function Monte-Carlo calculations [41] within uncertainty.' Specify which isotope(s) this statement refers to.
  5. [Main text, Final fit and energy extraction] The systematic uncertainty from line-shape modeling is stated only as 'on the order of 50 meV.' State explicitly how this number was derived from the hierarchical fitting stages in Table IV, including which pair of fits was compared.

Circularity Check

0 steps flagged

No significant circularity: the muonic transition energy is measured against external XRF calibrants and r_c is solved from a computed E(r_c) relation; the in-preparation ANN nuclear-polarization model is a verification gap, not a circular input.

full rationale

The claimed derivation chain is E(µ9Be:2p→1s) -> solve Eq. (16) for r_c. The experimental value Eq. (3) is anchored to independently tabulated La/Ba/Am calibration lines (Table I) and a measured 50.71(21) eV centroid offset (Eq. 2), so the measurement is not defined in terms of r_c. Eq. (16) is assembled from a QED parameterization (Eq. 6 from Ref. [54]), a NLEFT-based shape correction (Eq. 8), nucleon polarization (Eq. 9), and nuclear polarization (Eqs. 10-14). Its constant and coefficients are calculated, not fitted to the muonic energy; the photodisintegration data [60-62] used for the low-energy E1 strength are external to the muonic measurement. Consequently, equating Eq. (3) with Eq. (16) is a genuine inversion rather than a fit renamed as a prediction. Several cited theory inputs share authors with this paper (Refs. [40,54,56,57,59]), but they are not invoked as a uniqueness theorem and do not incorporate the target radius. The text itself flags the main caveat: 'The leading order term, ∆E(0)NP... can be predicted using a novel artificial neural network approach [59]' with Ref. [59] listed as 'in preparation,' and after the ∆E(1)NP calculation it states 'More details will be provided in a separate publication.' This is an omitted-proof/verification risk—a 0.2 eV error there shifts r_c by ≈0.003 fm—but no step of the derivation reduces, by the paper's own equations, to its own input. Secondary isotope-chain and mirror radii are explicitly assumption-dependent (Table II italics), not presented as first-principles predictions. Score 0 for circularity.

Axiom & Free-Parameter Ledger

5 free parameters · 7 axioms · 0 invented entities

No new physical entities are introduced. The central inputs that are not independently documented are the neural-network E1 strength model [59] and the NLO nuclear-polarization calculation deferred to a separate publication; these are modeling tools rather than new entities. The paper's fitted parameters are detector-calibration, empirical-lineshape, and nuclear-structure model inputs.

free parameters (5)
  • Pixel nonlinearity parameter n (per pixel) = 1.65e-4 to 1.95e-4 /keV
    Eq. (1) E(A_i)=m(A_i + n A_i^2); fitted to calibration lines in Table I. The correction is ≈0.3 eV at 33 keV, comparable to the statistical uncertainty.
  • Empirical lineshape parameters (Voigt core, exponential tails, step, background, amplitudes) = Γ_G≈23 eV, f_voigt≈0.89, b_rt≈25 eV, b_lt/b_rt≈2.1, f_lt/f_rt≈1.2, f_step≈1e-3 (Table IV)
    The 21-parameter SATLAS-2 fit uses these to extract the centroid difference; model variation contributes a 50 meV systematic.
  • Residual cross-contamination amplitudes = Two additional free parameters
    Account for calibration-line bleeding between coincidence and anti-coincidence spectra; part of the coupled fit.
  • Low-energy E1 strength function fit = Not quoted; dedicated fit to Refs. [60–62] below 5 MeV
    Used in ΔE_NP(0)=923+74−58 meV; higher energies come from the ANN prediction [59]. Feeds the 1.00(15) eV nuclear-polarization correction.
  • NLEFT systematic uncertainty estimate = (r_z/r_c)_NLEFT = 1.509(2)[2]
    Shape correction ΔE_shape=-0.07(4) eV uses the NLEFT charge-density ratio; the systematic is estimated as half the difference between N3LO and SU(4) interactions.
axioms (7)
  • domain assumption The theoretical muonic-atom QED parameterization, Eq. (6) from Ref. [54], correctly gives E_{2p-1S}(r_c) including finite-size, recoil, and QED effects.
    Used to convert the measured energy to radius; the resulting Eq. (16) carries uncertainty 0.16 eV. The central claim depends on this theory being complete.
  • domain assumption The tabulated La Kα1 energy 33442.12(27) eV [30] is accurate.
    The result is anchored to this external calibration value; it contributes 51% of the final variance. A bias here shifts r_c directly.
  • domain assumption The NLEFT charge distribution (pinhole algorithm) reliably gives the 9Be charge-density shape, specifically the ratio r_z/r_c = 1.509(2)[2].
    The shape correction ΔE_shape=-0.07(4) eV uses this ratio; a larger deviation from the real shape would shift the extracted radius.
  • ad hoc to paper The E1 dipole strength function model (low-energy fit to Refs. [60–62] + ANN prediction [59]) is accurate enough for ΔE_NP(0)=923+74−58 meV.
    Ref. [59] is 'in preparation' and not independently verifiable from this paper; the 74 meV uncertainty is included in the theory error.
  • domain assumption Inelastic three-photon exchange can be estimated as half the elastic contribution with 100% uncertainty (ΔE_3pE≈85(85) meV).
    Follows Ref. [8]; contributes to ΔE_NP uncertainty and hence to the radius error budget.
  • domain assumption The calculated hyperfine structure (Table III) is accurate; 'reasonable variations' do not significantly affect the centroid.
    Fitting the Be line as a hyperfine multiplet requires relative energies from the MDFGME code [52,53]; residual errors could bias the centroid, though the authors state they tested variations.
  • domain assumption Mirror-shift relation and negligible isospin symmetry breaking hold for the 10C and 10B* predictions.
    Used to derive r_c(10C)=2.671(14) fm and r_c(10B*)=2.565(13) fm; the authors flag these as assumption-dependent (italics in Table II).

reviewed 2026-08-02 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Nuclear Charge Radius of $^9$Be from Muonic Atom Spectroscopy Using a Microcalorimeter." pith.science (2026). https://pith.science/paper/7BGM4BES

@misc{pith2026260713690,
  author       = {Pith},
  title        = {Pith review of: Nuclear Charge Radius of $^9$Be from Muonic Atom Spectroscopy Using a Microcalorimeter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7BGM4BES}},
  note         = {Machine review of arXiv:2607.13690}
}
Share X Bluesky LinkedIn Reddit HN
abstract

The $2p\to1s$ transition energy in muonic $^9$Be was measured using a metallic magnetic calorimeter, resulting in $E_{2p\to 1s}=33\,391.48(34)\,$eV. The result is 30 times more precise than the previous best measurement and enables the extraction of the corresponding nuclear charge radius $r_c($$^9$Be$)=2.5506(51)\,$fm. It is $2.4$ times more precise than the commonly used value based on electron scattering and differs from it by $2.3$ times the combined uncertainties. This measurement represents the first determination of a nuclear charge radius using muonic x-ray spectroscopy with microcalorimeters.

Figures

Figures reproduced from arXiv: 2607.13690 by Andreas Abeln, Andreas Fleischmann, Andreas Knecht, Aziza Zendour, Ben Ohayon, C\'esar Godinho, Christian Enss, Daniel Hengstler, Daniel Kreuzberger, Daniel Unger, Frederik Wauters, Gon\c{c}alo Baptista, Johanna Walch, Jorge Machado, Katharina von Schoeler, Klaus Kirch, Loredana Gastaldo, Marie Deseyn, Michael Heines, Michael Roosa, Nancy Paul, Nir Barnea, Nitzan Goldberg, Noam Burger, Ofir Eizenberg, Paul Indelicato, Quentin Senetaire, Randolf Pohl, Shihang Shen, Shikha Rathi, Sonia Bacca, Stergiani Marina Vogiatzi, Thomas Elias Cocolios, Tim Egert, Tim Redelbach, Ulf-G. Mei{\ss}ner, Weiguang Jiang.

Figure 1
Figure 1. Figure 1: FIG. 1. Temperature correction for a single pixel. The [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Fitted histograms of the combined spectra from 35 pixels in the area of interest. The top figure shows x rays in time [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Recent Progress in Ab-Initio Nuclear Theory for Precision Physics Searches in Muonic Atoms and Superallowed $\beta$ Decays

    nucl-th 2026-07 accept novelty 2.0

    Ab initio nuclear theory for two-photon exchange in muonic atoms and the γW box in superallowed β decays shares one hadronic tensor, with recent light-nuclei results impacting charge radii, the helium isotope shift, and Vud.

Reference graph

Works this paper leans on

69 extracted references · 9 linked inside Pith · cited by 1 Pith paper

  1. [1]

    J. L. Friar, The structure of light nuclei and its effect on precise atomic measurements, inPrecision Physics of Simple Atomic Systems, edited by S. G. Karshenboim and V. B. Smirnov (Springer Berlin Heidelberg, Berlin, Heidelberg, 2003) pp. 59–79

    2003

  2. [2]

    Antognini, F

    A. Antognini, F. Kottmann, and R. Pohl, Laser spec- troscopy of light muonic atoms and the nuclear charge radii, SciPost Phys. Proc. , 021 (2021)

    2021

  3. [3]

    Gorchtein, V

    M. Gorchtein, V. Katyal, B. Ohayon, B. K. Sahoo, and C.-Y. Seng, Cabibbo-Kobayashi-Maskawa unitarity deficit reduction via finite nuclear size, Phys. Rev. Res. 7, L042002 (2025)

    2025

  4. [4]

    P. J. Mohr, D. B. Newell, B. N. Taylor, and E. Tiesinga, CODATA recommended values of the fundamental phys- ical constants: 2022, Rev. Mod. Phys.97, 025002 (2025)

    2022

  5. [5]

    V. A. Yerokhin and B. Ohayon, Model-independent de- termination of nuclear charge radii from Li-like ions, Phys. Rev. A113, 012804 (2026)

    2026

  6. [6]

    Fricke and K

    G. Fricke and K. Heilig, Nuclear Charge Radii 4-Be Beryllium: Datasheet from Landolt-Börnstein - Group I Elementary Particles, Nuclei and Atoms Volume 20: Nuclear Charge Radii (2004)

    2004

  7. [7]

    Ohayon, A

    B. Ohayon, A. Abeln, S. Bara, T. E. Cocolios, O. Eizenberg, A. Fleischmann, L. Gastaldo, C. God- inho, M. Heines, D. Hengstler, G. Hupin, P. Indeli- cato, K. Kirch, A. Knecht, D. Kreuzberger, J. Machado, P. Navratil, N. Paul, R. Pohl, D. Unger, S. M. Vo- giatzi, K. v. Schoeler, and F. Wauters, Towards Precision Muonic X-ray Measurements of Charge Radii of...

    2024

  8. [8]

    Pachucki, V

    K. Pachucki, V. Lensky, F. Hagelstein, S. S. Li Muli, S. Bacca, and R. Pohl, Comprehensive theory of the Lamb shift in light muonic atoms, Rev. Mod. Phys.96, 015001 (2024)

    2024

  9. [9]

    Ohayon, Critical evaluation of reference charge radii and applications in mirror nuclei, Atomic Data and Nu- clear Data Tables165, 101732 (2025)

    B. Ohayon, Critical evaluation of reference charge radii and applications in mirror nuclei, Atomic Data and Nu- clear Data Tables165, 101732 (2025)

    2025

  10. [10]

    Schmidt, M

    S. Schmidt, M. Willig, J. Haack, R. Horn, A. Adam- czak, M. A. Ahmed, F. D. Amaro, P. Amaro, F. Biraben, P. Carvalho, T.-L. Chen, L. M. P. Fernandes, T. Graf, M. Guerra, T. W. Hänsch, M. Hildebrandt, Y.-C. Huang, P. Indelicato, L. Julien, K. Kirch, A. Knecht, F. Kottmann, J. J. Krauth, Y.-W. Liu, J. Machado, M.Marszalek, C.M.B.Monteiro, F.Nez, J.Nuber, D....

    2018

  11. [11]

    L. A. Schaller, L. Schellenberg, A. Ruetschi, and H. Schneuwly, Nuclear charge radii from muonic x-ray transitions in beryllium, boron, carbon and nitrogen, Nu- clear Physics A343, 333 (1980)

    1980

  12. [12]

    Angeli, D

    I. Angeli, D. L. Balabanski, P. Dimitriou, K. T. Flana- gan, G. Georgiev, M. Gorchtein, P. Gùeye, F. Heiße, A. Knecht, K. Minamisono,et al., Towards better nuclear charge radii, arXiv preprint arXiv:2604.08985 (2026)

    Pith/arXiv arXiv 2026

  13. [13]

    Jansen, R

    J. Jansen, R. T. Peerdeman, and C. De Vries, Nuclear charge radii of12C and 9Be, Nuclear Physics A188, 337 (1972)

    1972

  14. [14]

    Fleischmann, C

    A. Fleischmann, C. Enss, and G. M. Seidel, Metallic magnetic calorimeters, inCryogenic Particle Detection (Springer Berlin Heidelberg, Berlin, Heidelberg, 2005) pp. 151–216

    2005

  15. [15]

    C. Enss, A. Fleischmann, K. Horst, J. Schönefeld, J. Soll- ner, J. S. Adams, Y.H. Huang, Y. H. Kim, andG. M. Sei- del, Metallic magnetic calorimeters for particle detection, Journal of Low Temperature Physics121, 137 (2000)

    2000

  16. [16]

    Fleischmann, M

    A. Fleischmann, M. Linck, T. Daniyarov, H. Rotzinger, C. Enss, and G. M. Seidel, Magnetic calorimeters for high resolution X-ray spectroscopy, Nuclear Instruments and Methods in Physics Research Section A: Acceler- ators, Spectrometers, Detectors and Associated Equip- ment520, 27 (2004)

    2004

  17. [17]

    C. Pies, S. Schäfer, S. Heuser, S. Kempf, A. Pabinger, J.-P. Porst, P. Ranitsch, N. Foerster, D. Hengstler, A. Kampkötter, T. Wolf, L. Gastaldo, A. Fleischmann, and C. Enss, maXs: Microcalorimeter Arrays for High- Resolution X-Ray Spectroscopy at GSI/FAIR, Journal of Low Temperature Physics167, 269 (2012)

    2012

  18. [18]

    Kempf, A

    S. Kempf, A. Fleischmann, L. Gastaldo, and C. Enss, Physics and applications of metallic magnetic calorime- ters, Journal of Low Temperature Physics193, 365 (2018)

    2018

  19. [19]

    Pfäfflein, S

    P. Pfäfflein, S. Allgeier, S. Bernitt, A. Fleischmann, M. Friedrich, C. Hahn, D. Hengstler, M. O. Herdrich, A. Kalinin, F. M. Kröger,et al., Integration of maXs- type microcalorimeter detectors for high-resolution X- ray spectroscopy into the experimental environment at the CRYRING@ESR electron cooler, Physica scripta97, 114005 (2022)

    2022

  20. [20]

    Hengstler, M

    D. Hengstler, M. Keller, C. Schötz, J. Geist, M. Krantz, S. Kempf, L. Gastaldo, A. Fleischmann, T. Gassner, G. Weber, R. Märtin, T. Stöhlker, and C. Enss, Towards FAIR: first measurements of metallic magnetic calorime- ters for high-resolution x-ray spectroscopy at GSI, Phys- ica Scripta2015, 014054 (2015)

    2015

  21. [21]

    Geist,Bestimmung der Isomerenergie von 229Thmit dem hochauflösenden Mikrokalorimeter-Array maXs30, Ph.d

    J. Geist,Bestimmung der Isomerenergie von 229Thmit dem hochauflösenden Mikrokalorimeter-Array maXs30, Ph.d. thesis, Kirchhoff Institute for Physics, Heidelberg (2020)

    2020

  22. [22]

    Unger, A

    D. Unger, A. Abeln, C. Enss, A. Fleischmann, D. Hengstler, S. Kempf, and L. Gastaldo, High-resolution for IAXO: MMC-based X-ray detectors, Journal of In- strumentation16(06), P06006

  23. [23]

    G. F. Knoll,Radiation Detection and Measurement, 4th ed. (John Wiley & Sons, Hoboken, NJ, 2010)

    2010

  24. [24]

    Clozza, F

    F. Clozza, F. Sgaramella, L. Abbene, F. Artibani, M. Bazzi, G. Borghi, D. Bosnar, M. Bragadireanu, A. Buttacavoli, M. Carminati,et al., Extended X-ray energy characterization of SIDDHARTA-2 large-area Sil- icon Drift Detectors up to 50 keV, Measurement Science and Technology37, 197001 (2026). 7

    2026

  25. [25]

    Unger, A

    D. Unger, A. Abeln, T. E. Cocolios, O. Eizenberg, C. Enss, A. Fleischmann, L. Gastaldo, C. Godinho, M. Heines, D. Hengstler,et al., MMC array to study x-ray transitions in muonic atoms, Journal of Low Tem- perature Physics216, 344 (2024)

    2024

  26. [26]

    Gerchow, S

    L. Gerchow, S. Biswas, G. Janka, C. Vigo, A. Knecht, S. M. Vogiatzi, N. Ritjoho, T. Prokscha, H. Luetkens, and A. Amato, GermanIum Array for Non-destructive Testing (GIANT) setup for muon-induced x-ray emission (MIXE) at the Paul Scherrer Institute, Review of Scien- tific Instruments94, 045106 (2023)

    2023

  27. [27]

    Knecht, A

    A. Knecht, A. Skawran, and S. M. Vogiatzi, Study of nu- clear properties with muonic atoms, The European Phys- ical Journal Plus135, 777 (2020)

    2020

  28. [28]

    102, 22399 Hamburg, Germany

    SIS3316 module from Struck Innovative Systeme GmbH, Harksheider Str. 102, 22399 Hamburg, Germany

  29. [29]

    Rodrigues, M

    M. Rodrigues, M. L. Zahir, M. Loidl, L. Chambon, Q. Drenne, M. Müller, S. Kempf, E. Nigron, and F. Had- dad, Measurements of absolute gamma-ray energies using an ultra-high energy resolution magnetic microcalorime- ter, arXiv preprint arXiv:2602.12836 (2026)

    arXiv 2026

  30. [30]

    R. D. Deslattes, E. G. J. Kessler, P. Indelicato, L. de Billy, E. Lindroth, J. Anton, J. S. Coursey, D. J. Schwab, C. Chang, R. Sukumar, K. Olsen, and R. A. Dragoset, X-ray Transition Energies (version 1.2) (2005), accessed: 2025-06-18

    2005

  31. [31]

    M. S. Basunia, Nuclear Data Sheets for A = 237, Nuclear Data Sheets107, 2323 (2006)

    2006

  32. [32]

    Seabold, J

    S. Seabold, J. Perktold,et al., Statsmodels: econometric and statistical modeling with python, scipy7, 92 (2010)

    2010

  33. [33]

    Sikorsky, J

    T. Sikorsky, J. Geist, D. Hengstler, S. Kempf, L. Gastaldo, C. Enss, C. Mokry, J. Runke, C. E. Düll- mann, P. Wobrauschek, K. Eberhardt, A. Fleischmann, L. von der Wense, and P. G. Thirolf, Measurement of the 229Thisomer energy with a magnetic microcalorimeter, Physical Review Letters125, 142503 (2020)

    2020

  34. [34]

    W. Gins, B. van den Borne, R. P. de Groote, and G. Neyens, SATLAS2: An update to the package for analysis of counting data, Computer Physics Communi- cations297, 109053 (2024)

    2024

  35. [35]

    H. J. Lipkin, Some simple features of the Mössbauer ef- fect, Annals of Physics9, 332 (1960)

    1960

  36. [36]

    Krieger, K

    A. Krieger, K. Blaum, M. L. Bissell, N. Frömmgen, C. Geppert, M. Hammen, K. Kreim, M. Kowalska, J. Krämer, T. Neff, R. Neugart, G. Neyens, W. Nörter- shäuser, C. Novotny, R. Sánchez, and D. T. Yordanov, Nuclear Charge Radius of 12Be, Phys. Rev. Lett.108, 142501 (2012)

    2012

  37. [37]

    Seng, Model-Independent Determination of Nu- clear Weak Form Factors and Implications for Standard Model Precision Tests, Phys

    C.-Y. Seng, Model-Independent Determination of Nu- clear Weak Form Factors and Implications for Standard Model Precision Tests, Phys. Rev. Lett.130, 152501 (2023)

    2023

  38. [38]

    T. A. Lähde and U.-G. Meißner,Nuclear Lattice Effective Field Theory: An introduction, Vol. 957 (Springer, 2019)

    2019

  39. [39]

    Elhatisariet al., Wavefunction matching for solving quantum many-body problems, Nature630, 59 (2024), arXiv:2210.17488 [nucl-th]

    S. Elhatisariet al., Wavefunction matching for solving quantum many-body problems, Nature630, 59 (2024), arXiv:2210.17488 [nucl-th]

    Pith/arXiv arXiv 2024

  40. [40]

    S. Shen, S. Elhatisari, D. Lee, U.-G. Meißner, and Z. Ren, Ab Initio Study of the Beryllium Isotopes7Be to 12Be, Phys. Rev. Lett.134, 162503 (2025), arXiv:2411.14935 [nucl-th]

    Pith/arXiv arXiv 2025

  41. [41]

    G. B. King, S. Bacca, G. Chambers-Wall, A. Gnech, S. Pastore, M. Piarulli, and R. B. Wiringa, Quantum Monte Carlo calculations of Zemach moments inA≤9 nuclei, arXiv preprint arXiv:2606.11153 (2026)

    Pith/arXiv arXiv 2026

  42. [42]

    G. B. King, J. Carlson, A. R. Flores, S. Gandolfi, E.Mereghetti, S.Pastore, M.Piarulli,andR.B.Wiringa, Quantum Monte Carlo calculation ofδ NS in 10C us- ing an effective field theory approach, arXiv preprint arXiv:2509.07310 (2025), arXiv:2509.07310

    Pith/arXiv arXiv 2025

  43. [43]

    Piarulli, R

    M. Piarulli, R. B. Wiringa, A. Lovato, G. B. King, and S. Pastore, Quantum Monte Carlo calculation ofδC in the superallowed beta decay of 10C, arXiv preprint arXiv:2605.14006 (2026)

    Pith/arXiv arXiv 2026

  44. [44]

    Seng and M

    C.-Y. Seng and M. Gorchtein, Data-driven reevaluation off tvalues in superallowedβdecays, Phys. Rev. C109, 045501 (2024)

    2024

  45. [45]

    B. He, M. Gorchtein, M. Heinz, B. Ohayon, L. Plat- ter, and C.-Y. Seng, Taming nuclear size and shape effects in superallowed beta-decay, arXiv preprint arXiv:2605.13985 (2026)

    Pith/arXiv arXiv 2026

  46. [46]

    thesis, Technische Universität Darmstadt, Darm- stadt, Germany (2020)

    B.Maaß,Laser Spectroscopy of the Boron Isotopic Chain, Ph.D. thesis, Technische Universität Darmstadt, Darm- stadt, Germany (2020)

    2020

  47. [47]

    Unger,High-Resolution X-ray Spectroscopy of Light Muonic Atoms, Ph.D

    D. Unger,High-Resolution X-ray Spectroscopy of Light Muonic Atoms, Ph.D. thesis, University of Heidelberg, Faculty of Physics and Astronomy, Kirchhoff Institute for Physics, Heidelberg, Germany (2025), dissertation (Dr. rer. nat.)

    2025

  48. [48]

    Hölzer, M

    G. Hölzer, M. Fritsch, M. Deutsch, J. Härtwig, and E. Förster,Kα 1,2 andKβ 1,3 x-ray emission lines of the 3dtransition metals, Phys. Rev. A56, 4554 (1997)

    1997

  49. [49]

    M. H. Mendenhall, A. Henins, L. T. Hudson, C. I. Szabo, D. Windover, and J. P. Cline, High-precision measure- ment of the x-ray CuKαspectrum, Journal of Physics B: Atomic, Molecular and Optical Physics50, 115004 (2017)

    2017

  50. [50]

    M. H. Mendenhall, L. T. Hudson, C. I. Szabo, A. Henins, and J. P. Cline, The molybdenum K-shell x-ray emission spectrum, Journal of Physics B: Atomic, Molecular and Optical Physics52, 215004 (2019)

    2019

  51. [51]

    M. O. Krause and J. H. Oliver, Natural widths of atomic K and L levels, KαX-ray lines and several KLL Auger lines, Journal of Physical and Chemical Reference Data 8, 329 (1979)

    1979

  52. [52]

    Indelicato, MDFGME: Multiconfiguration Dirac-Fock and General Matrix Elements Program (2026)

    P. Indelicato, MDFGME: Multiconfiguration Dirac-Fock and General Matrix Elements Program (2026)

    2026

  53. [53]

    Rathi, K

    S. Rathi, K. von Schoeler, P. Indelicato, and B. Ohayon, Reference Quadrupole Moments of Transition Elements from Lamb Shifts in Muonic Atoms, Phys. Rev. Lett. 137, 023001 (2026)

    2026

  54. [54]

    Rathi, U

    S. Rathi, U. D. Jentschura, P. J. Indelicato, and B. Ohayon, Binding energy of muonic beryllium: pertur- bative versus all-order calculations, Journal of Physics B: Atomic, Molecular and Optical Physics (2026)

    2026

  55. [55]

    Elhatisari, E

    S. Elhatisari, E. Epelbaum, H. Krebs, T. A. Lähde, D. Lee, N. Li, B.-n. Lu, U.-G. Meißner, and G. Rupak, Ab initio Calculations of the Isotopic Dependence of Nu- clear Clustering, Phys. Rev. Lett.119, 222505 (2017), arXiv:1702.05177 [nucl-th]

    Pith/arXiv arXiv 2017

  56. [56]

    C. Ji, S. Bacca, N. Barnea, O. J. Hernandez, and N. Nevo Dinur, Ab initio calculation of nuclear-structure corrections in muonic atoms, Journal of Physics G: Nu- clear and Particle Physics45, 093002 (2018)

    2018

  57. [57]

    O. J. Hernandez, C. Ji, S. Bacca, and N. Barnea, Probing uncertainties of nuclear structure corrections in light muonic atoms, Phys. Rev. C100, 064315 (2019), arXiv:1909.05717 [nucl-th]. 8

    Pith/arXiv arXiv 2019

  58. [58]

    Pachucki, V

    K. Pachucki, V. c. v. Patkóš, and V. A. Yerokhin, Three- photon-exchange nuclear structure correction in hydro- genic systems, Phys. Rev. A97, 062511 (2018)

    2018

  59. [59]

    Tim Egert and Weiguang Jiang and Sonia Bacca, Op- timizing artificial neural networks for nuclear-structure corrections in light muonic atoms (2026), in preparation

    2026

  60. [60]

    Utsunomiya, Y

    H. Utsunomiya, Y. Yonezawa, H. Akimune, T. Ya- magata, M. Ohta, M. Fujishiro, H. Toyokawa, and H. Ohgaki, Photodisintegration of9Bewith laser-induced compton backscatteredγrays, Phys. Rev. C63, 018801 (2000)

    2000

  61. [61]

    Utsunomiya, S

    H. Utsunomiya, S. Katayama, I. Gheorghe, S. Imai, H. Yamaguchi, D. Kahl, Y. Sakaguchi, T. Shima, K. Takahisa, and S. Miyamoto, Photodisintegration of 9Bethrough the1/2 + state and cluster dipole resonance, Phys. Rev. C92, 064323 (2015)

    2015

  62. [62]

    C. W. Arnold, T. B. Clegg, C. Iliadis, H. J. Karwowski, G. C. Rich, J. R. Tompkins, and C. R. Howell, Cross- sectionmeasurementof 9Be(γ, n)8Beandimplicationsfor α+α+n→ 9Be in therprocess, Phys. Rev. C85, 044605 (2012)

    2012

  63. [63]

    anl.gov/theory/research/density/, accessed: 2026- 06-29

    Argonne National Laboratory, Physics Division, The- ory Group, Single-Nucleon Densities,https://www.phy. anl.gov/theory/research/density/, accessed: 2026- 06-29

    2026

  64. [64]

    Linked" indicates the parameter was mathematically constrained to a master variable (typically the LaK α,1 transition)

    O. Eizenberg, Measurement of the Beryllium-9 Charge Radius via Muonic Atom Spectroscopy (2025). 9 END MATTER TABLE III. Calculated energies and intensities of hyperfine lines ofµ9Be: 2p→1speak (sorted by descending intensity). Energies are written relative to the center-of-gravity of the µ9Be: 2p→1speak. (Ji, Fi)→(J f , Ff ) Relative Energy [eV] Intensity...

    2025

  65. [65]

    S1, top left)

    Selectanupperχ 2 thresholdthatliesabovethever- tical dense lines corresponding to photonic peaks (see the example in Fig. S1, top left)

  66. [66]

    Within the main segment, a global outlier rejection for temperature- values was applied to exclude non-physical read- out errors

    Exclude the initial and final segments of the run that experienced instabilities. Within the main segment, a global outlier rejection for temperature- values was applied to exclude non-physical read- out errors. It was complemented by also removing deviations greater than3σfrom the local moving average as outliers. See (Fig. S1, bottom left)

  67. [67]

    Omit data from pixels exhibiting unstable baselines or irregular pulse shapes

  68. [68]

    S1, top right)

    To remove events from decay electrons that heat the entire array, apply a hold-off time of100ms after each cluster event (see Fig. S1, top right)

  69. [69]

    S1 (Bottom-Right))

    To prevent pile-up effects, apply a hold-off time of 540ms after a photon hits the same pixel (Fig. S1 (Bottom-Right)). The resulting rates are summarized in Table S1. FIG. S1. Examples of cuts described in the main text. TABLE S1. Summary of sequential data cuts, remaining counts, and rates summed across all active pixels of the MMC. # Cut Counts Rate (H...

This paper was first reviewed by deepseek-v4-flash on August 2, 2026.