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REVIEW 2 major objections 4 minor 42 references

Modern approach to muonic x-ray spectroscopy demonstrated through the measurement of stable Cl radii

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Modern muonic x-ray spectroscopy, with a Barrett-radius recipe and an energy-density-functional shape correction, determines the chlorine charge radii to 0.07% precision and replaces the electron-scattering values with results seven times…

desk verdict Careful first muonic radii for Cl with a useful new V2 method; the V2 uncertainty transfer is the main thing to probe in review. read the letter →

arxiv 2506.08804 v3 pith:DHRNHE4P submitted 2025-06-10 nucl-ex nucl-thphysics.atom-phphysics.data-an

K.A. Beyer , T.E. Cocolios , C. Costache , M. Deseyn , P. Demol , A. Doinaki , O. Eizenberg , M. Gorshteyn
show 20 more authors
M. Heines A. Herzáň P. Indelicato K. Kirch A. Knecht R. Lica V. Matousek E.A. Maugeri B. Ohayon N.S. Oreshkina W.W.M.M. Phyo R. Pohl S. Rathi W. Ryssens A. Turturica K. von Schoeler I.A. Valuev S.M. Vogiatzi F. Wauters A. Zendour
This is my paper · ORCID
classification nucl-exnucl-thphysics.atom-phphysics.data-an PACS 21.10.Ft36.10.-k
keywords nuclearchargeradiimuonicatomsx-rayspectroscopyenergydensityfunctionalsshapecorrectionBarrettradiusisotopeshiftschlorineisotopes
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

Modern muonic x-ray spectroscopy can determine absolute nuclear charge radii to about 0.07% precision, roughly seven times better than the older electron-scattering values it replaces. Demonstrating the method on the two stable chlorine isotopes, the paper reports $R(^{35}\mathrm{Cl}) = 3.3334(23)$ fm and $R(^{37}\mathrm{Cl}) = 3.3444(23)$ fm, which disagree with the 1980 electron-scattering radii by $3.2\sigma$ and $2.3\sigma$, and a differential mean square radius $\delta\langle r^2\rangle^{37,35} = +0.0771(66)\,\mathrm{fm}^2$ that is more than an order of magnitude more precise than the previous value. That differential is now precise enough to serve as the benchmark that laser spectroscopy of radioactive chlorine isotopes needs. The technical core is a new way to apply the nuclear shape correction using energy-density-functional charge distributions instead of electron scattering, so the same approach can be used for isotopes that have no high-quality scattering data.

What carries the argument

The load-bearing objects are the Barrett moment $\langle r^k e^{-\alpha r}\rangle$ with its equivalent radius $R_{k\alpha}$, and the shape-correction ratio $V_2 = R_{k\alpha}/R_{\mathrm{RMS}}$ that turns a model-independent muonic radius into an RMS charge radius. The Barrett recipe, which fits the parameters $k$ and $\alpha$ to the difference in the muon's initial- and final-state potentials, suppresses the dependence on the assumed charge distribution. The paper's new move is to compute $V_2$ from the monopole charge distributions of the BSkG4 energy density functional — a Skyrme-Hartree-Fock-Bogoliubov model fitted to observables across the nuclear chart — instead of from electron scattering, benchmarking the model against high-$q_{\max}$ scattering on nine neighboring nuclei ($^{31}$P, $^{32,34,36}$S, $^{39}$K, $^{40,48}$Ca), whose average $0.05\%$ deviation is adopted as the $V_2$ uncertainty for chlorine. A second supporting mechanism is the handling of correlated uncertainties: nuclear polarization is split into nuclear and nucleon parts with an implied $88.2\%$ inter-isotope correlation, and the $V_2$ factors are assumed $97.0\%$ correlated between isotopes, which is what makes the differential radius an order of magnitude more precise than the absolute radii.

What would settle it

Measure the charge radii of 35Cl and 37Cl again with a technique whose shape correction does not depend on BSkG4 — for example a high-momentum-transfer elastic electron scattering experiment on both isotopes, or a muonic measurement of a different isotope pair where high-$q_{\max}$ scattering independently fixes $V_2$. If the true chlorine $V_2$ error sits near the $0.071\%$ offset BSkG4 shows against the available low-$q_{\max}$ chlorine data rather than the assumed $0.05\%$, the absolute radii would shift by 2–3 attometers, a large fraction of the quoted 23-attometer uncertainty, and the differential claim could be tested directly by measuring the $97.0\%$ inter-isotope $V_2$ correlation through a third isotope pair.

Watch

Extended reading notes

Core claim

The paper's central claim is that a fully modernized muonic x-ray analysis — digitally acquired and precisely calibrated $np1s$ transition energies, QED corrections that include effects older work omitted (notably hadronic vacuum polarization and a relativistic finite-size self-energy), nuclear-polarization uncertainties benchmarked against microscopic calculations, and a Barrett-moment recipe that removes charge-distribution model dependence — yields charge radii accurate at the $0.07\%$ level. The final conversion from the model-independent Barrett radius $R_{k\alpha}$ to the RMS radius uses the shape-correction ratio $V_2 = R_{k\alpha}/R_{\mathrm{RMS}}$ computed from BSkG4 energy-density-functional charge distributions instead of electron scattering. Applied to the two stable chlorine isotopes this gives $R(^{35}\mathrm{Cl}) = 3.3334(23)$ fm and $R(^{37}\mathrm{Cl}) = 3.3444(23)$ fm, disagreeing with the 1980 electron-scattering values of $3.388(17)$ fm by $3.2\sigma$ and $2.3\sigma$, and a differential mean square radius $\delta\langle r^2\rangle^{37,35} = +0.0771(66)\,\mathrm{fm}^2$. The paper argues the discrepancy signals underestimated systematics in the old electron-scattering analysis, and that the new radii, used as mirror-pair inputs, bring the mirror-shift fit's reduced $\chi^2$ from $2.15$ to $1.01$, supporting that fit's use for predicting radii of exotic isotopes.

Load-bearing premise

The load-bearing premise is that the BSkG4 energy density functional predicts the shape of the chlorine charge distribution, through the $V_2$ correction, as reliably as it does for nine neighboring nuclei that have high-quality electron scattering, so a $0.05\%$ uncertainty drawn from that comparison is transferred to chlorine even though the only available chlorine scattering data is low-$q_{\max}$ and was not used to set the uncertainty.

Editorial extensions

If this is right

  • If the paper is right, muonic x-ray spectroscopy plus an energy-density-functional shape correction is accurate at the $0.07\%$ level for absolute radii in this mass region, and the 1980 electron-scattering chlorine values should be replaced by these numbers.
  • The differential radius $\delta\langle r^2\rangle^{37,35} = +0.0771(66)\,\mathrm{fm}^2$ is now precise enough to calibrate the isotope-shift factor for laser spectroscopy of radioactive chlorine isotopes, including $^{34}$Cl in superallowed beta-decay studies.
  • The EDF-based $V_2$ method removes the historical requirement of high-quality electron scattering, so the same pipeline can be applied to odd-mass and heavier nuclei, and to radioactive species where scattering is impossible.
  • The new chlorine radii, combined with the mirror partners $^{35}$Ar and $^{37}$Ca, lower the mirror-shift fit's reduced $\chi^2$ from $2.15$ to $1.01$, strengthening the empirical case that the mirror fit predicts radii of exotic mirror pairs.
  • The self-consistency check between measured $np1s$ energy differences and QED predictions certifies the energy scale, so the improved radii can serve as anchors for the charge-radius compilation in the $Z \approx 17$–$20$ region.

Reading between the lines

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

  • My inference: if the BSkG4 $V_2$ transfer holds for further mid-mass isotopes, the same pipeline could revive muonic x-ray spectroscopy as the standard absolute-radius anchor for exotic-beam laser spectroscopy chains, where no electron scattering will ever exist.
  • My inference: the $2.3$–$3.2\sigma$ disagreement with the 1980 scattering data raises the possibility that other compilation radii based on that era's low-$q_{\max}$ scattering carry similar unaccounted systematics; comparing differential radii from modern muonic measurements on several such isotope pairs would reveal the pattern.
  • My inference: the $97.0\%$ inter-isotope $V_2$ correlation is a testable prediction — a muonic measurement of a neighboring isotope pair (for example a sulfur pair) where high-$q_{\max}$ scattering exists would check whether differences of $V_2$ really cancel to that degree.
  • My inference: the mirror-shift consistency could be turned into a cross-check tool — for any future muonic radius pair, disagreement with the mirror trend would flag either the fit or the EDF shape correction, helping isolate where the method fails.
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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

2 major / 4 minor

Summary. The paper presents a muonic x-ray spectroscopy measurement of the stable chlorine isotopes 35Cl and 37Cl performed at the PSI πE1 beamline, and uses the extracted np1s transition energies (n = 2, 3, 4) to determine absolute nuclear charge radii and the differential mean square radius. The experimental analysis is state-of-the-art: per-detector hypermet line-shape fitting, bootstrap calibration with a detector-averaging bias check, a time-cut systematics correction, and a self-consistency check on radius-insensitive np1s energy differences. On the theory side, the paper combines MDFGME QED calculations, nuclear polarization with a new uncertainty treatment, Barrett-moment extraction, and a novel V2 shape correction obtained from BSkG4 energy-density-functional charge distributions. The central results are R(35Cl) = 3.3334(23) fm, R(37Cl) = 3.3444(23) fm, and δ⟨r²⟩^{37,35} = +0.0771(66) fm², which the authors compare with electron scattering literature and use to update a mirror shift fit. The paper claims a factor-of-seven improvement in absolute radius uncertainty and more than an order-of-magnitude improvement in the differential mean square radius.

Significance. If the extracted radii and uncertainties hold, this is an important result: it provides long-needed absolute charge radii for chlorine, benchmark values for laser spectroscopy of the Cl chain, and input for Vud, mirror-shift, and nuclear structure studies. The experimental work is careful and unusually transparent: the energy extraction includes a per-detector calibration with bootstrap uncertainty, a detector-averaging bias investigation, a high-statistics check on the time-cut induced shift, and a self-consistency comparison of np1s differences with QED. The paper also ships processed data on Zenodo, which supports reproducibility. The two load-bearing assumptions that need scrutiny are the transfer of the V2 uncertainty from neighboring nuclei to Cl and the assumed 97.0% inter-isotope V2 correlation; these are falsifiable calibration issues rather than internal inconsistencies, and they can be addressed by additional analysis within the scope of the manuscript.

major comments (2)
  1. [§IV E, Table X, Fig. 21] The 0.05% uncertainty assigned to the BSkG4 V2 correction for 35Cl and 37Cl is the average deviation of BSkG4 from high-qmax electron scattering on nine neighboring nuclei (31P, 32,34,36S, 39K, 40,48Ca), none of which is chlorine. The only direct Cl comparison, the low-qmax data of Ref. [49], gives V2 deviations of 0.071% (35Cl) and 0.059% (37Cl) from BSkG4, both outside the 0.05% band. The authors state in Fig. 21 that these points are 'in agreement', but agreement within the 0.15% uncertainty of the low-qmax measurement does not validate the much smaller EDF uncertainty. Since σV2 = 1.69 am is the dominant contribution to the absolute radius error budget (Table XIII) and the V2 term contributes 3.2 × 10⁻³ fm² to δ⟨r²⟩, a Cl-specific validation or a conservative V2 uncertainty based on the available low-qmax data is needed before the quoted error bars can be accepted.
  2. [§IV F, Eq. (12)–(13), Table XIII] The differential radius δ⟨r²⟩ = 0.0771(66) fm² relies on an assumed 97.0% correlation between the BSkG4 V2 factors of 35Cl and 37Cl. This correlation is inferred from neighboring nuclei and from the spread among BSkG2–BSkG4, with a stated minimal value of about 95%, but no Cl-specific check is available; ground-state configuration differences in odd-A nuclei could plausibly change the inter-isotope correlation. Because the V2 contribution to the δ⟨r²⟩ uncertainty is only 3.2 × 10⁻³ fm² at 97.0% correlation, the claimed 'more than one order of magnitude' improvement is sensitive to this number. Please provide a sensitivity analysis of δ⟨r²⟩ and its uncertainty for correlation values in a plausible range, or additional evidence that the neighboring-nucleus correlation applies to the Cl pair.
minor comments (4)
  1. [Abstract and Table XIV] The abstract quotes R(35Cl) = 3.3335(23) fm and R(37Cl) = 3.3445(23) fm, while Table XIV lists 3.3334(23) fm and 3.3444(23) fm; the rounding should be harmonized.
  2. [§IV A and Supplementary §II C] The main text states that the estimated QED uncertainty for the 1s state is about 3 eV, while the supplementary material says that adding the identified missing contributions in quadrature gives about 4 eV after rounding; please clarify which value is used in the error budget and make the statements consistent.
  3. [Fig. 21] The figure legend says the Cl values are 'in agreement' with BSkG4, but the two Cl points lie outside the blue band representing the average deviation; please state explicitly what the band represents and quote the 35Cl and 37Cl deviations in the text or caption.
  4. [Table III and Supplementary Table IX] The entry '06.6' for the 3p1s statistical-plus-calibration uncertainty of 35Cl has a leading zero, and the 4p1s total uncertainty for 35Cl is rounded to 12.5 eV in the main text but 12.6 eV in the supplementary material; these formatting and rounding inconsistencies should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: extraction chain is self-contained; V2 benchmark caveat is an accuracy concern, not circularity.

full rationale

The radius extraction chain is not circular. The measured muonic np1s transition energies are calibrated against independent gamma-ray lines (Table II), the QED and nuclear polarization corrections are computed with established codes and published data (MDFGME; Refs. 79-82), and the Barrett parameters k and alpha are obtained by fitting the muonic potential difference rather than the measured energies. The Barrett radii are converted to RMS radii using the V2 shape correction from BSkG4, an externally published energy-density functional, whose V2 predictions are benchmarked against high-qmax electron scattering on nine neighboring nuclei (Table X, Fig. 21); the BSkG4 V2 values are not fitted to the Cl muonic data. The extracted radii disagree with the 1980 electron-scattering values by 3.2 sigma and 2.3 sigma, confirming that the measurement carries independent information and is not forced by its inputs. Although several theoretical tools involve co-authors (BSkG models and nuclear polarization methods), they are independently published and benchmarked against external data, so the self-citation is not load-bearing in a circularity sense. The main caveat is that the 0.05% V2 uncertainty is transferred from a neighboring-nucleus benchmark while the direct low-qmax Cl comparison shows deviations slightly outside that band, and the assumed 97.0% inter-isotope V2 correlation is an assumption; these are falsifiable calibration concerns, not circular steps.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The central chain rests on the Barrett recipe's shape-model independence (validated internally to at most 2 eV), the transfer of NP uncertainty from a 40Ca Skyrme benchmark to odd-mass Cl (23%, with the authors conceding their Cl NP may be underestimated), the transfer of BSkG4 V2 accuracy (0.05%, the average benchmark deviation, not the maximum) to Cl whose only direct check sits 1.4 sigma outside that band, the assumed 88.2%/97.0% inter-isotope correlations that create the factor-6 shrinkage of the differential V2 uncertainty, a 3-4 eV QED truncation estimate, an assumed fully-populated 1s/2s screening configuration, and the reliability of the electron scattering V2 references used to validate BSkG4. No new physical entities are introduced; the Barrett radius and correlation fractions are analysis constructs.

free parameters (6)
  • Barrett parameters k and alpha = k = 2.0937(4), alpha = 0.0559(2) fm^-1 (35Cl); k = 2.0940(4), alpha = 0.0558(2) fm^-1 (37Cl)
    Fitted to the computed muon potential difference Vi - Vf = B r^k e^{-alpha r} in Section IV C (Table VII). Standard Barrett recipe; not fitted to the measured transition energies, but they set the scale of the extracted R_k,alpha.
  • NP correlated fraction fCorr = 90.4% (35Cl), 97.6% (37Cl), cross-isotope 88.2%
    Hand-set assumption in Section IV B (Table VI): low-lying states uncorrelated, giant resonances and nucleon polarization maximally correlated. Directly sets the NP contribution (0.68 am) to the differential radius uncertainty.
  • V2 inter-isotope correlation = 97.0%
    Assumed conservative estimate in Section IV E from e-scattering/model comparisons (97.4% on Cl data, 98.0% on 34,36S); reduces the V2 contribution to delta-r² from 42.0 to 3.2 x 10^-3 fm² (Table XIII).
  • V2 (BSkG4) relative uncertainty = 0.05%
    Average deviation of BSkG4 from high-qmax electron scattering over the nine-nucleus benchmark set, transferred to the Cl predictions (Section IV E).
  • NP relative uncertainty = 23%
    Maximum deviation of the adopted NP method from eleven Skyrme parametrizations on 40Ca, (209.8-170.5)/170.5 (Section IV B, supplementary Table XIII).
  • QED 1s uncertainty estimate = 3 eV (about 4 eV after quadrature)
    Estimated truncation error for uncalculated higher-order QED contributions on the 1s state (Section IV A, supplementary Section II C).
assumptions (7)
  • domain assumption A two-parameter Fermi charge distribution with fixed t = 2.3 fm adequately describes the nuclear charge for the QED and Barrett calculations.
    Section IV A: QED transition energies are computed on 2pF distributions with c varied over a 2% RMS range; residual charge-model dependence is claimed to be at most 2 eV via the Barrett recipe (Section IV C, Fig. 9).
  • domain assumption Nuclear polarization uncertainties for the odd-mass Cl isotopes can be quantified via the 40Ca Skyrme-benchmark spread (23%).
    Sections IV B and supplementary III: no microscopic NP calculation exists for 35,37Cl; 40Ca serves as the reliability anchor, and the authors state their Cl NP values are 'most likely to be somewhat underestimated'.
  • domain assumption BSkG4 predicts V2 for 35,37Cl with the 0.05% accuracy observed on the nine-nucleus benchmark set.
    Section IV E: the average BSkG4-vs-electron-scattering deviation is transferred to Cl; the only direct Cl anchor (low-qmax electron scattering, Ref. [49]) lies 0.071% away, i.e., 1.4 times the assigned uncertainty.
  • ad hoc to paper Inter-isotope correlations: V2 correlation 97.0%, NP correlation 88.2%.
    Sections IV B and IV E: correlations are assumed (LLS uncorrelated, GR and nucleon polarization maximally correlated; V2 correlation estimated from e-scattering/model comparisons and set to 97.0%). These assumptions shrink the delta-r² V2 contribution from 42.0 to 3.2 x 10^-3 fm².
  • domain assumption Uncalculated higher-order QED effects are bounded by 3-4 eV on the 1s state.
    Section IV A and supplementary II C: estimate from SE-VP (about 1 eV), eVP-NP (about 1.5 eV), three-loop (about 0.3 eV) effects plus unknown nuclear-self-energy and radius-definition terms.
  • domain assumption The high-qmax electron scattering datasets in the benchmark set provide reliable V2 references.
    Section IV E and supplementary V B: the 0.05% average deviation assumes the De Vries et al. compilation values used for 31P, 32,34,36S, 39K, 40,48Ca are unbiased references for the V2 ratio.
  • domain assumption Electron screening can be modeled with fully populated 1s and 2s electron orbitals during the muonic cascade.
    Section IV A: screening shifts the np1s transitions by -0.41 to -1.53 eV; the actual ionization state of the atom after muon capture is not measured.

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Pith. "Pith review of Modern approach to muonic x-ray spectroscopy demonstrated through the measurement of stable Cl radii." pith.science (2026). https://pith.science/paper/DHRNHE4P

@misc{pith2026250608804,
  author       = {Pith},
  title        = {Pith review of: Modern approach to muonic x-ray spectroscopy demonstrated through the measurement of stable Cl radii},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DHRNHE4P}},
  note         = {Machine review of arXiv:2506.08804}
}
abstract

Recent advances in muonic x-ray experiments have reinvigorated efforts in measurements of absolute nuclear charge radii. Here, a modern approach is presented, and demonstrated through determination of the charge radii of the two stable chlorine nuclides $^{35}$Cl and $^{37}$Cl. Knowledge of these radii has implications for fundamental studies in nuclear and atomic physics. For this purpose, a state-of-the-art experiment was performed at the $\pi$E1 beamline in the Paul Scherrer Institute (Switzerland), using a large-scale HPGe detector array in order to extract precise energies of the muonic $^{35}$Cl and $^{37}$Cl $np1s$ transitions. The nuclear charge radius extraction relies on modern calculations for QED effects and nuclear polarization with rigorous uncertainty quantification, including effects that were not accounted for in older studies. Additionally, we established a new method for applying the nuclear shape correction directly from energy density functionals, which are amenable to isotopes for which no high-quality electron scattering experiments are available. The resulting charge radii are $3.3335(23) fm$ for $^{35}$Cl and $3.3445(23) fm$ for $^{37}$Cl, thus improving the uncertainty of the available electron scattering values by a factor of seven. The correlation of several observables was evaluated between the different isotopes in order to produce a more precise value of the differential mean square charge radius $\delta \langle r^2 \rangle^{37, 35}=+0.0771(66) fm^{2}$. In this case, improvement of the uncertainty by more than one order of magnitude was achieved compared to the literature value. This precision is sufficient to use this differential as input for isotope shift factor determination.

Figures

Figures reproduced from arXiv: 2506.08804 by the authors.

Figure 1
Figure 1. FIG. 1: Flowchart of the radius extraction scheme. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Simplified experimental setup used for the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Prompt (top) and anticoincidence (bottom) [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (37 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Overlay between the calibration spectrum and [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Graphical representation of the energy [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Example of calibration residuals for one of the [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Experimental spectra and corresponding fits of [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Muonic 2 [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Mirror shift fit updated with chlorine radii [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 1
Figure 1. Figure 1: FIG. 1: Graphical representation of the used ELET algorithm. The dashed r [PITH_FULL_IMAGE:figures/full_fig_p022_1.png]
Figure 2
Figure 2. Figure 2: FIG. 2: Minimization procedure used for the ELET parameters. The final [PITH_FULL_IMAGE:figures/full_fig_p023_2.png]
Figure 3
Figure 3. Figure 3: FIG. 3: Comparison of uncorrelated (red) and anticoincidence (blue) sp [PITH_FULL_IMAGE:figures/full_fig_p024_3.png]
Figure 4
Figure 4. Figure 4: FIG. 4: Window optimization for the anticoincidence spectrum. The b [PITH_FULL_IMAGE:figures/full_fig_p025_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Residuals over time before gain drift correction in Ge02. The das [PITH_FULL_IMAGE:figures/full_fig_p026_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Residuals over time after gain drift correction in Ge02. The dash [PITH_FULL_IMAGE:figures/full_fig_p026_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Geant4 simulation of the step height as a function of energy for a 75% r [PITH_FULL_IMAGE:figures/full_fig_p028_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Fit for [PITH_FULL_IMAGE:figures/full_fig_p028_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Residuals (black points) and predicted error (shaded red regi [PITH_FULL_IMAGE:figures/full_fig_p029_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Time difference between germanium events and electron even [PITH_FULL_IMAGE:figures/full_fig_p030_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Time difference of germanium events with respect to the tri [PITH_FULL_IMAGE:figures/full_fig_p031_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: Cornerplot of fitresults for a 2 [PITH_FULL_IMAGE:figures/full_fig_p034_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13: Cornerplot of fitresults for a 2 [PITH_FULL_IMAGE:figures/full_fig_p035_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14: Spectra showing the effect of the time cut on the centroid pos [PITH_FULL_IMAGE:figures/full_fig_p036_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15: Weighted average of the 2 [PITH_FULL_IMAGE:figures/full_fig_p037_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16: Energy spectrum of 2 [PITH_FULL_IMAGE:figures/full_fig_p037_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17: Energy spectrum of 3 [PITH_FULL_IMAGE:figures/full_fig_p038_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18: Monte Carlo generated hyperfine structure spectrum for [PITH_FULL_IMAGE:figures/full_fig_p039_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19: Fit on the difference in potentials generated by the muon in the [PITH_FULL_IMAGE:figures/full_fig_p044_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20: Evaluation of the numerical uncertainty of extracted radii and [PITH_FULL_IMAGE:figures/full_fig_p045_20.png]
Figure 21
Figure 21. Figure 21: FIG. 21: Relative deviation between [PITH_FULL_IMAGE:figures/full_fig_p046_21.png]
Figure 22
Figure 22. Figure 22: FIG. 22: Integration error on the Barrett moment induced by the choice cu [PITH_FULL_IMAGE:figures/full_fig_p047_22.png]
Figure 23
Figure 23. Figure 23: FIG. 23: Emission energy as a function of Barrett radius for the 2 [PITH_FULL_IMAGE:figures/full_fig_p047_23.png]
Figure 24
Figure 24. Figure 24: FIG. 24: Description of the isotope shift with a linear model in ∆ [PITH_FULL_IMAGE:figures/full_fig_p048_24.png]
Figure 25
Figure 25. Figure 25: FIG. 25: Energy spectrum of 2 [PITH_FULL_IMAGE:figures/full_fig_p050_25.png]
Figure 26
Figure 26. Figure 26: FIG. 26: Energy spectrum of 3 [PITH_FULL_IMAGE:figures/full_fig_p050_26.png]
Figure 27
Figure 27. Figure 27: FIG. 27: Energy spectrum of 4p [PITH_FULL_IMAGE:figures/full_fig_p051_27.png]
Figure 28
Figure 28. Figure 28: FIG. 28: Energy spectrum of 2 [PITH_FULL_IMAGE:figures/full_fig_p051_28.png]
Figure 29
Figure 29. Figure 29: FIG. 29: Energy spectrum of 3 [PITH_FULL_IMAGE:figures/full_fig_p052_29.png]
Figure 30
Figure 30. Figure 30: FIG. 30: Energy spectrum of 4p [PITH_FULL_IMAGE:figures/full_fig_p052_30.png]
Figure 31
Figure 31. Figure 31: FIG. 31: Ratio of calibration uncertainty in the total uncertainty per de [PITH_FULL_IMAGE:figures/full_fig_p053_31.png]

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Works this paper leans on

42 extracted references · 41 canonical work pages

  1. [1]

    Coupled hypermet fitting The individual peaks are fitted with a hypermet function (Eq. ( 3)). f (E) = Nsig [fG ·g(E) +fT ·t(E) +s(E)] +b(E), (3) where fT = 1 −fG, g(E) = 1√ 2Ãà exp ( − 1 2 [E −µ à ]2) , t(E) = 1 2´ exp ( E −µ ´ + Ã2 2´2 ) erfc ( E −µ√ 2à + Ã√ 2´ ) , s(E) = A 2 erfc ( E −µ√ 2à ) , b(E) = Linear Chebychev with integral NBg =Ndata −Nsig. While...

    work page

  2. [2]

    If we proceed with stan dard least-squares fitting methods, it is assumed that the estimated centroid uncertainties are accurate

    Bootstrap fitting As a next step, a calibration fit was performed. If we proceed with stan dard least-squares fitting methods, it is assumed that the estimated centroid uncertainties are accurate. Whe n the calibration is statistics limited, this is a valid assumption. In many cases, however, the limiting aspect of th e calibration is an underlying non-linea...

    work page

  3. [3]

    The cuts applied for this dataset are:

    Rejection logic In order to clean up the prompt spectra, several rejections can be made . The cuts applied for this dataset are:

    work page

  4. [4]

    Muon pile-up protection cut

    work page

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    When muons arrive close to one another, a germanium event may be 11 100− 80− 60− 40− 20− 0 20 40 Time difference (ns) 0 20 40 60 80 100 120 140 160 180 200 310× Counts FIG

    Multiplicity cut The muon beam at PSI is approximately uniformly distributed in time (∼ 20 ns between proton pulses, O(1 ms) average time between subsequent muons). When muons arrive close to one another, a germanium event may be 11 100− 80− 60− 40− 20− 0 20 40 Time difference (ns) 0 20 40 60 80 100 120 140 160 180 200 310× Counts FIG. 10: Time difference ...

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Reviewed August 7, 2026 · model on record in the stance chip above.