Coulomb excitation of $^{124}$Te: Emerging collectivity and persisting seniority structure in the $6_1^+$ level

A. E. Stuchbery; A. Gargano; A. J. Mitchell; B. J. Coombes; C. Gautam; G. J. Lane; J. A. Woodside; L. J. Bignell; L. J. McKie; M. Reece

REVIEW 3 major objections 4 minor 78 references

Coulomb excitation of $^{124}$Te: Emerging collectivity and persisting seniority structure in the $6_1^+$ level

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Coulomb excitation of $^{124}$Te yields a first $B(E2;\,6_1^+\to 4_1^+)=27(9)$ W.u., far below vibrator predictions and matching the shell model, so the $6_1^+$ state keeps its two-proton $0g_{7/2}$ seniority structure while the lower state

desk verdict First B(E2;6+->4+) measurement for 124Te is a credible new datum, but the abstract oversells the shell-model agreement and the seniority conclusion is plausible, not demonstrated. read the letter →

arxiv 2508.09643 v1 pith:FIF7CEAL submitted 2025-08-13 nucl-ex nucl-th

M. Reece , B. J. Coombes , A. J. Mitchell , A. E. Stuchbery , G. J. Lane , A. Gargano , V. U. Bashu , L. J. Bignell

show 4 more authors

C. Gautam L. J. McKie N. J. Spinks J. A. Woodside

This is my paper · ORCID

classification nucl-exnucl-th PACS 25.70.De23.20.-g21.60.Cs

keywords Coulombexcitation124TeB(E2)transitionstrengthssenioritycollectivityshellmodel0g7/2two-protoncouplingGeneralCollective

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

$^{124}$Te sits at a transitional point between the collective structure expected near the neutron midshell ($N=66$) and the seniority-dominated structure at the $N=82$ shell closure. Using Coulomb excitation with $^{16}$O and $^{58}$Ni beams and particle–$\gamma$ coincidence spectroscopy, the authors measured the transition strength $B(E2;\,6_1^+\to 4_1^+)=27(9)$~W.u. for the first time in this nucleus. They argue that this value is far too small for a spherical vibrator or for General Collective Model fits, and that it agrees with large-basis shell-model calculations across the $^{124-134}$Te chain. Their conclusion is that the $6_1^+$ state of $^{124}$Te keeps a dominant two-proton $\pi(0g_{7/2})^2$ seniority structure while the $2_1^+$ and $4_1^+$ states of the same nucleus gain collective $E2$ strength. A sympathetic reader would care because this is a state-by-state demonstration that collectivity and seniority are not alternative global descriptions of a transitional nucleus but coexisting structures inside it.

What carries the argument

The argument is carried by one measured number set against two extreme model descriptions: the General Collective Model (a fully collective quadrupole-surface description that succeeds for $^{120}$Te but not $^{124}$Te) and large-basis shell-model calculations (protons and neutrons in the $0g_{7/2}$, $1d_{5/2}$, $1d_{3/2}$, $2s_{1/2}$, $0h_{11/2}$ orbits above a $^{100}$Sn core, with a CD-Bonn-derived effective interaction and empirical effective charges). The $E2$ matrix elements are extracted from the Coulomb-excitation yields with the semiclassical code GOSIA. The interpretive key is the seniority-$j^2$ limit: two valence protons coupled in the $0g_{7/2}$ orbit predict a characteristic se

What would settle it

Re-measure the $6_1^+\to 4_1^+$ yield with higher statistics or a cleaner beam–target combination: if $B(E2;\,6_1^+\to 4_1^+)$ rises well above the reported $27(9)$~W.u. toward the collective-model band, the seniority claim fails. The independent check is a $g$-factor measurement of the $6_1^+$ state, for which the shell model predicts $\approx +0.78$ while a collective assignment would sit near $Z/A \approx 0.42$.

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Extended reading notes

Core claim

The paper's central claim is that the newly measured $B(E2;\,6_1^+\to 4_1^+)=27(9)$~W.u. places the $6_1^+$ state of $^{124}$Te on the seniority side of the transition between the $N=82$ shell closure and the neutron midshell. General Collective Model fits that reproduce $^{120}$Te fail for $^{124}$Te, overestimating the measured $E2$ strengths; the $6_1^+$ decay lies well below the collective expectation. Large-basis shell-model calculations, in contrast, track the measured $6_1^+\to 4_1^+$ values for $^{124-134}$Te, and the authors interpret this as the persistence of the two-proton $\pi(0g_{7/2})^2$ seniority component in the $6_1^+$ wave function. For the same nucleus the shell model acc

Load-bearing premise

The seniority conclusion rests on a single weak gamma-ray line: the $6_1^+\to 4_1^+$ yield in the $^{58}$Ni-gated spectrum carries a 33% relative uncertainty, and the neighbouring steps it depends on — the partially obscured $4_2^+\to 2_1^+$ line and the unconstrained $2_2^+\to 2_1^+$ mixing ratio — are handled with adopted branching ratios, so a wrong yield assignment would move the matrix element and weaken the claim.

Editorial extensions

If this is right

The $6_1^+$ state of $^{124}$Te can no longer be counted among textbook vibrational excitations: its measured $E2$ decay strength is well below the spherical-vibrator and GCM values.
The seniority component of the $6_1^+$ wave function survives from the $N=82$ closure down to at least $^{124}$Te, two neutron pairs below midshell, while lower-lying states become collective — collectivity and seniority coexist in one nucleus, assigned state by state.
A fully collective description of the Te isotopes stops working by $^{124}$Te even though it still works at $^{120}$Te, pinning down the neutron range where the GCM loses predictive power for $E2$ observables.
The same analysis yields small $B(E2;\,0_2^+\to 2_1^+)$ and $B(E2;\,2_2^+\to 0_1^+)$ values that contradict the vibrator expectation (the $0_2^+\to 2_1^+$ strength should be twice the $2_1^+\to 0_1^+$ value, and the $2_2^+\to 0_1^+$ transition should be forbidden), supporting — with more data needed — the idea that the low-lying $0_2^+$ structure is not simply two-phonon.
The paper's shell-model $g$-factors imply a direct test: $g(6_1^+)$ should be about $+0.78$ (close to the $\pi g_{7/2}$ single-particle value) while $g(4_1^+)\approx +0.53$ and $g(2_1^+)\approx 0.30$; a $6_1^+$ $g$-factor measurement would confirm the seniority assignment at the level of angular-momentum coupling.

Reading between the lines

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

My inference: the sharpest falsifiable target the paper leaves open is the $g$-factor of the $6_1^+$ state — the shell model's $+0.78$ sits far from the collective rotor value $Z/A \approx 0.42$, so a single measurement cleanly separates the two pictures.
My inference: if the state-by-state decoupling is generic, the same signature should be sought in the neighbouring Cd ($Z=48$) and Sn ($Z=50$) isotopes, where the vibrator-versus-seniority debate has long been unsettled; the $^{124}$Te result predicts that higher-spin members of the 'phonon' multiplets can remain single-particle-like even where the $2_1^+$ is collective.
My inference: a repeat measurement with a different beam (avoiding the $^{58}$Ni contaminant that obscured the 1355-keV line) and higher statistics could cut the 33% uncertainty on the $6_1^+$ yield and decide whether the seniority dominance extends toward $^{122}$Te or whether the crossover to a collective $6_1^+$ lies between $^{124}$Te and $^{122}$Te.
My inference: transition strengths, not excitation energies, appear to be the reliable structural indicator in this region — energy spacings alone had labelled $^{124}$Te vibrator-like, and the $B(E2)$ data reassign the $6_1^+$ state without changing the level scheme.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague, here's my take on arXiv:2508.09643. The genuinely new thing is the first B(E2;6+->4+) for 124Te, 27(9) W.u., extracted from Coulomb excitation with the new CAESAR particle-gamma setup at ANU. That alone earns a serious referee: it's a real new datum in a nucleus sitting right at the collective/seniority boundary. The analysis follows the standard GOSIA path, with stopping powers, conversion coefficients, and safe energies disclosed, and the model comparison is well anchored: effective charges from 134Te and 132Sn, g_s quenching from g(2+) in 122Te, GCM parameters from 120Te. Nothing about the measurement chain looks cooked.

The soft spot is the gap between the abstract and the body. The abstract promises 'remarkably good agreement' with shell-model calculations for 124Te, but the body says the shell model accounts for 63(20)% of the measured 6+->4+ strength. That's a 1.9-sigma deficit — consistent at about the 2-sigma level, not remarkable. Since the seniority-persistence conclusion leans on that agreement, the headline should be softened to 'consistent with' rather than 'strong evidence for.' The measurement side has its own fragility: the 6+ yield carries 33% relative uncertainty, the 4_2+->2_1+ line was obscured and had to be inferred from a branching ratio, and the 2_2+->2_1+ mixing ratio is unconstrained. These are not red flags, but they mean the interpretation is plausible, not demonstrated. The authors essentially say this when they call for g-factor and E0 measurements; that caveat belongs in the abstract too. Two datasets are omitted from the systematic comparison with stated reasons; the exclusions sound defensible, but they make the 'good agreement for 124-134Te' claim harder to verify from the paper alone.

Bottom line: the paper ships a new, credible measurement and a fair interpretive frame. The seniority conclusion is a reasonable reading of the data, not an overreach once you correct the abstract. I'd send it to peer review and ask the authors to reconcile the abstract with the 63(20)% number and to show how the conclusion shifts under the branching-ratio and mixing-ratio uncertainties. It's citable for the datum, and worth a reading-group discussion on how abstract language outruns the body.

Referee Report

3 major / 4 minor

Summary. The paper reports a Coulomb-excitation study of 124Te performed at the ANU Heavy Ion Accelerator Facility using 58Ni and 16O beams with particle-γ coincidence detection. Transition yields are analyzed with the semiclassical code GOSIA, using disclosed SRIM stopping powers, BrIcc internal-conversion coefficients, and beam energies below the safe Coulomb-excitation limits. The principal new result is B(E2; 6_1^+ → 4_1^+) = 27(9) W.u., measured for the first time in 124Te. The authors compare this and other E2 strengths with large-basis shell-model calculations and with General Collective Model (GCM) fits. They argue that the 6_1^+ state retains a dominant π(0g_{7/2})^2 seniority structure even though 124Te lies near the neutron midshell, while the 2_1^+ and 4_1^+ states show emerging collectivity. The paper includes considerations of feeding, contaminants, branching-ratio inference, and model dependence, and explicitly notes remaining uncertainties, including the unconstrained 2_2^+→2_1^+ mixing ratio and the obscured 4_2^+→2_1^+ line.

Significance. If the central value and interpretation stand, this is a valuable first measurement of a transition strength that directly tests seniority-vs-collectivity competition in a transitional nucleus. The shell-model comparison is externally anchored: effective charges are fixed by 134Te and 132Sn, and the g-factor quenching by g(2_1^+) in 122Te, so the comparison is not circular. The GCM contrast also frames the discussion usefully. The main weakness is not the measurement chain per se but the internal inconsistency between the abstract's 'remarkably good agreement' and the body's report that the shell model reproduces only 63(20)% of the measured 6^+→4^+ strength in 124Te, placing the datum ~1.9σ above the model. Since that agreement is the paper's load-bearing interpretive evidence, the overstatement must be corrected and the sensitivity of the extraction to the inferred 4_2^+ yield should be quantified.

major comments (3)

[Abstract; Conclusions; Shell-model calculations] The abstract and conclusions state 'remarkably good agreement' between the measured B(E2; 6_1^+→4_1^+) = 27(9) W.u. and shell-model calculations for 124–134Te. In the body, however, the shell-model section reports that for 124Te the model accounts for only 63(20)% of the 6^+→4^+ strength, and groups 124Te with isotopes for which 'experimental E2 strengths rise above the shell-model values,' markedly so for the 6^+→4^+ transition in 124Te. Thus the datum is ~1.9σ above the prediction, not in remarkable agreement. This is a load-bearing inconsistency because the seniority-persistence conclusion is based on the claimed close agreement. Please revise the abstract/conclusions and re-frame the interpretation as consistent with a large-but-incomplete seniority component, or else demonstrate quantitatively why the 63(20)% reproduction should be considered 'remarkable.'
[Section II/III, Fig. 3, Table III] The 4_2^+→2_1^+ (1355 keV) yield is not directly measured because the line is obscured by the Doppler-shifted 1454-keV 2_1^+→0_1^+ transition in 58Ni; its intensity is inferred from the 4_2^+→4_1^+ line and the adopted branching ratio [16,34]. This inferred yield enters the GOSIA fit and could influence the extracted matrix elements. The manuscript should provide a quantitative sensitivity test: vary the branching ratio and the inferred yield within their uncertainties and show the resulting range for B(E2; 6_1^+→4_1^+), or otherwise justify that the 6^+ matrix element is insensitive to this input. Without such a test, the robustness of the central datum is not fully established.
[Shell-model calculations, Fig. 9] The text notes that the near-j^2-limit behavior of the B(E2) ratios B_{4/2} and B_{6/2} is 'puzzling and likely fortuitous' because the computed g-factor ratios disagree with the j^2 limit. Yet the seniority conclusion is later presented as 'strong evidence' partly on the basis of these ratios. This internal tension should be resolved explicitly: either the ratios are not used as evidence (and the argument rests on the absolute 6^+→4^+ strength), or their fortuitous nature needs to be reconciled with the conclusion. As written, a reader cannot tell how much weight to place on Fig. 9 in the central interpretation.

minor comments (4)

[Section II, Table III] Please state explicitly the adopted numerical value and uncertainty of the 4_2^+ branching ratio used to infer the 1355-keV yield, rather than only referencing [16,34].
[Section III or IV] The phrase 'j2 model' appears before it is defined; define j^2 coupling and the relation to π(0g_{7/2})^2 seniority at first use.
[Abstract/Conclusions] The phrase 'significantly below that expected for a spherical vibrator' should be accompanied by the quantitative vibrator prediction and the comparison uncertainty, so the reader can judge the significance without digging through tables.
[Conclusions] The final paragraph says 'it is clear that... the two-proton seniority structure... persists in the 6_1^+ state to at least 124Te.' Given the 33% uncertainty on the 6^+ value and the 1.9σ shell-model offset, 'clear' is too strong; consider 'the data are consistent with' or 'support' a dominant seniority component.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the new B(E2; 6_1+→4_1+) is measured and the model comparisons are anchored to 134Te, 132Sn, 122Te, and 120Te, not to the 124Te datum. The abstract's 'remarkably good agreement' is overstated versus the body's 63(20)% shell-model reproduction, but that is an internal-consistency issue, not circularity.

full rationale

The paper's central quantity, B(E2; 6_1+→4_1+) = 27(9) W.u., is a measured value extracted by GOSIA from Coulomb-excitation yields; it is not derived from any of the models it is compared with. The shell-model comparison is externally anchored: the paper states that effective charges e_pi = 1.7e and e_nu = 0.9e were 'determined empirically from the semimagic nuclei 134Te and 132Sn, respectively', and that the g_s quench is 'fixed by using the experimental g-factor value of the 2_1+ state in 122Te'. Thus the 124Te B(E2; 6_1+→4_1+) is an independent prediction, not a fitted input. The GCM comparison is similarly anchored to 120Te. The seniority conclusion draws additional support from parameter-free ratios (B4/2, B6/2), in which effective charges cancel, and from a predicted g(6_1+) = +0.78, neither of which reduces to the measured 124Te strength. The self-citations to Refs. [10], [72], and [74] are not load-bearing in a circular sense: they report external data and parameter fits (134Te, 132Sn, 122Te g-factor), not an assertion of the target result. The clearest weakness is rhetorical rather than circular: the abstract claims 'remarkably good agreement' for 124–134Te, but the body reports that for 124Te the shell model accounts for only 63(20)% of the 6_1+→4_1+ strength, i.e., the model lies ~1.9σ below the data. That is an overstatement/internal inconsistency, not a circular reduction. A minor circularity-adjacent caveat is that 134Te—one endpoint of the stated agreement range—is the nucleus used to fix e_pi, so agreement at that endpoint is partly inherited from the fit; the 124Te result itself, which carries the paper's conclusion, is an independent measurement-and-prediction comparison.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central content is a measured B(E2), so the ledger mainly records modeling inputs that carry the interpretation. The shell model relies on two effective charges fitted to semimagic neighbors (134Te, 132Sn) and a g_s quench fitted to g(2_1+) in 122Te; the GCM benchmark uses potential parameters fitted to 120Te. On the analysis side, the obscured 4_2+ to 2_1+ yield is recovered through an adopted branching ratio, and the 2_2+ to 2_1+ mixing ratio is left unconstrained. No new entities (particles, forces, dimensions) are introduced; the pi g7/2^2 seniority component is a standard shell-model configuration.

free parameters (5)

shell-model effective charges e_pi, e_nu = e_pi = 1.7e, e_nu = 0.9e
Determined empirically from semimagic 134Te and 132Sn (Ref. [72]), then applied to 124Te. The 124Te B(E2) predictions scale with these values; they are fitted, not derived, and anchor the central comparison.
g-factor g_s quenching = g_s = 0.75 g_s^free for protons and neutrons
Fixed by the experimental g(2_1+) = 0.351(18) in 122Te (Ref. [74]); used for the shell-model g-factor predictions that the authors say would test the seniority interpretation.
GCM potential parameters = fitted to 120Te; values not itemized in text
The GCM fits reproduce 120Te; the potential parameters are fitted to 120Te level data before comparison to 124Te. The claim that 124Te cannot be described by the GCM rests on this fitting procedure.
branching ratio (4_2+ to 2_1+) : (4_2+ to 4_1+) = adopted from Refs. [16,34]
Used to infer the obscured 1355-keV 4_2+ to 2_1+ yield from the observed 4_2+ to 4_1+ transition; feeds the GOSIA fit.
delta(2_2+ to 2_1+) mixing ratio = unconstrained; several assumed values
B(E2; 2_2+ to 0_1+) results vary with the assumed mixing ratio; the paper reports small values for each assumed ratio, an acknowledged ambiguity.

assumptions (4)

domain assumption Semiclassical Coulomb-excitation theory (GOSIA) is valid at the safe energies used (191.5 MeV 58Ni, 47 MeV 16O)
Invoked in the GOSIA analysis section; safe-energy values from Cline [47] define the sub-Coulomb condition. If the condition were violated, extracted B(E2) values would shift.
domain assumption Adopted level scheme and spin-parity assignments for 124Te (Ref. [16]) are correct, including the 6_1+ assignment
The 8 identified transitions and the headline 6_1+ to 4_1+ identification rely on this scheme, cited in Results: 'Eight transitions from the adopted level scheme [16] were identified.'
standard math The CD-Bonn-derived shell-model Hamiltonian (Coraggio et al. [71]) with a 100Sn core and 0g7/2, 1d5/2, 1d3/2, 2s1/2, 0h11/2 orbits is a valid effective interaction for 124-134Te
Used for all shell-model B(E2) and g-factor predictions; the paper notes excitation energies systematically exceed experiment toward 124Te, so the interaction has known residual inaccuracy.
domain assumption A GCM potential fitted to 120Te is the appropriate collective-model benchmark; its failure for 124Te indicates absent collectivity rather than an ill-posed fit
The GCM comparison tests 124Te against a parametrization that works for 120Te; if the fit for 124Te were under-constrained, the negative conclusion would be weaker.

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Cite this review

Pith. "Pith review of Coulomb excitation of $^{124}$Te: Emerging collectivity and persisting seniority structure in the $6_1^+$ level." pith.science (2026). https://pith.science/paper/FIF7CEAL

@misc{pith2026250809643,
  author       = {Pith},
  title        = {Pith review of: Coulomb excitation of $^124$Te: Emerging collectivity and persisting seniority structure in the $6_1^+$ level},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FIF7CEAL}},
  note         = {Machine review of arXiv:2508.09643}
}

abstract

The low-lying energy spectra of even-even tellurium isotopes near midshell have long been interpreted as `textbook' examples of vibrational collective motion. However, in many cases electric-quadrupole observables, which are a particularly sensitive probe of collectivity, remain undetermined. Coulomb-excitation measurements were performed to measure transition strengths connecting the ground and low-excitation states in $^{124}$Te. This isotope lies at a transitional point between collective structure near the neutron midshell and seniority structures near the $N=82$ shell. A transition strength, $B(E2; 6_1^+ \to 4_1^+)$, of 27(9)~W.u. was measured for the $6^+_1\rightarrow4^+_1$ transition for the first time in this nucleus; this value is significantly below that expected for a spherical vibrator, as well as other collective models. We examine the transition strengths in $^{124}$Te and its neighbors by comparison with large-basis shell-model calculations and by comparison with General Collective Model (GCM) fits. A GCM description of $^{120}$Te agrees with experimental $E2$ transition strengths, but no comparable description of $^{124}$Te is possible with the GCM. In contrast, there is remarkably good agreement between the $B(E2; 6_1^+ \to 4_1^+)$ values and shell-model calculations for $^{124-134}$Te. It appears that, despite approaching midshell, $^{124}$Te retains a seniority structure for the $6^+_1$ level, i.e. a significant $\pi 0g_{7/2}^2$ contribution. This persistence of the shell structure at the $6^+_1$ state is in contrast to the $B(E2)$ values of the lower-excitation $2^+_1$ and $4^+_1$ states in $^{124}$Te, and neighboring $^{120}$Te and $^{122}$Te, for which the collectivity becomes enhanced as more neutrons are removed from $N=82$.

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