Typical new spherical-like $\gamma$-soft spectra in $^{104,106,108}$Pd

Tao Wang

REVIEW 4 major objections 5 minor 1 cited by

Typical new spherical-like $\gamma$-soft spectra in $^{104,106,108}$Pd

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that 106Pd is a typical spherical-like γ-soft nucleus, with low-lying states up to the 10+ state under 4000 keV reproduced by an extended interacting-boson model with SU(3) higher-order interactions.

desk verdict A systematic fit with some real hits, but the central claim rests on an unsupported assumption about negligible normal–intruder coupling; the abstract overstates what the evidence shows. read the letter →

arxiv 2501.10925 v4 pith:MCFA6FKR submitted 2025-01-19 nucl-th

Tao Wang This is my paper · ORCID

classification nucl-th MSC 81V35 PACS 21.60.Fw21.10.Ky23.20.-g27.60.+j

keywords spherical-likegamma-softnucleusinteractingbosonmodelSU(3)higher-orderinteractions106PdCdpuzzlequadrupolemomentsB(E2)valuesshapecoexistence

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 106Pd, long treated as a near-spherical phonon-vibrational nucleus, is in fact a new type of collective object: a spherical-like γ-soft nucleus. It argues that the low-lying levels up to the 10+ state below 4000 keV, the B(E2) transition strengths, and the quadrupole moments of 106Pd all match the spectra of an extended interacting-boson-model Hamiltonian with higher-order SU(3) interactions. The distinctive fingerprint is a pattern in which the 0+2, 2+2, 4+1 triplet appears like two-phonon states, yet no 0+3 sits near the 6+1, 4+2, 3+1, 2+3 states, and a 0+3 appears near the third group at roughly twice the 0+2 energy. If this reading is right, the standard phonon-excitation picture of spherical nuclei fails in the Cd-Pd region, and quadrupole deformation is fundamental even in these near-magic nuclei.

What carries the argument

The machinery is an extended interacting boson model Hamiltonian $\hat H = c\{(1-\eta)\hat n_d + \eta[-\hat C_2[SU(3)]/(2N) + \alpha \hat C_3[SU(3)]/(2N^2) + \beta \hat C_2^2[SU(3)]/(2N^3) + \gamma \Omega/(2N^2) + \delta \Lambda/(2N^3)]\}$, plus an $\hat L^2$ term, diagonalized in a $U(6)\supset SU(3)\supset SO(3)$ basis. SU(3) symmetry dominates: the second- and third-order Casimir operators describe prolate and oblate deformation, and their combination produces a low-lying spectrum that mimics phonon triplets but with a characteristic missing 0+3 near the second group and a 0+3 near the third group at roughly twice the 0+2 energy. The SU(3) decomposition of the 0+ states in the new model is shown to differ from both the U(5) and O(6) limits, which is how the paper distinguishes the new mode from both spherical vibration and standard γ-soft rotation.

What would settle it

Precision measurement of the B(E2) values for the 0+5 and 2+3 states in 106Pd would settle the claim: the new model predicts B(E2;0+5→2+3)=60.5 W.u. and B(E2;2+3→0+2)=37.8 W.u., while the IBM-2 fits give 23.2 and 12.3 W.u., respectively. Resolving these branches with a Coulomb-excitation or gamma-spectroscopy experiment would distinguish the two pictures.

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Extended reading notes

Core claim

The central claim is that direct experimental evidence for the proposed spherical-like γ-soft nucleus exists in 106Pd. The author shows that the complete low-lying normal-state spectrum up to the 10+1 state under 4000 keV, including the third and fourth group levels 8+1, 6+2, 5+1, 4+3, 4+4, 2+4, 0+3 and 10+1, 8+1, 7+1, 6+3, 5+2, 6+4, 4+5, 3+2, 2+5, 0+4, is reproduced by the new model with a single set of fitted parameters. The theoretical B(E2) values and quadrupole moments agree with experiment at a good level, and the paper claims the agreement is better than that of the IBM-2 calculations. The neighboring nuclei 104Pd and 108Pd are also discussed, with 108Pd behaving as a softer, near-critical nucleus between the new γ-soft mode and the prolate shape, and 104Pd requiring intruder configuration mixing. The author concludes that these results completely disprove the possibility that the Cd-Pd nuclei are spherical phonon vibrators.

Load-bearing premise

The load-bearing premise is that the coupling between the normal states and the intruder states in 106Pd can be ignored, based on the small B(E2;0+3→2+1) value near 2.41 W.u.; if that coupling were significant, the fitted normal levels would be shifted and the comparison to the pure normal-state model would collapse.

Editorial extensions

If this is right

The two-times relationship between the 0+3 and 0+2 energies becomes a testable fingerprint: nuclei showing a 0+ state near twice the 0+2 energy, with no 0+3 near the second group, can be classified as spherical-like γ-soft candidates.
The Cd-Pd region would be reinterpreted: the traditional spherical phonon-vibrational description should be replaced by γ-soft rotation with spherical-like low-lying spectra, and the Cd puzzle dissolves.
For 104Pd, the model requires explicit normal-intruder configuration mixing; future calculations including intruder states should reduce B(E2;0+2→2+1) toward experiment.
108Pd emerges as the critical nucleus for a shape phase transition between the new γ-soft phase and prolate shapes, which explains its softer quadrupole moments and larger odd-even staggering.
The success of higher-order SU(3) terms suggests that quadrupole deformation, not spherical vibration, is the organizing principle of low-lying collective spectra in medium-mass nuclei.

Reading between the lines

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

A decisive cross-check would be to search for the predicted 0+5 and 2+3 branching pattern in 108Pd or in a cadmium isotope whose intruder states are pushed high enough to expose the normal spectrum, since one nucleus alone may not rule out all phonon-based alternatives.
Because the emergent fingerprint is energy-based, it can be searched for in large nuclear data sets: nuclei whose 0+ spectra show a gap followed by a 0+ state at twice the first excited 0+ energy.
The proposed proton-neutron extension of the model is likely to resolve the odd-even γ-band staggering discrepancy and could be tested on 104Pd and 108Pd by comparing neutron and proton boson numbers.
If the two-times 0+ relationship holds in other nuclei with N=7 bosons, such as 110Cd and 118Cd, the new mode is not an isolated case but a systematic family.

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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

The one thing to know: this is not the direct evidence it claims to be. The paper does do a systematic fit of 106Pd levels and B(E2) values with a modified IBM Hamiltonian, and it scores some genuine hits – the energy pattern in Fig. 3 is indeed reminiscent of the predicted γ-soft spectrum, and the quadrupole moment anomaly in the 4+1 state across Pd isotopes is a real, non-trivial observation that IBM-2 misses. But the load-bearing step is the assumption that the normal–intruder coupling in 106Pd is negligible, and that assumption is inferred almost entirely from one number: B(E2;0+3→2+1)≈2.41 W.u. In a two-state mixing picture, that small value does not imply small mixing; it can easily arise from destructive interference or from the unperturbed 0+3 having little E2 strength to 2+1. The stress-test estimate of |β|≈0.26 is enough to shift levels by hundreds of keV, which would undermine the clean comparison in Fig. 3. The paper even acknowledges that 104Pd needs configuration mixing because B(E2;0+3→2+1)>25 W.u., but gives no criterion for where small becomes negligible.

The second soft spot is the B(E2;0+2→2+2) discrepancy: theory gives 84.6 W.u. against the measured 19(+7,-3) W.u. That is not a small deviation, and the author's discussion – listing other nuclei with large values – does not explain why 106Pd itself is off by a factor of four. Combined with assigning three unobserved blue levels as 'real' based on band structure, the 'verification' is considerably weaker than the abstract suggests. The conclusion that this 'completely disproves' spherical phonon excitations is an overclaim; at most, the data are consistent with the author's model.

That said, the paper is not a hack job. The calculations are reproducible in principle, the parameter choices are stated, and the author is explicit about where the fit is poor. The comparison with IBM-2 is useful, and the Q(4+1) anomaly is a nice point in the model's favor. The problem is the gap between what the evidence supports and what the abstract claims.

Who is this for? Someone working on Cd-Pd shape coexistence who wants a concrete alternative to the phonon picture. It deserves a serious referee, not a desk reject – but the referee should ask for a quantitative treatment of the mixing (or a clear argument why it is truly negligible) and a rewriting of the conclusions to match the evidence. I would not cite it as a settled result in the next year, but I would keep it on the table as a provocative alternative.

Recommendation: send to peer review, with the expectation of major revision if the mixing issue can be addressed; otherwise it should not be published as a 'confirmation'.

Referee Report

4 major / 5 minor

Summary. The manuscript argues that 106Pd is a "typical spherical-like γ-soft nucleus" described by an extended interacting-boson-model Hamiltonian with SU(3) higher-order terms. After fitting parameters (η, γ, δ, c, f, e) to the low-lying energies and selected B(E2) values, the paper reports agreement of the calculated level scheme with experimental data up to the 10+1 state, interprets the 0+5 state via a two-times energy relationship, and extends the analysis to 104Pd and 108Pd. It concludes that the existence of the new spherical-like γ-soft mode is confirmed and that phonon excitations in the Cd–Pd region are "completely disproved."

Significance. The proposed mode is an alternative to the traditional spherical-vibrator picture in the Cd–Pd nuclei, and if the claim were established it would be of considerable interest. The paper provides a broad set of comparisons (level energies, B(E2) values, quadrupole moments) and makes concrete predictions for unobserved states, which is a useful feature. However, the verification is weakened by the use of fitted parameters, the unjustified neglect of normal–intruder mixing, and the inclusion of unobserved levels as confirmed states; the comparison with the phonon picture is not quantitative. The paper does not provide reproducible code or machine-checked proofs, but the tabulated comparisons are a service to the community.

major comments (4)

[Section II] The conclusion that normal–intruder coupling "can be also ignored" in 106Pd rests solely on the small value B(E2;0+3→2+1)≈2.41 W.u. This inference is not quantitatively supported: in a two-state mixing picture with B(E2;0+2→2+1)≈35 W.u., the observed interband strength can be produced by a mixing amplitude |β|≈sqrt(2.41/35)≈0.26, which is not negligible, and the small interband B(E2) can also arise from destructive interference even for large mixing. Section V admits that B(E2;0+3→2+1)>25 W.u. in 104Pd calls for configuration mixing, but no threshold is given for when mixing is negligible; therefore the unmixed comparison in Fig. 3 is not an established verification of the normal-state spectrum.
[Table I and Section IV] The theoretical B(E2;0+2→2+2)=84.6 W.u. disagrees with the adopted experimental value 19(+7,-3) W.u. by more than a factor of four, a deviation far outside the uncertainty quoted for the data. The subsequent comparison with larger values in 108Pd and Cd nuclei does not explain this specific discrepancy in 106Pd; as presented, this undercuts the statement that there is "no complete inconsistency" and limits the strength of the B(E2) verification claim.
[Section IV and Fig. 3] Three of the levels used in the comparison (the blue 8+, 7+, and 6+ states) are not experimentally observed and are placed "according to the features of the level bands", and the 0+5 state is identified at least in part because the model requires a two-times relationship with the 0+2 state. The claimed confirmation of the third and fourth group states therefore relies on assignments that are not independent of the model. Independent experimental identification of these states is needed before the spectrum can be said to be verified.
[Abstract and Section VII] The statement that these results "completely disprove the possibility of the phonon excitations of the spherical nucleus in the Cd-Pd nuclei region" is not supported by any quantitative comparison with a phonon (e.g., U(5)-type) description of the same data. A model with several fitted parameters that reproduces part of a level scheme cannot by itself exclude an alternative model; the conclusion should be severely softened or the phonon model should be tested directly.

minor comments (5)

[Throughout] There are typographical errors, for example "pro posed" in the abstract, "Nucl. Phya." in Ref. [20], and "quadruple moments" in Section VI; these should be corrected.
[Fig. 3] The experimental status of the blue levels (observed vs. predicted) should be stated unambiguously in the caption, since the text says these levels have not been found yet.
[Table I] The table would be more informative if the theoretical uncertainties and the references for each experimental column were reported consistently; currently the source labels a–d are not all mapped in every row.
[Section III] The value α=3N/(2N+3) appears to be fixed, but the paper does not explain why this choice is made for the fits; the sensitivity of the results to this choice should be commented on.
[References] Several references are to "submitted" or "in preparation" items (Refs. [38], [39], [49], [50]); these should be replaced by published versions or clearly identified as preprints.

Circularity Check

2 steps flagged · score 8.0 of 10

Central 106Pd 'spherical-like γ-soft' claim is built from fitted parameters and a self-definitional 0+ assignment, so its confirmation partly reduces to the model's own assumptions.

fitted input called prediction [Section III (Hamiltonian) and Section IV (Results for 106Pd), after Eq. (1) / around Fig. 3]
"The η is determined to allow the energies of the 0+ states to agree with the experimental data. The γ and δ are to match the value of B(E2; 0+2 → 2+1). ... The theoretical five lowest 0+ states all have experimental correspondences."

These parameters are adjusted to the 0+ spectrum and one B(E2) of 106Pd itself; the same 0+ states and B(E2) are then offered as 'verification' and 'confirmation' of the new mode. In particular, the paper's 'fit below will show that this is not a coincidence' cannot do that for the 0+5 two-times state, because η was chosen to make the 0+ energies agree and the two-times feature is a property of the author's model. Thus the central 0+ comparison in Fig. 3 is, for the pattern being claimed, a report of the fit rather than an independent prediction.
self definitional [Section II, paragraph following Fig. 2(b)]
"Importantly, compared with 118Cd, the 0+4 state does not belong to the normal states too, which is not mentioned in previous studies. ... If the normal states of 106Pd show the new spherical-like γ-soft spectra, there must be a 0+ state with energy that is twice the one of the 0+2 state (two-times relationship). I indeed find this 0+5 state of 106Pd in Fig. 2(b). The fit below will show that this is not a coincidence."

The paper removes observed 0+3 and 0+4 from the normal set partly because 0+4 'does not belong to the normal states'—an assignment not previously established and not derived from independent data in this paper. It then selects 0+5 as the state required by the model's two-times relationship. Because the model is used both to define the expected pattern and to sort the experimental states, the later agreement is self-consistent by construction. This is the decisive step that turns the fit into 'direct evidence' for the new spherical-like γ-soft nucleus.

full rationale

The paper is not wholly without independent content: it uses external ENSDF and published B(E2) data, fixes one parameter set and applies it to 104Pd and 108Pd, and reports quadrupole moments and several B(E2) values not directly used in the fit. Those comparisons could in principle falsify the model, and the normal–intruder decoupling question (small B(E2;0+3→2+1)=2.41 W.u. versus mixing) is a physics/correctness concern rather than a circularity. However, the central 106Pd identification depends on (i) fitting η to the 0+ energies and γ, δ to a B(E2) of the same nucleus, and (ii) declaring the experimental 0+4 to be an intruder and choosing 0+5 as the model's two-times state. These steps make the main 'verification' partly circular: the model's pattern is used to select the normal-state spectrum, and the selected spectrum is then used to confirm the model. Hence score 8 rather than 6.

Assumptions & free parameters 6 free parameters · 4 assumptions · 1 invented entities

The central claim rests on a model with six fitted parameters per nucleus, an assumed decoupling of intruder states, and the assignment of unmeasured levels. The invented 'new mode' has no independent evidence outside the same fitting framework.

free parameters (6)

eta = 0.47 for 106,104,108Pd
Adjusted to make the energies of 0+ states agree with experimental data (Section III).
gamma = 1.728
Chosen to match B(E2; 0+2 to 2+1) in 106Pd (Section III).
delta = 1.34
Also fit to B(E2) data (Section III).
c = 1177.42 keV (106Pd), 1317.46 keV (104Pd), 1152.21 keV (108Pd)
Global energy scale, fit to level energies for each nucleus.
f = 37.04 keV (106Pd), 45.12 keV (104Pd), 27.40 keV (108Pd)
Fit to reproduce odd-even staggering and level spacings.
e = 2.027, 2.082, 1.950 (W.u.)^(1/2) for 106Pd, 104Pd, 108Pd
Effective charge fit to reproduce absolute B(E2) values for each nucleus.

assumptions (4)

ad hoc to paper The IBM with the Hamiltonian in Eq. (1) is an adequate description of low-lying collective states.
The Hamiltonian is constructed specifically to produce the spherical-like gamma-soft spectra; it is not derived from a more fundamental theory in this paper.
standard math The SU(3) basis diagonalization code (Refs. 17, 56) is correct.
The paper assumes the code accurately diagonalizes the Hamiltonian.
domain assumption The coupling between normal and intruder states is negligible in 106Pd.
Section II states this based on the small B(E2; 0+3 to 2+1) value.
domain assumption The assignment of unobserved levels (blue levels in Fig. 3) is correct.
Section IV assumes the existence of 8+, 7+, 6+ levels based on band features, despite no experimental confirmation.

invented entities (1)

New spherical-like gamma-soft nuclear mode
purpose: To explain the Cd puzzle and provide an alternative to spherical phonon excitations.
The mode's existence is inferred from the author's model and the fit to 106Pd; no independent falsifiable prediction is provided that does not rely on the same model and fitted parameters.

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Cite this review

Pith. "Pith review of Typical new spherical-like $\gamma$-soft spectra in $^{104,106,108}$Pd." pith.science (2026). https://pith.science/paper/MCFA6FKR

@misc{pith2026250110925,
  author       = {Pith},
  title        = {Pith review of: Typical new spherical-like $\gamma$-soft spectra in $^104,106,108$Pd},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MCFA6FKR}},
  note         = {Machine review of arXiv:2501.10925}
}

abstract

To solve the Cd puzzle (spherical nucleus puzzle), I have proposed a new spherical-like $\gamma$-soft nucleus. Since shape coexistence often occurs in such nuclei, explicit spherical-like $\gamma$-soft spectra are not easily identified. In this paper, I finally find the direct evidence for the existence of the new spherical-like $\gamma$-soft nucleus. $^{106}$Pd is in fact a typical spherical-like $\gamma$-soft nucleus. The low-lying parts, up to the $10_{1}^{+}$ state, under 4000 keV, of the spherical-like $\gamma$-soft spectra are verified. The B(E2) values and the quadrupole moments of the low-lying levels are also studied. By comparison, the new spherical-like $\gamma$-soft mode is confirmed. Moreover $^{104,108}$Pd are also discussed. These results completely disprove the possibility of the phonon excitations of the spherical nucleus in the Cd-Pd nuclei region.

Figures

Figures reproduced from arXiv: 2501.10925 by the authors.

**Figure 1.** depicts the new spherical-like γ-soft spectra for N = 7, which is the boson number of 106Pd. In Ref. [43], the spectra for N = 6 is also shown. For the low-lying part up to the 12+ 1 state, the new spherical-like γ-soft spectra are nearly the same, and resemble the phonon excitations of the spherical nucleus. The characteristics of the spectra are: (1) similar to the two-phonon excitations in the spherical nucleus,… view at source ↗

**Figure 2.** FIG. 2. (a) The lowest six 0 [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 4.** FIG. 4. The experimental and theoretical results of the low [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: FIG. 5. The experimental and theoretical results of the low [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6. The evolutional behaviors of the quadrupole moment [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

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Reviewed August 10, 2026 · model on record in the stance chip above.

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