Nuclear size, electric monopole transitions, and the location of $0^+_2$ states

B. Maheshwari; K. Nomura; P. Van Isacker

REVIEW 3 major objections 4 minor 57 references

Nuclear size, electric monopole transitions, and the location of $0^+_2$ states

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

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

Pith's one-line read The paper claims that one configuration-mixing mechanism, fixed by energy levels, reproduces both the calcium charge-radius isotopic shift and the electric monopole strengths $\rho^2(E0)$, establishing a direct size–E0 correlation.

desk verdict First IBM-CM treatment of the Ca charge-radius puzzle reproduces the 48Ca dip and links radii to E0 strengths, but the central correlation rests on an untested κ_int scaling assumption that a referee should probe. read the letter →

arxiv 2502.04999 v1 pith:LOPE4QGN submitted 2025-02-07 nucl-th

B. Maheshwari , K. Nomura , P. Van Isacker This is my paper · ORCID

classification nucl-th

keywords interactingbosonmodelconfigurationmixingchargeradiiisotopicshiftelectricmonopoletransition0+statescalciumisotopesshapecoexistence

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

Charge radii of the calcium isotopes behave oddly: $^{40}$Ca and $^{48}$Ca have nearly equal radii, $^{44}$Ca is a local maximum, and the radius rises steeply beyond $^{48}$Ca. This paper argues that the interacting boson model with configuration mixing (IBM-CM), with wave functions fixed by fitting energy levels, reproduces this isotopic shift, in particular the $^{48}$Ca dip and the $^{50}$Ca rise. It then uses the same deformation parameters that fix the radii to compute electric monopole strengths $\rho^2(E0;0_2^+\to0_1^+)$; the results agree with data for $^{44}$Ca and $^{48}$Ca. If correct, the argument establishes a direct correlation between nuclear size and monopole transitions and makes unmeasured $E0$ strengths predictable from measured radii.

What carries the argument

The central machinery is the IBM-CM Hamiltonian $\hat H' = \hat H_{\rm reg} + (\hat H_{\rm int}+\Delta)+\hat V_{\rm mix}$ acting in the direct sum of the regular $[n_b]$ and intruder $[n_b+2]$ boson spaces, together with the charge-radius operator $\hat T(r^2) = \langle r^2\rangle_c + \kappa\,\hat n_b + \eta\,\hat n_d/\hat n_b$. The load-bearing identity is the linear-scaling constraint $\kappa_{\rm int} = \frac{n_b}{n_b+2}\kappa_{\rm reg}$, which keeps the radius linear when moving from the regular to the intruder space. Once the wave functions are fixed from energy levels, the isotopic-shift formula and the E0 formula express both observables in terms of the same $\eta_{\rm reg}$ and $\eta_{\rm int}$, so the radius and the monopole strength are locked together by construction. The mixing probabilities of the ground and $0_2^+$ states carry the physical content: the $^{48}$Ca ground state is 78% intruder and $^{50}$Ca is 98% intruder.

What would settle it

Measure the electric monopole strength $\rho^2(E0;0_2^+\to0_1^+)$ in $^{46}$Ca: the model's parameters are already fixed by the Ca isotopic shifts, so Eq. (6) gives a definite value, and a significant deviation would show the claimed size–E0 correlation is not quantitative.

Watch

Extended reading notes

Core claim

The central claim is that the location of the $0_2^+$ state and the size of the nucleus are not independent: both are governed by the mixing between the regular $[n_b]$ (0p-0h) and intruder $[n_b+2]$ (2p-2h) boson spaces. In IBM-CM, wave functions fitted to level energies give ground-state mixing probabilities in which $^{48}$Ca is 78% intruder and $^{50}$Ca is 98% intruder, and the radius operator with the imposed linear-scaling constraint then predicts the Ca isotopic shift, including the $^{48}$Ca dip. Taking the same $\eta_{\rm reg}$ and $\eta_{\rm int}$ from the isotopic-shift fit, the E0 formula yields $\rho^2(E0)$ values for $^{44}$Ca and $^{48}$Ca in reasonable agreement with experiment; thus the paper establishes a direct correlation between nuclear size and electric monopole transitions. The mechanism is that the $^{48}$Ca ground state is strongly intruder-dominated but blocked, giving a small radius, while $^{50}$Ca is almost pure intruder, giving the steep rise; the same wave functions explain why the $0_2^+$ state is isomeric in $^{40}$Ca but less so in $^{48}$Ca. Calculations for the Ar and Ti isotopes show a similar, though weaker, connection.

Load-bearing premise

The load-bearing premise is that the radius contribution of the intruder $[n_b+2]$ configuration scales with the regular $[n_b]$ configuration through the imposed linear relation $\kappa_{\rm int} = \frac{n_b}{n_b+2}\kappa_{\rm reg}$; if a microscopic calculation showed otherwise, the fitted $\eta$ values and the derived $\rho^2(E0)$ strengths would lose their meaning.

Editorial extensions

If this is right

If the correlation is correct, the $^{48}$Ca radius dip and the $^{50}$Ca rise require no new physics beyond two-configuration mixing; the same wave functions that fit energies already produce them.
Because the same $\eta_{\rm reg}$ and $\eta_{\rm int}$ enter the radius and E0 formulas, measured charge radii become predictions for unmeasured $\rho^2(E0)$ values, for example in $^{46}$Ca.
The $0_2^+$ energy becomes a size-relevant observable: interactions that misplace the $0_2^+$ state, as the paper notes for the $^{36,38}$Ca shell-model results, will also fail on radii.
The binding-energy argument yields a qualitative rule for other chains: isotopes with similar binding energy per nucleon, such as $^{40}$Ca and $^{48}$Ca, should have similar radii, while unequal pairs such as $^{38}$Ar and $^{46}$Ar should differ.

Reading between the lines

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

A testable extension is to apply the same parameter-sharing to other mass regions with known charge radii and E0 data, such as the tin or lead isotopes; if the correlation persists there, it is a structural feature of configuration mixing rather than a Ca-specific fit.
The linear-scaling constraint is a modeling assumption; a shell-model calculation that computed the intruder 2p-2h space's radius directly could verify or falsify the $\kappa_{\rm int}$ relation without new experiment.
The paper's failure to describe $^{42}$Ca suggests the two-configuration truncation may be too restrictive there; adding 4p-4h admixtures or a different intruder Hamiltonian could be the next step.
A sharp, implied prediction is the sign and magnitude of $\rho^2(E0)$ in $^{46}$Ca, which is measurable with current electron-conversion techniques and would provide a clean test of the size–E0 link.

Share X Bluesky LinkedIn Reddit HN

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

Here's the short version: this is the first IBM-CM calculation of the Ca charge-radius puzzle, and it gets the qualitative structure right—the 48Ca dip, the 50Ca kink—by letting intruder mixing drive the size change. It also uses the same η parameters fitted to radii to compute ρ²(E0), and the E0 data for 44,48Ca come out in reasonable agreement. That partial independence is the strongest point: the E0 values are not fitted, they follow from the radius fit and the wave functions.

What's actually new is the application, not the framework. The IBM-CM machinery comes from the authors' earlier spectral study of Ca isotopes, and the radius operator is standard. But connecting the size puzzle to E0 strengths in this way is a genuine step, and the paper is honest about where it fails: 38,42Ca and 44Ti are not explained, and the 42Ca E0 deviates from data.

The soft spot that matters is the κ_int = nb/(nb+2) κ_reg scaling in Eq. (5), cited to ref [33]. This is what makes the κ part of the radius operator configuration-independent and what removes it from the E0 matrix element in Eq. (6). The η's fitted to isotopic shifts then determine ρ²(E0). If the intruder 2p–2h configuration doesn't obey that exact linear scaling, the κ terms re-enter the E0 matrix element and the fitted η values no longer pin down the E0 strength. The paper gives no microscopic derivation or empirical test for this scaling. That doesn't kill the paper—the 44,48Ca E0 agreement is real—but it does mean the central 'direct correlation' claim is conditional on an ad hoc assumption, not established. A referee should ask for a sensitivity test: vary κ_int around the scaling and show how much ρ²(E0) moves.

Minor issues: the agreement with isotopic shifts is visual, with no error bars or χ² metric; the IBM-1 code is unpublished (ref [29]); and the per-chain fitting of κ and η means the radius prediction is partly a fit, as the reader's report notes. None of these are disqualifying for a phenomenological paper.

Bottom line: this deserves a serious referee. It is a solid, useful step for the Ca radius puzzle, and the E0 link is worth testing further. The main thing I'd want from a revision is a robustness analysis of the κ_int scaling and a quantitative goodness-of-fit. I'd bring it to a reading group, and I'd cite it for the IBM-CM radii application.

Referee Report

3 major / 4 minor

Summary. The paper uses the interacting boson model with configuration mixing (IBM-CM) to study the even-even Ca, Ar, and Ti isotopic chains. Hamiltonian parameters are fitted to low-energy spectra, and the resulting wave functions are combined with the radius operator of Eq. (4) to compute charge-radius isotopic shifts, including the scaling relation κ_int = nb/(nb+2) κ_reg between regular and intruder configurations. The same η parameters are then used in Eq. (6) to compute ρ²(E0; 0+2 → 0+1) for the Ca isotopes. The calculation reproduces the isotopic-shift dip at 48Ca and the rise at 50Ca, and gives E0 strengths in agreement with data for 44Ca and 48Ca. The paper also reports NSM energy comparisons and discusses the role of binding energy and shell gaps in the Ar and Ti chains.

Significance. If the proposed correlation could be rigorously established, it would provide a simple phenomenological bridge between ground-state charge radii and 0+2 → 0+1 electric monopole strengths in shape-coexisting nuclei, which is of practical interest for interpreting isomerism and configuration mixing. The paper is transparent about its caveats, and the E0 comparison for 44,48Ca is genuinely nontrivial because those E0 data are not used in the fits. The main weaknesses are that the radius comparison is partly circular, since the same isotopic-shift data determine the fitted parameters, and that the E0 formula depends on an unverified scaling relation for the intruder bosons. The Ar and Ti results are presented as exploratory and are not backed by E0 data. Overall, the manuscript is a reasonable phenomenological study, but the central claim needs to be supported by additional justification and quantitative assessment.

major comments (3)

[Sec. II, Eqs. (4)-(6)] The cancellation of the κ contribution to ρ(E0; 0+2 → 0+1) rests on the exact relation κ_int = nb/(nb+2) κ_reg introduced before Eq. (5). Under this relation, κ_reg nb P_reg + κ_int(nb+2) P_int is proportional to the identity, so the κ term has zero matrix element between orthogonal 0+ states. The paper cites ref. [33] for this scaling but gives no microscopic derivation or empirical test for the 2p-2h intruder bosons. If the scaling is violated, a term proportional to (κ_reg nb − κ_int(nb+2)) times the regular-space overlap enters Eq. (6), and the η values fitted to isotopic shifts no longer determine the E0 strength by themselves. Because the claimed radius–E0 correlation is the central result, the authors should either justify the scaling specifically for the intruder configuration or present a sensitivity analysis with respect to deviations from it.
[Sec. III A, Fig. 4(a); Sec. III B, Fig. 7] The agreement in the radius plots is not an independent test of the model. For each chain, κ_reg, η_reg, and η_int are fitted to the isotopic-shift data displayed in those figures, so the comparison largely reflects the fitting procedure. The paper is honest about the fitting, but the abstract and conclusions should not present the radius agreement as independent evidence for the correlation without distinguishing it from the non-fitted E0 results in Fig. 5. To make the assessment quantitative, the authors should add a residual or rms deviation and state explicitly that only the E0 strengths in 44,48Ca, which were not used in any fit, provide independent support.
[Sec. III A, Fig. 5; Sec. III B] The successful E0 points are only two (44Ca and 48Ca), and the model fails for 42Ca in both radius and E0, for 38Ca and 42Ca in radius, and for 44Ti in radius. These failures are not marginal but are attributed to structural reasons, such as the 'limited number of active bosons' in Sec. III A. The claim that a 'direct correlation' is established is therefore too strong. The authors should either give a systematic condition for when the correlation is expected to hold or reduce the claim to one of a correlation suggested by the 44,48Ca data and the IBM-CM mechanism.

minor comments (4)

[Eqs. (4) and (5)] Equation (5) has a factor 2 multiplying κ_reg nb, whereas Eq. (4) has κ_reg nb. The origin of this factor, presumably the number of nucleons per boson pair, should be stated explicitly so the two equations do not appear inconsistent.
[Sec. III A] The sentence 'The calculated results are in an overall agreement with the data for the long chain of Ca isotopes' is difficult to reconcile with the immediately following admission that the isotopic shifts of 38Ca and 42Ca are not explained; please qualify the claim.
[Sec. III A] The text refers to the experimental 2+1 level of 52Ca at 2.563 MeV while also saying that experimental data for 52Ca are quite limited; please provide the source or clarify that this value is taken from ENSDF.
[Table II] Several levels in Table II are marked with asterisks for unconfirmed spin-parity assignments, but the text does not discuss how this affects the 50Ca comparison; a brief note would help the reader.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: E0 strengths are predicted from η parameters fitted to radii, and the E0 data are not used in the fit.

full rationale

The paper fits κ_reg, η_reg, and η_int to the Ca isotopic-shift data (Sec. III A: 'The parameter κreg = 0.0005 fm2 is adjusted to the overall slope of the isotopic shift... ηreg = 8.69 fm2 and ηint = 0.59 fm2'), then uses those same η values in Eq. (6) to compute ρ2(E0). This is a parameter-transfer prediction, not a circular reduction: the E0 strengths also depend on the off-diagonal matrix elements ⟨0+1|n_d|0+2⟩ in the regular and intruder spaces, which are not fixed by the ground-state radii used in the fit, and the E0 data for 40,42,44,48Ca are never used in the fit. The reported agreement for 44,48Ca therefore provides independent, cross-observable support for the assumed common operator. The relation κ_int = nb/(nb+2) κ_reg, cited to Ref. [33] (which shares an author with the present paper), is stated as a consistency condition to preserve linearity of the radius and to eliminate the κ contribution from the E0 matrix element. It is an unverified model assumption, but it is not circular: the E0 predictions are not equivalent to that assumption, and the quoted condition is explicitly derived from the stated linearity requirement rather than from the E0 data. The self-citation to Ref. [19] supplies Hamiltonian parameters fitted to level energies, not to radii or E0, so it is standard prior work rather than load-bearing circularity. Overall the derivation chain is self-contained with respect to external data; the only caveat is that the radius agreement for the fitted isotopes is a fit, not an independent test, but that is not presented as a prediction.

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

The central claim rests on three kinds of fitted input: radius operator parameters per isotopic chain, Hamiltonian parameters per isotope, and structural assumptions (boson mapping, two-space mixing, linear radius scaling, 40Ca core). The E0 calculation adds no new free parameters but inherits all of these.

free parameters (4)

Radius operator parameters for Ca: κ_reg=0.0005 fm², η_reg=8.69 fm², η_int=0.59 fm² = κ_reg=0.0005, η_reg=8.69, η_int=0.59 fm²
Fitted to the Ca isotopic shift data (Sec. III A); κ_reg adjusts the overall slope and η values control the main features. These same η values are used in Eq. (6) for E0 strengths.
Radius operator parameters for Ar: κ_reg=0.024 fm², η_reg=5.47 fm², η_int=0.01 fm² = κ_reg=0.024, η_reg=5.47, η_int=0.01 fm²
Fixed fitted values for 40-46Ar, then used to estimate 32-38Ar (Sec. III B).
Radius operator parameters for Ti: κ_reg=-0.0002 fm², η_reg=4.1 fm², η_int=-1.5 fm² = κ_reg=-0.0002, η_reg=4.1, η_int=-1.5 fm²
Used for 44-50Ti (Sec. III B, Fig. 7c); the calculation fails for 44Ti.
IBM-CM Hamiltonian parameters (ϵ, a1, a2, χ, α=β, Δ) per isotope = Table I values
All parameters in Table I are fitted to low-lying energy levels (mostly 2+1 and 0+2), as stated in Sec. II; the wave functions from this fit are then used unchanged for radius and E0 calculations.

assumptions (4)

domain assumption Boson mapping: valence nucleon pairs are represented as s and d bosons with nb = nπ + nν counted from the 40Ca core.
This is the standard IBM assumption and determines the boson numbers used for all isotopes (Sec. II, Eq. (1) and surrounding text).
domain assumption Configuration mixing includes only regular [nb] and intruder [nb+2] spaces, with mixing operator V_mix of Eq. (3) and α=β fixed per chain (except 48Ca).
The restriction to two spaces and the equality α=β are modeling choices stated in Sec. II; they are not derived from a microscopic theory.
ad hoc to paper Charge radius operator depends only on nb and nd/nb, Eq. (4), with κ_int = nb/(nb+2) κ_reg to enforce linearity.
Introduced in Sec. II before Eq. (5) to maintain linear increase of radius with valence pair number; this is an assumption about the intruder space without independent microscopic justification.
domain assumption 40Ca is the inert core for Ca, Ar, and Ti chains; neutron-deficient 36,38Ca and 32-38Ar bosons are interpreted as hole pairs with negative κ/η.
Sec. III A; this enables a common IBM description but is not microscopically derived and affects the sign and size of the radius contributions.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Nuclear size, electric monopole transitions, and the location of $0^+_2$ states." pith.science (2026). https://pith.science/paper/LOPE4QGN

@misc{pith2026250204999,
  author       = {Pith},
  title        = {Pith review of: Nuclear size, electric monopole transitions, and the location of $0^+_2$ states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LOPE4QGN}},
  note         = {Machine review of arXiv:2502.04999}
}

abstract

The work addresses the isotopic shift of nuclear radii for the even-even $^{36-52}$Ca isotopes using the interacting boson model (IBM) that includes the mixing from normal and intruder configurations. We obtain a good agreement between the calculated and experimental data, particularly for the dip at $^{48}$Ca. A direct correlation between nuclear size and electric monopole transitions is established to compute the electric monopole transition strengths, $\rho^2(E0)$. We further study the isotopic shift for the even-even $^{32-46}$Ar and $^{44-50}$Ti isotopes.

Figures

Figures reproduced from arXiv: 2502.04999 by the authors.

**Figure 2.** FIG. 2. (Color online) Empirical binding energy per nucleon [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. (Color online) Experimental [24] and calculated en [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. (Color online) (a) Experimental [7, 8] and calcu [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5. (Color online) Experimental [25] and calculated [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6. (Color online) Experimental [24] and calculated en [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

57 extracted references · 52 canonical work pages

[33]

Elliott and J.A

J.P. Elliott and J.A. Evans, Phys. Lett. B 101, 216 (1981)

work page 1981
[29]

Arima and F

A. Arima and F. Iachello, Phys. Rev. Lett. 35, 1069 (1975)

work page 1975
[1]

Goeppert-Mayer and J

M. Goeppert-Mayer and J. H. D. Jensen, Elementary the- ory of nuclear shell structure (Wiley, New York, 1955)

work page 1955
[2]

Exotic Nuclear Structure and Dy- namics

are not. Figure 2 shows the empirical binding energy per nu- cleon (BE/A) for even-even Ca, Ar, and Ti isotopes [42]. The binding energy per nucleon is rather flat for the Ca isotopes between doubly-magic 40Ca and 48Ca, and it 5 FIG. 3. (Color online) Experimental [24] and calculated en- ergies E of the yrast 2 + 1 and yrare 0 + 2 levels in the even-even ...

work page 1975
[3]

Ozawa, T

A. Ozawa, T. Kobayashi, T. Suzuki, K. Yoshida, and I. Tanihata, Phys. Rev. Lett. 84, 5493 (2000)

work page 2000
[4]

Otsuka, A

T. Otsuka, A. Gade, O. Sorlin, T. Suzuki, and Y. Utsuno, Rev. Mod. Phys. 92, 015002 (2020)

2020
[5]

Sorlin and M.-G

O. Sorlin and M.-G. Porquet, Prog. in Part. Nucl. Phys. 61, 602 (2008)

work page 2008
[6]

Otsuka, R

T. Otsuka, R. Fujimoto, Y. Utsuno, B. A. Brown, M. Honma, and T. Mizusaki, Phys. Rev. Lett. 87, 082502 (2001)

2001

Show all 57 references

[7]

A. J. Miller et al. , Nature Phys. 15, 432 (2019)

2019
[8]

Otsuka, T

T. Otsuka, T. Suzuki, R. Fujimoto, H. Grawe, and Y. Akaishi, Phys. Rev. Lett. 95, 232502 (2005)

2005
[9]

Iachello and A

F. Iachello and A. Arima, The interacting boson model (Cambridge University Press, 1987)

1987
[10]

R. F. Garcia Ruiz et al. , Nat. Phys. 12, 594 (2016)

2016
[11]

Zerguine, P

S. Zerguine, P. Van Isacker, and A. Bouldjedri, Phys. Rev. C 85, 034331 (2012)

2012
[12]

Zerguine, P

S. Zerguine, P. Van Isacker, A. Bouldjedri, and S. Heinze, Phys. Rev. Lett. 101, 022502 (2008)

2008
[13]

Nomura, R

K. Nomura, R. Rodr ´ ıguez-Guzm´ an, and L. M. Robledo, Phys. Rev. C 87, 064313 (2013)

2013
[14]

Van Isacker, Nucl

P. Van Isacker, Nucl. Data Sheets 120, 119 (2014). 9

2014
[15]

Mizusaki, T

T. Mizusaki, T. Otsuka, M. Honma, and B. A. Brown, Phys. Rev. C 63, 044306 (2001)

2001
[16]

Force et al

C. Force et al. , Phys. Rev. Lett. 105, 102501 (2010)

2010
[17]

https://www-phynu.cea.fr/science_en_ligne/ carte_potentiels_microscopiques/carte_potentiel_ nucleaire_eng.htm
[18]

Delaroche, M

J.-P. Delaroche, M. Girod, J. Libert, H. Goutte, S. Hi- laire, S. P´ eru, N. Pillet, and G. F. Bertsch, Phys. Rev. C 81, 014303 (2010)

2010
[19]

Maheshwari and K

B. Maheshwari and K. Nomura, J. Phys. G: Nucl. Part. Phys. 51, 095101 (2024)

2024
[20]

The measurements sup- port the doubly-magic nature of 40Ca and 48Ca, which have almost equal charge radii [8]

are now known over a long range, from proton-rich 36Ca [7] to neutron-rich 52Ca [8]. The measurements sup- port the doubly-magic nature of 40Ca and 48Ca, which have almost equal charge radii [8]. The local maximum of the charge radius of 44Ca and its very steep increase beyond...

2025 arXiv
[21]

Heyde and J.L

K. Heyde and J.L. Wood, Rev. Mod. Phys. 83, 1467 (2011)

2011
[22]

Garrett, M

P. Garrett, M. Zielinska, and E. Clement, Prog. Part. and Nucl. Phys. 124, 103931 (2022)

2022
[23]

Leoni, B

S. Leoni, B. Fornal, A. Bracco, Y. Tsunoda, and T. Ot- suka, Prog. Part. and Nucl. Phys. 139, 104119 (2024)

2024
[24]

Gorodetzky, N

S. Gorodetzky, N. Schulz, J. Chevallier, and A. C. Knip- per, J. Phys. (Paris) 27, 521 (1966)

1966
[25]

Schulz, Ann

N. Schulz, Ann. Phys. (Paris) 2, 81 (1967)

1967
[26]

nndc.bnl.gov/ensdf/

Evaluated Nuclear Structure Data File, https://www. nndc.bnl.gov/ensdf/
[27]

Kibedi, A

T. Kibedi, A. B. Garnsworthy, and J. L. Wood, Prog. Part. Nucl. Phys. 123, 103930 (2022)

2022
[28]

P. D. Duval and B. R. Barrett, Phys. Lett. B 100, 223 (1981)

1981
[30]

Otsuka, A

T. Otsuka, A. Arima, and F. Iachello, Nucl. Phys. A 309, 1 (1978)

1978
[31]

Van Isacker, IBM-1 code, unpublished

P. Van Isacker, IBM-1 code, unpublished
[32]

Elliott and A.P

J.P. Elliott and A.P. White, Phys. Lett. B97, 169 (1980)

1980
[34]

Lac, J.P

V.S. Lac, J.P. Elliott, J.A. Evans and G.L. Long, Nucl. Phys. A 587, 101 (1995)

1995
[35]

M. K. Harder, K. T. Tang, and P. Van Isacker, Phys. Lett. B 405, 25 (1997)

1997
[36]

Angeli, and K

I. Angeli, and K. P. Marinova, Atomic Data and Nuclear Data Tables 99, 69 (2013)

2013
[37]

Shimizu, T

N. Shimizu, T. Mizusaki, Y. Utsuno, and Y. Tsunoda, Comp. Phys. Comm. 244, 372 (2019)

2019
[38]

Utsuno, T

Y. Utsuno, T. Otsuka, B. A. Brown, M. Honma, T. Mizusaki, and N. Shimizu, Phys. Rev. C 86, 051301 (R) (2012)

2012
[39]

Poves, J

A. Poves, J. Sanchez-Solano, E. Caurier, and F. Nowacki, Nucl. Phys. A 694, 157 (2001)

2001
[40]

Speidel et al

K.-H. Speidel et al. , Phys. Rev. C 78, 017304 (2008)

2008
[41]

Blaum, W

K. Blaum, W. Geithner, J. Lassen, P. Lievens, K. Mari- nova, and R. Neugart, Nucl. Phys. A 799, 30 (2008)

2008
[42]

H. D. Wohlfahrt, E. B. Shera, M. V. Hoehn, Y. Ya- mazaki, and R. M. Steffen, Phys. Rev. C 23, 533 (1981)

1981
[43]

Anastassov, Yu

A. Anastassov, Yu. P. Gangrsky, K. P. Marinova, B. N. Markov, and S. G. Zemlyanoi, Z. Phys. D 30, 275 (1994)

1994
[44]

National Nuclear Data Center, https://www.nndc.bnl. gov/
[45]

https://www.chem.ualberta.ca/~massspec/atomic_ mass_abund.pdf
[46]

A. R. L. Kennington et al., Phys. Rev. C 103(3), 035805 (2021)

2021
[47]

Lalanne et al

L. Lalanne et al. , Phys. Rev. Lett. 129, 122501 (2022)

2022
[48]

Caurier, K

E. Caurier, K. Langanke, G. Mart ´ ınez-Pinedo, F. Nowacki, and P. Vogel, Phys. Lett. B 522, 240 (2001)

2001
[49]

Otsuka,Y

T. Otsuka,Y. Tsunoda, T. Abe, N. Shimizu, and P. Van Duppen, Phys. Rev. Lett. 123, 222502 (2019)

2019
[50]

R. A. Ricci and P. R. Maurenzig, Riv. Nuovo Cimento 1, 291 (1969)

1969
[51]

Retamosa, E

J. Retamosa, E. Caurier, F. Nowacki, and A. Poves, Phys. Rev. C 55, 1266 (1997)

1997
[52]

Bhuyan et al

M. Bhuyan et al. , J. Phys. G: Nucl. Part. Phys. 48, 075105 (2021)

2021
[53]

Goriely, F

S. Goriely, F. Tondeur, and J. M. Pearson, Atomic Data and Nuclear Data Tables 77, 311 (2001)

2001
[54]

K. H. Souw, J. C. Adloff, D. Disdier, and P. Chevallier, Phys. Rev. C 12, 1103 (1975)

1975
[55]

Ernst et al

R. Ernst et al. , Phys. Rev. Lett. 84, 416 (2000)

2000
[56]

Adhikari et al

D. Adhikari et al. (CREX), Phys. Rev. Lett. 129, 042501 (2022)

2022
[57]

Adhikari et al

D. Adhikari et al. (PREX), Phys. Rev. Lett. 126, 172502 (2021)

2021

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [33]

Elliott and J.A

J.P. Elliott and J.A. Evans, Phys. Lett. B 101, 216 (1981)

work page 1981

[2] [29]

Arima and F

A. Arima and F. Iachello, Phys. Rev. Lett. 35, 1069 (1975)

work page 1975

[3] [1]

Goeppert-Mayer and J

M. Goeppert-Mayer and J. H. D. Jensen, Elementary the- ory of nuclear shell structure (Wiley, New York, 1955)

work page 1955

[4] [2]

Exotic Nuclear Structure and Dy- namics

are not. Figure 2 shows the empirical binding energy per nu- cleon (BE/A) for even-even Ca, Ar, and Ti isotopes [42]. The binding energy per nucleon is rather flat for the Ca isotopes between doubly-magic 40Ca and 48Ca, and it 5 FIG. 3. (Color online) Experimental [24] and calculated en- ergies E of the yrast 2 + 1 and yrare 0 + 2 levels in the even-even ...

work page 1975

[5] [3]

Ozawa, T

A. Ozawa, T. Kobayashi, T. Suzuki, K. Yoshida, and I. Tanihata, Phys. Rev. Lett. 84, 5493 (2000)

work page 2000

[6] [4]

Otsuka, A

T. Otsuka, A. Gade, O. Sorlin, T. Suzuki, and Y. Utsuno, Rev. Mod. Phys. 92, 015002 (2020)

2020

[7] [5]

Sorlin and M.-G

O. Sorlin and M.-G. Porquet, Prog. in Part. Nucl. Phys. 61, 602 (2008)

work page 2008

[8] [6]

Otsuka, R

T. Otsuka, R. Fujimoto, Y. Utsuno, B. A. Brown, M. Honma, and T. Mizusaki, Phys. Rev. Lett. 87, 082502 (2001)

2001