Pith. sign in

REVIEW 3 major objections 6 minor 55 references

Charge radii of calcium isotopes within relativistic configuration-interaction density functional theory

T0 review · 3 major / 6 minor · reviewed 2026-07-10 · grok-4.5

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

Pith's one-line read Beyond-mean-field correlations in ReCD soften calcium potential energy surfaces, shift them to deformed shapes matching B(E2) data, and improve charge radii including the 40-48 equality, large 52Ca, and odd-even staggering.

desk verdict Solid ReCD application to Ca radii: soft deformed PESs and better β are real; the plotted radii drop the mixing sum in Eq. (12), so the OES claim is only partly rigorous. read the letter →

arxiv 2607.08478 v1 pith:AECBK3HL submitted 2026-07-09 nucl-th

Teng Qu , Bo Li , Yakun Wang This is my paper · ORCID
classification nucl-th
keywords chargeradiicalciumisotopesbeyond-mean-fieldcorrelationsReCDtheoryquadrupoledeformationodd-evenstaggeringpotentialenergysurfacesangular-momentumprojection
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

Calcium charge radii show puzzling patterns: nearly equal sizes for 40Ca and 48Ca, an unexpectedly large 52Ca, and strong odd-even staggering. Mean-field calculations leave the nuclei nearly spherical and understate several of those radii. This paper applies relativistic configuration-interaction density functional (ReCD) theory, which restores rotational symmetry and mixes quasiparticle configurations, treating even-even and odd-A isotopes alike. The correlations soften the potential energy surfaces and move the minima from spherical mean-field solutions to deformed shapes whose quadrupole parameters match experimental B(E2) values far better. Because charge radii grow with deformation, the ReCD radii are larger and reproduce the 40-48 equality, the large 52Ca radius, and an improved staggering pattern, especially the enhanced sizes of 42Ca and 44Ca. Secondary minima appear for some odd-A isotopes, and the authors note that fuller shape mixing may still be needed for residual staggering features.

What carries the argument

Relativistic configuration-interaction density functional (ReCD) theory: many-body wave functions are angular-momentum-projected superpositions of quasiparticle configurations built on triaxial relativistic Hartree-Bogoliubov states (PC-PK1 functional plus separable pairing), with even-even and odd-A nuclei treated on the same footing; charge radii are evaluated from the resulting projected densities.

What would settle it

Compute charge radii from the complete ReCD wave functions after explicit particle-number projection (or an equivalent number-conserving treatment) and check whether the improvement over mean-field radii, the 40-48 equality, the large 52Ca size, and the odd-even staggering survive; or measure or re-extract beta for 42Ca and 44Ca and test whether the ReCD minima still match.

Watch

Extended reading notes

Core claim

Beyond-mean-field correlations included in ReCD theory (angular-momentum projection plus quasiparticle configuration mixing) significantly soften the potential energy surfaces of calcium isotopes, shift their energy minima from nearly spherical mean-field shapes to deformed configurations whose beta values agree with B(E2) data, and thereby produce charge radii that are generally larger than mean-field results and that reproduce the near equality of 40Ca and 48Ca, the large 52Ca radius, and improved odd-even staggering, particularly for 42Ca and 44Ca.

Load-bearing premise

Charge radii are computed after dropping the sum over excited quasiparticle configurations because particle-number expectation values drift from Z=20 once mixing is included; the reported radii therefore rest only on projected reference configurations rather than the full mixed wave functions.

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.

Referee Report

3 major / 6 minor

Summary. The manuscript applies relativistic configuration-interaction density functional (ReCD) theory to the charge radii of 39–52Ca. Beyond-mean-field correlations are included via three-dimensional angular-momentum projection and quasiparticle configuration mixing on top of triaxial relativistic Hartree–Bogoliubov solutions with PC-PK1 and a finite-range separable pairing force. The authors report that these correlations soften the potential energy surfaces and move the energy minima from nearly spherical mean-field shapes to deformed (often triaxial) configurations, with quadrupole deformations β in substantially better agreement with B(E2)-based data than TRHB. Because charge radii are sensitive to deformation, the ReCD radii are generally larger than mean-field values and reproduce the near equality of the 40Ca and 48Ca radii, the large 52Ca radius, and an improved odd–even staggering, especially the enhanced radii of 42Ca and 44Ca. Secondary local minima on the PESs of several odd-A isotopes are noted, and incomplete shape mixing is suggested as a possible remaining source of the residual OES discrepancy.

Significance. If the deformation-driven mechanism is robust, the work provides a microscopic, parameter-free (with respect to Ca radii) account of several long-standing anomalies in the calcium charge-radius chain within a single framework that treats even-even and odd-A nuclei on the same footing. That is a clear advance over mean-field DFT and over prior beyond-mean-field studies restricted to even-even systems. The PES evolution (Figs. 1–2) and the β comparison to experiment (Fig. 3) are concrete, falsifiable results that support the physical picture. The paper is also explicit about remaining limitations (particle-number restoration, shape mixing), which is a strength. The main scientific value therefore hinges on whether the reported radii are faithful expectation values of the ReCD ground states or only of projected reference configurations at the ReCD minima.

major comments (3)
  1. Sec. IV, after Eq. (12): The charge radii plotted as “ReCD” are not expectation values of the full mixed wave functions of Eq. (3). Because ⟨ΨI|N̂p|ΨI⟩ deviates from Z=20 once configuration mixing is included, the authors omit the sum over excited quasiparticle configurations κ in Eq. (12) and evaluate radii from projected reference configurations only. The central claim that configuration mixing improves the radii (and the OES for 42,44Ca) therefore rests on an uncontrolled approximation. The manuscript should either (i) quantify the size of the omitted κ contributions (e.g., by comparing radii with and without the sum at fixed particle-number constraint, or by a limited particle-number projection), or (ii) rephrase the radius results as “projected reference configurations at ReCD minima” and clearly separate the deformation-driven effect from true configuration-mixing contributions to
  2. Sec. IV and Fig. 4: Related to the above, the residual failure at 43Ca and the partial OES description are attributed to missing shape mixing between global and secondary minima (Fig. 2). That interpretation is plausible and honestly stated, but it also means the present calculation does not yet fully test the ReCD framework for the most distinctive feature of the Ca chain (the pronounced OES). At minimum, the paper should report the energy differences between the global and secondary minima for 41,43,47Ca and the corresponding radius differences (already partly shown as open triangles in Fig. 4) in a table, so that the reader can judge how much room shape mixing has to correct the OES. A short estimate of the effect of mixing two nearby minima would strengthen the outlook claim.
  3. Sec. III: The configuration space is restricted to 0/2qp (even-even) and 1qp (odd-A) with Ecut=3.5 MeV, on the grounds that 3/4qp contributions to ground-state radii are “expected to be negligible.” For soft PESs and for odd-A nuclei with nearby secondary minima, that expectation is not obvious. A brief convergence check—e.g., radii and β for 42,43,44Ca with a higher Ecut or with selected 3qp/4qp configurations—would make the truncation claim load-bearing rather than asserted.
minor comments (6)
  1. Eq. (1) and the text after Eq. (12): The constant 0.64 fm² is standard for the proton finite size, but the paper should state whether neutron finite-size or spin-orbit contributions to the charge radius are neglected, for comparison with other recent RDFT work on Ca radii.
  2. Fig. 1 caption and panels: The energy scale and contour spacing are not stated in the caption; adding them would make the “softening” claim easier to read quantitatively.
  3. Fig. 3: Experimental β values are taken from B(E2) data [55]. A short remark that these are effective, model-dependent deformations (and not static mean-field β) would help non-specialist readers understand why mean-field β=0 is not a fair comparison target.
  4. Sec. II, Eqs. (4)–(5): The configuration lists are clear, but the text could note explicitly that for odd-A the blocked orbital ν0 is the lowest quasineutron at each (β,γ), and whether reordering of blocked orbitals is checked near the secondary minima.
  5. References: The recent RDFT+5DCH study of even-even Ca radii (Ref. [36]) is cited; a one-sentence comparison of the present β and rc trends with that work for 40–48Ca would place the ReCD results in context.
  6. Typographical: “DET AILS” and “RESUL TS” in section headings appear to contain stray spaces; “coefficients” uses a non-ASCII ligature that may not render uniformly.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: fixed external functional/pairing, ReCD method applied to external Ca radius and B(E2) benchmarks without refitting or definitional forcing of the target observables.

full rationale

The derivation chain is a standard application of an established beyond-mean-field method (angular-momentum projection + quasiparticle configuration mixing on TRHB states from PC-PK1 + separable pairing G=728 MeV fm^{3}). Parameters are taken from the literature and not adjusted to the calcium charge-radius data or OES that the paper claims to describe. Equilibrium deformations and radii are computed and then compared to independent experimental B(E2)-derived β values and measured charge radii; the improvements relative to pure TRHB are therefore not forced by construction or by a fit. Self-citations are confined to the prior definition and validation of the ReCD framework itself and do not supply a uniqueness theorem or ansatz that dictates the Ca results. The acknowledged approximation of dropping the κ sum when evaluating radii (to avoid particle-number contamination) is a technical limitation of the present implementation, not a circular reduction of the prediction to its inputs. Consequently the central claims remain independently testable against external data.

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

The central claim rests on the standard PC-PK1 + separable pairing mean-field, the ReCD projection/mixing machinery, a hand-chosen configuration cutoff, and an ad-hoc truncation of the radius expectation value that drops configuration mixing. No new particles or forces are invented; free parameters are mostly inherited or truncation choices rather than fits to the Ca radii themselves.

free parameters (4)
  • qp excitation-energy cutoff Ecut = 3.5 MeV
    Truncates configuration space; set to 3.5 MeV by hand with a convergence claim but not derived from a uniqueness principle.
  • harmonic-oscillator major shells = 10
    Basis size for TRHB; fixed at 10 shells without systematic continuum/basis-error quantification for radii.
  • separable pairing strength G = 728 MeV·fm³
    Taken as the standard value from Tian et al.; not refitted here but still a free parameter of the model that affects deformation and radii.
  • PC-PK1 functional parameters = as in Zhao et al. (2010)
    Global EDF parameters fixed in prior work; inherited inputs that control mean-field sphericity and radii before beyond-mean-field corrections.
assumptions (5)
  • domain assumption ReCD many-body states from AMP + qp configuration mixing on TRHB vacua adequately capture rotational and vibrational zero-point correlations relevant to ground-state charge radii.
    Invoked throughout Sec. II–IV as the justification that ReCD is the right beyond-mean-field tool for radii and OES.
  • ad hoc to paper Ground-state radii receive negligible contributions from three- and four-quasiparticle configurations, so only 0/2qp (even-even) and 1qp (odd-A) spaces are needed.
    Stated in Sec. III as an expectation, not demonstrated with a controlled enlargement of the space for radii.
  • domain assumption Charge radius formula rc = sqrt(<r_p^2> + 0.64 fm) with finite proton size 0.64 is adequate.
    Eq. (1); standard nuclear-structure practice.
  • ad hoc to paper Lagrange multipliers −λ_p(N_p−Z)−λ_n(N_n−N) sufficiently restore mean particle numbers for energy, and omitting κ in the radius kernel is an acceptable substitute for full particle-number projection.
    Sec. II Eq. (8) and Sec. IV discussion after Eq. (12); the second part is paper-specific and load-bearing for reported radii.
  • domain assumption Experimental β extracted from B(E2) can be compared directly to PES-minimum β from ReCD as a reliability check of the surfaces.
    Fig. 3 and surrounding text; common but model-dependent comparison.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Charge radii of calcium isotopes within relativistic configuration-interaction density functional theory." pith.science (2026). https://pith.science/paper/AECBK3HL

@misc{pith2026260708478,
  author       = {Pith},
  title        = {Pith review of: Charge radii of calcium isotopes within relativistic configuration-interaction density functional theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AECBK3HL}},
  note         = {Machine review of arXiv:2607.08478}
}
abstract

The charge radii of calcium isotopes are investigated within the framework of relativistic configuration-interaction density functional (ReCD) theory. The ReCD theory microscopically incorporates beyond-mean-field correlations through rotational symmetry restoration and configuration mixing among quasiparticle excited states, and treats even-even and odd-$A$ isotopes on the same footing. It is found that beyond-mean-field correlations significantly soften the potential energy surfaces of calcium isotopes and shift the energy minima from nearly spherical mean-field solutions to deformed shapes. The quadrupole deformation parameters predicted by the ReCD theory show much better agreement with the available experimental data than the mean-field results, supporting the reliability of the calculated potential energy surfaces and highlighting the important role of beyond-mean-field correlations. Owing to the sensitive dependence of charge radii on nuclear deformation, the charge radii obtained within the ReCD framework are generally larger than the mean-field predictions. The nearly identical charge radii of $^{40}\mathrm{Ca}$ and $^{48}\mathrm{Ca}$, as well as the unexpectedly large charge radius of $^{52}\mathrm{Ca}$, are well reproduced. Compared with the mean-field calculations, the description of the odd-even staggering is improved, especially for the enhanced charge radii of $^{42}\mathrm{Ca}$ and $^{44}\mathrm{Ca}$. It is also worth noting that secondary local minima appear in the ReCD-based potential energy surfaces of the odd-$A$ calcium isotopes $^{41,43,47}\mathrm{Ca}$. The present results suggest that shape mixing between different local minima, which is not fully included in the present calculation, may further improve the description of the pronounced odd-even staggering observed in calcium isotopes.

Figures

Figures reproduced from arXiv: 2607.08478 by the authors.

Figure 1
Figure 1. (b). Such a soft PES implies that additional correlation effects can further modify the equilibrium shape. Indeed, after including additional correlations induced by qp config￾uration mixing within the full ReCD framework, the minimum of the PES is further shifted to a larger deformation, located at (β, γ) = (0.21, 25◦ ), as shown in [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages

  1. [1]

    Kreim, M

    K. Kreim, M. L. Bissell, J. Papuga, K. Blaum, M. De Rydt, R. F. Garcia Ruiz, S. Goriely, H. Heylen, M. Kowalska, R. Neugart, G. Neyens, W. Nörtershäuser, M. M. Rajabali, R. Sánchez Alarcón, H. H. Stroke, and D. T. Yordanov, Nuclear charge radii of potassium isotopes beyond N = 28, Phys. Lett. B 731, 97 (2014)

    work page 2014

  2. [2]

    Angeli and K

    I. Angeli and K. P. Marinova, Correlations of nuclear charge radii with other nuclear observ- ables, J. Phys. G: Nucl. Part. Phys. 42, 055108 (2015) . 13

    work page 2015

  3. [3]

    Gorges, L

    C. Gorges, L. V. Rodríguez, D. L. Balabanski, M. L. Bissell, K. Blaum, B. Cheal, R. F. Garcia Ruiz, G. Georgiev, W. Gins, H. Heylen, A. Kanellakopoulos, S. Kaufmann, M. Kowal- ska, V. Lagaki, S. Lechner, B. Maaß, S. Malbrunot-Ettenauer, W. Nazarewicz, R. Neugart, G. Neyens, W. Nörtershäuser, P.-G. Reinhard, S. Sailer, R. Sánchez, S. Schmidt, L. Wehner, C....

    work page 2019

  4. [4]

    B. A. Marsh, T. Day Goodacre, S. Sels, Y. Tsunoda, B. Andel, A. N. Andreyev, N. A. Al- thubiti, D. Atanasov, A. E. Barzakh, J. Billowes, K. Blaum, T. E. Cocolios, J. G. Cubiss, J. Dobaczewski, G. J. Farooq-Smith, D. V. Fedorov, V. N. Fedosseev, K. T. Flanagan, L. P. Gaffney, L. Ghys, M. Huyse, S. Kreim, D. Lunney, K. M. Lynch, V. Manea, Y. Martinez Palenz...

    work page 2018

  5. [5]

    Barzakh, A

    A. Barzakh, A. N. Andreyev, C. Raison, J. G. Cubiss, P. Van Duppen, S. Péru, S. Hilaire, S. Goriely, B. Andel, S. Antalic, M. Al Monthery, J. C. Berengut, J. Bieroń, M. L. Bissell, A. Borschevsky, K. Chrysalidis, T. E. Cocolios, T. Day Goodacre, J.-P. Dognon, M. Elan- tkowska, E. Eliav, G. J. Farooq-Smith, D. V. Fedorov, V. N. Fedosseev, L. P. Gaffney, R....

    work page 2021

  6. [6]

    X. F. Yang, C. Wraith, L. Xie, C. Babcock, J. Billowes, M. L. Bissell, K. Blaum, B. Cheal, K. T. Flanagan, R. F. Garcia Ruiz, W. Gins, C. Gorges, L. K. Grob, H. Heylen, S. Kaufmann, M. Kowalska, J. Kraemer, S. Malbrunot-Ettenauer, R. Neugart, G. Neyens, W. Nörtershäuser, J. Papuga, R. Sánchez, and D. T. Yordanov, Isomer shift and magnetic moment of the lo...

    work page 2016

  7. [7]

    Hagen, A

    G. Hagen, A. Ekström, C. Forssén, G. R. Jansen, W. Nazarewicz, T. Papenbrock, K. A. Wendt, S. Bacca, N. Barnea, B. Carlsson, C. Drischler, K. Hebeler, M. Hjorth-Jensen, M. Miorelli, G. Orlandini, A. Schwenk, and J. Simonis, Neutron and weak-charge distributions of the 48Ca nucleus, Nat. Phys. 12, 186 (2016)

    work page 2016

  8. [8]

    Yang and J

    J. Yang and J. Piekarewicz, Difference in proton radii of mirror nuclei as a possible surrogate for the neutron skin, Phys. Rev. C 97, 014314 (2018)

    work page 2018

Show all 55 references
  1. [9]

    M. K. Gaidarov, I. Moumene, A. N. Antonov, D. N. Kadrev, P. Sarriguren, and E. Moya de Guerra, Proton and neutron skins and symmetry energy of mirror nuclei, Nucl. Phys. A 1004, 122061 (2020)

    2020

  2. [10]

    Nörtershäuser, D

    W. Nörtershäuser, D. Tiedemann, M. Žáková, Z. Andjelkovic, K. Blaum, M. L. Bissell, R. Cazan, G. W. F. Drake, Ch. Geppert, M. Kowalska, J. Krämer, A. Krieger, R. Neugart, R. Sánchez, F. Schmidt-Kaler, Z.-C. Yan, D. T. Yordanov, and C. Zimmermann, Nuclear charge radii of 7,9,10...

    2009

  3. [11]

    Reinhard and W

    P.-G. Reinhard and W. Nazarewicz, Nuclear charge and neutron radii and nuclear matter: Trend analysis in Skyrme density-functional-theory approach, Phys. Rev. C 93, 051303 (2016)

    2016

  4. [12]

    B. A. Brown, Mirror charge radii and the neutron equation of state, Phys. Rev. Lett. 119, 122502 (2017)

    2017

  5. [13]

    S. V. Pineda, K. König, D. M. Rossi, B. A. Brown, A. Incorvati, J. Lantis, K. Minamisono, W. Nörtershäuser, J. Piekarewicz, R. Powel, and F. Sommer, Charge radius of neutron- deficient 54Ni and symmetry energy constraints using the difference in mirror pair charge radii, Phys....

    2021

  6. [14]

    König, J

    K. König, J. C. Berengut, A. Borschevsky, A. Brinson, B. A. Brown, A. Dockery, S. Elhatisari, E. Eliav, R. F. G. Ruiz, J. D. Holt, B.-S. Hu, J. Karthein, D. Lee, Y.-Z. Ma, U.-G. Meißner, K. Minamisono, A. V. Oleynichenko, S. V. Pineda, S. D. Prosnyak, M. L. Reitsma, L. V. Skri...

    2024

  7. [15]

    De Vries, C

    H. De Vries, C. De Jager, and C. De Vries, Nuclear charge-density-distribution parameters from elastic electron scattering, At. Data Nucl. Data Tables 36, 495 (1987) . 15

    1987

  8. [16]

    Antwis, S

    L. Antwis, S. Bara, C. Bruhn, T. E. Cocolios, M. Deseyn, A. Doinaki, C. E. Düllmann, J. Fletcher, M. Heines, R. Heller, P. Indelicato, U. Kentsch, T. Kieck, K. Kirch, A. Knecht, E. A. Maugeri, M. Niikura, A. Ouf, L. M. C. Pereira, W. W. M. M. Phyo, R. Pohl, D. Renisch, N. Ritj...

    2025

  9. [17]

    X. F. Yang, S. J. Wang, S. G. Wilkins, and R. F. G. Ruiz, Laser spectroscopy for the study of exotic nuclei, Prog. Part. Nucl. Phys. 129, 104005 (2023)

    2023

  10. [18]

    Angeli and K

    I. Angeli and K. P. Marinova, Table of experimental nuclear ground state charge radii: An update, At. Data Nucl. Data Tables 99, 69 (2013)

    2013

  11. [19]

    Mårtensson-Pendrill, A

    A.-M. Mårtensson-Pendrill, A. Ynnerman, H. Warston, L. Vermeeren, R. E. Silverans, A. Klein, R. Neugart, C. Schulz, P. Lievens, and The ISOLDE Collaboration, Isotope shifts and nuclear-charge radii in singly ionized 40−48Ca, Phys. Rev. A 45, 4675 (1992)

    1992

  12. [20]

    Vermeeren, R

    L. Vermeeren, R. E. Silverans, P. Lievens, A. Klein, R. Neugart, Ch. Schulz, and F. Buchinger, Ultrasensitive radioactive detection of collinear-laser optical pumping: Measurement of the nuclear charge radius of 50Ca, Phys. Rev. Lett. 68, 1679 (1992)

    1992

  13. [21]

    Vermeeren, P

    L. Vermeeren, P. Lievens, R. E. Silverans, U. Georg, M. Keim, A. Klein, R. Neugart, M. Neu- roth, F. Buchinger, and the ISOLDE Collaboration, The mean square nuclear charge radius of 39Ca, J. Phys. G: Nucl. Part. Phys. 22, 1517 (1996)

    1996

  14. [22]

    R. F. Garcia Ruiz, M. L. Bissell, K. Blaum, A. Ekström, N. Frömmgen, G. Hagen, M. Hammen, K. Hebeler, J. D. Holt, G. R. Jansen, M. Kowalska, K. Kreim, W. Nazarewicz, R. Neugart, G. Neyens, W. Nörtershäuser, T. Papenbrock, J. Papuga, A. Schwenk, J. Simonis, K. A. Wendt, and D. ...

    2016

  15. [23]

    A. J. Miller, K. Minamisono, A. Klose, D. Garand, C. Kujawa, J. D. Lantis, Y. Liu, B. Maaß, P. F. Mantica, W. Nazarewicz, W. Nörtershäuser, S. V. Pineda, P.-G. Reinhard, D. M. Rossi, F. Sommer, C. Sumithrarachchi, A. Teigelhöfer, and J. Watkins, Proton superfluidity and charge...

    2019

  16. [24]

    Wienholtz, D

    F. Wienholtz, D. Beck, K. Blaum, Ch. Borgmann, M. Breitenfeldt, R. B. Cakirli, S. George, F. Herfurth, J. D. Holt, M. Kowalska, S. Kreim, D. Lunney, V. Manea, J. Menéndez, D. Nei- dherr, M. Rosenbusch, L. Schweikhard, A. Schwenk, J. Simonis, J. Stanja, R. N. Wolf, and 16 K. Zu...

    2013

  17. [25]

    Wang and T

    N. Wang and T. Li, Shell and isospin effects in nuclear charge radii, Phys. Rev. C 88, 011301 (2013)

    2013

  18. [26]

    Caurier, K

    E. Caurier, K. Langanke, G. Martínez-Pinedo, F. Nowacki, and P. Vogel, Shell model descrip- tion of isotope shifts in calcium, Phys. Lett. B 522, 240 (2001)

    2001

  19. [27]

    Nakada, Further evidence for three-nucleon spin-orbit interaction in isotope shifts of nuclei with magic proton numbers, Phys

    H. Nakada, Further evidence for three-nucleon spin-orbit interaction in isotope shifts of nuclei with magic proton numbers, Phys. Rev. C 92, 044307 (2015)

    2015

  20. [28]

    V. Somà, P. Navrátil, F. Raimondi, C. Barbieri, and T. Duguet, Novel chiral Hamiltonian and observables in light and medium-mass nuclei, Phys. Rev. C 101, 014318 (2020)

    2020

  21. [29]

    Heinz, T

    M. Heinz, T. Miyagi, S. R. Stroberg, A. Tichai, K. Hebeler, and A. Schwenk, Improved structure of calcium isotopes from ab initio calculations, Phys. Rev. C 111, 034311 (2025)

    2025

  22. [30]

    Reinhard and W

    P.-G. Reinhard and W. Nazarewicz, Toward a global description of nuclear charge radii: Ex- ploring the Fayans energy density functional, Phys. Rev. C 95, 064328 (2017)

    2017

  23. [31]

    U. C. Perera, A. V. Afanasjev, and P. Ring, Charge radii in covariant density functional theory: A global view, Phys. Rev. C 104, 064313 (2021)

    2021

  24. [32]

    An, L.-S

    R. An, L.-S. Geng, and S.-S. Zhang, Novel ansatz for charge radii in density functional theories, Phys. Rev. C 102, 024307 (2020)

    2020

  25. [33]

    R. An, X. Jiang, N. Tang, L.-G. Cao, and F.-S. Zhang, Improved description of nuclear charge radii: Global trends beyond N = 28 shell closure, Phys. Rev. C 109, 064302 (2024)

    2024

  26. [34]

    Bohr and B

    A. Bohr and B. R. Mottelson, Nuclear Structure, Vol. II (Benjamin, New York, 1975)

    1975

  27. [35]

    Ideguchi, D

    E. Ideguchi, D. G. Sarantites, W. Reviol, A. V. Afanasjev, M. Devlin, C. Baktash, R. V. F. Janssens, D. Rudolph, A. Axelsson, M. P. Carpenter, A. Galindo-Uribarri, D. R. LaFosse, T. Lauritsen, F. Lerma, C. J. Lister, P. Reiter, D. Seweryniak, M. Weiszflog, and J. N. Wilson, Su...

    2001

  28. [36]

    H. H. Xie, J. Li, Y. L. Yang, and P. W. Zhao, Charge radii of calcium isotopes within relativistic density functional theory: The finite size of the nucleon and quadrupole shape-fluctuation effects, Phys. Rev. C 112, L021303 (2025)

    2025

  29. [37]

    P. W. Zhao, P. Ring, and J. Meng, Configuration interaction in symmetry-conserving covariant density functional theory, Phys. Rev. C 94, 041301 (2016) . 17

    2016

  30. [38]

    Y. K. Wang, P. W. Zhao, and J. Meng, Configuration-interaction projected density functional theory: Effects of four-quasiparticle configurations and time-odd interactions, Phys. Rev. C 105, 054311 (2022)

    2022

  31. [39]

    Y. Wang, P. Zhao, and J. Meng, Relativistic configuration-interaction density functional the- ory: Nonaxial effects on nuclear ββ decay, Sci. Bull. 69, 2017 (2024)

    2017

  32. [40]

    Y. Wang, P. Zhao, and J. Meng, Correlation between neutrinoless double- β decay and double Gamow-Teller transitions, Phys. Lett. B 855, 138796 (2024)

    2024

  33. [41]

    Y. Wang, P. Zhao, and J. Meng, Nuclear chiral rotation within relativistic configuration- interaction density functional theory, Phys. Lett. B 848, 138346 (2024)

    2024

  34. [42]

    T. Qu, Y. K. Wang, and P. W. Zhao, Relativistic configuration-interaction density functional theory for nuclear wobbling motion, Phys. Rev. C 111, 064309 (2025)

    2025

  35. [43]

    J. Meng, H. Toki, S. Zhou, S. Zhang, W. Long, and L. Geng, Relativistic continuum Hartree Bogoliubov theory for ground-state properties of exotic nuclei, Prog. Part. Nucl. Phys. 57, 470 (2006)

    2006

  36. [44]

    Nikšić, N

    T. Nikšić, N. Paar, D. Vretenar, and P. Ring, DIRHB—a relativistic self-consistent mean-field framework for atomic nuclei, Comput. Phys. Commun. 185, 1808 (2014)

    2014

  37. [45]

    Ring and P

    P. Ring and P. Schuck, The nuclear many-body problem (Springer Science & Business Media, 2004)

    2004

  38. [46]

    Meng, ed., Relativistic Density Functional for Nuclear Structure , International Review of Nuclear Physics, Vol

    J. Meng, ed., Relativistic Density Functional for Nuclear Structure , International Review of Nuclear Physics, Vol. 10 (World Scientific, Singapore, 2016) pp. 1–699

    2016

  39. [47]

    B. G. Carlsson and J. Rotureau, New and practical formulation for overlaps of bogoliubov vacua, Phys. Rev. Lett. 126, 172501 (2021)

    2021

  40. [48]

    Q. L. Hu, Z. C. Gao, and Y. Chen, Matrix elements of one-body and two-body operators between arbitrary HFB multi-quasiparticle states, Phys. Lett. B 734, 162 (2014)

    2014

  41. [49]

    Bonche, J

    P. Bonche, J. Dobaczewski, H. Flocard, P.-H. Heenen, and J. Meyer, Analysis of the generator coordinate method in a study of shape isomerism in 194Hg, Nucl. Phys. A 510, 466 (1990)

    1990

  42. [50]

    K. Hara, A. Hayashi, and P. Ring, Exact angular momentum projection of cranked Hartree- Fock-Bogoliubov wave functions, Nucl. Phys. A 385, 14 (1982)

    1982

  43. [51]

    Y. Tian, Z. Y. Ma, and P. Ring, A finite range pairing force for density functional theory in superfluid nuclei, Phys. Lett. B 676, 44 (2009) . 18

    2009

  44. [52]

    P. W. Zhao, Z. P. Li, J. M. Yao, and J. Meng, New parametrization for the nuclear covariant energy density functional with a point-coupling interaction, Phys. Rev. C 82, 054319 (2010)

    2010

  45. [53]

    http://nuclearmap.jcnp.org/

  46. [54]

    Y. L. Yang, Y. K. Wang, P. W. Zhao, and Z. P. Li, Nuclear landscape in a mapped collective Hamiltonian from covariant density functional theory, Phys. Rev. C 104, 054312 (2021)

    2021

  47. [55]

    National Nuclear Data Center, Nudat 3: Interactive chart of nuclides , Brookhaven National Laboratory (2023), accessed: 2024-07-10. 19

    2023

Pith tools

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