Gamow-Teller strength in 48 Ca and 78 Ni with the charge-exchange subtracted second random-phase approximation
GGamow-Teller strength in Ca and Ni with the charge-exchange subtracted secondrandom-phase approximation
D. Gambacurta, M. Grasso, and J. Engel INFN-LNS, Laboratori Nazionali del Sud, 95123 Catania, Italy Universit´e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France Department of Physics and Astronomy, CB 3255,University of Noth Carolina, Chapel Hill, North Carolina 27599-3255, USA
We develop a fully self-consistent subtracted second random-phase approximation for charge-exchange processes with Skyrme energy-density functionals. As a first application, we study Gamow-Teller excitations in the doubly-magic nucleus Ca, the lightest double- β emitter that could be usedin an experiment, and in Ni, the single-beta-decay rate of which is known. The amount of Gamow-Teller strength below 20 or 30 MeV is considerably smaller than in other energy-density-functionalcalculations and agrees better with experiment in Ca, as does the beta-decay rate in Ni. Theseimportant results, obtained without ad hoc quenching factors, are due to the presence of two-particle– two-hole configurations. Their density progressively increases with excitation energy, leading toa long high-energy tail in the spectrum, a fact that may have implications for the computation ofnuclear matrix elements for neutrinoless double- β decay in the same framework. Charge-exchange (CE) excitations [1, 2] such as theGamow-Teller (GT) resonance are closely linked to elec-tron capture and β decay, which play important rolesin nuclear astrophysics [3, 4]. They also aid the con-struction of nuclear effective interactions, for which theyconstrain couplings in the spin-isospin channel. Finally,they are relevant to the nuclear physics that affects neu-trinoless double- β (0 νββ ) decay, in which two neutronschange into two protons [5–7]. The experimental obser-vation of this rare process would be a breakthrough forfundamental physics; it would mean that neutrinos areMajorana particles and would imply new phenomena be-yond the Standard Model that could be related to thematter-antimatter asymmetry in the universe. The rateof 0 νββ decay depends on nuclear matrix elements thatmay only be determined theoretically. At present, predic-tions for these matrix elements differ from one model tothe next by factors of two or three, an amount that is toolarge to allow the efficient planning and interpretation ofexperiments.Any model that hopes to describe double- β decay mustbe able to predict the distribution of GT strength be-cause the GT operator, multiplied by the axial-vectorcoupling constant g A , is the leading contribution to theoperator that governs β decay. Recent work [8–10] thatcorrelates calculations of double-GT and 0 νββ matrixelements even suggests that one could deduce the lat-ter from double-CE experiments [11, 12]. Most double– β emitters that could be used in experiments are stilltoo complex to easily treat from first principles with ab-initio methods (though recently, such methods were ap-plied to Ca [13–15], and to Ge and Se [15]). Morephenomenological approaches are therefore still impor-tant, even necessary [16–26]. However, these theoreticalschemes do not correctly describe the available data forGT excitations and β -decay half-lives and must resort to ad hoc “quenching factors” to obtain reasonable resultsfor GT strength below 20 or 30 MeV of excitation energy.In Ref. [27], for example, models based on the random- phase approximation (RPA) overestimate the strengthsignificantly. This kind of over-prediction is usually as-cribed to missing physics, for example the ∆ excitation[28] or complex configurations such as two-particle – two-hole (2p2h) excitations [29–31]. The results in Ref. [27]indicate that higher-order correlations are needed beyondthose in the RPA, which is essentially a time-dependentversion of mean-field theory. Ab-initio work with operators and currents from chiraleffective field theory has recently had some success in ex-plaining the quenching in β decay. Reference [32] showedthat correlations omitted from the shell model and frommean-field-based calculations, together with two-bodyweak currents, account for most of that quenching. Ref-erence [33] showed that the same effects quench the inte-grated β strength function. But weak two-body currentsplay no obvious role in charge-exchange transitions, andso the implications of this last result for our work are notclear. Similarly, starting from realistic potentials, theauthors of Ref. [34] used many-body perturbation theoryto derive effective shell-model operators that implicitlyinclude correlations from outside shell-model spaces, ob-taining the correct quenching of GT strength in severalnuclei of interest for double- β experiments, but failing todo so in the lightest of these, Ca. They also had diffi-culty in that nucleus with two-neutrino double- β decay,a very closely related process.Finally, EDF-based models that go beyond mean-fieldtheory have been proposed for CE excitations, for exam-ple in both relativistic (see Refs. [35, 36] for the most re-cent developments) and nonrelativistic (see for instanceRef. [37]) particle-vibration-coupling models. The pre-dicted integrated strengths are always better than in the(Q)RPA. Again, however, the improvement is minor for Ca. Reference [37] shows that the GT strength below20 MeV continues to be significantly overestimated inthat nucleus, even when beyond-mean-field correlationsare included. In this paper a better description of theGT − strength — measured in charge-exchange reactions a r X i v : . [ nu c l - t h ] S e p by adding a proton and removing a neutron — in Ca,and of the GT β decay of Ni, are achieved with a sub-tracted second RPA (SSRPA).Many-body theorists employ RPA-based schemes ex-tensively in atomic, solid-state, and nuclear physics, aswell as in quantum chemistry. Extensions are usefulwherever beyond-mean-field correlations play an impor-tant role. Second RPA, which includes 2p2h configura-tions for a richer description of the fragmentation andwidths of excited states, has thus made its way from nu-clear physics, where it was born, to mesoscopic physics[38] and chemistry [39].Versions of the RPA for CE processes were introducedseveral decades ago; references [40–42] contain useful dis-cussions of the stability of the Hartree-Fock solution withrespect to isospin excitations. The non–trivial step ofconstructing a full CE second RPA was taken in later,in Ref. [30]. There, one can find expressions for theHamiltonian matrix, which contains a 1p1h sector char-acterized by the matrices called A and B , a sectorthat mixes 1p1h and 2p2h configurations, with the ma-trices A and B , and a pure 2p2h sector, with thematrices A and B . Because the diagonalization oflarge dense matrices was impossible at that time, theHamiltonian matrix was drastically simplified by neglect-ing the interaction among 2p2h configurations. Thatstep allowed the full SRPA diagonalization to be replacedby an RPA-type computation with an energy-dependentHamiltonian. Much more recently, a self-energy subtrac-tion procedure was designed for extensions of RPA [43]and implemented in charge-conserving second RPA [44–49] to make the treatment of excitations consistent withground-state density-functional theory, guarantee Thou-less stability, and eliminate ultra-violet divergences.The CE second RPA developed here is the first that isfully self consistent and includes a subtraction procedurethat, just as in the charge-conserving case, corrects theresponse to make it consistent with ground-state density-functional theory at zero frequency. We apply it in to-gether with the Skyrme interaction [50–52] SGII [53, 54].The subtraction procedure requires the inversion of thelarge matrix A . To make the problem tractable, weconsistently cut off 2p2h configurations at 40 MeV, bothin the diagonalization of A and its inversion, havingverified that results do not change significantly when thecutoff is raised beyond that level.GT strength is constrained by the Ikeda sum rule,which relates the integrated strengths S to the numberof neutrons N and protons Z in the nucleus: S GT − − S GT + = 3( N − Z ) . (1)The sum rule is model independent under the condition ofcompleteness of states and given the properties of isospinoperators. It holds in RPA approaches and their exten-sions if, as in this work, the quasiboson approximationapplies.In nuclei with a significant neutron excess, such as Caand Ni, the GT − strength is much larger than GT + S t r e ng t h ExpSSRPARPA
20 25 30
Excitation energy (MeV) S t r e ng t h Figure 1. Experimental GT − plus isovector spin-monopolestrength in MeV − [55] and discrete RPA and SSRPAstrength distributions (no units) obtained with the Skyrmeparameterization SGII, for Ca . The RPA strength hasbeen divided by nine and the SSRPA strength by two so thatthe discrete distributions can be displayed on the same figureas the continuous experimental distribution (see text). Theinsert shows the energy region between 20 and 30 MeV. strength, measured by adding a neutron and removing aproton. The excitation operator for GT − transitions canbe written as ˆ O − = A (cid:88) i =1 (cid:88) µ σ µ ( i ) τ − ( i ) (2)where A is the number of nucleons, τ ( i ) − is the isospin-lowering operator τ − ≡ t x − it y for the i th nucleon, and σ µ ( i ) is the corresponding spin operator. Because S GT − is so much larger than S GT + the Ikeda sum rule is essen-tially a measure of the total GT − strength (as we willsee later numerically).We begin with the case of Ca. Reference [55] reportsthe results of a Ca(p, n) and Ti(n, p) experimentsat a beam energy of 300 MeV at the Research Centerfor Nuclear Physics in Osaka. The total GT − strengthbelow 30 MeV (which probably includes some contribu-tions from isovector spin-monopole excitations) is only 64 ±
9% of that given by the Ikeda sum rule. The locationof this “missing strength” has long been a mystery fornuclear physics.Compared to other EDF approaches (both mean-fieldand beyond-mean-field) ours better predicts the strengthdistributions, so that we obtain a much more accuratevalue for the sum of the strength up to 20 or 30 MeV,without resorting to quenching factors. The 2p2h con-figurations, which increase in density with excitation en-ergy, lead to a long high-energy tail that draws strengthfrom lower energies. The “missing strength” is thusspread out over a large range at higher energies, mak-ing it hard to discriminate from background. S t r e ng t h ExpSSRPARPA C u m u lt a ti v e s u m ExpSSRPARPA (a) (b)
Figure 2. (a) GT − strength distributions in MeV − . Theexperimental points are extracted from Ref. [55]. The RPAand SSRPA responses, computed with the parameterizationSGII, are folded with a Lorentzian having a width of 1 MeV.(b) Cumulative strengths up to 30 MeV. Figure 1 shows the experimental strength extractedfrom Ref. [55]. One of the most important features ofthe SSRPA is its ability to describe the width and frag-mentation of excitation spectra. This asset is visible inthe figure, which contains both the RPA and SSRPAdiscrete-strength distributions. Because the experimen-tal strength is a continuous function of energy, it has dif-ferent units from the discrete theoretical strengths, andthe absolute strength values are thus not comparable.However, by plotting the discrete spectra one can com-pare the location and fragmentation of the main peaks,without generating any artificial spreading by folding. Tobetter display the results in the figure, we rescale theRPA and SSRPA discrete strengths so that their respec-tive highest peaks have approximately the same height asthe corresponding experimental peak. To achieve this, wedivide the RPA strength by nine and multiply the SSRPAstrength by two.The SSRPA strength is indeed quite fragmented, par-ticularly in the region between 6 and 16 MeV, wherethree groups of peaks are concentrated around 8, 11, and14 MeV, in accordance with the experimental distribu-tion of peaks. One may also observe another group ofmuch weaker peaks concentrated around 17 MeV, whichcorresponds to the location of the highest-energy exper-imental peak. Finally, a very dense high-energy SSRPAtail is visible in the insert, which focuses on the energyregion between 20 and 30 MeV. Such a tail is completelyabsent from the RPA spectrum, which is composed of afew well separated peaks and misses the complex struc-ture of the experimental strength. The long high-energytail is indeed the explanation for the missing strength atlower energies.The very lowest-energy part of the SSRPA spectrumis less satisfactory than the rest, with a main peak pre-dicted at about 5 MeV; the lowest experimental peak, by contrast, is located at 3 MeV. The SSRPA does pre-dict some fragmented strength is in the region around 3MeV, however. The RPA discrete spectrum in Fig. 1, bycontrast, shows only a single visible peak at 4 MeV.To more directly compare the theoretical and experi-mental strengths, we have folded our response functionstogether with a Lorentzian distribution of width 1 MeV.Panel (a) of Fig. 2 presents the folded RPA and SSRPAstrength distributions along with the experimental dis-tribution. Panel (b) shows the cumulative strength as afunction of energy up to 30 MeV. As we have already seenin the discrete spectra, the SSRPA reproduces the GTdistribution quite well, with the exception of the lowest-energy peak. The RPA, on the other hand, cannot re-produce the complex structure of the spectrum.The most striking result is in panel (b). The SSRPAcumulative strength is greatly reduced from that of theRPA and is in much better agreement with the experi-mental value, even at low energies. Furthermore, the SS-RPA curve is smooth and follows the experimental profile(owing to the physical description of widths and frag-mentation, and to the subtraction procedure, which isneeded to place centroids at the correct energies), exceptbeyond 20 MeV, where the tail is a little too high, TheRPA curve, by contrast, shows steps because of its veryfew well separated peaks. The improvement with respectto the RPA is more significant than in other beyond-mean-field approaches. In Ref. [37], for example, thesame Skyrme interaction SGII produces more than 20units of strength below 20 MeV.The ratio between the experimental and the theoreti-cal integrated strength below 20 MeV is 0.58 for the RPAand 0.93 for the SSRPA, showing that quenching fac-tors are not needed in SSRPA. In the particle–vibration–coupling calculations of Ref. [37], this ratio is ≤ β + strength at that energy is only 0.10, so we reproduce theIkeda sum of 24 to within about 1%.To generalize our analysis, and to show that the de-crease in strength below 20 MeV is an intrinsic effectof the SSRPA and not an artifact of a specific param-eterization, we show in Fig. 3 RPA and SSRPA resultsobtained with three different parameterizations — SGII,SkM ∗ [56], and SkMP [57] — together with their ex-perimental counterparts. Panels (a), (b), and (c) showthe cumulative strengths up to 20 MeV, whereas panel(d) displays the percentages of the Ikeda sum rule fromstrength below 30 MeV. For this last quantity, Ref. [55]has reported both the experimental value and its uncer-tainty. In all three upper panels the integrated strengthin the SSRPA is smaller than in the RPA, especially forSGII and SkM ∗ . With these two parameterizations, thepercentages of the Ikeda sum in panel (d) are quite closeto the experimental value. The upper panels show thatthe detailed structure of the experimental spectrum isbest reproduced by the parameterization SGII; in panel(a) the curve follows the experimental profile very closely.We turn now to Ni. Though the GT spectrum ofthis nucleus has never been measured, its β -decay life-time is known [58]. Our GT − strength, integrated up to20 MeV, appears in Fig. 4(a) from both the RPA and SS-RPA (this time in a diagonal approximation so that wecan invert the larger A in this heavier nucleus). Thefigure also displays two other theoretical values for theintegrated strength at 20 MeV, from the beyond-mean-field calculations of Refs. [35] and [59] (for the latter,with the same interaction SGII as we use). Our value issignificantly lower than those of both the RPA and theother two beyond-mean-field calculations.Finally, we use our computed GT strength to obtainthe β -decay half-life of Ni. Because correlations make ad-hoc quenching unnecessary in our approach, we usethe bare value 1.28 for g A , the weak axial-vector cou-pling constant. Our result appears in Fig. 4(b), togetherwith the experimental value [58]. The SSRPA half-lifeis 0.19 seconds, very close to the experimental half-life.The RPA half-life, by contrast, is 9.51 seconds with thebare value of g A . The predictions of Refs. [35] (in themost complete scheme that includes polarization effectsrelated to CE phonons) and [59] (with the interactionSGII) are 0.04 and 0.69 seconds respectively, again with-out g A renormalization. Both these results are fartherfrom the experimental half-life than ours.In summary, we have presented a crucial improve- I k e d a s u m r u l e % I n t e g r a t e d s t r e ng t h Squares: RPA Diamonds: SSRPA
SGII SkM* SkMPExp (b)
SGII SkM* SkMPExp (a) (c)(d)
Exp Exp
Figure 3. (a), (b), (c) Strengths integrated up to 20 MeV. Theblack symbols represent the experimental values [55]. Thesolid (dashed) lines correspond to the SSRPA (RPA) resultsobtained with the parameterizations SGII (a), SkM ∗ (b), andSkMP (c). (d) Experimental percentage of the Ikeda sumrule below 30 MeV extracted from Ref. [55] (red dashed hori-zontal line), and its associated uncertainty (grey area). RPAand SSRPA percentages obtained with the parameterizationsSGII, SkM ∗ , and SkMP are also shown. -5 0 5 10 15 20 E ( MeV) C u m u l a ti v e S u m RPASSRPA T / ( s ec ) Ref. [59]Ref. [35]
SSRPAExp(a) (b)
Ref. [35]Ref. [59]
RPA
Figure 4. (a) Cumulative sum for different models (see leg-end and text) for the nucleus Ni; (b) β -decay half-life for Ni predicted by SSRPA, compared with predictions of othermodels and the experimental value [58]. The yellow band rep-resents the experimental uncertainty. ment in the description of the GT strength for the light-est double- β emitter, the nucleus Ca, and the heav-ier nucleus Ni. This achievement was made possibleby the beyond-mean-field EDF-based CE-SSRPA, devel-oped here for the first time. We have used the approachto compute GT − strengths with the Skyrme interactionSGII, and shown that it reproduces the complex frag-mented spectrum in Ca much better than does theRPA. Our most important result is that the total SSRPAstrength below 20-30 MeV is much smaller than in othermean-field and beyond-mean-field EDF models, and inbetter agreement with the corresponding experimentalvalues, without the use of ad hoc quenching factors. Byworking with two additional Skyrme parameterizationswe showed that these successes are due primarily to ourmany-body method, the key ingredient of which is theexplicit inclusion of 2p2h configurations. Their densitystrongly increases with the excitation energy, leading toa high-energy tail in the spectrum. We showed that thesame effects reduce the GT strength in the nucleus Ni,so that the predicted β -decay half-life agrees better withexperiment than do other beyond-mean-field models.The ability to describe CE strength may have a strongimpact in astrophysical scenarios where GT resonancesplay an important role. It also promises to improve EDF-based calculations of the nuclear matrix elements govern-ing 0 νββ decay, a process at the intersection of severalscientific domains. Our less accurate reproduction of theweak low-energy part of the Ca spectrum is unlikely tohave a large effect on the 0 νββ matrix element, whichcontains an unweighted sum over states at all energies.We plan to apply our approach to open–shell nuclei byincluding pairing correlations of both the usual isovectortype and the isoscalar type that are important for β anddouble- β decay [26, 60–62]. ACKNOWLEDGMENTS
M.G. acknowledges funding from the European UnionHorizon 2020 research and innovation program under Grant No. 654002 and from the IN2P3-CNRS BRIDGE-EDF project. J.E. acknowledges support from the U.S.Department of Energy (DOE), Office of Science, underGrant No. DE-FG02-97ER41019. [1] F. Osterfeld, Rev. Mod. Phys. , 491 (1992).[2] M. Ichimura, H. Sakai, and T. Wakasa, Prog. Part. Nucl.Phys. , 446 (2006).[3] K. Langanke and G. Mart´ınez-Pinedo, Rev. Mod. Phys. , 819 (2003).[4] H.-T. Janka, K. Langanke, A. Marek, G. Martnez-Pinedo, and B. Mller, Physics Reports , 38 (2007),the Hans Bethe Centennial Volume 1906-2006.[5] F. T. Avignone, S. R. Elliott, and J. Engel, Rev. Mod.Phys. , 481 (2008).[6] J. Engel and J. Men´endez, Reports on Progress in Physics , 046301 (2017).[7] J. Men´endez, Journal of Physics G: Nuclear and ParticlePhysics , 014003 (2017).[8] N. Shimizu, J. Menndez, and K. Yako, Phys. Rev. Lett. , 142502 (2018).[9] E. Santopinto, H. Garc´ıa-Tecocoatzi, R. I. Maga˜naVsevolodovna, and J. Ferretti (NUMEN Collaboration),Heavy-ion double-charge-exchange and its relation toneutrinoless double- β decay, Phys. Rev. C , 061601(2018).[10] V. d. S. Ferreira, A. R. Samana, F. Krmpoti´c, andM. Chiapparini, Nuclear structure model for double-charge-exchange processes, Phys. Rev. C , 044314(2020).[11] F. Cappuzzello et al., Eur. Phys. J. A , 72 (2018).[12] H. Lenske, F. Cappuzzello, M. Cavallaro, andM. Colonna, Prog. Part. Nucl. Phys. , 103716 (2019).[13] J. M. Yao, B. Bally, J. Engel, R. Wirth, T. R. Rodr´ıguez,and H. Hergert, Phys. Rev. Lett. , 232501 (2020).[14] S. J. Novario, P. Gysbers, J. Engel, G. Hagen, G. R.Jansen, T. D. Morris, P. Navrtil, T. Papenbrock, andS. Quaglioni, Coupled-cluster calculations of neutrinolessdouble-beta decay in ca (2020), arXiv:2008.09696 [nucl-th].[15] A. Belley, C. G. Payne, S. R. Stroberg, T. Miyagi,and J. D. Holt, Ab initio neutrinoless double-beta de-cay matrix elements for 48ca, 76ge, and 82se (2020),arXiv:2008.06588 [nucl-th].[16] J. Menendez, A. Poves, E. Caurier, and F. Nowacki, Nucl.Phys. A , 139 (2009).[17] Y. Iwata, N. Shimizu, T. Otsuka, Y. Utsuno, J. Menndez,M. Honma, and T. Abe, Phys. Rev. Lett. , 112502(2016), [Erratum: Phys.Rev.Lett. 117, 179902 (2016)].[18] M. Horoi and A. Neacsu, Phys. Rev. C , 024308 (2016).[19] J. Barea, J. Kotila, and F. Iachello, Phys. Rev. C ,034304 (2015).[20] T. R. Rodr´ıguez and G. Mart´ınez-Pinedo, Phys. Rev.Lett. , 252503 (2010).[21] N. L. Vaquero, T. R. Rodr´ıguez, and J. L. Egido, Phys.Rev. Lett. , 142501 (2013).[22] J. Yao, L. Song, K. Hagino, P. Ring, and J. Meng, Phys.Rev. C , 024316 (2015).[23] J. Yao and J. Engel, Phys. Rev. C , 014306 (2016).[24] F. ˇ Simkovic, V. Rodin, A. Faessler, and P. Vogel, Phys. Rev. C , 045501 (2013).[25] J. Hyvrinen and J. Suhonen, Phys. Rev. C , 024613(2015).[26] M. Mustonen and J. Engel, Phys. Rev. C , 064302(2013).[27] L.-G. Cao, S.-S. Zhang, and H. Sagawa, Phys. Rev. C , 054324 (2019).[28] A. Bohr and B. R. Mottelson, On the role of the δ reso-nance in the effective spin-dependent moments of nuclei,Physics Letters B , 10 (1981).[29] G. Bertsch and I. Hamamoto, Phys. Rev. C , 1323(1982).[30] S. Ait-Tahar and D. Brink, Nucl. Phys. A , 765(1993).[31] A. Klein, W. G. Love, and N. Auerbach, Continuumcharge-exchange spectra and the quenching of gamow-teller strength, Phys. Rev. C , 710 (1985).[32] P. Gysbers et al., Nature Phys. , 428 (2019).[33] A. Ekstrm, G. R. Jansen, K. A. Wendt, G. Hagen, T. Pa-penbrock, S. Bacca, B. Carlsson, and D. Gazit, Phys.Rev. Lett. , 262504 (2014).[34] L. Coraggio, L. De Angelis, T. Fukui, A. Gargano,N. Itaco, and F. Nowacki, Phys. Rev. C , 014316(2019).[35] C. Robin and E. Litvinova, Phys. Rev. C , 051301(2018).[36] C. Robin and E. Litvinova, Phys. Rev. Lett. , 202501(2019).[37] Y. F. Niu, G. Col`o, and E. Vigezzi, Phys. Rev. C ,054328 (2014).[38] D. Gambacurta and F. Catara, Collective excitations inmetallic clusters within the second random phase ap-proximation, Journal of Physics: Conference Series ,012012 (2009).[39] D. Peng, Y. Yang, P. Zhang, and W. Yang, Restrictedsecond random phase approximations and tamm-dancoffapproximations for electronic excitation energy calcu-lations, The Journal of Chemical Physics , 214102(2014).[40] N. Auerbach, A. Klein, and N. Van Giai, The giant dipoleresonance and its charge-exchange analogs, Physics Let-ters B , 347 (1981).[41] N. Auerbach, L. Zamick, and A. Klein, Rpa calculationsof isovector spin-flip excitations, Physics Letters B ,256 (1982).[42] A. Lane and J. Martorell, The random phase approxi-mation: Its role in restoring symmetries lacking in thehartree-fock approximation, Annals of Physics , 273(1980).[43] V. Tselyaev, Phys. Rev. C , 054301 (2013).[44] D. Gambacurta, M. Grasso, and J. Engel, Phys. Rev. C , 034303 (2015).[45] D. Gambacurta, M. Grasso, and O. Vasseur, Phys. Lett.B , 163 (2018).[46] O. Vasseur, D. Gambacurta, and M. Grasso, Phys. Rev. C , 044313 (2018).[47] M. Grasso, D. Gambacurta, and O. Vasseur, Phys. Rev.C , 051303 (2018).[48] D. Gambacurta, M. Grasso, and O. Sorlin, Phys. Rev. C , 014317 (2019).[49] M. Grasso and D. Gambacurta, Phys. Rev. C ,064314 (2020).[50] T. Skyrme, Phil. Mag. , 1043 (1956).[51] T. Skyrme, Nucl. Phys. , 615 (1959).[52] D. Vautherin and D. Brink, Phys. Rev. C , 626 (1972).[53] N. van Giai and H. Sagawa, Phys. Lett. B , 379(1981).[54] N. V. Giai and H. Sagawa, Nuclear Physics A , 1(1981).[55] K. Yako et al., Phys. Rev. Lett. , 012503 (2009).[56] J. Bartel, P. Quentin, M. Brack, C. Guet, and H.-B.Hakansson, Nucl. Phys. A , 79 (1982). [57] L. Bennour, P.-H. Heenen, P. Bonche, J. Dobaczewski,and H. Flocard, Phys. Rev. C , 2834 (1989).[58] P. T. Hosmer, H. Schatz, A. Aprahamian, O. Arndt,R. R. C. Clement, A. Estrade, K.-L. Kratz, S. N. Liddick,P. F. Mantica, W. F. Mueller, F. Montes, A. C. Mor-ton, M. Ouellette, E. Pellegrini, B. Pfeiffer, P. Reeder,P. Santi, M. Steiner, A. Stolz, B. E. Tomlin, W. B. Wal-ters, and A. W¨ohr, Half-life of the doubly magic r -processnucleus Ni, Phys. Rev. Lett. , 112501 (2005).[59] Y. F. Niu, Z. M. Niu, G. Col`o, and E. Vigezzi, Particle-vibration coupling effect on the β decay of magic nuclei,Phys. Rev. Lett. , 142501 (2015).[60] M. Mustonen, T. Shafer, Z. Zenginerler, and J. Engel,Phys. Rev. C , 024308 (2014).[61] C. Bai, H. Sagawa, G. Col, Y. Fujita, H. Zhang,X. Zhang, and F. Xu, Phys. Rev. C , 054335 (2014).[62] Y. Niu, G. Colo, E. Vigezzi, C. Bai, and H. Sagawa, Phys.Rev. C94