Nuclear charge-exchange excitations based on relativistic density-dependent point-coupling model
aa r X i v : . [ nu c l - t h ] D ec Nuclear charge-exchange excitations based on relativisticdensity-dependent point-coupling model
D. Vale ∗ Gimnazija i strukovna ˇskola Jurja Dobrile Pazin,ˇSetaliˇste Pazinske gimnazije 11, Pazin 52000, CroatiaOsnovna ˇskola Vodnjan/Scuola elementare Dignano,ˇZuka 6/Via delle ginestre 6, Vodnjan/Dignano 52215, Croatia andDepartment of Physics, Faculty of Science,University of Zagreb, Bijeniˇcka cesta 32, 10000 Zagreb, Croatia
Y. F. Niu
School of Nuclear Science and Technology,Lanzhou University, Lanzhou 730000, China,ELI-NP, Horia Hulubei National Institute for Physics and Nuclear Engineering,30 Reactorului Street, RO-077125, Bucharest-Magurele, Romania
N. Paar † Department of Physics, Faculty of Science,University of Zagreb, Bijeniˇcka cesta 32, 10000 Zagreb, Croatia (Dated: December 23, 2020) bstract Spin-isospin transitions in nuclei away from the valley of stability are essential for the descriptionof astrophysically relevant weak interaction processes. While they remain mainly beyond thereach of experiment, theoretical modeling provides important insight into their properties. Inorder to describe the spin-isospin response, the proton-neutron relativistic quasiparticle randomphase approximation (PN-RQRPA) is formulated using the relativistic density-dependent pointcoupling interaction, and separable pairing interaction in both the T = 1 and T = 0 pairingchannels. By implementing recently established DD-PCX interaction with improved isovectorproperties relevant for the description of nuclei with neutron-to-proton number asymmetry, theisobaric analog resonances (IAR) and Gamow-Teller resonances (GTR) have been investigated.In contrast to other models that usually underestimate the IAR excitation energies in Sn isotopechain, the present model accurately reproduces the experimental data, while the GTR propertiesdepend on the isoscalar pairing interaction strength. This framework provides not only an improveddescription of the spin-isospin response in nuclei, but it also allows future large scale calculationsof charge-exchange excitations and weak interaction processes in stellar environment. PACS numbers: 21.30.Fe, 21.60.Jz, 24.30.Cz, 25.40.Kv ∗ [email protected] † [email protected] . INTRODUCTION Charge-exchange excitations in atomic nuclei correspond to a class of nuclear transitionscomposed of the particle-hole configurations that contain the exchange of the nucleon charge,described by the isospin projection lowering (increasing) operator τ − ( τ + ). The fundamentalcharge-exchange excitation is the isobaric analog resonance (IAR) [1–8], with no changes inquantum numbers ∆ J = ∆ L = ∆ S = 0, thus the IAR corresponds to a collective excitationwith J π = 0 + . The Gamow-Teller resonance (GTR) represents another relevant charge-exchange mode, characterized by J π = 1 + , i.e., it corresponds to spin-flip excitations withoutchanging the orbital motion, ∆ S = 1, ∆ L = 0.As it has been emphasized in Ref. [9], recent interest in the GTR studies is motivated byits importance for understanding the spin and spin-isospin dependence of modern effectiveinteractions [10–17], nuclear beta decay [18–26], beta delayed neutron emission [27], as wellas double beta decay [28–36]. In addition, accurate description of GT ± transitions, includingboth in stable and exotic nuclei, is relevant for the description of a variety of astrophysicallyrelevant weak interaction processes [37–41], electron capture in presupernova stars [42–44],r-process [45, 46] and neutrino-nucleus interaction of relevance for neutrino detectors andneutrino nucleosynthesis in stellar environment [47–53].The properties of charge-exchange modes of excitation have extensively been studied[54, 55]. Following theoretical prediction [56], in 1975 the GTR has been experimentallyconfirmed in ( p, n ) reactions [57]. The GTR represents one of the most extensively inves-tigated collective excitation in nuclear physics, both experimentally and theoretically (e.g.,see Refs. [10, 54, 55, 57–86]). More details about experimental studies of spin-isospin exci-tations are also reviewed in Ref. [55]. Recent studies of the GTR in the framework based onrelativistic energy density functional include relativistic quasiparticle random phase approx-imation (RQRPA) [6], the relativistic RPA based on the relativistic Hartree-Fock (RHF)[87], relativistic QRPA based on point coupling model with nonlinear interactions [25, 43],and relativistic QRPA formulated using the relativistic Hatree–Fock–Bogoliubov (RHFB)model for the ground state [88]. In Ref. [89] the nuclear density functional framework, basedon chiral dynamics and the symmetry breaking pattern of low-energy QCD, has been used toformulate the proton–neutron QRPA to investigate the role of chiral pion–nucleon dynamicsin the description of charge-exchange excitations.3t present, the knowledge about charge-exchange transitions in nuclei away from thevalley of stability is rather limited, and mainly beyond the reach of experiment. Since thesenuclei are especially important for their astrophysical relevance in stellar evolution and nucle-osynthesis, it is crucial to develop microscopic theoretical approaches, that allow quantitativeand systematic analyses of the transition strength distributions of unstable nuclei. In orderto assess the overview into systematical model uncertainties in modeling charge-exchangeexcitation phenomena, it is important to address their properties from various approaches,by implementing different theory frameworks and effective nuclear interactions.In Ref. [6] charge-exchange excitations have been studied in the framework based onthe relativistic nuclear energy density functional (EDF), within the approach that unifiesthe treatment of mean-field and pairing correlations, relativistic quasiparticle random phaseapproximation (RQRPA) formulated in the canonical single-nucleon basis of the relativisticHartree-Bogoliubov (RHB) model. In this implementation the relativistic EDF with explicitdensity dependence of the meson-nucleon couplings is used, that provides an improved de-scription of asymmetric nuclear matter, neutron matter and nuclei far from stability. Thepairing correlations were described by the pairing part of the finite range Gogny interaction[90, 91]. However, the EDFs have usually been parameterized with the experimental dataon the ground state properties, supplemented with the pseudo-observables on nuclear mat-ter properties. In the case of density dependent meson-exchange interactions, the neutronskin thickness in Pb has been introduced as an additional constraint on the isovectorchannel of the effective interaction. However, the results from measurements of the neutron-skin thickness are usually model-dependent, and the pseudo-observables on nuclear matterare often rather arbitrary. Recently, a novel EDF parameterization has been establishedbased on the relativistic point coupling interaction, by using in the χ minimization the nu-clear ground state properties (binding energies, charge radii, pairing gaps) together with theproperties of collective excitations in nuclei, isoscalar giant monopole resonance energy anddipole polarizability [92]. In this way an effective interaction DD-PCX has been establishedwith improved isovector properties, that is successful not only in the description of nuclearground state, but also of the excitation phenomena, incompressibility of nuclear matter andthe symmetry energy close to the saturation density [92]. The improved isovector channelfor the DD-PCX interaction is especially important not only for the symmetry energy ofthe nuclear equation of state, but also for the description of ground-state and excitation4roperties of N = Z nuclei. Clearly, this is very important for the implementation of theEDF based models to exotic nuclei, as well as for applications in nuclear astrophysics.In this work we establish the proton-neutron RQRPA in the canonical single-nucleon basisof the RHB model based on density-dependent relativistic point coupling interaction. Ourstudy represents the first implementation of the relativistic point coupling interaction withdensity dependent vertex functions in formulating the RQRPA for the description of charge-exchange excitations. In addition, the treatment of the pairing correlations is also improved,by implementing the separable pairing force that allows accurate and efficient calculations ofthe pairing properties [93]. By using recently established DD-PCX interaction with improvedproperties that are essential for description of nuclei away from the valley of stability [92],the proton-neutron RQRPA established in this work will be employed in the investigationof the properties of collective charge-exchange excitations, IAR and GTR.Clearly, a study of both the IAR and GTR properties represents an important benchmarktest for novel theoretical approaches established not only for description of charge-excitationmodes, but also for modeling a variety of astrophysically relevant processes in stellar envi-ronment. Therefore, in the present work that introduces a microscopic approach to describecharge-exchange excitations based on density-dependent relativistic point coupling interac-tion, the novel theory framework will be employed in the analyses of charge-exchange modes,the IAR and GTR, both for magic nuclei, as well as for open-shell nuclei to probe the effectof the pairing correlations.In Sec. II the formalism of the proton-neutron RQRPA based on the density dependentpoint coupling interaction is introduced. In Sec. III the model is employed in studies ofcharge-exchange modes of excitation, the IAR and the GTR. The conclusions of this workare summarized in Section IV. II. PROTON-NEUTRON RQRPA BASED ON RELATIVISTIC POINT COU-PLING INTERACTION
In the previous implementation in the relativistic framework, the RQRPA has been estab-lished on the ground of relativistic Hartree-Bogoliubov (RHB) model, based on the effectiveLagrangian with density-dependent meson-exchange interaction terms [6]. Therein the pair-ing correlations have been described by the pairing part of the Gogny interaction [90, 91].5n the present study, the RQRPA is established using the relativistic point coupling in-teraction, while the pairing correlations are described by the separable pairing force fromRef. [93]. Since the full RQRPA equations are rather complicated, in the present study wesolve the respective equations in the canonical basis, where the Hartree-Bogoliubov wavefunctions can be expressed in the form of the BCS-like wave functions. More details onthe implementation of the canonical basis in the RHB model and general formalism of thePN-RQRPA equations in the canonical basis are given in Ref. [6]. The focus of this work isthe implementation of the relativistic point coupling interaction in deriving the PN-RQRPAequations. The nuclear ground state properties are described in the RHB model for thepoint coupling interaction, described in detail in Ref. [94].Starting from the 0 + ground state of a spherical even-even nucleus, transitions to J π excited state of the corresponding odd-odd daughter nucleus are considered, using the charge-exchange operator O JM . The general form of the PN-RQRPA equations read [6], A J B J B ∗ J A ∗ J X λJ Y λJ = E λ − X λJ Y λJ , (1)where the A and B matrices are defined in the canonical basis, A Jpn,p ′ n ′ = H pp ′ δ nn ′ + H nn ′ δ pp ′ + ( u p v n u p ′ v n ′ + v p u n v p ′ u n ′ ) V phJpn ′ np ′ + ( u p u n u p ′ u n ′ + v p v n v p ′ v n ′ ) V ppJpnp ′ n ′ B Jpn,p ′ n ′ = ( u p v n v p ′ u n ′ + v p u n u p ′ v n ′ ) V phJpn ′ np ′ − ( u p u n v p ′ v n ′ + v p v n u p ′ u n ′ ) V ppJpnp ′ n ′ . (2)The proton and neutron quasiparticle canonical states are denoted by p , p ′ , and n , n ′ ,respectively. V ph is the proton-neutron particle-hole residual interaction, and V pp is thecorresponding particle-particle interaction, and u and v denote the occupation amplitudesof the respective states. Since the canonical basis does not diagonalize the Dirac single-nucleon mean-field Hamiltonian, the off-diagonal matrix elements H nn ′ and H pp ′ are alsoincluded in the A matrix, as given in Ref. [6]. E λ denote the excitation energy, while X λJ and Y λJ are the corresponding forward- and backward-going QRPA amplitudes, respectively.By solving the eigenvalue problem (1), the reduced transition strength can be obtainedbetween the ground state of the even-even ( N, Z ) nucleus and the excited state of the odd-odd ( N + 1 , Z −
1) or ( N − , Z + 1) nucleus, using the corresponding transition operators6 JM in both channels, B − λJ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X pn < p ||O J || n > (cid:0) X λJpn u p v n + Y λJpn v p u n (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3) B + λJ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X pn ( − j p + j n + J < n ||O J || p > (cid:0) X λJnp v p u n + Y λJnp u p v n (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (4)For the presentation purposes, the discrete strength distribution is folded by the Lorentzianfunction of the width Γ = 1 MeV, R ( E ) ± = X λ B ± λJ π Γ / E − E λ ± ) + (Γ / . (5)In the implementation of the relativistic point coupling interaction, the spin-isospin de-pendent terms in the residual interaction of the PN-RQRPA are induced by the isovector-vector and pseudovector terms. In comparison, for the finite range meson-exchange interac-tion, these terms were obtained from the ρ -and π -meson exchange, respectively [6]. In thepresent study, the PN-RQRPA residual interaction terms V abcd are derived from the effectiveLagrangian density for the point coupling interaction.The isovector-vector part of PN-RQRPA contains only non-rearrangement terms of theresidual two-body interaction. For the respective spacelike components we obtain, V (TVs) abcd = − Z d r Z d r ψ † a ( ~r ) ( ~τ γ γ i ) (1) ψ c ( ~r ) α T V ( ρ ) δ ( ~r − ~r ) ψ † b ( ~r ) (cid:0) ~τ γ γ i (cid:1) (2) ψ d ( ~r ) , (6)and timelike components are given by V (TVt) abcd = Z d r Z d r ψ † a ( ~r ) ~τ (1) ψ c ( ~r ) α T V ( ρ ) δ ( ~r − ~r ) ψ † b ( ~r ) ~τ (2) ψ d ( ~r ) , (7)where the coupling α T V ( ρ ) is a function of baryon (vector) density. For Dirac spinor givenby, ψ am a ( r ) = f a ( r )Ω κ a m a (Ω) ig a ( r )Ω ¯ κ a m a (Ω) , (8)where a stands for all quantum numbers, with the exception of the projection of total angularmomentum m a . Quantum number κ is defined as κ = − ( l + 1) for j = l + 1 / κ = l for j = l − /
2, while ¯ l = 2 j − l corresponds to the lower component orbital angular momentum[95]. The spacelike part of the matrix elements obtained by the angular momentum coupling7s given V (TVs) Jabcd = 2(2 J + 1) X L Z drr α T V ( ρ ) (cid:20) f a ( r ) g c ( r ) h (1 / l a ) j a || [ σ S Y L ] J || (1 / l c ) j c i− g a ( r ) f c ( r ) h (1 / l a ) j a || [ σ S Y L ] J || (1 / l c ) j c i (cid:21)(cid:20) f b ( r ) g d ( r ) h (1 / l d ) j d || [ σ S Y L ] J || (1 / l b ) j b i− g b ( r ) f d ( r ) h (1 / l d ) j d || [ σ S Y L ] J || (1 / l b ) j b i (cid:21) , (9)and the timelike part is V ( T V t ) Jabcd = 22 J + 1 Z drr α T V ( ρ ) (cid:20) f a ( r ) f c ( r ) + g a ( r ) g c ( r ) (cid:21)(cid:20) f b ( r ) f d ( r ) + g b ( r ) g d ( r ) (cid:21) × h (1 / l a ) j a || Y J || (1 / l c ) j c ih (1 / l d ) j d || Y J || (1 / l b ) j b i . (10)When compared to standard R(Q)RPA matrix elements, in particular corresponding directterm, there exists an additional factor of 2 in the numerator of Eqs. (9) and (10) due tothe difference in the isospin part of the matrix element (see Appendix A). The isovector-pseudovector part of the point coupling interaction is given by V P V = − α P V δ ( ~r − ~r ) ( γ γ γ µ ~τ ) (1) ( γ γ γ µ ~τ ) (2) , (11)where α P V denotes the strength parameter of the interaction. Since α P V remains a freeparameter of the model that cannot be constrained by the ground state properties, its valueshould be constrained by the experimental data on charge-exchange excitations. For thecorresponding timelike part of the pseudovector matrix elements we obtain, V PV(t)J abcd = 2 α P V J + 1 Z drr (cid:20) f a ( r ) g c ( r ) − g a ( r ) f c ( r ) (cid:21)(cid:20) f b ( r ) g d ( r ) − g b ( r ) f d ( r ) (cid:21) × h (1 / l a ) j a || Y J || (1 / l c ) j c ih (1 / l d ) j d || Y J || (1 / l c ) j c i . (12)We note that the timelike matrix elements are non-zero only in the case of unnaturalparity transitions, e.g., for Gamow-Teller transitions. The spacelike part of the isovector-pseudovector matrix elements results, V P V ( s ) Jabcd = 2 α P V J + 1 X L Z drr (cid:20) f a ( r ) f c ( r ) h (1 / l a ) j a || [ σ S Y L ] J || (1 / l c ) j c i + g a ( r ) g c ( r ) h (1 / l a ) j a || [ σ S Y L ] J || (1 / l c ) j c i (cid:21)(cid:20) f b ( r ) f d ( r ) h (1 / l d ) j d || [ σ S Y L ] J || (1 / l b ) j b i + g b ( r ) g d ( r ) h (1 / l d ) j d || [ σ S Y L ] J || (1 / l b ) j b i (cid:21) . (13)8n order to constrain the value of the pseudovector coupling α P V , we follow the same pro-cedure as used in the case of relativistic functionals with meson-exchange [6], i.e., α P V isadjusted to reproduce the experimental value of excitation energy for Gamow-Teller reso-nance in
Pb, E = 19 . α P V = 0.734 for DD-PC1, and α P V = 0.621 for DD-PCX point coupling interaction, and use these values systematically inall further investigations.The PN-RQRPA also includes the pairing correlations. In most of the previous applica-tions of the RHB+RQRPA model, the pairing correlations have been described by the paringpart of the Gogny force D1S [90, 91]. This interaction has already been used in the RHBcalculations of various ground state properties in nuclei [99]. Since the calculations basedon the finite range Gogny force require considerable computational effort, in the presentformulation of the PN-RQRPA the separable form of the pairing interaction is used [93].In Ref. [93] Y. Tian et al. introduced separable pairing interaction in the gap equation ofsymmetric nuclear matter in S channel:∆( k ) = − Z ∞ k ′ dk ′ π h k | V S sep | k ′ i ∆( k ′ )2 E ( k ′ ) , (14)where h k | V S sep | k ′ i = − G p ( k ) p ( k ′ ) , (15)with Gaussian ansatz p ( k ) = e − a k . (16)The two parameters a and G were adjusted to density dependence of the gap at the Fermisurface in nuclear matter, calculated with the Gogny force [93]. After transformation of thepairing force from momentum to coordinate space one obtains, V ( ~r , ~r , ~r ′ , ~r ′ ) = − G δ ( ~R − ~R ) G ( r ) G ( r ′ ) 1 − ˆ P σ ~r = 1 / √ ~r − ~r ) and ~R = 1 / √ ~r + ~r ). G ( r ) is the Fourier transform of p ( k ), G ( r ) = e − r / (2 a ) (4 πa ) / . (18)Thus, the pairing force has finite range, and due to the presence of the factor δ ( ~R − ~R ) itpreserves the translational invariance [93]. Due to coordinate transformation from laboratoryto center of mass system and relative coordinates we need to use Talmi-Moschinsky brackets, | n l , n l ; λµ i = X NLnl M NLnln l n l | N L, nl ; λµ i . (19)9y employing the basis of spherical harmonic oscillator,˜ I n = √ π Z R nl ( r ) G ( r ) r dr = 12 / π / b / (1 − α ) n (1 + α ) n +3 / √ n + 12 n n ! , (20)the coupled matrix element for T = 1 pairing is given by, V ( pair ) JMabcd = − G ˆ j a ˆ j b ˆ j c ˆ j d ( − l b + l d + j a + j c l a j a / j b l b J l c j c / j d l d J X Nnn ′ ˜ I n ˜ I n ′ M NJn n a l a n b l b M NJn ′ n c l c n d l d . (21)Furthermore, for Gamow-Teller transitions in open shell nuclei we need to extend Eq. (21)to include both T = 0 and T = 1 channels. Therefore, we introduce natural extension ofthe pairing: V ( pair ) JMabcd = − G ˆ j a ˆ j b ˆ j c ˆ j d X LS X T (cid:16) − S ′ + T +1 (cid:17) ˜ f ( S, T ) ˆ S ˆ L l b / j b l a / j a L S J l d / j d l c / j c L S J X nn ′ ˜ I n ˜ I ′ n M NLn n a l a n b l b M NLn ′ n c l c n d l d . (22)This is non-vanishing only for S = 0 and T = 1 or S = 1 and T = 0 pairing. Therefore,˜ f ( S = 0 , T = 1) = 1 case corresponds to the Eq. (21), while the case ˜ f ( S = 1 , T =0) = V pp . See Appendix B for detailed derivation. We don’t know a priori the value ofthe isoscalar proton-neutron pairing strength parameter V pp . It may be somewhat reducedor enhanced compared to the case T = 1( S = 0), which is only present at the RHB level,and should be deduced from experimental data on excitations or charge-exchange processesin open shell nuclei.In comparison to the nuclear ground state based on the RHB, the PN-RQRPA residualinteraction includes an additional channel, described by the pseudovector term and addi-tional T = 0 ( S = 1) pairing term for Gamow-Teller transitions in open shell nuclei, thatare not present in the ground state calculations. Apart from this, the PN-RQRPA intro-duced in this work is self-consistent, i.e., the same interactions, both in the particle-holeand particle-particle channels, are used in the RHB equation that determines the canonical10uasiparticle basis, and in the PN-RQRPA. In both channels, the same strength parametersof the interactions are used in the RHB and RQRPA calculations.Similar to the previous implementations of the PN-RQRPA [6], the two-quasiparticleconfiguration space includes states with both nucleons in the discrete bound levels, stateswith one nucleon in the bound levels and one nucleon in the continuum, and also stateswith both nucleons in the continuum. The RQRPA configuration space also includes pair-configurations formed from the fully or partially occupied states of positive energy and theempty negative-energy states from the Dirac sea [6]. As pointed out in Ref. [6], the inclusionof configurations built from occupied positive-energy states and empty negative-energy statesis essential for the consistency of the model, as well as to reproduce the model independentsum rules. III. RESULTSA. The isobaric analog resonance
The PN-R(Q)RPA based on point coupling interactions introduced in Sec. II is firstimplemented in the case of J π = 0 + charge-exchange transition, isobaric analog resonance(IAR). It is induced by the Fermi isospin-flip operator,ˆ T Fβ ± = A X i =1 τ ± . (23)Figure 1 shows the transition strength distributions for the IAR in closed shell nuclei Ca, Zr and
Pb, calculated with the PN-RRPA using two density-dependent point couplinginteractions, DD-PCX and DD-PC1, and density-dependent meson-exchange effective inter-action DD-ME2. As expected, for each nucleus the response to the Fermi operator results ina pronounced single IAR peak. The IAR peak energy and transition strength display rathermoderate model dependence. The most pronounced spread of the IAR excitation energiesfor different interactions, about 1 MeV, is obtained for the heaviest system,
Pb, while for Ca and Zr differences are smaller. The results of model calculations are compared withthe experimental data for IAR excitation energies, denoted by arrows, obtained from ( p, n )scattering on Ca [100], Zr [101, 102], and
Pb [97]. Good agreement of the PN-RRPAresults with the experimental data is obtained. In all three cases the calculated transition11
DD-PCXDD-PC1DD-ME2 R [ / M e V ] E [MeV] Ca Zr Pb FIG. 1. The PN-RRPA isobaric analog resonance transition strength distribution for for Ca, Zrand
Pb, calculated using DD-PCX, DD-PC1, and DD-ME2 functionals. strengths of the IAR fulfil the Fermi non-energy weighted sum rule, consistent with Ref. [6].Next we explore the evolution of the IAR within the Sn isotope chain, for A =104 − , , , Sn. Model calculations are based on the PN-RQRPA with DD-PCX interaction.The results without and with the T = 1 proton-neutron pairing in the residual PN-RQRPAinteraction are shown separately, in comparison to the experimental data from a systematicstudy of the ( He,t) charge-exchange reaction in stable Sn isotopes [4]. As shown in Fig.3, the full PN-RQRPA calculations result in a pronounced single IAR peak, with the exci-tation energy that is in excellent agreement with the experimental data for , , Sn [4].Complete treatment of paring correlations both in the RHB and PN-RQRPA is essential fordescription of the IAR [6]. This is illustrated in Fig. 3, where the strength functions are alsoshown without including the T = 1 proton-neutron residual pairing interaction, i.e., onlythe the ph -channel of the RQRPA residual interaction is included. Without the contribu-tions of the pp -channel, pronounced fragmentation of the transition strength is obtained for12 full pairingno pn-pairing R [ / M e V ] E[MeV] Sn Sn Sn Sn PN-RQRPA(DD-PCX)
IAR
FIG. 2. The isobaric analog resonance transition strength distribution for , , , Sn, calcu-lated with the PN-RQRPA using DD-PCX interaction.The results without (dashed line) and with(solid line) proton-neutron pairing in the residual PN-RQRPA interaction are shown separately, incomparison to the experimental data from Ref. [4], denoted by arrows. , Sn, and the excitation energies are overestimated. By including the attractive proton-neutron pairing interaction, the transition strength becomes redistributed toward a singlepronounced IAR peak, that is consistent with the expectation of a narrow resonance peakfrom the experimental study [4]. More pronounced effect of the residual pairing interactionis obtained for , , Sn, while for
Sn that is near the neutron closed shell the effect israther small.Figure 3 shows the evolution of the IAR excitation energy for the isotopic chain − Sn,with the proton-neutron pairing included in the PN-RQRPA. The results are shown for thepoint coupling interactions DD-PCX and DD-PC1. For comparison, the IAR excitationenergies from the previous study based on DD-ME1 interaction [6] and the experimentaldata from Ref. [4] are displayed. As one can observe in the figure, recently establishedinteraction DD-PCX reproduces the experimental data with high accuracy, while the DD-PC1 and DD-ME1 interactions provide systematically lower energies. We note that DD-PCX parameterization has been established using additional constraints on nuclear collectivetransitions that resulted with improved isovector properties, essential for the description13
00 104 108 112 116 120 124 128 132
A1313.51414.5 E I A R [ M e V ] DD-PCXDD-PC1DD-ME1EXP.
FIG. 3. The PN-RQRPA isobaric analog resonance excitation energy for the isotope chain − Sn, with the proton-neutron pairing included. Calculations are based on the point cou-pling interaction DD-PCX and DD-PC1. The results from the previous study based on DD-ME1interaction [6] and the experimental data from Ref. [4] are shown for comparison. of nuclear ground state, excitation phenomena, and nuclear matter properties around thesaturation density [92]. Clearly, the PN-RQRPA framework based on DD-PCX interactionintroduced in this work represents a considerable progress in comparison to other approaches.
B. The Gamow-Teller resonance
The Gamow-Teller transitions involve both the spin and isospin degrees of freedom. Inthe charge-exchange excitation spectra, these transitions mainly concentrate in a pronouncedresonance peak - Gamow-Teller resonance (GTR), representing a coherent superposition of J π = 1 + proton-particle – neutron-hole transitions of neutrons from orbitals with j = l + into protons in orbitals with j = l − . The GT transitions are excited by the spin-isospinoperator T GTβ ± = A X i =1 Σ τ ± . (24)14igure 4 shows the GT − transition strength distribution for closed shell nuclei Ca, Zr and
Pb, calculated with the PN-RQRPA using DD-PCX and DD-PC1 interactions. For com-parison, the results of the meson exchange functional with the DD-ME2 parameterizationare also shown. The experimental values for the main GT − peak are denoted with arrowsfor Ca [100], Zr [101, 102], and
Pb [96–98]. The transition stength distributions aredominated by the main Gamow-Teller resonance peak, that is composed from direct spin-fliptransitions, ( j = l + → j = l − ). In addition, pronounced low-energy GT − strength isobtained, composed from the core-polarization spin-flip ( j = l ± → j = l ± ), and backspin-flip transitions ( j = l − → j = l + ). As it is well known, quantitative description ofthe low-energy GT − strength is essential in modelling beta decay half-lives [23, 103]. Whenusing three different effective interactions as shown in Fig. 4, the spread of values of theGT − excitation energies within ≈ Pb, reasonable agreementwith experimental data is obtained for Ca and Zr without additional adjustments of theeffective interaction. As it has already been discussed in previous studies, the Ikeda sum rulefor GT transition strength is fully reproduced in a complete calculation that includes boththe configurations formed from occupied states in the Fermi sea and empty negative-energystates in the Dirac sea [6, 104–107].In the following, the PN-RQRPA based on point coupling interaction DD-PCX is em-ployed in the study of GT − transitions in Sn istotope chain. For open shell nuclei, in additionto the separable pairing interaction included in the RHB, the PN-RQRPA residual interac-tion also includes the isoscalar proton-neutron pairing as introduced in Sec. II. Since thispairing interaction channel is not present within the RHB, its strength parameter V can beconstrained by the experimental data, e.g., on GT − excitation energies or beta decay half-lives. Rather than providing the optimal value of V , in the present analysis we explore thepairing properties and sensitivity of the GT − transitions by systematically varying V . InFig.5, the GT − strength distribution is shown for the isotopes , , , Sn, calculated withthe PN-RQRPA using DD-PCX interaction. The isoscalar proton-neutron pairing interac-tion strength parameter is varied within the range of V =0-1.3 MeV. At higher excitationenergies, a pronounced GT resonance peak is obtained, except for Sn, where the mainpeak is split. In all cases, pronounced low-energy GT − strength is obtained, spreading over15 DD-PCXDD-PC1DD-ME2 R [ / M e V ] E [MeV] Ca Zr Pb FIG. 4. The GT − strength distribution for Ca, Zr and
Pb, calculated using the DD-PCX,DD-PC1, and DD-ME2 functionals. The experimental values of the main GT − peak for Ca [100], Zr [102] and
Pb [4, 96, 97] are denoted with arrows. the energy range of ≈ νj = l + → πj = l − )dominate the high-energy region, the low-energy strength is dominated by core-polarizationspin-flip ( νj = l ± → πj = l ± ), and back spin-flip ( νj = l − → πj = l + ) transitions.Considering the dependence of the GT − strength on the T = 0 proton-neutron pairing, onecan observe in Fig.5 that the main GTR peak appears insensitive to this pairing interactionchannel. In the low-energy region, pronounced sensitivity of the GT − spectra on V is ob-tained, i.e. with increased V the strength is shifted toward lower energies. This result is inagreement with previous studies based on different effective interactions both in the ph and pp channels [6].In Fig. 6 the PN-RQRPA results for the GT − direct spin-flip transition excitation energycentroid with the respect to the IAR energy are shown for the chain of even-even isotopes − Sn. The available experimental data obtained from Sn( He,t)Sb charge-exchangereactions are shown for comparison [4]. The PN-RQRPA calculations are performed forthe range of values of the isoscalar proton-neutron pairing interaction strength parameter,16 R [ / M e V ] E[MeV] V = 0.0V = 0.4V = 0.7V = 1.0V = 1.3 Sn Sn Sn Sn PN-RQRPA(DD-PCX) GT - FIG. 5. The GT − strength distribution for , , , Sn, calculated with the PN-RQRPA usingDD-PCX interaction, for the range of values of the isoscalar proton-neutron pairing interactionstrength parameter, V =0-1.3. Experimental values for the main GT − peaks for Sn,
Sn and
Sn are denoted with arrows [4]. V =0.4-2.5. The point coupling interaction DD-PCX is used and the results based on theDD-ME2 interaction are also displayed [6]. One can observe that the proton-neutron pairinginteraction systematically reduces the GT-IAR energy splittings along the Sn isotope chain,resulting in good agreement for V ≈ V arerequired to reproduce the GT-IAR energy splittings. In Refs. [6, 108], it has been emphasizedthat the energy difference between the GTR and the IAS reflects the magnitude of theeffective spin-orbit potential. As one can see in Fig. 6, the GT-IAR energy splittings reducein neutron-rich Sn isotopes toward zero value, reflecting considerable reduction of the spin-orbit potential and the corresponding increase of the neutron skin thickness r n − r p [108].Therefore, as pointed out in Ref. [108], the energy difference E GT − E IAS , obtained from theexperiment, could be used to determine the value of neutron skin thickness in a consistentframework that can simultaneously describe the charge-exchange excitation properties andthe r n − r p value, such as the RHB+PN-RQRPA approach.17
10 120 130 A -2024681012 E G T R - E I A R [ M e V ] DD-PCX (V =0.4)DD-PCX (V =0.7)DD-PCX (V =1.3)DD-PCX (V =1.9)DD-PCX (V =2.5)DD-ME2EXP. Sn FIG. 6. The PN-RQRPA excitation energy for GT − direct spin-flip transitions for − Sn,for the range of values of the isoscalar proton-neutron pairing interaction strength parameter V .Calculations are based on the density dependent point coupling interaction DD-PCX and theresults from the previous study based on the DD-ME2 interaction [6] and the experimental datafrom Ref. [4] are shown for comparison. IV. CONCLUSION
In this work we have formulated a consistent framework for description of nuclear charge-exchange transitions based on the proton-neutron relativistic quasiparticle random phaseapproximation in the canonical single-nucleon basis of the relativistic Hartree-Bogoliubovmodel, using density-dependent relativistic point coupling interactions. The implementationof recently established DD-PCX interaction[92], adjusted not only to the nuclear groundstate properties, but also with the symmetry energy of the nuclear equation of state and theincompressibility of nuclear matter constrained using collective excitation data, allows im-proved description of ground-state and excitation properties of nuclei far from stability, thatis important for the studies of exotic nuclear structure and dynamics, as well as for applica-tions in nuclear astrophysics. The introduced formalism based on point coupling interactionsis also relevant from the practical point of view, because it allows efficient systematic large-scale calculations of nuclear properties and processes of relevance for the nucleosynthesis and18tellar evolution modeling. In the current formulation of the RHB+PN-RQRPA, the pairingcorrelations are implemented by the separable pairing force [93] that allows accurate andefficient calculations of the pairing properties. The PN-RQRPA includes both the T = 1and T = 0 pairing channels. While the T = 1 channel corresponds to the pairing interactionconstrained at the ground state level, T = 0 proton-neutron pairing strength parameter canbe determined from the experimental data on charge-exchange transitions and beta decayhalf lives. In order to validate the PN-R(Q)RPA framework introduced in this work, spinand isospin excitations - isobaric analog resonances and Gamow-Teller transitions - havebeen investigated in several closed-shell nuclei and Sn isotope chain. The results show verygood agreement with the experimental data, representing an improvement compared to pre-vious studies based on the relativistic nuclear energy density functionals. When comparedto other theoretical approaches that usually underestimate the IAR excitation energies inSn isotope chain, the present model using DD-PCX interaction accurately reproduces theexperimental data. Therefore, the framework introduced in this work represents an impor-tant contribution for the future studies of astrophysically relevant nuclear excitations andweak interaction processes, in particular β -decay in neutron-rich nuclei, electron capture inpresupernova stage of massive stars, and neutrino-nucleus reactions relevant for the synthe-sis of elements in the universe and stellar evolution. Especially important is the extensionof theory framework introduced in this work to include both the finite temperature effectstogether with nuclear pairing and apply it for the description of electron capture and betadecay at finite temperature characteristic for stellar environment [109, 110]. This work iscurrently in progress [26, 44]. ACKNOWLEDGMENTS
Stimulating discussions with Ante Ravli´c are gratefully acknowledged. This work is sup-ported by the QuantiXLie Centre of Excellence, a project co financed by the CroatianGovernment and European Union through the European Regional Development Fund, theCompetitiveness and Cohesion Operational Programme (KK.01.1.1.01).19 ppendix A: Particle-hole matrix elements
For the isovector-pseudovector part of particle-hole channel in the PN-RQRPA we usethe contact type of interaction with constant coupling, V P V = − α P V [ γ γ ν ~τ ] [ γ γ ν ~τ ] δ ( ~r − ~r ) . (A1)The uncoupled matrix element of isovector-pseudovector interaction is given by h a b | V | c d i = − α P V Z drr X Lλ (cid:20) Z dr r Z d Ω ¯ ψ a ( r , Ω )[ γ γ ν ~τ ] (1) Y ∗ Lλ (Ω ) δ ( r − r ) ψ c ( r , Ω ) (cid:21)(cid:20) Z dr r Z d Ω ¯ ψ b ( r , Ω )[ γ γ ν ~τ ] (2) Y Lλ (Ω ) δ ( r − r ) ψ d ( r , Ω ) (cid:21) , (A2)where we used expansion of delta function in spherical coordinates [111], δ ( ~r − ~r ) = δ ( r − r ) r r X λµ Y ∗ λµ (Ω ) Y λµ (Ω ) (A3)= Z drr δ ( r − r ) δ ( r − r ) r r r X λµ Y ∗ λµ (Ω ) Y λµ (Ω ) . (A4)Indices a , b , c and d refer to all quantum numbers involved, while the Dirac’s conjugate isdefined in standard way as ¯ ψ a = ψ + a γ . (A5)If we ignore for a moment the isospin part of the wave function (corresponding quantumnumbers isospin t and its third projection t ) one can rewrite Eq. (A2) in the followingform: h am a bm b | V | cm c dm d i PV = − α P V Z drr X λµ ( − µ X ν (cid:2) Q νλµ ( r, Ω ) (cid:3) ac [ Q λ − µ ; ν ( r, Ω )] bd (A6)where the index ν refers to Dirac matrix γ ν ( γ ν ) in the vertex and we used (see Ref. [95] forthe sherical case of spinor): (cid:2) Q νλµ ( r, Ω i ) (cid:3) am a cm c = Z d Ω i (cid:16) f n a κ a ( r )Ω † κ a m a (Ω i ) − ig n a κ a ( r )Ω † ¯ κ a m a (Ω i ) (cid:17) [ γ γ γ ν ] ( i ) × Y λµ (Ω i ) f n c κ c ( r )Ω κ a m c (Ω i ) ig n c κ c ( r )Ω ¯ κ c m c (Ω i ) , (A7)20here the subscripts 1 and 2 refer to Ω i = 1,2 = ( θ i , φ i ), while radial parts of bi-spinors f ( r )and g ( r ) are assumed to be real in our case. From Eq. (A7) one can easily obtain timelike (cid:2) Q λµ ( r, Ω i ) (cid:3) am a cm c = − (cid:20) i f n a κ a ( r ) g n c κ c ( r ) h (1 / l a ) j a m a | Y λµ (Ω i ) | (1 / l c ) j c m c i− i g n a κ a ( r ) f n c κ c ( r ) h (1 / l a ) j a m a | Y λµ (Ω i ) | (1 / l c ) j c m c i (cid:21) (A8)and spacelike components ( k = 1 , (cid:2) Q kλµ ( r, Ω i ) (cid:3) am a cm c = − (cid:20) f n a κ a ( r ) f n c κ c ( r ) h (1 / l a ) j a m a | σ k Y λµ (Ω ) | (1 / l c ) j c m c i + g n a κ a ( r ) g n c κ c ( r ) h (1 / l a ) j a m a | σ k Y λµ (Ω ) | (1 / l c ) j c m c i (cid:21) . (A9)In order to couple particle and hole operator, i.e. states, into good total angular momentum J(and projection M) in the matrix elements of the residual particle-hole two body interaction,one needs to start from [112] | ( ph − ) J M i = h c † p h † h i JM | i (A10)= X m p m h C JMj p m p j h m h c † pm p h † hm h | i (A11)= X m p m h ( − j h − m h C JMj p m p j h − m h a † pm p a h m h | i , (A12)from which J J -coupled particle-hole matrix element follows directly h ac − | V | b − d i JM = X m a m c ( − j c − m c C J Mj a m a jc − m c X m b m d ( − j b − m b C J Mj d m d j b − m b h am a bm b | V | cm c dm d i . (A13)Using the Wigner-Eckart theorem for angular parts in Eqs. (A8) and (A9) after somemanipulations with Clebsch-Gordan coefficients one obtains the ph matrix elements in J J -coupled form. Therefore, after including isospin part of the matrix elements, which givesa factor of 2, which we have ignored for the moment, for the timelike part of pseudovectorcoupling we obtain: V ( P V t ) Jabcd = 2 α P V J + 1 h (1 / l a ) j a || Y J (Ω ) || (1 / l c ) j c ih (1 / l d ) j d || Y J (Ω ) || (1 / l b ) j b i Z drr (cid:20) f a ( r ) g c ( r ) − g a ( r ) f c ( r ) (cid:21)(cid:20) f b ( r ) g d ( r ) − g b ( r ) f d ( r ) (cid:21) (A14)21nd for spacelike part V P V ( s ) Jabcd = 2 α P V J + 1 X L Z drr (cid:20) f a ( r ) f c ( r ) h (1 / l a ) j a || [ σ S Y L ] J || (1 / l c ) j c i + g a ( r ) g c ( r ) h (1 / l a ) j a || [ σ S Y L ] J || (1 / l c ) j c i (cid:21)(cid:20) f b ( r ) f d ( r ) h (1 / l d ) j d || [ σ S Y L ] J || (1 / l b ) j b i + g b ( r ) g d ( r ) h (1 / l d ) j d || [ σ S Y L ] J || (1 / l b ) j b i (cid:21) , (A15)where the reducible angular part of matrix elements can be written in terms of 3jm symbols: h (1 / l a ) j a || Y J || (1 / l c ) j c i = 1 + ( − l a + l c + J j a ˆ j c ˆ J √ π ( − j a − / j a J j c − / / , (A16)and h (1 / l a ) j a || [ σ S Y L ] J || (1 / l c ) j c i = 1 + ( − l a + l c + J j a ˆ j c ˆ J ˆ L √ π ( − l a + L " ( − l c + j c +1 / L J j a J j c − / / − √ L J − j a J j c / − / . (A17)Note that the timelike pseudovector matrix elements are non-zero only in the case of unnat-ural parity transitions, like Gamow-Teller transition. The isovector-vector part of two-bodyinteraction is given by V V = α V [ ρ V ( r )] [ γ µ ~τ ] [ γ µ ~τ ] δ ( ~r − ~r ) . (A18)There are no additional rearrangement terms due to isospin restriction of the PN-R(Q)RPA,i.e. changing the nucleon from neutron to proton or vice versa and properties of the pointcoupling functional itself. Therefore, one should start from Eq. (A18) and follow the sameprocedure described before, from Eqs. (A2) to (A15), replacing Eqs. (A8) and (A9) with (cid:2) Q kλµ ( r, Ω i ) (cid:3) am a cm c = (cid:20) if n a κ a ( r ) g n c κ c ( r ) h (1 / l a ) j a m a | σ k Y L Λ (Ω i ) | (1 / l c ) j c m c i− ig n a κ a ( r ) f n c κ c ( r ) h (1 / l a ) j a m a | σ k Y L Λ (Ω i ) | (1 / l c ) j c m c i (cid:21) (A19)and (cid:2) Q λµ ( r, Ω i ) (cid:3) am a cm c = (cid:20) f n a κ a ( r ) f n c κ c ( r ) h (1 / l a ) j a m a | Y λµ (Ω i ) | (1 / l c ) j c m c i + g n a κ a ( r ) g n c κ c ( r ) h (1 / l a ) j a m a | Y λµ (Ω i ) | (1 / l c ) j c m c i (cid:21) (A20)22n order to obtain J J -coupled ph matrix elements for this channel, i.e. spacelike part V ( T V s ) Jabcd = 22 J + 1 X L Z drr α T V [ ρ V ( r )] (cid:20) f a ( r ) g c ( r ) h (1 / l a ) j a || [ σ S Y L ] J || (1 / l c ) j c i− g a ( r ) f c ( r ) h (1 / l a ) j a || [ σ S Y L ] J || (1 / l c ) j c i (cid:21)(cid:20) f b ( r ) g d ( r ) h (1 / l d ) || [ σ S Y L ] J || (1 / l b ) j b i− g b ( r ) f d ( r ) h (1 / l d ) j d || [ σ S Y L ] J || (1 / l b ) j b i (cid:21) (A21)and timelike part V ( T V t ) Jabcd = 22 J + 1 Z drr α T V [ ρ V ( r )] (cid:20) f a ( r ) f c ( r ) + g a ( r ) g c ( r )) (cid:21)(cid:20) f b ( r ) f d ( r ) + g b ( r ) g d ( r ) (cid:21) × h (1 / l a ) j a || Y J || (1 / l c ) j c ih (1 / l d ) j d || Y J || (1 / l b ) j b i . (A22) Appendix B: Natural extension of separable pairing
In order to extend the standard S = 0 ( T = 1) pairing, which was used in the RHBand RQRPA, one needs to start from general expression for particle-particle matrix elementin the uncoupled form: V abcd = h n a (1 / l a ) j a n b (1 / l b ) j b | V (cid:16) − ˆ P r ˆ P σ ˆ P τ (cid:17) | n c (1 / l c ) j c n d (1 / l d ) j d i (B1)where ˆ P r , ˆ P σ and ˆ P τ represent the exchange of relative spatial coordinate, spin and isospin,respectively. For action of the spin exchange operator ˆ P σ we simply have P σ | (1 / (1) / (2) ) SM S (cid:11) = ( − S +1 | (1 / (1) / (2) ) SM S (cid:11) , (B2)which affects just the order of coupling, i.e. the phase ( − S +1 of symmetry transformationof the Clebsch-Gordan coefficient. Furthermore, one may construct P σ mathematically inthe following way: ˆ P σ = 1 + ~σ · ~σ , (B3)and isospin exchange operator in analogous way. However, the action of ˆ P r affects onlyrelative spatial coordinates, ˆ P r | N LM L i = | N LM L i , (B4)ˆ P r | nlm l i = ( − l | nlm l i , (B5)23herefore, as first step one needs to do the LS recoupling in bra and ket independently V (pair) Jabcd = ˆ j a ˆ j b ˆ j c ˆ j d X λS X λ ′ S ′ ˆ λ ˆ S ˆ λ ′ ˆ S ′ s a l a j a s b l b j b S λ J s c l c j c s d l d j d S ′ λ ′ J X T M T X T ′ M T ′ C T M T / − / / / C T ′ M T ′ / − / / / X M S M ′ S C JMSM S λµ C JMS ′ M ′ S λ ′ µ ′ h (1 / / T M T |h ( s a s b ) SM S ( l a l b ) λµ | V (1 − ˆ P r ˆ P σ ˆ P τ ) | ( s c s d ) S ′ M ′ S ( l c l d ) λ ′ µ ′ i| (1 / / T ′ M T ′ i . (B6)In the second step one needs to substitute V with generic separable form, similar to Eq.(17) but now without any kind of projection, V ( ~r , ~r , ~r ′ , ~r ′ ) = − G δ ( ~R − ~R ) G ( r ) G ( r ′ ) (B7)into Eq. (B6) and transform the laboratory coordinates ~r and ~r ( ~r ′ and ~r ′ ) in bra ( ket ) intothe center of mass ~R ( ~R ′ ) and relative coordinates ~r ( ~r ′ ), i.e. the so-called Talmi-Moschinskytransformation. Therefore we need to evaluate X T M T X T ′ M T ′ C T M T / − / / / C T ′ M T ′ / − / / / h (1 / / T M T |h (1 / / SM S ( l a l b ) λµ | V (1 − ˆ P r ˆ P σ ˆ P τ ) | (1 / / S ′ M S ′ ( l c l d ) λ ′ µ ′ i| (1 / / T ′ M T ′ i == − πG δ SS ′ δ M S M S ′ δ λL δ µm L δ λ ′ L δ µ ′ m L ′ X NLnn ′ I n I n ′ M NLn n a l a n b l b M NLn ′ n c l c n d l d X T (cid:16) − S ′ + T +1 (cid:17) (B8)Constraining ourself to the proton-neutron case only, note that the only nonvanishing casesof the isospin coupling are T = 0 and M T = 0 or T = 1 and M T = 0, which both lead tofactor 1 / V JMabcd = − G ˆ j a ˆ j b ˆ j c ˆ j d X LS X T (cid:16) − S ′ + T +1 (cid:17) ˜ f ( S, T ) ˆ S ˆ L l b / j b l a / j a L S J l d / j d l c / j c L S J X nn ′ ˜ I n ˜ I ′ n M NLn n a l a n b l b M NLn ′ n c l c n d l d . (B9)24owever, we also add multiplication with function ˜ f ( S, T ) which should take into accountsomewhat smaller or enhanced effect of the pairing in T = 0 case,˜ f ( S, T ) = , for S = 0 , T = 1 V pp , for S = 1 , T = 00 , the rest (B10)while ˜ I n = √ πI n is spatial integral of Gaussian G ( r ) derived analytically in the nextsection of Appendix. Appendix C: Analytical solution of radial part
Radial part of the eigenfunction of spherical 3D harmonic oscillator is given by R nl ( r, b ) = b − / R nl ( ξ ) = b − / N nl ξ l L l +1 / n ( ξ ) e − ξ / , (C1)where b represents oscillatory length, while n corresponds to the nodes number (in thisnotation we don’t take into account 0 and ∞ ). Possible values are n = 0 , , ... , and ξ = r/b . Normalization factor N nl is given by N nl = (cid:18) n !Γ( l + n + 3 / (cid:19) / (C2)In the case of half-number arguments gamma function is given byΓ (cid:18)
12 + n (cid:19) = 1 · · · · · (2 n − n √ π = (2 n − n √ π, (C3)We are interested in the analytical solution of the following integral I n = Z R nl ( r ) G ( r ) r dr (C4)By inserting G(r) from Eq. (18) in Eq. (C4) we obtain I n = b / N n (4 πa ) / Z ∞ exp (cid:20) − ξ (cid:18) b a + 1 (cid:19)(cid:21) L / n ( ξ ) ξ dξ. (C5)After substitution α = a b , ξ = η → ξdξ = dη , (C6)25q. (C5) can be rewritten as I n = b − / N n (4 πα ) / Z ∞ exp (cid:20) − η + η (cid:18) − α (cid:19)(cid:21) L (1 / n ( η ) √ ηdη. (C7)Generating function for Laguerre polynomials with α = 1 / l = 0) is given by expression[113] 1(1 − z ) / exp (cid:18) xzz − (cid:19) = ∞ X n =0 L / n ( x ) z n . (C8)By using substitution 12 (cid:18) − α (cid:19) = zz − → z = 1 − α α (C9)and Eq. (C8), we can rewrite Eq. (C7) in the following form, I n = b − / N n (4 πα ) /
12 (1 − z ) / ∞ X m =0 z m Z ∞ dηη / e − η L (1 / n ( η ) L (1 / m ( η ) . (C10)The integral in Eq.(C10) is just orthogonality condition for Laguerre’s polynomials [113], Z ∞ x α e − α L αn ( x ) L αm ( x ) dx = Γ( n + α + 1) n ! δ nm . (C11)Therefore, Eq. (C10) reduces to I n = 12 b − / N n (4 πα ) / (1 − z ) / ∞ X m =0 z m Γ( m + 3 / m ! δ nm . (C12)Using Eq. (C3), inserting normalization factor (Eq. (C2)) and expressing everything interms of α and n , after some mathematical manipulations, it can further be simplified to I n = 12 / π / b / (1 − α ) n (1 + α ) n +3 / p (2 n + 1)!2 n n ! . (C13) [1] N. Auerbach, J. H¨ufner, A. K. Kerman, and C. M. Shakin, Rev. Mod. Phys. , 48 (1972).[2] N. Auerbach, Physics Reports , 273 (1983).[3] O. A. Rumyantsev and M. H. Urin, Phys. Rev. C , 537 (1994).[4] K. Pham, J. J¨anecke, D. A. Roberts, M. N. Harakeh, G. P. A. Berg, S. Chang, J. Liu, E. J.Stephenson, B. F. Davis, H. Akimune, and M. Fujiwara, Phys. Rev. C , 526 (1995).
5] G. Col`o, H. Sagawa, N. Van Giai, P. F. Bortignon, and T. Suzuki,Phys. Rev. C , 3049 (1998).[6] N. Paar, T. Nikˇsi´c, D. Vretenar, and P. Ring, Phys. Rev. C , 054303 (2004).[7] S. Fracasso and G. Col`o, Phys. Rev. C , 064310 (2005).[8] X. Roca-Maza, G. Col`o, and H. Sagawa, Phys. Rev. C , 031306 (2012).[9] X. Roca-Maza and N. Paar, Progress in Particle and Nuclear Physics , 96 (2018).[10] M. Bender, J. Dobaczewski, J. Engel, and W. Nazarewicz, Phys. Rev. C , 054322 (2002).[11] T. Marketin, G. Mart´ınez-Pinedo, N. Paar, and D. Vretenar,Phys. Rev. C , 054313 (2012).[12] E. Litvinova, B. A. Brown, D. L. Fang, T. Marketin, and R. G. T. Zegers,Physics Letters B , 307 (2014).[13] A. Ekstr¨om, G. R. Jansen, K. A. Wendt, G. Hagen, T. Papenbrock, S. Bacca, B. Carlsson,and D. Gazit, Phys. Rev. Lett. , 262504 (2014).[14] V. Zelevinsky, N. Auerbach, and B. M. Loc, Phys. Rev. C , 044319 (2017).[15] H. Morita and Y. Kanada-En’yo, Phys. Rev. C , 044318 (2017).[16] J.-U. Nabi, M. Ishfaq, M. B¨oy¨ukata, and M. Riaz, Nuclear Physics A , 1 (2017).[17] E. Ha and M.-K. Cheoun, The European Physical Journal A , 26 (2017).[18] J. Engel, P. Vogel, and M. R. Zirnbauer, Phys. Rev. C , 731 (1988).[19] I. Borzov, S. Fayans, and E. Trykov, Nuclear Physics A , 335 (1995).[20] J. Engel, M. Bender, J. Dobaczewski, W. Nazarewicz, and R. Surman,Phys. Rev. C , 014302 (1999).[21] I. N. Borzov, Phys. Rev. C , 025802 (2003).[22] M. Madurga, S. V. Paulauskas, R. Grzywacz, D. Miller, D. W. Bardayan, J. C. Batchelder,N. T. Brewer, J. A. Cizewski, A. Fija lkowska, C. J. Gross, M. E. Howard, S. V. Ilyushkin,B. Manning, M. Matoˇs, A. J. Mendez, K. Miernik, S. W. Padgett, W. A. Peters, B. C. Rasco,A. Ratkiewicz, K. P. Rykaczewski, D. W. Stracener, E. H. Wang, M. Woli ´nska Cichocka,and E. F. Zganjar, Phys. Rev. Lett. , 092502 (2016).[23] T. Marketin, L. Huther, and G. Mart´ınez-Pinedo, Phys. Rev. C , 025805 (2016).[24] B. Moon, C.-B. Moon, A. Odahara, R. Lozeva, P.-A. S¨oderstr¨om, F. Browne, C. Yuan,A. Yagi, B. Hong, H. S. Jung, P. Lee, C. S. Lee, S. Nishimura, P. Doornenbal, G. Lorusso,T. Sumikama, H. Watanabe, I. Kojouharov, T. Isobe, H. Baba, H. Sakurai, R. Daido, . Fang, H. Nishibata, Z. Patel, S. Rice, L. Sinclair, J. Wu, Z. Y. Xu, R. Yokoyama,T. Kubo, N. Inabe, H. Suzuki, N. Fukuda, D. Kameda, H. Takeda, D. S. Ahn, Y. Shimizu,D. Murai, F. L. Bello Garrote, J. M. Daugas, F. Didierjean, E. Ideguchi, T. Ishigaki,S. Morimoto, M. Niikura, I. Nishizuka, T. Komatsubara, Y. K. Kwon, and K. Tshoo,Phys. Rev. C , 014325 (2017).[25] Z. Y. Wang, Y. F. Niu, Z. M. Niu, and J. Y. Guo,Journal of Physics G: Nuclear and Particle Physics , 045108 (2016).[26] A. Ravli´c, E. Y¨uksel, Y. F. Niu, and N. Paar, arXiv:2010.06394 [nucl-th] (2020).[27] R. Caballero-Folch, C. Domingo-Pardo, J. Agramunt, A. Algora, F. Ameil, A. Arcones,Y. Ayyad, J. Benlliure, I. N. Borzov, M. Bowry, F. Calvi˜no, D. Cano-Ott, G. Cort´es,T. Davinson, I. Dillmann, A. Estrade, A. Evdokimov, T. Faestermann, F. Farinon,D. Galaviz, A. R. Garc´ıa, H. Geissel, W. Gelletly, R. Gernh¨auser, M. B. G´omez-Hornillos,C. Guerrero, M. Heil, C. Hinke, R. Kn¨obel, I. Kojouharov, J. Kurcewicz, N. Kurz, Y. A.Litvinov, L. Maier, J. Marganiec, T. Marketin, M. Marta, T. Mart´ınez, G. Mart´ınez-Pinedo,F. Montes, I. Mukha, D. R. Napoli, C. Nociforo, C. Paradela, S. Pietri, Z. Podoly´ak, A. Proc-hazka, S. Rice, A. Riego, B. Rubio, H. Schaffner, C. Scheidenberger, K. Smith, E. Sokol,K. Steiger, B. Sun, J. L. Ta´ın, M. Takechi, D. Testov, H. Weick, E. Wilson, J. S. Winfield,R. Wood, P. Woods, and A. Yeremin, Phys. Rev. Lett. , 012501 (2016).[28] A. Faessler and F. Simkovic, Journal of Physics G: Nuclear and Particle Physics , 2139 (1998).[29] J. Suhonen and O. Civitarese, Physics Reports , 123 (1998).[30] F. ˇSimkovic, R. Hod´ak, A. Faessler, and P. Vogel, Phys. Rev. C , 015502 (2011).[31] J. Men´endez, D. Gazit, and A. Schwenk, Phys. Rev. Lett. , 062501 (2011).[32] J. D. Vergados, H. Ejiri, and F. ˇSimkovic, Reports on Progress in Physics , 106301 (2012).[33] J. Men´endez, T. R. Rodr´ıguez, G. Mart´ınez-Pinedo, and A. Poves,Phys. Rev. C , 024311 (2014).[34] D. c. v. ˇStef´anik, F. ˇSimkovic, and A. Faessler, Phys. Rev. C , 064311 (2015).[35] D. Navas-Nicol´as and P. Sarriguren, Phys. Rev. C , 024317 (2015).[36] D. S. Delion and J. Suhonen, Phys. Rev. C , 034330 (2017).[37] K. Langanke and G. Martinez-Pinedo, Nuclear Physics A , 481 (2000).[38] K. Langanke and G. Mart´ınez-Pinedo, Rev. Mod. Phys. , 819 (2003).
39] H. T. Janka, K. Langanke, A. Marek, G. Mart´ınez-Pinedo, and B. M¨uller,
The Hans BetheCentennial Volume 1906-2006 , Physics Reports , 38 (2007).[40] S. Noji, R. G. T. Zegers, S. M. Austin, T. Baugher, D. Bazin, B. A. Brown, C. M. Campbell,A. L. Cole, H. J. Doster, A. Gade, C. J. Guess, S. Gupta, G. W. Hitt, C. Langer, S. Lipschutz,E. Lunderberg, R. Meharchand, Z. Meisel, G. Perdikakis, J. Pereira, F. Recchia, H. Schatz,M. Scott, S. R. Stroberg, C. Sullivan, L. Valdez, C. Walz, D. Weisshaar, S. J. Williams, andK. Wimmer, Phys. Rev. Lett. , 252501 (2014).[41] N. Paar, T. Marketin, D. Vale, and D. Vrete-nar, International Journal of Modern Physics E , 045807 (2011).[43] Z. M. Niu, Y. F. Niu, Q. Liu, H. Z. Liang, and J. Y. Guo, Phys. Rev. C , 051303 (2013).[44] A. Ravli´c, E. Y¨uksel, Y. F. Niu, G. Col´o, E. Khan, and N. Paar,arXiv:2006.08803 [nucl-th] (2020).[45] M. Arnould, S. Goriely, and K. Takahashi, Physics Reports , 97 (2007).[46] K. Mori, M. A. Famiano, T. Kajino, T. Suzuki, J. Hidaka, M. Honma, K. Iwamoto,K. Nomoto, and T. Otsuka, The Astrophysical Journal , 179 (2016).[47] H. Ejiri, Physics Reports , 265 (2000).[48] T. Suzuki and H. Sagawa, Nuclear Physics A , 446 (2003).[49] D. Frekers, H. Ejiri, H. Akimune, T. Adachi, B. Bilgier, B. A. Brown, B. T. Cleve-land, H. Fujita, Y. Fujita, M. Fujiwara, E. Ganio˘glu, V. N. Gavrin, E. W. Grewe, C. J.Guess, M. N. Harakeh, K. Hatanaka, R. Hodak, M. Holl, C. Iwamoto, N. T. Khai, H. C.Kozer, A. Lennarz, A. Okamoto, H. Okamura, P. P. Povinec, P. Puppe, F. ˇSimkovic,G. Susoy, T. Suzuki, A. Tamii, J. H. Thies, J. Van de Walle, and R. G. T. Zegers,Physics Letters B , 134 (2011).[50] M.-K. Cheoun, E. Ha, S. Y. Lee, K. S. Kim, W. Y. So, and T. Kajino,Phys. Rev. C , 028501 (2010).[51] N. Paar, D. Vretenar, T. Marketin, and P. Ring, Phys. Rev. C , 024608 (2008).[52] N. Paar, T. Suzuki, M. Honma, T. Marketin, and D. Vretenar,Phys. Rev. C , 047305 (2011).
53] M. Karako¸c, R. G. T. Zegers, B. A. Brown, Y. Fujita, T. Adachi, I. Boztosun, H. Fujita,M. Csatl´os, J. M. Deaven, C. J. Guess, J. Guly´as, K. Hatanaka, K. Hirota, D. Ishikawa,A. Krasznahorkay, H. Matsubara, R. Meharchand, F. Molina, H. Okamura, H. J. Ong,G. Perdikakis, C. Scholl, Y. Shimbara, G. Susoy, T. Suzuki, A. Tamii, J. H. Thies, andJ. Zenihiro, Phys. Rev. C , 064313 (2014).[54] F. Osterfeld, Rev. Mod. Phys. , 491 (1992).[55] Y. Fujita, B. Rubio, and W. Gelletly, Progress in Particle and Nuclear Physics , 549 (2011).[56] K. Ikeda, S. Fujii, and J. I. Fujita, Physics Letters , 271 (1963).[57] R. R. Doering, A. Galonsky, D. M. Patterson, and G. F. Bertsch,Phys. Rev. Lett. , 1691 (1975).[58] J. A. Halbleib and R. A. Sorensen, Nuclear Physics A , 542 (1967).[59] I. S. Towner and F. C. Khanna, Phys. Rev. Lett. , 51 (1979).[60] C. D. Goodman, C. A. Goulding, M. B. Greenfield, J. Rapaport, D. E. Bainum, C. C. Foster,W. G. Love, and F. Petrovich, Phys. Rev. Lett. , 1755 (1980).[61] G. Brown and M. Rho, Nuclear Physics A , 397 (1981).[62] G. F. Bertsch and I. Hamamoto, Phys. Rev. C , 1323 (1982).[63] K. Nakayama, A. Pio Gale˜ao, and F. Krmpoti´c, Physics Letters B , 217 (1982).[64] N. Van Giai and H. Sagawa, Physics Letters B , 379 (1981).[65] C. Gaarde, Nuclear Physics A , 127 (1983).[66] V. A. Kuzmin and V. G. Soloviev, Journal of Physics G: Nuclear Physics , 1507 (1984).[67] G. Col`o, N. Van Giai, P. F. Bortignon, and R. A. Broglia, Phys. Rev. C , 1496 (1994).[68] I. Hamamoto and H. Sagawa, Phys. Rev. C , R960 (1993).[69] B. A. Brown and K. Rykaczewski, Phys. Rev. C , R2270 (1994).[70] T. Suzuki and T. Otsuka, Phys. Rev. C , 847 (1997).[71] C. De Conti, A. Gale˜ao, and F. Krmpoti´c, Physics Letters B , 14 (1998).[72] E. Caurier, K. Langanke, G. Mart´ınez-Pinedo, and F. Nowacki,Nuclear Physics A , 439 (1999).[73] K. Langanke, E. Kolbe, and D. J. Dean, Phys. Rev. C , 032801 (2001).[74] A. Algora, B. Rubio, D. Cano-Ott, J. L. Ta´ın, A. Gadea, J. Agramunt, M. Gierlik, M. Karny,Z. Janas, A. P lochocki, K. Rykaczewski, J. Szerypo, R. Collatz, J. Gerl, M. G´orska, H. Grawe,M. Hellstr¨om, Z. Hu, R. Kirchner, M. Rejmund, E. Roeckl, M. Shibata, L. Batist, and . Blomqvist (GSI Euroball Collaboration), Phys. Rev. C , 034301 (2003).[75] Y. Kalmykov, T. Adachi, G. P. A. Berg, H. Fujita, K. Fujita, Y. Fujita, K. Hatanaka,J. Kamiya, K. Nakanishi, P. von Neumann-Cosel, V. Y. Ponomarev, A. Richter, N. Sakamoto,Y. Sakemi, A. Shevchenko, Y. Shimbara, Y. Shimizu, F. D. Smit, T. Wakasa, J. Wambach,and M. Yosoi, Phys. Rev. Lett. , 012502 (2006).[76] C. L. Bai, H. Sagawa, H. Q. Zhang, X. Z. Zhang, G. Col`o, and F. R. Xu,Physics Letters B , 28 (2009).[77] Y. S. Lutostansky, Physics of Atomic Nuclei , 1176 (2011).[78] M. Sasano, G. Perdikakis, R. G. T. Zegers, S. M. Austin, D. Bazin, B. A. Brown, C. Caesar,A. L. Cole, J. M. Deaven, N. Ferrante, C. J. Guess, G. W. Hitt, R. Meharchand, F. Montes,J. Palardy, A. Prinke, L. A. Riley, H. Sakai, M. Scott, A. Stolz, L. Valdez, and K. Yako,Phys. Rev. Lett. , 202501 (2011).[79] Y. F. Niu, G. Col`o, M. Brenna, P. F. Bortignon, and J. Meng,Phys. Rev. C , 034314 (2012).[80] E. Ha and M.-K. Cheoun, Phys. Rev. C , 017603 (2013).[81] X. Roca-Maza, G. Col`o, and H. Sagawa, Physica Scripta , 014011 (2013).[82] M. Martini, S. P´eru, and S. Goriely, Phys. Rev. C , 044306 (2014).[83] E. Ha and M.-K. Cheoun, Phys. Rev. C , 054320 (2016).[84] Y. F. Niu, G. Col`o, E. Vigezzi, C. L. Bai, and H. Sagawa, Phys. Rev. C , 064328 (2016).[85] H. Z. Liang, H. Sagawa, M. Sasano, T. Suzuki, and M. Honma,Phys. Rev. C , 014311 (2018).[86] J. Yasuda, M. Sasano, R. G. T. Zegers, H. Baba, D. Bazin, W. Chao, M. Dozono, N. Fukuda,N. Inabe, T. Isobe, G. Jhang, D. Kameda, M. Kaneko, K. Kisamori, M. Kobayashi,N. Kobayashi, T. Kobayashi, S. Koyama, Y. Kondo, A. J. Krasznahorkay, T. Kubo, Y. Kub-ota, M. Kurata-Nishimura, C. S. Lee, J. W. Lee, Y. Matsuda, E. Milman, S. Michimasa,T. Motobayashi, D. Muecher, T. Murakami, T. Nakamura, N. Nakatsuka, S. Ota, H. Otsu,V. Panin, W. Powell, S. Reichert, S. Sakaguchi, H. Sakai, M. Sako, H. Sato, Y. Shimizu,M. Shikata, S. Shimoura, L. Stuhl, T. Sumikama, H. Suzuki, S. Tangwancharoen, M. Takaki,H. Takeda, T. Tako, Y. Togano, H. Tokieda, J. Tsubota, T. Uesaka, T. Wakasa, K. Yako,K. Yoneda, and J. Zenihiro, Phys. Rev. Lett. , 132501 (2018).[87] H. Liang, N. Van Giai, and J. Meng, Phys. Rev. Lett. , 122502 (2008).
88] Z. M. Niu, Y. F. Niu, H. Z. Liang, W. H. Long, and J. Meng,Phys. Rev. C , 044301 (2017).[89] P. Finelli, N. Kaiser, D. Vretenar, and W. Weise, Nuclear Physics A , 57 (2007).[90] J. Decharg´e and D. Gogny, Phys. Rev. C , 1568 (1980).[91] J. Berger, M. Girod, and D. Gogny, Computer Physics Communications , 365 (1991).[92] E. Y¨uksel, T. Marketin, and N. Paar, Phys. Rev. C , 034318 (2019).[93] Y. Tian, Z.-Y. Ma, and P. Ring, Phys. Rev. C , 064301 (2009).[94] T. Nikˇsi´c, N. Paar, D. Vretenar, and P. Ring,Computer Physics Communications , 1808 (2014).[95] W. Greiner and D. Bromley, Relativistic Quantum Mechanics. Wave Equations (Springer,2000).[96] D. J. Horen, C. D. Goodman, C. C. Foster, C. A. Goulding, M. B. Greenfield, J. Ra-paport, D. E. Bainum, E. Sugarbaker, T. G. Masterson, F. Petrovich, and W. G. Love,Physics Letters B , 27 (1980).[97] H. Akimune, I. Daito, Y. Fujita, M. Fujiwara, M. B. Greenfield, M. N. Harakeh, T. Ino-mata, J. J¨anecke, K. Katori, S. Nakayama, H. Sakai, Y. Sakemi, M. Tanaka, and M. Yosoi,Phys. Rev. C , 604 (1995).[98] A. Krasznahorkay, H. Akimune, M. Fujiwara, M. N. Harakeh, J. J¨anecke, V. A. Rodin, M. H.Urin, and M. Yosoi, Phys. Rev. C , 067302 (2001).[99] D. Vretenar, A. Afanasjev, G. Lalazissis, and P. Ring, Physics Reports , 101 (2005).[100] B. D. Anderson, T. Chittrakarn, A. R. Baldwin, C. Lebo, R. Madey, P. C. Tandy, J. W.Watson, B. A. Brown, and C. C. Foster, Phys. Rev. C , 1161 (1985).[101] D. E. Bainum, J. Rapaport, C. D. Goodman, D. J. Horen, C. C. Foster, M. B. Greenfield,and C. A. Goulding, Phys. Rev. Lett. , 1751 (1980).[102] T. Wakasa, H. Sakai, H. Okamura, H. Otsu, S. Fujita, S. Ishida, N. Sakamoto, T. Uesaka,Y. Satou, M. B. Greenfield, and K. Hatanaka, Phys. Rev. C , 2909 (1997).[103] T. Marketin, D. Vretenar, and P. Ring, Phys. Rev. C , 024304 (2007).[104] H. Kurasawa, T. Suzuki, and N. V. Giai, Phys. Rev. Lett. , 062501 (2003).[105] H. Kurasawa, T. Suzuki, and N. Van Giai, Phys. Rev. C , 064311 (2003).[106] Z. Ma, B. Chen, and N. e. a. Van Giai, Eur. Phys. J. A , 429 (2004).[107] H. Kurasawa and T. Suzuki, Phys. Rev. C , 014306 (2004). , 262502 (2003).[109] E. Y¨uksel, G. Col`o, E. Khan, Y. F. Niu, and K. Bozkurt, Phys. Rev. C , 024303 (2017).[110] E. Y¨uksel, N. Paar, G. Col`o, E. Khan, and Y. F. Niu, Phys. Rev. C , 044305 (2020).[111] P. Ring and P. Schuck, The Nuclear Many-Body Problem , Physics and astronomy onlinelibrary (Berlin, Springer, 2004).[112] J. Suhonen,
From Nucleons to Nucleus: Concepts of Microscopic Nuclear Theory , Theoreti-cal and Mathematical Physics (Springer Berlin Heidelberg, 2007).[113] M. Abramowitz and I. Stegun,
Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables ,Applied mathematics series (Dover Publications, 1965).,Applied mathematics series (Dover Publications, 1965).