Spin-isospin response of deformed neutron-rich nuclei in a self-consistent Skyrme energy-density-functional approach

We develop a new framework of the self-consistent deformed proton-neutron quasiparticle-random-phase approximation (pnQRPA), formulated in the Hartree-Fock-Bogoliubov (HFB) single-quasiparticle basis. The same Skyrme force is used in both the HFB and pnQRPA calculations except in the proton-neutron particle-particle channel, where an S=1 contact force is employed. Numerical application is performed for Gamow-Teller (GT) strength distributions and $\beta$-decay rates in the deformed neutron-rich Zr isotopes located around the path of the rapid-neutron-capture process nucleosynthesis. It is found that the GT strength distributions are fragmented due to deformation. Furthermore we find that the momentum-dependent terms in the particle-hole residual interaction leads to a stronger collectivity of the GT giant resonance. The T=0 pairing enhances the low-lying strengths cooperatively with the T=1 pairing correlation, which shortens the $\beta$-decay half lives by at most an order of magnitude. The new calculation scheme reproduces well the observed isotopic dependence of the $\beta$-decay half lives of deformed $^{100-110}$Zr isotopes.


I. INTRODUCTION
Study of unstable nuclei has been one of the major subjects in nuclear physics for a couple of decades. Collective mode of excitation emerged in the response of the nucleus to an external field is a manifestation of the interaction among nucleons. Thus, the spin-isospin channel of the interaction or the spin-isospin part of the energy-density functional (EDF), which is crusial for understanding and predicting the properties of unstable nuclei and asymmetric nuclear matter, has been much studied through especially the Gamow-Teller (GT) strength distributions [1,2].
The GT strength distribution has been extensively investigated experimentally and theoretically not only because of interests in nuclear structure but also because β-decay half lives set a time scale of the rapid-neutroncapture process (r-process), and hence determine the production of heavy elements in the universe [3]. The rprocess path is far away from the stability line, and involves neutron-rich nuclei. They are weakly bound and many of them are expected to be deformed according to the systematic Skyrme-EDF calculation [4].
Collective modes of spin-isospin excitation in nuclei are described microscopically by the proton-neutron randomphase approximation (pnRPA) or the proton-neutron quasiparticle-RPA (pnQRPA) including the pairing correlations on top of the self-consistent Hartree-Fock (HF) or HF-Bogoliubov (HFB) mean fields employing the nuclear EDF. There have been many attempts to investigate the spin-isospin modes of excitation in stable and unstable nuclei [5]. These studies are largely restricted to spherical systems, and the collective modes in deformed nuclei remain mostly unexplored.
The spin-isospin responses of deformed nuclei have been extensively investigated by the Madrid group [6][7][8] in connection to the studies of beta decay and double-beta decay in a Skyrme-pnQRPA model. The method employed in these preceding works relies on the BCS pairing instead of the HFB pairing, and the residual interactions are treated in a separable approximation. The BCS approximation for pairing is inappropriate for describing the weakly bound nuclei due to the unphysical nucleon gas problem [9]. Furthermore, collectivity and details of the strength distribution are sensitive to both the shell structure around the Fermi levels and the residual interactions. Quite recently, in Ref. [10], the fully self-consistent Skyrme-pnQRPA model was established in an HFB single-canonical basis and was applied to the study of double-beta decay.
Recently, β-decay half lives of neutron-rich Kr to Tc isotopes with A ≃ 110 located on the boundary of the rprocess path were newly measured at RIKEN RIBF [11]. The ground state properties such as deformation and superfluidity in neutron-rich Zr isotopes up to the drip line had been studied by employing the Skyrme-HFB method, and it had been predicted that Zr isotopes around A = 110 are well deformed in the ground states [12].
In the present article, to investigate the Gamow-Teller mode of excitation and β-decay properties in the deformed neutron-rich Zr isotopes, we construct a new framework of the calculation scheme employing the Skyrme EDF self-consistently in both the static and the dynamic levels. Furthermore, to describe properly the pairing correlations in weakly bound systems and coupling to the continuum states, the HFB equations are solved in the real space. This framework is extended based on the deformed like-particle QRPA method developed in Ref. [13].
The article is organized as follows: In Sec. II, the deformed Skyrme-HFB + pnQRPA method for describing the spin-isospin responses is explained. In Sec. III, results of the numerical analysis of the giant resonance in the neutron-rich Zr isotopes are presented. Discussion on effects of the T = 0 pairing is included. Finally, summary is given in Sec. IV.

A. Microscopic calculation of spin-isospin modes of excitation in deformed nuclei
To describe the nuclear deformation and the pairing correlations in the ground state, simultaneously, in good account of the continuum, we solve the HFB equations [9,14] in coordinate space using cylindrical coordinates r = (ρ, z, φ). We assume axial and reflection symmetries.
Here, the superscript q denotes ν (neutron, t z = 1/2) or π (proton, t z = −1/2). The mean-field Hamiltonian h is derived from the Skyrme EDF. The pairing fieldh is treated by using the density-dependent contact interaction [15], where ̺ 0 (r) denotes the isoscalar density and P σ the spin exchange operator.
Since we consider the even-even mother (target) nuclei only, the time-reversal symmetry is assumed. A nucleon creation operatorψ † q (rσ) at the position r with the intrinsic spin σ is then written in terms of the quasiparticle (qp) wave functions aŝ The notation ϕ(rσ) is defined by ϕ(rσ) = −2σϕ(r − σ).
Using the quasiparticle basis obtained as a selfconsistent solution of the HFB equations (1), we solve the pnQRPA equation withĤ ′ =Ĥ − λ νNν − λ πNπ . The charge-changing QRPA phonon operators are defined aŝ whereâᾱ ,q is a quasiparticle annihilation operator of the time-reversed state of α.
In the present calculation, we solve the pnQRPA equation (4) in the matrix formulation Using the qp wave functions ϕ 1 (rσ) and ϕ 2 (rσ), the solutions of the coordinate-space HFB equation (1), the matrix elements of (6) are written as Here, the time-reversed state is defined as and d1 stands for σ1 dr 1 . If one assumes the effective interaction for the particlehole (p-h) channel is local,v ph is written as and V ph is derived from the Skyrme EDF; . The coefficients in Eq. (11) can be found in the appendix of Ref. [16] or in Ref [17].
Assuming the proton-neutron particle-particle (p-p) effective interaction is local similarly to the p-h channel, we can writev pp as The residual interaction in the p-p channel V pp could be derived from the proton-neutron pairing EDF. However, it is not well established yet. What we need in our framework is an interaction between the proton-neutron particle-particle (p-p), and hole-hole (h-h) pairs. In the present calculation, we consider the p-p (h-h) interaction between the T = 0, S = 1 pair only and take v(r) = v 0 as a constant for simplicity. Here, P τ denotes the isospin exchange operator. The GT ± transition strengths to the state i with angular momentum K(K = 0, ±1) are calculated as under the quasi-boson approximation. The HFB vacuum is denoted as |HFB , and |αβ =â † α,νâ † β,π |HFB is a 2qp excited state. The GT ± operators are given bŷ The transition-strength distributions as functions of the excitation energy E * with respect to ground state of the odd-odd daughter nucleus are calculated as The smearing width γ is introduced to make the strength distributions easier to read. E 0 denotes the lowest quasiparticle energy of protons and neutrons. When either or both pairing gaps vanish, we take the lowest occupied neutron and unoccupied proton states for the t − channel. It is noted that the spin-parity of the state with E 0 is, in general, different from 1 + .

B. Details of the numerical calculation
We employ the SkM* [18] and SLy4 [19] EDFs for the mean-filed Hamiltonian and the residual interaction for the p-h channel. The pairing strength parameter t ′ 0 is determined so as to approximately reproduce the experimental pairing gap of 120 Sn (∆ exp = 1.245 MeV) as in Ref. [20], where the giant monopole resonance in the deformed neuron-rich Zr isotopes was investigated. The strengths t ′ 0 = −240 and −290 MeV fm 3 for the mixedtype interaction (t ′ 3 = −18.75t ′ 0 ) [21] lead to the neutron pairing gap ∆ ν = 1.20 and 1.24 MeV in 120 Sn with the SkM* and SLy4 EDFs, respectively. The strength parameter v 0 for the T = 0 pairing interaction can be considered as a free parameter, because it dose not affect the ground state properties, and it is active only in the dynamic level. Our procedure to determine it is to fit approximately the β-decay lifetime of 100 Zr (T exp 1/2 = 7.1 s [22]). The strengths v 0 = −395 and −320 MeV fm 3 give the calculated β-decay half life T 1/2 = 7.08 s and 7.63 s with the SkM* and SLy4 EDFs, respectively.
Because of the assumption of the axially symmetric potential, the z−component of the qp angular momentum, Ω, is a good quantum number. Assuming timereversal symmetry and reflection symmetry with respect to the x − y plane, we have only to solve Eq. (1) for positive Ω and positive z. We use the lattice mesh size ∆ρ = ∆z = 0.6 fm and a box boundary condition at ρ max = 14.7 fm, z max = 14.4 fm to discretize the continuum states. The differential operators are represented by use of the 13-point formula of finite difference method. The quasiparticle energy cutoff is chosen at E qp,cut = 60 MeV and the quasiparticle states up to Ω π = 31/2 ± are included.
We introduce an additional truncation for the pn-QRPA calculation, in terms of the 2qp energy as E α + E β ≤ 60 MeV. This reduces the number of 2qp states to, for instance, about 30 000 for the K π = 0 + excitation. The number of 2qp states included in the calculation is large enough to satisfy the Ikeda sum-rule values to an accuracy of 1%. The calculation of the QRPA matrix elements in the qp basis, and diagonalization of the QRPA matrix are performed in the parallel computers as in Ref. [23].
As a test calculation, our method is applied to the isobaric analogue state (IAS) in 90 Zr. When the Coulomb potential is discarded, that is, the electron charge e is set zero, the IAS appears at ∼ 0.2 MeV excitation energy with respect to the ground state of 90 Zr with the SkM* EDF and the 2qp energy cutoff described above. For the deformed 110 Zr case, we obtained the IAS at ∼ 0.4 MeV. This implies that our calculation scheme satisfies the self-consistency between the static and the dynamic calculations [24]. It is noted here that the mean energy of the IAS is given by with the sum-rule method for the separable interaction [1]. Thus, without the Coulomb potential, the excitation energy would be zero if and only if the isospin is a good quantum number [24]. The finite excitation energy obtained here is due to the spurious isospin mixing in the HFB approximation for N = Z nuclei. As stated in Introduction, a similar calculation of the self-consistent HFB + pnQRPA for axially deformed nuclei has been recently reported [10]. They adopt the canonical-basis representation and introduce a further truncation according to the occupation probabilities of 2qp excitations. In contrast, we adopt the qp representation and truncation simply due to the 2qp energies.    In Tables I and II, we summarize the ground state properties of the Zr isotopes calculated with the SkM* and SLy4 EDFs combined with the mixed-type pairing interaction. The ground state of 98 Zr is spherical, and we see a sudden onset of deformation in the Zr isotopes with N ≥ 60. Both Skyrme EDFs give the similar deformations and root-mean-square radii of the ground states in nuclei under investigation. In the present article, we investigate the GT excitation mainly in deformed nuclei. However, to see the deformation effect, we include 98 Zr as a reference.
The For each isotopes in the tables, the neutron chemical potential is shallower, and the proton chemical potential is instead deeper with SLy4 than with SkM*. This gives larger Q β values of β-decay with the SLy4 EDF because the differences in E 0 are not very large. The Q β value is given by under the independent quasi-particle approximation.
Here ∆M n−H = 0.78227 MeV is the mass difference between a neutron and a hydrogen atom. Noted that Q β is given by ∆M n−H − min(ε unocc. π − ε occ. ν ) in terms of the single-particle energies for unpaired systems if we choose E 0 as described above.
B. GT giant resonance Figure 1 shows the strength distributions associated with the GT − operator (17) without the T = 0 pairing in the Zr isotopes as functions of the excitation energy with respect to the daughter nuclei. We also show in this figure the contribution of the K = 0 and 1 components to the total strength. As we will discuss in Sec. III C, we see a tiny amount of energy change due to T = 0 pairing for the GT giant resonance (GR). We need to multiply the strengths shown in the figure by g 2 A /4π or (g 2 A ) eff /4π including the quenching effect to obtain the B(GT − ) values in Eq. (14).
To quantify the excitation energy of the GR, we introduce the centroid energy which is frequently used in the experimental analysis, defined by where m k is a k−th moment of the transition-strength distribution in an energy interval of [E a , E b ] MeV; with Fig. 2(a), we show the centroid energies of the GTGR and the isobaric analogue resonance (IAR). Since we have only a single peak of the IAR, we take the energy interval to evaluate the centroid energy as E a = 0 and E b = 40 MeV. For evaluation of the centroid energy of the GTGR, we take E a = 15 and E b = 30 MeV.
The centroid energies obtained with the SkM* and SLy4 EDFs of the GTGR are similar to each other. Properties of a GR are liked to the nuclear matter properties. In nuclear matter, the interaction strength in the GT channel is sensitive to the Landau-Migdal (LM) parameter g ′ 0 [25]. The SkM* and SLy4 EDFs give the similar values of g ′ 0 ; 0.94 and 0.90, respectively. Thus, the collectivity of the GTGR calculated may be similar. These values of g ′ 0 are large and comparable with g ′ 0 = 0.93 of the SGII interaction [25,26], which is designed to describe the spin-isospin excitations. Noted that, however, they are much smaller than the empirical value g ′ 0 (exp) ≃ 1.8 [1]. We are going to discuss the deformation effects. In the spherical 98 Zr nucleus, the centroid energy of the GTGR is about 20 MeV, and it appears as a narrow peak. Besides the GTGR, we see a low-energy resonance structure in the energy region of 5 − 15 MeV. When the system gets deformed, strengths both of the GTGR and of the low-energy resonance are fragmented. The deformation splitting between the K = 0 and 1 states of the GTGR is at most 1 MeV. So, the splitting effect is washed out by the smearing width γ. This is consistent with the finding in Ref. [27], where the schematic residual interaction was employed. The spreading effect Γ ↓ not taken in the present calculation may be larger than 1 MeV so that it is difficult to observe the deformation splitting of the GTGR experimentally.
As increasing the neutron number, the centroid energy of the GTGR monotonically increases. This characteristic feature of increase in the excitation energy as a function of the neutron (mass) number is also found in Ref. [28]. In 112 Zr, the centroid energy reaches about 25 MeV. We find also a monotonic increase in the centroid energy of the IAR in the deformed systems. Seen is a general feature that the energies of the IAR calculated with SLy4 are higher than those calculated with SkM*. It is noted that there has been a discussion on the correlation between the symmetry energy and the energy of the IAR [29]. Indeed, the symmetry coefficient of SLy4 (32.00 MeV) is larger than that of SkM* (30.03 MeV).
With the sum-rule method for the separable interaction [1], the energy difference of the GTGR and IAR is given by where ∆E ls is an average value of the spin-orbit splitting, andκ στ andκ τ are the coupling constants of the spin-isospin and isospin residual forces in the separable Hamiltonian. The result shown in Fig. 2(b) suggests that our microscopic calculation obeys this simple relation in a good approximation as long as the deformed systems are considered. The slope parametersκ στ −κ τ fitted for SkM* and SLy4 in the deformed Zr isotopes are −16.7 and −15.6 MeV, respectively. The slope parameters microscopically obtained here are not far from the systematic value of −14.5 MeV [30]. Figure 3 shows the strength distributions in some selected isotopes calculated with the SLy4 EDF. In the deformed isotopes other than 104 Zr, we see the similar features discussed below. Here, we compare the QRPA results with those obtained in the LM approximation, and those obtained without the residual interactions. In the LM approximation, we treat the p-h residual interaction as instead of (11). Here, N 0 is the density of states and the LM parameters f ′ 0 , g ′ 0 are deduced from the same Skyrme force which generates the mean field [25]. The Fermi momentum k F appearing in the LM parameters is evaluated in the local density approximation.
In both nuclei, one of which is spherical and the other is deformed, we see two prominent peaks at around 6 and 14 MeV in the unperturbed 2qp transition-strength distribution. A difference to be noticed is that the strengths are fragmented in 104 Zr due to deformation at the meanfield level. Associated with the repulsive p-h residual interaction, most of the strengths are absorbed by the GTGR, and the resonance peak is shifted higher in energy. The energy shift due to the RPA correlation is much larger in 104 Zr than in 98 Zr. It is pointed out that the energy and collectivity of the GTGR are changed when omitting the momentum-dependent terms in the residual interaction for the SLy5 EDF in a framework of the spherical HF-BCS + pnQRPA [17]. We clearly see here that the momentum dependence in the p-h residual interaction has a significant effect in generating collectivity of the GTGR for the SkM* and SLy4 EDFs.  Figure 4 shows the strength distributions associated with the GT − operator (17) with the SLy4 EDF combined with and without the T = 0 pairing interaction in 104 Zr. We see only a tiny amount of energy change due to T = 0 pairing for the GTGR. The change in the centroid energy of the GTGR due to T = 0 pairing is 0.14 MeV. This result indicates that the GTGR is built almost entirely of the p-h excitations.
In Fig. 4, we also show the result in Ref. [28], where the separable forces were employed for the residual interactions. As is discussed in the previous subsection, the momentum-dependent terms in the p-h residual interac- tion plays an important role in generating the collectivity. The peak energy calculated in Ref. [28] is lower than our result by about 1.5 MeV.
Compared to the GTGR, the low-lying GT strength distribution is affected appreciably by the T = 0 pairing interaction due to the following mechanism showing up from the structure of the matrix elements of the pnQRPA Hamiltonian (7) and (8). The proton orbitals around the Fermi level are partially occupied due to the T = 1 pairing correlations. Thus, the neutrons in the hole(-like) orbital can have a chance to decay into the proton orbitals through the p-h residual interaction and the T = 0 p-p interaction simultaneously. Similarly, when the neutrons are paired, the neutrons around the Fermi level can decay to protons in the particle(-like) orbitals through the p-h and p-p residual interactions. We are considering here the t − channel, but it also holds for the t + channel. Figure 5 shows the strength distribution in the lowexcitation energy region in the deformed Zr isotopes. In this figure, the smearing width γ is set 0.1 MeV. The lowlying states are sensitive to the shell structure around the Fermi levels, however, we see some generic features; the peak position is shifted lower in energy and at the same time the transition strength increases.
In a well-deformed nuclei, the asymptotic quantum numbers of a single-particle orbital are approximately good quantum numbers. Though the selection rules based on the Nilsson wave functions are broken in a loosely bound system as pointed out in Ref. [31], they serve as a zeroth order guideline for understanding the structure of the excitation modes. For the GT − operator, the nonvanishing matrix elements in N > Z nuclei are given as [27] | In the strength distribution of 100 Zr calculated with SkM*, we see a prominent peak at E * ≃ −1 MeV. The QRPA frequency of the lowest K = 1 state is ω = 0.33 MeV, and the sum of backward-going amplitude squared Y 2 is 0.49. This eigenstate is predominantly generated by a ν[422]3/2 ⊗ π[422]5/2 excitation. Noted that this is a h-h type excitation because the occupation probability of a π[422]5/2 orbital is 0.78. It indicates that this mode has large transition strengths for the proton-neutron-pair creation/annihilation operators as well.
In 106 Zr, we see an appreciable effect of the T = 0 pairing interaction on the low-lying state. With SkM*, the K π = 1 + eigenstate at ω = 1.  16. Therefore, the the low-lying mode in 110 Zr calculated with SLy4 is dominantly a p-h type excitation and the p-p residual interaction does not play a significant role.
From this analysis, we come to the following conclusion: The number of 2qp excitations generating the lowlying mode is small in the Zr isotopes under consideration. The K π = 1 + state possessing an appreciable strength is generated by mainly a 2qp excitation satisfying the selection rule (25). The effect of the p-p residual interaction in the low-lying mode thus depends on the location of the chemical potential, whether it is a p-h type or a p-p type excitation.
We also show the results in Ref. [28] in Fig. 5. Although the GTGR are predicted lower in energy in a separable approximation than by our calculations, the lowlying strength distributions are not very different. In a separable approximation, they have some small strengths in the energy region of 0 − 1 MeV in all the isotopes.
The low-lying GT strength distribution strongly affects the β-decay rate. Thus, we can clearly see the effect of T = 0 pairing in the β-decay life time. The β-decay half life T 1/2 can be calculated with the Fermi Golden rule as [32], where D = 6163.4 s and we set (g A /g V ) eff = 1 rather than its actual value of 1.26 to account for the quenching of spin matrix in nuclei [33]. The Fermi integral f (Z, Q β − E * i ) in (26) including screening and finite-size effects is given by with where γ = 1 − (αZ) 2 , ν = αZW/p, α is the fine structure constant, R is the nuclear radius. W is the total energy of β particle, W 0 is the total energy available in m e c 2 units, and p = √ W 2 − 1 is the momentum in m e c units [32]. Here, the energy released in the transition from the ground state of the target nucleus to an excited state in the daughter nucleus is given approximately by [26]  The results in Ref. [28] and experimental data [11,22,34,35] are also shown. Figure 6 shows the the β-decay half lives of the Zr isotopes thus calculated with the SkM* and SLy4 EDFs combined with and without the T = 0 pairing interaction. We see that the attractive T = 0 pairing interaction shortens substantially the β-decay half lives.
Without the T = 0 pairing interaction, the half lives calculated with SkM* are about 2 − 20 times longer than those with SLy4. About a half of the differences is due to the smaller Q β values calculated with SkM* than with SLy4. Thus, we need a stronger p-p interaction for SkM* to reproduce the observed half life of 100 Zr. Then, the half lives calculated with the two EDFs together with the T = 0 pairing interaction come closer to each other exept in 102,104 Zr. As mentioned in Sec. III A, each of the isotopes has the different pairing properties depending on the Skyrme-EDF employed.
In Fig. 6, we include the results in Ref. [28] together with the available experimental data [11,22,34,35]. The recent experiment data obtained at RIKEN RIBF show the short half lives in the 110 Zr region [11]. They are reproduced well by the calculation in Ref. [28], while it overestimates the half lives of the lighter Zr isotopes. A good reproduction of the half lives in the A ∼ 110 region may be due to the presence of the low-lying strengths. Our calculations, in particular employing SLy4, reproduce well the observed half lives systematically in 100−110 Zr.
The low-lying GT strengths relevant to the β-decay rate are a quite delicate quantity because they emerge as a consequence of cancellation between a repulsive ph residual interaction and an attractive T = 0 pairing interaction, both of which are largely uncertain in a nuclear EDF method. We need a more reliable EDF together with the proton-neutron pairing interaction that is able to describe well the spin-isospin excitations systematically in order to make further steps toward a selfconsistent and systematic description of β-decay of nuclei involved in the r-process nucleosynthesis. A virtue of our new framework developed in the present article is that it is suitable for a systematic calculation of the spin-isospin responses of nuclei because it is applicable to a nucleus with an arbitrary mass number whichever it is spherical or deformed, deeply bound or weakly bound in a reasonable calculation time with a help of the massively parallel computers, once the EDF and the proton-neutron pairing interaction are given. Through the systematic calculations employing several parameter sets of interaction and their comparison with available experimental data or observations, we can put constraints on the spin-isospin part of the new EDF and the proton-neutron pairing interaction. Then, we can proceed to a more reliable calculation.

IV. SUMMARY
We have developed the fully-self-consistent framework to calculate the spin-isospin collective modes of excitation in nuclei using the Skyrme EDF. We solve the deformed HFB equations on a grid in coordinate space. This enables us to investigate the excitation modes in nuclei off the stability line with an arbitrary mass.
Numerical applications have been performed for the Gamow-Teller excitation in the deformed neutron-rich Zr isotopes. We found a small amount of fragmentation due to deformation in the GT transition-strength distribution. The momentum-dependent terms in the p-h residual interaction play an important role in generating the collectivity. An attractive T = 0 pairing interaction has little influence on the energy of the GT giant resonance, while lowers the energies and enhances the GT strengths in the low energy region. The effect of the T = 0 pairing interaction in the low-lying mode depends sensitively on the location of the Fermi level of neutrons.
β-decay rates depend primarily on the Q β value, the residual interactions for both the p-h and the p-p channels, and the shell structures. The framework developed in this article treats self-consistently these key ingredients on the same footing. Once the strength of the T = 0 pairing interaction is determined so as to reproduce the observed β-decay half-life of 100 Zr, our calculation scheme produces well the isotopic dependence of the half lives up to 110 Zr as was recently observed at RIKEN RIBF.
Systematic calculations with the HFB + pnQRPA for nuclei in a whole nuclear chart help us not only to understand and to predict new types of collective modes of excitation in unstable nuclei, and to provide the microscopic inputs for the astrophysical simulation but also to shed light on the nuclear EDF of new generations.

Acknowledgments
The author thanks P. Sarriguren for providing him with the GT strength distributions to be compared. He also thanks K. Matsuyanagi, N. Van Giai, H. Z. Liang and F. Minato for stimulating discussions. This work was supported by KAKENHI Grant Nos. 23740223 and 25287065. The numerical calculations were performed on SR16000 at the Yukawa Institute for Theoretical Physics, Kyoto University and on T2K-Tsukuba, at the Center for Computational Sciences, University of Tsukuba. Part of the results is obtained by using the K computer at the RIKEN Advanced Institute for Computational Science and by pursuing HPCI Systems Research Projects (Proposal Number hp120192).