Suddenly shortened half-lives beyond $^{78}$Ni: $N=50$ magic number and high-energy non-unique first-forbidden transitions

TL;DR

This study uses nuclear density-functional theory to explain the sudden shortening of half-lives in neutron-rich Ni isotopes beyond N=50, highlighting the role of shell gaps and high-energy forbidden transitions.

Contribution

It provides a microscopic explanation for the half-life shortening in Ni isotopes beyond N=50 using Skyrme EDF and pnQRPA, including forbidden transitions.

Findings

Reproduces the observed half-life shortening with Skyrme functionals.

Identifies the shell gap at N=50 as a key factor.

Highlights the importance of forbidden transitions in decay rates.

Abstract

$\beta$-decay rates play a decisive role in understanding the nucleosynthesis of heavy elements and are governed by microscopic nuclear-structure information. A sudden shortening of the half-lives of Ni isotopes beyond $N=50$ was observed at the RIKEN-RIBF. This is considered due to the persistence of the neutron magic number $N=50$ in the very neutron-rich Ni isotopes. By systematically studying the $\beta$-decay rates and strength distributions in the neutron-rich Ni isotopes around $N=50$, I try to understand the microscopic mechanism for the observed sudden shortening of the half-lives. The $\beta$-strength distributions in the neutron-rich nuclei are described in the framework of nuclear density-functional theory. I employ the Skyrme energy-density functionals (EDF) in the Hartree-Fock-Bogoliubov calculation for the ground states and in the proton-neutron Quasiparticle Random-Phase Approximation (pnQRPA) for the transitions. Not only the allowed but the first-forbidden (FF) transitions are considered. The experimentally observed sudden shortening of the half-lives beyond $N=50$ is reproduced well by the calculations employing the Skyrme SkM* and SLy4 functionals. The sudden shortening of the half-lives is due to the shell gap at $N=50$ and cooperatively with the high-energy transitions to the low-lying $0^-$ and $1^-$ states in the daughter nuclei. The onset of FF transitions pointed out around $N=82$ and 126 is preserved in the lower-mass nuclei around $N=50$. This study suggests that needed is a microscopic calculation where the shell structure in neutron-rich nuclei and its associated effects on the FF transitions are selfconsistenly taken into account for predicting $\beta$-decay rates of exotic nuclei in unknown region.

Tables

Table 1. Table 1: Summary of O A ( 𝒓 , 𝝈 ) [ V O ( 𝒓 ) ] {}^{A}\!O(\bm{r},\bm{\sigma})[^{V}\!O(\bm{r})] in the operators O ^ J ​ L ​ K A ​ ( V ) superscript superscript ^ 𝑂 𝐽 𝐿 𝐾 𝐴 𝑉 {}^{A(V)}\!\hat{O}^{JLK} needed for the matrix elements of the FF transitions, where m n subscript 𝑚 n m_{\mathrm{n}} is the mass of the nucleon. The factor Θ K subscript Θ 𝐾 \Theta_{K} arises from the transformation from the intrinsic to the laboratory frames of reference  boh75 : Θ K = 1 subscript Θ 𝐾 1 \Theta_{K}=1 and 2 2 \sqrt{2} for K = 0 𝐾 0 K=0 and K ≠ 0 𝐾 0 K\neq 0 , respectively.

${}_{}{}^A\hat{O}_{}^{000}$	${}_{}{}^A\hat{O}_{}^{J1K}$	${}_{}{}^V\hat{O}_{}^{10K}$	${}_{}{}^V\hat{O}_{}^{11K}$
$\frac{𝝈\cdot \nabla }{m_n}$	$\sqrt{\frac{4\pi }{3}}r{\left[Y_1\otimes 𝝈\right]}_K^J{\Theta }_K$	$\frac{{\nabla }_K}{m_n}{\Theta }_K$	$\sqrt{\frac{4\pi }{3}}r{Y}_{1K}{\Theta }_K$

Equations

$$\left(\begin{array}{cc}{h}^q(\mathbf{r}\sigma )-{\lambda }^q& {\tilde{h}}^q(\mathbf{r}\sigma )\\ {\tilde{h}}^q(\mathbf{r}\sigma )& -(h^q(\mathbf{r}\sigma )-{\lambda }^q)\end{array}\right)\left(\begin{array}{c}{\phi }_{1,\alpha }^q(\mathbf{r}\sigma )\\ {\phi }_{2,\alpha }^q(\mathbf{r}\sigma )\end{array}\right)=E_{\alpha }\left(\begin{array}{c}{\phi }_{1,\alpha }^q(\mathbf{r}\sigma )\\ {\phi }_{2,\alpha }^q(\mathbf{r}\sigma )\end{array}\right),$$

$$|i⟩$$

$${\hat{\Gamma }}_i^{†}$$

$${}^A{\hat{O}}_{\pm }$$

$${}^V{\hat{O}}_{\pm }$$

$$\frac{1}{t_{1/2}}$$

$$f$$

$$F(Z,W)=2(1+\gamma )(2pR{)}^{-2(1-\gamma )}{e}^{\pi \nu }{\left|\frac{\Gamma (\gamma +\mathrm{i}\nu )}{\Gamma (2\gamma +1)}\right|}^2,$$

$$W_0$$

$$=m_e{c}^2-E_{\mathrm{T},i}+\Delta M_{n-H},$$

$$C_0=|⟨1⟩{|}^2+{\lambda }^2|⟨\mathbf{\sigma }⟩{|}^2,$$

$$f{t}_{1/2}$$

$$f_0$$

$$\overline{C(W)}=\frac{f}{f_0}$$

$$C(W)=k+kaW+kb/W+kc{W}^2,$$

$$k$$

$$kb$$

$$k=$$

$$+\frac{1}{18}{W}_0^2(2x+u{)}^2-\frac{1}{18}{\lambda }_2(2x-u{)}^2,$$

$$ka=$$

$$kb=$$

$$kc=$$

$$k$$

$$ka$$

$$kc$$

$$V$$

$${\zeta }_0$$

$$Y$$

$${\zeta }_1$$

$$w$$

$$x$$

$$u$$

$$z$$

$${\xi }^{\prime }v$$

$${\xi }^{\prime }y$$

$$\begin{array}[]{ll}1-\dfrac{1}{5}\left(\dfrac{r}{R}\right)^{2},&0<r<R,\\ \dfrac{R}{r}-\dfrac{1}{5}\left(\dfrac{r}{R}\right)^{3},&r>R,\end{array}$$

Full text

Purpose: By systematically studying the $\beta$-decay rates and strength distributions in the neutron-rich Ni isotopes around $N=50$, I try to understand the microscopic mechanism for the observed sudden shortening of the half-lives.

Methods: The $\beta$-strength distributions in the neutron-rich nuclei are described in the framework of nuclear density-functional theory. I employ the Skyrme energy-density functionals (EDF) in the Hartree-Fock-Bogoliubov calculation for the ground states and in the proton-neutron Quasiparticle Random-Phase Approximation (pnQRPA) for the transitions. Not only the allowed but the first-forbidden (FF) transitions are considered.

Results: The experimentally observed sudden shortening of the half-lives beyond $N=50$ is reproduced well by the calculations employing the Skyrme SkM* and SLy4 functionals in contrast to the monotonic shortening predicted in the preceding calculation using the SkO’ functional.

Conclusions: The sudden shortening of the half-lives beyond $N=50$ in the neutron-rich Ni isotopes is due to the shell gap at $N=50$ and cooperatively with the high-energy transitions to the low-lying $0^{-}$ and $1^{-}$ states in the daughter nuclei. The onset of FF transitions pointed out around $N=82$ and 126 is preserved in the lower-mass nuclei around $N=50$. This study suggests that needed is a microscopic calculation where the shell structure in neutron-rich nuclei and its associated effects on the FF transitions are selfconsistenly taken into account for predicting $\beta$-decay rates of exotic nuclei in unknown region.

I Introduction

Magic numbers are a key quantity to understand the quantum many-body systems such as atomic nuclei. This is because the single-particle motion in a mean-field potential and the associated shell structure are an important concept in nuclear structure. The local stability with the specific combinations of neutrons and protons is naturally understood with the help of the magic numbers. The canonical values were established in the systematic studies of nuclei along the $\beta$ stability line. In the recent studies of neutron-rich nuclei, however, it was found that some of the magic numbers disappear and the new ones show up instead sor08 . Exploring the evolution of the shell structures and elucidating the underlying mechanism as functions of the neutron and proton numbers have been a fundamental research topic in the field of nuclear physics ots18 .

The neutron magic number $N=50$ around the very neutron-rich nucleus 78Ni $(N/Z\sim 1.8)$ has attracted a considerable interest not only in a view of nuclear structure but in a point of the $r$-process nucleosynthesis: 78Ni can be a doubly-magic nucleus and an important waiting point serving as a bottleneck in the synthesis of heavy elements hos05 . Therefore, there have been a numerous number of experimental efforts on if the $N=50$ magic number survives in 78Ni hak08 ; wel17 ; bar08 ; por12 ; zha04 ; pad05 ; van07 ; leb09 ; gad10 ; shi16 ; aoi10 ; tan14 ; xu14 .

Though single-particle energies in a spherical mean field are not a direct observable, two-nucleon and one-nucleon separation energies are often considered as a corresponding signature. A high-precision nuclear mass measurement at the IGISOL on the $Z=30-33$ nuclei with a mass number around $80-90$ revealed a reduction of the $N=50$ shell gap energy towards Ge ($Z=32$) and an increase at Ga $(Z=31)$ hak08 , where the experimental shell gap energy was defined by the difference of the two-neutron separation energies as $\Delta_{n}(N)=S_{2n}(N)-S_{2n}(N+2)$. By comparing with some mean-field calculations and mass models, Hakala et al. in Ref. hak08 obtained an indication of the persistent $N=50$ shell gap in 78Ni. The recent measurement on 79Cu at the CERN-ISOLDE revealed a reduction of the $N=48$ shell gap, suggesting indirectly the enhanced $N=50$ shell gap wel17 . Another mass measurement on the Zn $(Z=30)$ isotopes including 81Zn confirmed that the $N=50$ shell gap is maintained for the Zn isotopes by looking at the one-neutron separation energies, and supports the indication above bar08 . Porquet and Sorlin pointed out that the linear fits to the one-neutron separation energies lead to a diminished $N=50$ shell gap energy in 78Ni, where the collective effect at $Z=32$ was carefully examined por12 . On the other hand, the large-scale shell model calculation explaining these observations predicts a persistent shell closure in 78Ni sie12 .

The low-lying quadrupole state is sensitive to the softness of a spherical nucleus against quadrupole deformation. Thus, the excitation energy of the first-excited $2^{+}$ state and the transition matrix element are also a possible signature of the rigidity at magic numbers. A systematic investigation on the low- and medium-spin states in the $N=50$ isotones with $Z=32-37$ suggests a constant $N=50$ shell gap by comparing to the shell model calculation zha04 . The lowering of transition probability $B(E2)$ in the Ge isotopes up to 82Ge also indicates that the $N=50$ shell closure remains in the neutron-rich nuclei pad05 ; leb09 ; gad10 . The higher energy of the $2^{+}_{1}$ state and the lower $B(E2)$ value measured in 80Zn van07 and the recently observed $4^{+}_{1}$ state shi16 further confirmed the persistent $N=50$ shell closure in 78Ni. However, looking at the Ni isotopes, the $E(2_{1}^{+})$ value decreases up to $N=48$ and the $B(E2)$ value increases up to $N=46$ aoi10 . Thus, the ‘direct’ measurement of the $E(2_{1}^{+})$ and $B(E2)$ in 78Ni is strongly desired tan14 , while the recent theoretical calculations predict the doubly-closed structure in 78Ni hag16 ; now16 .

Even though the production yields are small and the experiments on the mass or the low-lying quadrupole state are difficult to perform, the measurement of $\beta$-decay half-lives $T_{1/2}$ is feasible. Thus, there have been some attempts to study the structure of very neutron-rich nuclei via half-lives. The $\beta$-decay $T_{1/2}$ were discussed in terms of the shell effects, in particular the nuclear deformation effect, in the S and Cl nuclei with $N=28$ sor93 and in the $N\simeq 40$ nuclei around 64Cr dau11 . In these works, the measured $T_{1/2}$ were compared with the results of model calculations such as the FRDM-QRPA mol03 or its earlier version. The authors in Refs. sor93 ; dau11 discussed how much of the deformation parameter is reasonable to explain the observation, and pointed out the shortcoming of the QRPA in which the deformation parameters for the mother and daughter nuclei are the same.

Xu et al. carried out a systematic measurement of the $\beta$-decay $T_{1/2}$ of 20 neutron-rich nuclei in the 78Ni region at the RIKEN-RIBF xu14 . They found a sudden shortening of $T_{1/2}$ in the Ni isotopes beyond $N=50$. When the $\beta$-decay $Q_{\beta}$ value is sufficiently high, $T_{1/2}$ is well approximated by a fifth-power dependence on $Q_{\beta}$. Experimentally, the monotonic shortening of $\log_{10}T_{1/2}$ of the Ni isotopes as a function of the neutron number was observed below $N=50$, indicating that the $Q_{\beta}$ value gradually increases in the isotopic chain below 78Ni. It was argued that the sudden shortening of $T_{1/2}$ beyond $N=50$ is due to the sudden increase in $Q_{\beta}$ because the nuclear deformation is not expected to occur: A large shell gap at $N=50$ gives us a dramatic increase in $Q_{\beta}$. As mentioned in Ref. xu14 , the neutrons outside the $N=50$ gap may have a contribution to the $\beta$-decay via the forbidden transitions. However, Borzov predicted that the first-forbidden (FF) transitions play only a minor role in the half-life of 79,80Ni, while it is important for describing the $\beta$-decay properties around $N\simeq 82$ and 126, and the $\beta$-delayed neutron emission probability around 78Ni bor03 . Therefore, not only the shell gap but also the details of the nuclear wave functions are needed to investigate for understanding the origin of the observed sudden shortening of $T_{1/2}$ in the Ni isotopes.

In the present article, I study systematically the $\beta$-decay $T_{1/2}$ in the neutron-rich Ni isotopes around 78Ni. Then, I try to understand the microscopic mechanism for the observed sudden shortening of the half-lives. To this end, the $\beta$-strength distributions in the neutron-rich nuclei are described in a microscopic framework of nuclear density-functional theory. Here, not only the allowed but the FF transitions are considered on the same footing.

This article is organized in the following way: The theoretical frameworks for describing the ground state and the nuclear matrix elements needed for the $\beta$-decay rates are given in Sec. II. And details of the numerical calculation are also given. However, a part of the details on the matrix elements needed for the FF transitions are given separately in Appendix. Section III is devoted to the numerical results and discussion based on the microscopic calculation. Then, summary is given in Sec. IV.

II Theoretical framework

II.1 HFB and pnQRPA for the nuclear matrix elements

In a framework of the nuclear energy-density functional (EDF) method I employed, the ground state of mother (target) nucleus is described by solving the Hartree-Fock-Bogoliubov (HFB) equation dob84 ; bul80 :

[TABLE]

where the mean field and the pair field are given by the functional derivative of the EDF with respect to the density and the pair density, respectively. The superscript $q$ denotes $\nu$ (neutron, $t_{z}=1/2$) or $\pi$ (proton, $t_{z}=-1/2$).

The excited states $|i\rangle$ of the daughter nucleus are described as a one-phonon excitation built on the ground state $|0\rangle$ of the mother (target) nucleus:

[TABLE]

where $\hat{a}^{\dagger}_{\nu}(\hat{a}^{\dagger}_{\pi})$ and $\hat{a}_{\nu}(\hat{a}_{\pi})$ are the neutron (proton) quasiparticle creation and annihilation operators. The phonon states, the amplitudes $X^{i},Y^{i}$ and the vibrational frequency $\omega_{i}$, are obtained in the proton-neutron quasiparticle random-phase approximation (pnQRPA).

The local one-body operators for the nuclear matrix elements relevant to the $\beta$-decay rates of the axial-vector and vector type transitions are written by

[TABLE]

respectively, where $\bm{\sigma}=(\sigma_{-1},\sigma_{0},\sigma_{+1})$ denotes the spherical components of the Pauli spin matrices, and $\tau_{\pm}=(\tau_{x}\pm\mathrm{i}\tau_{y})$ are the isospin-ladder operators. The nucleon field operators $\hat{\psi}^{\dagger}$ and $\hat{\psi}$ are expressed in terms of the quasiparticle operators $\hat{a}^{\dagger}$ and $\hat{a}$ through the generalized Bogoliubov transformation. I consider the allowed and first-forbidden (FF) transitions in the present calculation. For the allowed transitions, the operators needed are just for ${}^{A}\!O(\bm{r},\bm{\sigma})=\bm{\sigma}$ and ${}^{V}\!O(\bm{r})=1$, while for the FF transitions, one needs the operators for those containing $rY_{1}$ and those appearing due to the relativistic correction as summarized in Table 1. Here, the operators ${}^{A}\!\hat{O}^{J1K}$ and ${}^{V}\!\hat{O}^{11K}$ correspond to the charge-exchange rank-$J$ spin-dipole and dipole operators, respectively. The nuclear transition matrix elements $\langle i|\hat{O}|0\rangle$ are evaluated in the standard quasi-boson approximation as $\langle\mathrm{HFB}|[\hat{\Gamma}_{i},\hat{O}]|\mathrm{HFB}\rangle$.

II.2 numerical calculations

I solved the coordinate-space HFB equations in the cylindrical coordinates $\bm{r}=(\rho,z,\phi)$ with a mesh size of $\Delta\rho=\Delta z=0.6$ fm and a box boundary condition at $(\rho_{\mathrm{max}},z_{\mathrm{max}})=(14.7,14.4)$ fm. The qp states were truncated according to the qp energy cutoff at 60 MeV, and the qp states up to the magnetic quantum number $\Omega=23/2$ with positive and negative parities were included.

The two-body interactions for the pnQRPA equation were derived self-consistently from the EDF. I introduced the truncation for the two-quasiparticle (2qp) configurations in the QRPA calculation, in terms of the 2qp-energy as 60 MeV. More details of the calculation scheme are given in Ref. yos13 .

For the normal (particle-hole) part of the EDF, I employed mainly the SkM* functional bar82 and secondarily the SLy4 functional cha98 . For the pairing energy, I adopted the one in Ref. yam09 that depends on both the isoscalar and isovector densities, in addition to the pairing density, with the parameters given in Table III of Ref. yam09 . The same pairing EDF was employed for the $T=1$ pn-pairing interaction in the pnQRPA calculation, while the linear term in the isovector density was dropped. The $T=0$ pairing interaction was also included in the present pnQRPA calculation, with the same strength as the $T=1$ pairing interaction. Though I did not optimize the pairing strengths, the characteristic isotopic dependence was reproduced well by the present work.

II.3 calculation of the $\beta$-decay rates

The $\beta$-decay rate $\lambda_{\beta}$ and the partial half-life $t_{1/2}$ including the allowed and FF transitions can be calculated as beh82 ; sch66 ,

[TABLE]

where I used $D=6147$ s for the constant, and $C(W)$ is the shape factor containing the nuclear matrix elements as described below. The Fermi function $F(Z,W)$ in (7) including the effect of the Coulomb distortion on the electron wave function is given by

[TABLE]

where $\gamma=\sqrt{1-(\alpha Z)^{2}}$, $\nu=\alpha ZW/p$, $\alpha$ is the fine structure constant, and $R$ is the nuclear radius calculated as $1.2\times A^{1/3}$ fm. $W$ is the total energy of the electron, $W_{0}$ is the total energy available;

[TABLE]

and $p=\sqrt{W^{2}-1}$ is the momentum. $\Delta M_{n-H}$=0.782 MeV is a mass difference between a neutron and a hydrogen atom, and $E_{\mathrm{T},i}$ the excitation energy with respect to the ground-state of the mother (target) nucleus, and $\lambda_{\pi(\nu)}$ the chemical potential of protons (neutrons). I use natural units $\hbar=c=m_{e}=1$. The unit of length is the reduced electron Compton wavelength, $\lambdabar_{e}=386.16$ fm.

The shape factor for the allowed transitions is given as

[TABLE]

where $\langle\cdot\rangle$ denotes the nuclear transition matrix element between the ground and excited states $|i\rangle$ for the isospin lowering operator, and $\lambda=-(g_{A}/g_{V})=1.2701(25)$ is the ratio of weak axial and vector coupling constants. I used the quenching factor for the allowed Gamow-Teller transitions, $q=(1/1.27)^{2}=0.62$ or equivalently $\lambda=1$, as commonly used in the pnQRPA framework eng99 ; mol03 ; mus16 ; mar16 ; sar18 . Since the shape factor $C_{0}$ is independent of the energy $W$, it is convenient to define the phase space factor, or so-called the integrated Fermi function $f_{0}$ as

[TABLE]

When the FF transitions take part in the $\beta$-decay process, I need to consider the explicit dependence of the shape factors on the electron energy as the details are given in Appendix A. To discuss the FF transition strengths distribution and compare with the allowed transitions relevant to the $\beta$-decay, it is useful to define the averaged shape factor as beh82

[TABLE]

so that the partial half-life is related to $\overline{C(W)}$ as $D/f_{0}t_{1/2}$ similarly to the allowed transitions.

III Results and discussion

Figure 1 shows the calculated $\beta$-decay half-lives of the neutron-rich even-$N$ Ni isotopes by using the SkM* functional. One clearly sees a sudden shortening of the half-lives beyond $N=50$. Up to $N=50$ and above $N=52$, the calculation shows a monotonic decrease in the half-lives. One can thus expect something singular to have happened between $N=50$ and 52. In what follows, I am going to discuss the mechanism for this sudden shortening of the half-lives beyond $N=50$.

Before discussing the isotopic dependence of the calculated half-lives, I briefly mention the results obtained in the preceding pnQRPA calculations, which are also shown in Fig. 1. The calculation employing the relativistic functional D3C* mar16 gives the similar result to that using the SkM* functional, while the other microscopic calculations shown here mus16 ; bor05 produce the monotonic decrease. Taking a closer look at the result of the FRDM mol03 , where the FF transitions are taken into account by the gross theory, one sees that the model predicts a sudden shortening beyond $N=54$. The calculations except that using DF3 bor05 , where the large quenching factor $q=0.81$ was used, overestimate the observed half-lives. Though it is beyond the scope of the present work, there is room for a further investigation on the beyond-RPA effects such as in a particle-vibration coupling scheme as discussed in Ref. niu16 .

I show in Fig. 2 the calculated $\beta$-decay half-lives obtained in the allowed (A) approximation and those where the FF transitions are also taken into account (A+FF). The calculations in the A approximation only produce the monotonic shortening, while those in the A+FF approximation give a sudden shortening at $N=52$. Not only the calculation using SkM* but that using the SLy4 functional produce the sudden shortening. Thus, the FF transitions are necessary to explain the observed sudden shortening of the $\beta$-decay half-lives. The calculation scheme in Ref. mus16 is analogous to the present one in the sense that the Skyrme-type EDF was employed for the HFB+pnQRPA calculation and the FF transitions were taken into account in a microscopic way. The only difference to the present calculation is that the SkO’ functional rei99 was used there. I am going to investigate the reason why the SkM*, SLy4, and D3C* functionals produce the sudden shortening, while SkO’ only produces the monotonic one for the $\beta$-decay half-lives, then unravel the mechanism for the sudden shortening beyond $N=50$.

Figure 3 displays the FF contribution to the computed total $\beta$-decay rate. The calculations using the SkM*, SLy4, and D3C* functionals show that the FF transitions largely contribute to the $\beta$-decay rate at $N=52$ while the FF contribution is low at $N=50$. However, the calculation using SkO’ only shows the monotonic increase in the FF contribution as the neutron number increases from 46 to 52, and the much less FF contribution than the others for $N\geq 54$. Therefore, the onset of the FF transitions beyond $N=50$ is a key to the understanding of the mechanism for the sudden shortening of the $\beta$-decay half-lives in the neutron-rich Ni isotopes.

The onset of the FF transitions beyond $N=50$ is simply understood by the appearance of the low-energy $-1\hbar\omega_{0}$ excitation associated with the shell structure in neutron-rich nuclei yos17 . In the light nuclei around $A\simeq 40$, the similar mechanism for the onset of the FF transitions beyond $N=28$ was also discussed yos18 . Figure 4 shows the fraction of the summed strengths of the dipole ($\Delta J^{\pi}=1^{-}$) and the spin dipole ($\Delta J^{\pi}=0^{-},1^{-},$ and $2^{-}$) transitions in low energy below $E_{\mathrm{T}}<0$. The upper figure in Fig. 4 corresponds to Fig. 3(b) in Ref. yos17 . As the neutrons start to occupy the $2d_{5/2}$ and $2d_{3/2}$ orbitals, the negative-parity $\nu d_{5/2},d_{3/2}\to\pi p_{3/2},p_{1/2}$ excitations are possible to occur. From this figure, one sees that the $0^{-}$ and $1^{-}$ excitations are crucial for the sudden decease in $T_{1/2}$ between $N=50$ and 52. The contribution of the $2^{-}$ excitation is anti-correlated with the decrease in $T_{1/2}$. Figure 5 displays the shape factors and the averaged shape factors for the allowed and the FF transitions from 76-82Ni. As can be seen in Fig. 5(b), there show up several negative-parity states possessing an appreciable strength.

I am going to discuss further the microscopic origin of the sudden onset of the FF transitions beyond $N=50$. I show in Fig. 6 the contributions of the FF multipoles to the total $\beta$-decay rate obtained by using SkM*. One clearly sees that the $0^{-}$ and $1^{-}$ transitions are dominant beyond $N=50$. This can be understood by the location of the Fermi level of neutrons. Up to $N=50$, the $g_{9/2}$ is the only positive-parity occupied orbital. So, the $\nu g_{9/2}\to\pi f_{5/2}$ FF transition with $\Delta J^{\pi}=2^{-}$ is only possible to occur. When the Fermi level moves to the $d_{5/2}$ orbital, the partially occupied neutrons in the $d_{5/2}$ orbital can participate in the $\nu d_{5/2}\to\pi f_{5/2}$ excitation with $\Delta J^{\pi}=0^{-}$ and the $\nu d_{5/2}\to\pi p_{3/2}$ excitation with $\Delta J^{\pi}=1^{-}$ besides the $\nu d_{5/2}\to\pi p_{1/2}$ excitation with $\Delta J^{\pi}=2^{-}$ transition. One sees these excitations in the averaged shape factors for the FF transitions displayed in Fig. 5(b). The low-lying prominent peaks in 80,82Ni correspond to the $\Delta J^{\pi}=0^{-}$ and $1^{-}$ excitations. Furthermore, in the relatively higher energy region $E_{\mathrm{T}}\sim-5$ MeV, one sees the $\nu g_{9/2}\to\pi f_{5/2}$ excitation.

One more important thing for the sudden onset of the FF transitions beyond $N=50$ is the magnitude of the relative energy between the FF and allowed transitions. One sees in Fig. 5(a) a prominent peak in the shape factor around $E_{\mathrm{T}}\sim-5$ MeV, which is above the FF transition by about 5 MeV. This is predominantly constructed by the $\nu p_{1/2}\to\pi p_{3/2}$ excitation with the RPA amplitude $X^{2}-Y^{2}$ being greater than 0.9. The transition strength for this allowed transition is much larger than that for the FF transitions. Since the phase space factor $f_{0}$ roughly behaves as $W_{0}^{5}$, the FF transitions are able to overcome the contribution from the allowed transitions if they are high enough in energy, and are well apart from the allowed transitions. In the present case, the FF transitions gain a factor $\sim(10/5)^{5}=32$ from the phase space factor. Therefore, the contribution from the FF transitions at $N=52$ dominates that from the allowed transitions.

The low-lying states relevant to the $\beta$-decay rate are weakly collective in the present calculation for the Ni isotopes. Thus, I can discuss qualitatively the competitive roles played by the allowed and FF transitions in terms of the single-particle levels around the Fermi levels. I show in Fig. 7 the single-particle energies of neutrons and protons relative to that of the $\nu 2p_{1/2}$ and $\pi 1f_{7/2}$ orbitals, respectively calculated using the SkM*, SLy4, SkO’, and D3C* functionals. As discussed above, the $\nu g_{9/2}\to\pi f_{5/2}$ excitation is only available up to $N=50$ for the FF transition, and the $\nu p_{1/2}\to\pi p_{3/2}$ excitation dominantly contributes to the allowed transitions. So, the size of $N=40$ gap and the relative location between the $\pi f_{5/2}$ and $\pi p_{3/2}$ levels govern the $\beta$-decay rate for $N\leq 50$. As the shell gap at $N=40$ is small, the energy difference between the allowed and FF transitions is small. Thus, the FF contribution is small. In the calculation using SkM*, since the $\pi f_{5/2}$ level is located above the $\pi p_{3/2}$ level by 1.5 MeV, the FF contribution is strongly suppressed. On the other hand, as seen in the calculations with the SLy4, D3C*, and SkO’ functionals, the FF contribution is larger as the $N=40$ gap increases.

Beyond $N=50$, the multiple FF transitions involving the $\nu 2d_{5/2}$ orbital are possible to occur. The larger the sum of the gap energy of the $N=40$ and 50, the larger the FF contribution is. This is because the energy difference between the allowed and FF excitations is large. For a sudden onset of the FF contribution above $N=50$, preferable is the situation in which an $N=50$ gap is large and an $N=40$ gap is instead small.

IV Summary

I have carried out a systematic calculation of the $\beta$-decay rates for the neutron-rich Ni isotopes around $N=50$ by means of the fully selfconsistent proton-neutron-QRPA with the Skyrme EDFs. The experimentally observed sudden shortening of the half-lives beyond $N=50$ was reproduced well by the calculations employing the Skyrme SkM* and SLy4 functionals. I found that the onset of the first-forbidden (FF) transitions plays a decisive role for the sudden shortening of the half-life in 80Ni. This is due to a small sub-shell gap at $N=40$ and a large shell gap at $N=50$; the former suppresses the contribution from the FF transitions below $N=50$ and the latter enhances it above $N=50$. The present study reveals that a microscopic calculation taking the shell structure in neutron-rich nuclei and its associated effects on the FF transitions taken into account is necessary for predicting the $\beta$-decay rates of nuclei far from stability.

Acknowledgements.

The author thanks J. Engel and T. Marketin for valuable communications, and N. Van Giai for discussions. This work was supported by the JSPS KAKENHI (Grant No. 16K17687), and the JSPS-NSFC Bilateral Program for Joint Research Project on “Nuclear mass and life for unraveling mysteries of the r-process”. The numerical calculations were performed on CRAY XC40 at the Yukawa Institute for Theoretical Physics, Kyoto University, and on COMA (PACS-IX) at the Center for Computational Sciences, University of Tsukuba.

Appendix A First-forbidden $\beta$-decay rates

The shape factor for the FF transitions is energy dependent and given as sch66

[TABLE]

where the coefficients $k,ka,kb$, and $kc$ depend on the nuclear matrix elements and the maximum electron energy $W_{0}$. The non-vanishing coefficients are

[TABLE]

for rank 0,

[TABLE]

for rank 1, and

[TABLE]

for rank 2, respectively with $V,Y,\zeta_{0}$, and $\zeta_{1}$ being defined by

[TABLE]

Here, the dimensionless parameter $\xi$ is defined as $\xi=\alpha Z/2R$. The Coulomb functions $\mu_{1}$ and $\lambda_{2}$ are defined in terms of electron wave functions and depend on the electron momentum sch66 . These values are close to unity, so I used the approximations $\mu_{1}=1$ and $\lambda_{2}=1$ as usually adopted war88 .

The matrix elements are related to the form factors ${}^{A(V)}\!F_{Jls}$ as beh82

[TABLE]

where $\lambda=-(g_{A}/g_{V})=1.27$. For the FF transitions, no quenching factors were introduced for simplicity in the present calculation as in Ref. mus16 . The primed matrix elements $w^{\prime},x^{\prime}$, and $u^{\prime}$ are calculated with the operator multiplied by

[TABLE]

where $r=\sqrt{\rho^{2}+z^{2}}$ in the cylindrical coordinates.