Introduction

The multipole response and appearance of pygmy dipole resonance (PDR) in finite nuclei far from the β-stability line have become hot issues in nuclear physics [1, 2]. Pygmy dipole resonance, which corresponds to the collective motion between the neutron skin and saturated core, has gained considerable attention because of its important applications in nuclear astrophysics and nuclear physics [3-7]. For example, the PDR found in the isovector giant dipole resonance could have a pronounced effect on the neutron capture rate in r-process nucleosynthesis. The properties of PDR are also used to constrain the equation of state of asymmetric nuclear matter [8-12], it plays similar role as the neutron skin in nuclear physics [13, 14].

PDR has been widely studied over the years, both experimentally and theoretically, using various methods [15-28]. In contrast, pygmy monopole resonance (PMR) has been much less analyzed experimentally in neutron-rich nuclei, and it has been theoretically predicted in neutron-rich Mg [29, 30], Ca [30-32], Ni [30, 33, 34], Sn, and Pb [35-37] isotopes. The calculations were based mainly on discretized quasiparticle random phase approximation (QRPA) or the finite amplitude method. It has been shown that PMR may significantly reduce the incompressibility in the nucleus with pronounced neutron excess, which could provide a more general and deeper understanding of nuclear incompressibility in isospin asymmetric systems [35]. Therefore, the measurement of isoscalar giant monopole resonances (ISGMR) and confirming the existence of PMR in neutron-rich nuclei are very important in nuclear physics. Following the suggestion of theoretical results, the ISGMR was measured by Vandebrouck et al. in the neutron-rich nucleus ⁶⁸Ni using inelastic α scattering at 50A MeV in inverse kinematics with the active target MAYA at GANIL [38, 39]. The pygmy monopole strength was observed at 12.9 MeV in addition to isoscalar giant monopole resonance. However, in the case of giant monopole resonance, excitations are usually built on the 2ℏω particle-hole configurations, which indicates that the particle states involved in all low-energy monopole excitations may be embedded in the continuum. Thus, correct treatment of the continuum in a neutron-rich nucleus is required to explain the experimental results.

In Refs. [40, 41], the nonrelativistic and relativistic continuum random phase approximations (CRPA) with Green’s function method were used to calculate the monopole strength distributions of ^68,78Ni, and the calculations indicated that there was no pronounced monopole state below the excitation energy of 20 MeV. Instead, a shoulder structure appeared in the low-energy region. This suggests that the discretized RPA may not be applicable to the calculation of the monopole response in ⁶⁸Ni, which should be replaced by the CRPA with Green’s function method. CRPA calculations show that there is no PMR for ^68,78Ni. However, it is unclear whether PMR exists in more neutron-rich Ni isotopes. In this work, we focus on the evolution of ISGMR in neutron-rich Ni isotopes, particularly with respect to its low-energy strength.

The PMR in an open-shell nucleus cannot be accurately described by the CRPA with Green’s function method because the pairing correlation is not considered. Hagino and Sagawa formulated a continuum quasiparticle random phase approximation (CQRPA) for open-shell nuclei in the coordinate space representation in Ref. [42]. The nucleon-nucleon interactions for the ground state adopted the Woods-Saxon type. For the residual interactions in the CQRPA calculations, they used the t₀ and t₃ parts of the Skyrme residual interactions. In this study, we extend the CQRPA model in Ref. [42] in a more consistent manner and applied it to study the ISGMR in neutron-rich Ni isotopes. In the new CQRPA model, the Schrödinger equation with a Woods-Saxon mean-field potential was replaced by the Hartree-Fock mean field theory with the standard Skyrme interaction in the ground state calculations. The Landau-Migdal forms of residual interactions derived from the Skyrme energy density functional (EDF) are adopted in the CQRPA calculations.

The remainder of this paper is organized as follows. In Sect. 2, we briefly introduce our theoretical framework. In Sect. 3, the CQRPA monopole strength distributions were investigated. The low-energy strengths of more neutron-rich Ni isotopes were studied carefully to explore the PMR. Finally, Sect. 4 provides summary and perspective.

Theoretical framework

In this work, the Skyrme Hartree-Fock + Bardeen-Cooper-Schrieffer (HF+BCS) and CQRPA methods were employed to study pygmy monopole resonance in neutron-rich nickel isotopes. The Skyrme interaction is expressed as an effective zero-range force between nucleons with density- and momentum-dependent terms, which has been successfully applied in the description of various nuclear properties [43, 44]. In this study, the Skyrme force SLy5 [45] was adopted for ground and excited state calculations. The pairing correlation is generated by a density-dependent zero-range force $V_{\text{pair}}\left(r_1\mathrm{,}{r}_2\right)=V_0\left[1-\eta \left(\frac{\rho (r)}{{\rho }_0}\right)\right]\delta \left(r_1-r_2\right)\mathrm{,}$ (1) where $\rho (r)$ is the particle density and ρ₀=0.16 fm^-3 is the density at nuclear saturation. η was set to 0.5, corresponding to a mixed pairing interaction. The pairing strength V₀ is fixed to be 483.5 MeV⋅fm³ by reproducing the empirical neutron gap in ⁷⁴Ni (Δ_n=1.262 MeV) [46, 47]. This value was then extended to calculations of other nickel isotopes.

The CQRPA model is briefly reviewed as follows. Further details are provided in Ref. [42]. The CQRPA response function ${\Pi }_{\text{CQRPA}}$ is governed by the Bethe-Salpeter (B-S) equation, its formalism generalized to the nuclear systems is given by ${\Pi }_{\text{CQRPA}}={\Pi }_0+{\Pi }_0{V}_{\text{res}}{\Pi }_{\text{CQRPA}}\mathrm{,}$ (2) where ${\prod }_0\left(r\mathrm{,}{r}^{\prime };E\right)$ is the unperturbed response function, which can be given by $\begin{array}{c}{\Pi }_0\left(r\mathrm{,}{r}^{\prime };E\right)=-{\displaystyle \sum _{\alpha \le \beta }}{D}_{\alpha \beta }\left(r\right)D_{\alpha \beta }\left(r^{\prime }\right)(\frac{1}{E_{\alpha }+E_{\beta }-E-i\eta }\\ +\frac{1}{E_{\alpha }+E_{\beta }+E-i\eta })\\ -{\displaystyle \sum _{\alpha }}{\varphi }_{\alpha }\left(r\right){\varphi }_{\alpha }\left(r^{\prime }\right)v_{\alpha }^2{\displaystyle \sum _{j_k{l}_k}}{\langle j_{\alpha }{l}_{\alpha }\Vert Y_L\Vert j_k{l}_k\rangle }^2\frac{1}{2L+1}\\ \times \{\langle r|\frac{1}{E_{\alpha }+\widehat{h}-\lambda -E-i\eta }\\ +\frac{1}{E_{\alpha }+\widehat{h}-\lambda +E-i\eta }|r^{\prime }\rangle \\ -{\displaystyle \sum _{\beta }}{\delta }_{j_k\mathrm{,}{j}_{\beta }}{\delta }_{l_k\mathrm{,}{l}_{\beta }}{\varphi }_{\beta }\left(r\right){\varphi }_{\beta }\left(r^{\prime }\right)\\ \left(\frac{1}{E_{\alpha }+{\epsilon }_{\beta }-\lambda -E-i\eta }+\frac{1}{E_{\alpha }+{\epsilon }_{\beta }-\lambda +E-i\eta }\right)\}，\end{array}$ (3) where $v_{\alpha }^2$, Eα, and the occupation probability, quasiparticle energy, and radial wavefunction of the quasiparticle state α, respectively. D_αβ(r) is the matrix element expressed as $D_{\alpha \beta }\left(r\right)={\varphi }_{\alpha }\left(r\right){\varphi }_{\beta }\left(r\right)\langle j_{\alpha }{l}_{\alpha }\left|\right|Y_L\left|\right|j_{\beta }{l}_{\beta }\rangle \frac{u_{\alpha }{v}_{\beta }+{(-)}^L{v}_{\alpha }{u}_{\beta }}{\sqrt{2L+1}}{\left(1+{\delta }_{\alpha \mathrm{,}\beta }\right)}^{-1/2}\mathrm{.}$ (4) In Eq. (3), the first term represents the two-quasiparticle excitations within the pairing active space, whereas the second term corresponds to the transitions from the inside to the outside of the pairing active space. The terms, V_res in Eq. (2) are the residual interactions in the B-S equation, which are from the second derivative of the Skyrme EDF with respect to the proton and neutron densities and are expressed by the Landau-Migdal parameters [48] in this study.

The monopole strength distribution S(E) of the system to an external field $V_{\text{ext}}\left(r\right)=r^2{Y}_{\text{LM}}\left(\widehat{r}\right)$ is then given by $S(E)=\frac{1}{\pi }\text{Im}{\displaystyle \int }\text{d}r\text{d}{r}^{\prime }{V}_{\text{ext}}\left(r\right)\Pi \left(r\mathrm{,}{r}^{\prime };E\right)V_{\text{ext}}\left(r^{\prime }\right)\mathrm{.}$ (5) After that, various moments can be calculated by means of the equation $m_k={\displaystyle \int }{E}^{k}S(E)\text{d}E\mathrm{,}$ (6) then one can obtain the constrained energy E_con and centroid energy E_cen $E_{\text{con}}=\sqrt{\frac{m_1}{m_{-1}}}\mathrm{,}{E}_{\text{cen}}=\frac{m_1}{m_0}\mathrm{.}$ (7) Besides, the ratio of mk for the low-energy (LE) PMR to the total strength, namely $R\equiv \frac{{\displaystyle {\int }_0^{E_{\text{LE}}}}{E}^{k}S(E)\text{d}E}{{\displaystyle {\int }_0^{E_{\text{max}}}}{E}^{k}S(E)\text{d}E}\mathrm{,}$ (8) is defined to quantify the evolution of the PMR with neutron excess, where E_LE is set to 11 MeV and E_max is equal to 40 MeV.

Results and discussions

First, we briefly discuss the ground-state properties of the nickel isotopes. The ground-state properties of finite nuclei are depicted using the HF+BCS method [50-52]. The HF+BCS equation is solved in coordinate space, where the radial size is set to 20 fm, which guarantees that the results under study are stable.

Table 1 shows the binding energies per nucleon, the neutron (proton) separation energies, charge radii and neutron Fermi energies in even-even ^68-84Ni isotopes calculated by using the SLy5 Skyrme interaction, meanwhile the calculated results are compared with the corresponding experimental data.

It can be seen that the binding energies per nucleon decrease with increasing mass number, and the calculated values can reproduce the measurements well. The neutron separation energies of ^68-82Ni predicted by the Skyrme EDF are somewhat smaller than the experimental data, but the calculated results can reproduce the data tendency with respect to the mass number well. In Table 1, it is shown that the theoretical proton separation energies S_p are in good agreement with the experimental data. We also show the calculated charge radii of even-even ^68-84Ni isotopes in the table, which increase with increasing mass number. For the studied nuclei, only two nuclei, ^68,70Ni, had experimental charge radii data. These results were well reproduced by the calculations. The calculated neutron Fermi energies are presented in the last column of Table 1, one can see that the neutron Fermi energies are approaching to zero when the nuclei are becoming more and more unstable.

For neutron-rich nickel nuclei, the discretized RPA has been proven to be unreliable, whereas Green’s function technique can properly take into account the contribution from the continuum [40, 41]. Therefore, CQRPA was adopted to explore the PMR in more neutron-rich Ni isotopes in the present study. As mentioned above, the residual interactions in the CQRPA calculations adopt the Migdal form. This means that the interactions used in CQRPA are not the same as those used in the ground-state calculations. We adjusted the residual interactions to ensure that a spurious isoscalar dipole state appeared at zero excitation energy, and the value of the renormalization factor was approximately 0.8.

The CQRPA monopole strength distributions for ^70-84Ni are shown in Fig. 1. One can see the monopole strengths for ^70-78Ni increased monotonically from the particle threshold to the ISGMR peak at approximately 21 MeV. This implies that a shoulder structure appears in the low-energy region for ^70-78Ni. This is consistent with the conclusions of Refs. [40, 41], there is no PMR for ^70-78Ni. However, starting from ⁸⁰Ni, the particle threshold becomes much lower, and an obvious PMR emerges in the energy region between 2.5 and 11 MeV. With the increasing of mass number, the low-energy strength becomes more and more strong.

Fig. 1Full-screen View

(Color online) The CQRPA monopole strength distributions for ^70-84Ni predicted by the SLy5 Skyrme interaction

We separated the low-energy PMR from the giant monopole resonance at an excitation energy of E=11 MeV, and calculated the ratios of the low-energy strength to the whole ISGMR strength for the non-energy-weighted sum rule m₀, inverse energy-weighted sum rule m_-1 and energy-weighted sum rule m₁, respectively. As illustrated in Fig. 2(a), the ratios of m₀ (orange circles), m_-1 (purple squares), and m₁ (black stars) of the low-energy strength are almost zero until mass number A=78. From ⁸⁰Ni, the three ratios increased significantly, and the values became larger in more neutron-rich nuclei. This suggests that the contribution of the PMR below 11 MeV increases with increasing mass number. The centroid energies E_cen (green pentagons) and constrained energies E_con (pink triangles) are plotted as functions of the mass number in Fig. 2(b), which is similar to Fig. 2(a), the values of E_cen and E_con remain constant when the mass number is not greater than 78, whereas the two energies are significantly decreased from ⁸⁰Ni to ⁸⁴Ni because of the appearance and enhancement of the low-energy monopole strengths. The values of E_cen are somewhat higher than those of E_con along the Ni isotopic chain, especially for ^80-84Ni.

Fig. 2Full-screen View

(Color online) (a) The ratios R of m₀, m_-1, and m₁ for the even-even nickel isotopes from ⁶⁸Ni to ⁸⁴Ni. (b) The centroid energies E_cen and constrained energies E_con in ^68-84Ni

In this paragraph, quasiparticle excitations in the low-energy region will be carefully investigated because these excitations may contribute significantly to the PMR strengths. The unperturbed and CQRPA monopole strengths of ^80-84Ni are shown in Fig. 3(a)-(c). For the giant monopole resonance, the CQRPA strengths are shifted down to a lower energy compared to the distribution of unperturbed strengths because the attractive residual interactions play an important role in isoscalar monopole excitation. The PMR strengths were slightly reduced compared to the unperturbed strengths, but the locations were almost unchanged. It can be seen that the low-energy strengths are highly sensitive to neutron excess. It was found that the low-energy strength below 20 MeV shown in the figures is made of the excitations mainly contributed by neutron states around the Fermi level, including 1g_9/2, 2d_5/2, 3s_1/2, and 2d_3/2. The corresponding single-particle energies E_s.p., gaps Δ, quasiparticle energies E_q.p. and occupation probabilities v² are listed in Table 2. We noticed that the single-particle energies of the four neutron states become increasingly bound with an increase in the neutron excess. It is shown that the gaps of the four neutron states are rather stable at approximately 0.6 MeV. The neutron states 2d_5/2 are just above or below the Fermi energies; therefore, the quasiparticle energies are relatively small. As for states 1g_9/2, 3s_1/2 and 2d_3/2, they are a little far from the Fermi energies, and their quasiparticle energies are relatively large except for state 3s_1/2 in ⁸⁴Ni, because its single-particle energy is much closer to the Fermi energy. One can see that the occupation probabilities of 1g_9/2 are almost 1.0, leading to relatively stable excitations. Other partially occupied orbits (2d_5/2, 3s_1/2 and 2d_3/2) changed their occupation probabilities when the neutron excess was increased. The corresponding unperturbed neutron threshold strengths, contributed by the excitation of neutrons around the Fermi surface to the continuum, are gradually enhanced[see Fig. 3(d)-(f)]: The occupancy probabilities of 2d_5/2 are increased much more than the other states with the filling of neutrons, from 0.33 in ⁸⁰Ni increased to 0.92 in ⁸⁴Ni. Therefore, the increase in low-energy strengths in ^80-84Ni is mainly due to the contribution of a stronger threshold strength of 2d_5/2.

Fig. 3Full-screen View

(Color online) Figures (a)-(c): The unperturbed and CQRPA monopole strength distributions for ^80-84Ni predicted by the SLy5 Skyrme interaction. Figures (d)-(f): Some unperturbed neutron threshold strengths, which contribute appreciably to the total unperturbed strength below 20 MeV in ^80-84Ni, are shown for respective occupied orbits, (1g_9/2)^-1, (2d_5/2)^-1, (3s_1/2)^-1, and (2d_3/2)^-1

Summary and Perspectives

In the present study, we extended the CQRPA approach in Ref. [42] in a more consistent manner, in which the model includes the Skyrme interaction for both ground and excited state calculations. Then, the consistent Skyrme HF+BCS and CQRPA models were applied to explore the emergence, evolution, and origin of low-energy monopole strengths along the even-even Ni isotopes. Shoulder structures at low-energy region for ^70-78Ni are found, which are similar to the conclusions in Refs.[40, 41]. However, the situation changed dramatically with the occupation of the weakly bound neutron orbitals. Indeed, starting from ⁸⁰Ni, pronounced pygmy monopole strengths were clearly identified. The origin of the low-energy monopole strength is attributed to neutron excitations from the weakly bound orbitals into the continuum, including neutron states 1g_9/2, 2d_5/2, 3s_1/2 and 2d_3/2. The changes in the ratios of low-energy strengths to total ISGMR strengths for m_-1, m₀ and m₁ as well as the centroid and constrained energies along Ni isotopes, are also discussed, and the changes are more obvious when the mass numbers are larger than 78, which are attributed mainly to the emergence of low-energy strengths. Eventually, the experimental data of PMR in neutron-rich nuclei are obviously inadequate, more efforts from the experimental investigations of PMR shall be made to confirm or disprove the predictions from models in the future.