Nuclear multipole responses from chiral effective field theory interaction

Giant resonances (GRs) in atomic nuclei are the most collective excitations in which many nucleons participate in a joint motion with various multipolarities and different spin-isospin quantum numbers. GRs are relevant to many physics problems, ranging from finite nuclei to infinite nuclear matter to neutron stars and supernovae Glendenning (1997); Harakeh and van der Woude (2001). The isoscalar giant monopole resonance (ISGMR) and isoscalar giant dipole resonance (ISGDR) provide direct way to probe the incompressibility of nuclei and nuclear matters Harakeh and van der Woude (2001); Garg and Colò (2018); Colò et al. (2014). The electric dipole polarizability $\alpha_{\rm D}$ quantifies the behavior of dipole response and is related to the neutron distribution Reinhard and Nazarewicz (2010); Hagen et al. (2016). The GRs of nuclei far from the valley of stability provide particular information on the structures of exotic nuclei and the equation of state (EOS) of neutron-rich matter.

The ISGMR in neutron-rich nuclei can be used to probe the density dependence of the symmetry energy because it is sensitive to the incompressibility of nuclear matter Fattoyev and Piekarewicz (2012); Piekarewicz and Centelles (2009); Hebeler and Schwenk (2014); Colò et al. (2014). For neutron-rich nuclei, there exist low-lying electric dipole responses with weak strengths, named pygmy dipole resonance (PDR) Brzosko et al. (1969); Savran et al. (2013). They are interpreted as the dipole oscillations of excess neutrons against a core made by all other nucleons Mohan et al. (1971). The PDR can be used to determine the neutron-skin thickness and the parameters of the EOS. Neutron capture cross sections in the astrophysical $r$-process are also impacted by PDRs Goriely (1998); Savran et al. (2013). Studying the evolution of GRs along an isotopic chain is useful in both experiment and theory for understanding of the isotopes and EOS. Nickel isotopes provide an excellent laboratory for the investigation of the evolution. In ⁵⁶Ni, Monrozeau et al. Monrozeau et al. (2008) implemented the first measurement of the isoscalar response. Recently, the multipole response strengths in the neutron-rich ⁶⁸Ni were observed Rossi et al. (2013); Vandebrouck et al. (2014, 2015). ⁷⁸Ni with an extreme neutron-proton asymmetry (28 protons and 50 neutrons) is claimed to be a doubly magic nucleus Taniuchi et al. (2019). It was commented that there is a competition between spherical and deformed shapes, which is challenging the current theory Taniuchi et al. (2019).

A variety of nonrelativistic and relativistic mean-field models have been successfully applied to the GRs of nuclei (see, e.g., Paar et al. (2007); Bai et al. (2010); Van Giai and Sagawa (1981a); Wang et al. (2017); Sagawa and Bertulani (1996); Litvinova and Wibowo (2018) and references therein). Calculations based on realistic nuclear forces have also made considerable progresses in the descriptions of the multipole responses, such as few-body approach Gazit et al. (2006); Bacca et al. (2015), no-core shell model Quaglioni and Navrátil (2007); Baker (2019), self-consistent Green’s function (SCGF) Raimondi and Barbieri (2019), coupled cluster combined with the Lorentz integral transform Bacca et al. (2013); Miorelli et al. (2016) and Hartree-Fock plus random-phase approximation (HF-RPA) Paar et al. (2006); Papakonstantinou et al. (2007); Wu et al. (2018).

In theory, the collective responses of nuclei are directly related to certain properties of the underlying nuclear force. The roles of the tensor force and three-nucleon force (3NF) have recently been highlighted in nuclear structure calculations. It has been claimed that the tensor force has a significant contribution to charge-exchange excitation Bai et al. (2009, 2010); Minato and Bai (2013). The 3NF has been known playing an important role in the first-principles calculations of nuclear matters and structures Sammarruca et al. (2012); Ekström et al. (2015); Jurgenson et al. (2009); Maris et al. (2011); Hergert et al. (2013); Ekström et al. (2014); Carlson et al. (2015); Ma et al. (2019). The chiral effective field theory (EFT) provides a robust framework to construct nucleon-nucleon interaction based on quantum chromodynamics Machleidt and Entem (2011). An important advantage of the chiral EFT is that it creates two- and three-nucleon forces on an equal footing. The chiral EFT interaction provides a good platform for analyzing the effects of the tensor force and 3NFs.

In the previous work Wu et al. (2018), we calculated the monopole, dipole and quadrupole resonances of the closed-shell nuclei ⁴He,^16,22,24O and ^40,48Ca using the chiral EFT NNLO_sat interaction within the Hartree-Fock plus RPA (HF-RPA) approach. The HF-RPA can reproduce experimental multipole resonances reasonably. In this work, we extend the calculations to heavier nuclei, ^56,68,78Ni, with particular focus on the roles of the tensor force and 3NF of the underlying realistic nuclear force in the giant resonances.

The intrinsic Hamiltonian of the $A$-nucleon system reads

where $V^{\rm NN}_{ij}$ is the two-body nucleon-nucleon ($\rm NN$) interaction, and $V^{\rm 3NF}_{ijk}$ is the three-nucleon force (3NF). The chiral EFT two-body interaction N³LO developed by Entem and Machleidt Entem and Machleidt (2003) is used. We include the 3NF via the in-medium two-body potential V${}^{\rm 3NF_{\rm eff}}$ that was derived from the chiral N²LO 3NF by integrating one nucleon over the Fermi sea (i.e., up to the Fermi momentum $k_{\rm F}$) in symmetric nuclear matter Holt et al. (2009); *PhysRevC.81.024002; Hebeler and Schwenk (2010). The extra low-energy constants for the chiral effective N²LO 3NF are $c_{\rm D}$ = $-$0.2 and $c_{\rm E}$ = 0.735 with the effective cutoff $\Lambda$ = 500 MeV and the Fermi momentum $k_{\rm F}$ = 0.95 fm^-1, which are the same as given in Ref. Hagen et al. (2012). With the chiral NN interaction at N³LO Entem and Machleidt (2003) and the effective in-medium 3NF_eff at N²LO, the coupled cluster calculations have well described binding energies and low-lying excitation energies of heavy $pf$-shell nuclei Hagen et al. (2012).

The chiral NN+3NF_eff is expressed in 13 major harmonic oscillator (HO) shells with the commonly used oscillator frequency $\hbar\omega$ = 24 MeV Hagen et al. (2012); Hu et al. (2019); Henderson et al. (2018). With the interaction established thus, we perform the HF-RPA calculations for the isoscalar monopole, isoscalar dipole, isovector dipole and isoscalar quadrupole resonances of the closed-shell nuclei ^56,68,78Ni. The detail of the HF-RPA approach can be found in our previous article Wu et al. (2018). The $E1$ photo-absorption cross section $\sigma(E)$ and electric dipole polarizability $\alpha_{D}$ are interesting observables for nuclear giant responses Raimondi and Barbieri (2019); Miorelli et al. (2016), which were not calculated in our previous paper on the He, O and Ca isotopes Wu et al. (2018). The $\sigma(E)$ and $\alpha_{D}$ measure the responses to the isovector dipole resonance. We have

and

where $\alpha$ is fine-structure constant and $E$ is the excitation energy of the resonance. $R(E)$ is the response strength distribution of the $E1$ transition,

where $B(E1,0\to\nu)$ is the reduced electric dipole transition probability Wu et al. (2018). The summation is over all possible particle-hole excitation modes Wu et al. (2018). $\hbar\Omega_{\nu}$ stands for the excitation energy of an excitation mode which is obtained in the HF-RPA calculation Wu et al. (2018). The discrete stength distributions $R(E)$ is smoothed by using the Lorentzian function,

where the width of $\Gamma$ = 2 MeV Paar et al. (2006); Papakonstantinou et al. (2007); Wu et al. (2018) is used.

Figure 1 gives the HF-RPA results for ⁵⁶Ni, compared with available experimental data Monrozeau et al. (2008). In the experiment Monrozeau et al. (2008), the centroid energies of the ISGMR and isoscalar giant quadrupole resonance (ISGQR) were measured at 19.3$\pm$0.5 MeV and 16.2$\pm$0.5 MeV via the ⁵⁶Ni($d$,$d^{\prime}$) reaction. The data provide useful information to test theoretical approaches and can be used to extract the nuclear matter incompressibility Harakeh and van der Woude (2001); Garg and Colò (2018); Colò et al. (2014). In order to dissect the roles of the different components of the interaction, we use the spin-tensor decomposition method Elliott et al. (1968); Kirson (1973); Umeya and Muto (2006) to decompose the interaction into central, tensor, and spin-orbit parts. The calculation without tensor force means that the tensor terms in NN+3NF_eff are taken away by using the spin-tensor decomposition method. As shown in Fig. 1, the tensor force and 3NF have no significant effect on the IS monopole $0^{+}$ resonance, while their effects on other multipole resonances are seen clearly. The tensor force shifts the energy of the low-lying $2^{+}$ state by 2 MeV, and other IS quadrupole $2^{+}$ and IS dipole $1^{-}$ peaks are shifted within 5% (the percent of the peak-energy shift over the peak energy). It changes the centroid energy of the IV dipole $1^{-}$ resonance by 16%. It seems that the 3NF effects are more significant in the IS quadrupole $2^{+}$ and IS dipole $1^{-}$ resonances. The calculation without 3NF cannot clearly give the peak at $\sim$ 1 MeV in the IS quadrupole $2^{+}$ response distribution. The calculated RPA wave function shows that the low-lying $2^{+}$ state is caused by several one-particle one-hole excitations with the neutron or proton excited from $0f_{7/2}$ to $0f_{5/2}$ or $1p_{3/2}$. The peak seems to correspond to the experimental 2${}^{+}_{1}$ excited state at $\sim$ 2.7 MeV nnd . For the $2^{+}$ state, the present HF-RPA gives a reduced E2 transition probability $B(E2,0^{+}\to 2^{+})$ = 0.454 e²b², larger than the experimental datum of 0.060(12) e²b² Kraus et al. (1994). The discrepancy between the calculations and data would originate from missing higher-order correlations in RPA Wu et al. (2018).

We find that the 3NF effect on giant resonances is more pronounced in light nuclei. In Fig. 2, we show the calculations of the IV dipole resonances for the closed-shell nuclei, ⁴He, ¹⁶O and ^40,48Ca. The nuclei were investigated in our previous paper using the chiral NNLO_sat with the 3NF being normal ordered at a two-body level Wu et al. (2018). Here, we recalculate their IV dipole resonances and analyze the 3NF effect by using the 3NF_eff. Obtained results using the 3NF_eff are similar to those given by using NNLO_sat Wu et al. (2018). We see that the 3NF effects are significant, particularly in ⁴He and ¹⁶O. Since we have no NNLO_sat interaction matrix elements for the heavy Ni isotopes, the N³LO(NN) and NNLO(3NF_eff) are used in the present calculations.

In Table 1, we calculate the dipole polarizability $\alpha_{\rm D}$ for the nuclei, compared with data Pachucki and Moro (2007); Ahrens et al. (1975); Birkhan et al. (2017), coupled-cluster (CC) Miorelli et al. (2016); Hagen et al. (2016) and density functional theory (DFT) Colò et al. (2013) calculations. In the CC calculations Miorelli et al. (2016); Hagen et al. (2016), the chiral NNLO_sat(NN+3NF) interaction was used. For the DFT results, the self-consistent RPA calculations with Skyrme forces SG$\rm II$ Van Giai and Sagawa (1981b), SkM* Bartel et al. (1982), SkP Dobaczewski et al. (1984), Sk255 Agrawal et al. (2003), SLy4 Chabanat et al. (1998); *1998441, Sly5 Chabanat et al. (1998); *1998441 and LNS Cao et al. (2006) were performed by using the skyrme_-rpa code Colò et al. (2013). The use of different Skyrme forces gives a range of the $\alpha_{\rm D}$ values, shown in Table 1. The present and DFT calculations overestimate slightly the dipole polarizability $\alpha_{\rm D}$ in ⁴He, ¹⁶O and ⁴⁸Ca, while in ⁴⁰Ca the $\alpha_{\rm D}$ is slightly underestimated in the present and CC calculations. The tensor force has no significant effect on the electric dipole polarizability $\alpha_{\rm D}$, as shown in Table 1.

Figure 3 shows the results for ⁶⁸Ni. It is seen that the effect of tensor force is similar to that in ⁵⁸Ni. In the neutron-rich ⁶⁸Ni, the isovector PDR (IVPDR) and isovector GDR (IVGDR) were observed recently by Rossi et al. Rossi et al. (2013) with peaks located at 9.55(17) MeV and 17.1(2) MeV, respectively. We see that the calculation with 3NF can give the IV $1^{-}$ PDR peaked at energy $\approx$ 10 MeV, while the calculation without 3NF does not show a clear PDR peak. However, the reduced electric dipole transition probability $B(E1)$ calculated without 3NF shows a possible weak PDR at energy $\approx$ 10 MeV. There is a strong resonance peak at $\approx$ 10 MeV in the IS $1^{-}$ channel, which overlaps with the weak IV $1^{-}$ peak. This indicates a mixing nature of isoscalar and isovector resonances in the dipole channel at low energy. The mixing was seen in our previous calculations Wu et al. (2018) for the neutron-rich ^22,24O. Experimentally, the IS dipole resonance can be incurred by inelastic scatterings with an isoscalar particle (e.g., $\alpha$ particle), while electromagnetic excitations (usually by electron scattering) give the total strength of the IS and IV resonances. The recent experiment Vandebrouck et al. (2015) has shown the possible IS dipole resonances in the energy range of $\approx$ 11$-$29 MeV (see Fig. 3).

The experiment by Rossi et al. Rossi et al. (2013) presents the first measurement of the dipole polarizability $\alpha_{\text{D}}$ in an unstable neutron-rich nucleus. Using the measured $\alpha_{D}$, the authors Rossi et al. (2013) deduced a neutron-skin thickness of 0.17(2) fm in ⁶⁸Ni by taking a nearly linear relation between $\alpha_{D}$ and neutron-skin thickness guided by the relativistic RPA calculation Piekarewicz (2011). The data provide further constraint on the isospin-asymmetric part of the EOS.

Table 2 lists the calculated isovector dipole polarizability, compared with the data Rossi et al. (2013), SCGF Raimondi and Barbieri (2019) and DFT Colò et al. (2013) calculations. The calculated dipole polarizability agrees well with the data Rossi et al. (2013) and SCGF Raimondi and Barbieri (2019). In the SCGF calculations, NNLO_sat(NN+3NF) was used. There are two IV dipole resonance peaks at the low energies of 10.6 and 11.6 MeV, similar to the SCGF conclusions. The mean-field calculation Khan et al. (2011) predicted a possible soft monopole excitation in ⁶⁸Ni. However, we do not see the soft monopole mode in the present calculations, while there is a low-lying IS quadrupole response peak at the energy of $\approx$ 4.2 MeV. The low-energy IS quadrupole peak would correspond to a soft resonance involving few-particle few-hole excitations or even one-particle one-hole excitation as happening in ^22,24O Wu et al. (2018). These may need to be verified further by experiments and other models. In Table 2, we also show the results of ⁷⁸Ni as prediction.

The sum rule of the response strength distribution can be used to analyze the interaction in the momentum dependence and isospin exchange Erler et al. (2011). The energy-weighted sum rule (EWSR) is defined by Orlandini and Traini (1991)

where $\kappa$ is the so-called enhancement factor which can be obtained by integrating the strength function Wu et al. (2018). As shown in Table 2, the present calculation with 3NF_eff gives that, in ⁶⁸Ni, the strength below 12 MeV exhausts 5.2% of the classical Thomas-Reiche-Kuhn (TRK) energy-weighted sum rule (EWSR), compared with the experimentally extracted value of $\approx 5\%$ Wieland et al. (2009) or 2.8(5)% Rossi et al. (2013). Note that the PDR peak at 10.64 MeV contributes about 2.0% of the EWSR, and the 11.55 MeV peak contributes about 3.2%. We predict that it is about 5.8% in ⁷⁸Ni. Table 2 also gives the DFT results, as discussed in Table 1. The DFT calculations give smaller EWSR but larger $\alpha_{\rm D}$ than the present calculations in ^68,78Ni.

Figure 4 displays the calculated ISGMR, ISGQR, ISGDR, and IVGDR for ⁷⁸Ni, predicting a IV $1^{-}$ PDR peaked at 11 MeV. We see that the PDR is enhanced in the calculation with 3NF. The calculated centroid energies of the isovector PDR and GDR are given in Table 2. The isovector GDR energy in ⁷⁸Ni is 1.5 MeV lower than the one in ⁶⁸Ni. The dipole polarizability $\alpha_{\rm D}$ is also predicted in Table 2. ⁷⁸Ni has been believed to be a doubly magic nucleus, with a high $2^{+}_{1}$ excitation energy at 2.6 MeV Taniuchi et al. (2019). The present HF-RPA calculation with 3NF gives 2.90 MeV for the $2_{1}^{+}$ excited state, which is consistent with the ab-initio coupled cluster and in-medium similarity renormalization group calculations given in Ref. Taniuchi et al. (2019).

From the calculations discussed above, we see that 3NF effects on collective multipole resonances are meaningful, particularly for light nuclei. The 3NF (including tensor ingredients) plays a crucial role in the formations of the PDRs in ^68,78Ni. Fig. 5 plots the binding energies calculated within the RPA framework. In the binding energy calculation, the correction from the second-order perturbation Wu et al. (2018); Hu et al. (2016) is considered. We see that the tensor force provides more than 50% of the binding energy. If tensor components were taken away from the realistic interaction, we should not be able to describe the ground states of nuclei correctly. The 3NF also has a significant effect on binding energy, and improves the calculation, see Fig. 5. We find that the second-order many-body perturbation correction is not converged in ⁵⁶Ni if only two-body interaction is considered in the calculation. This is why in Fig. 5 the ⁵⁶Ni binding energy is missing in the NN-only cure.

In realistic nuclear forces, the tensor component is large. For example, in the chiral EFT leading-order term that consists of one-pion exchange and contact interactions, the tensor force has the same strength as the central force in the one-pion exchange Machleidt and Entem (2011); Tanihata (2013); Sagawa and Colò (2014). The leading-order term should be most important for the calculations of binding energy and other observables. The result that the tensor force provides a large proportion of the binding energy is quite general in calculations based on realistic forces. In Refs. Tanihata (2013); Sakai et al. (1974), the calculation with the Hamada-Johnston and Tamagaki interaction shows that about 50% of the ⁴He potential energy comes from the tensor force. In the Green-function Monte-Carlo calculations Pieper and Wiringa (2001), the one-pion exchange in the AV18 interaction provides 70%-80% of the nuclear potential energy for light nuclei. With the present method and interaction, we have also calculated the binding energies of ⁴He, ¹⁶O and ^40,48Ca, giving the similar result, i.e., the tensor force provides about 50% of the binding energy.

Starting from the chiral effective field theory interaction N³LO(NN)+N²LO(3NF_eff), we have performed the RPA calculations within the Hartree-Fock (HF) approach, to investigate the monopole, dipole and quadrupole resonances in ^56,68,78Ni. The ground-state energies have also been calculated by incorporating the second-order perturbation correction into the HF-RPA energy. The present HF-RPA calculations reproduce reasonably the multipole resonances observed in ^56,68Ni and their binding energies as well. The pygmy dipole resonance, the dipole polarizability $\alpha_{\rm D}$ and the sum rule in ⁶⁸Ni have been discussed, and compared with experimental data available. The properties of ⁷⁸Ni have been predicted.

We dissect the 3NF and tensor terms of the realistic interaction, to see their roles in the multipole resonances of nuclei. Although the tensor force may be important for charge-exchanged collective resonances, it is not such significant for the electric giant resonances. However, the tensor force provides more than half of binding energy using the chiral EFT interaction. The three-body force has a nonnegligible effect on multipole resonances, particularly on the formation of the pygmy resonance. The tensor force and three-body force hardly affect the isoscalar monopole giant resonance. In possible future work, we will extend the present framework to charge-exchange excitations.