Self-consistent Hartree-Fock and RPA Green's function method indicate no pygmy resonance in the monopole response of neutron-rich Ni isotopes

The random phase approximation (RPA) monopole strength and the unperturbed particle-hole excitation strength are studied, in which the strength in the continuum is properly treated without discretizing unbound particle spectra. The model is the self-consistent Hartree-Fock calculation and the RPA Green's function method with Skyrme interactions. Numerical examples are the Ni isotopes, especially ${}_{28}^{68}{\mathrm{Ni}}_{40}$, in which an experimental observation of a low-lying peak with an appreciable amount of monopole strength at $12.9\pm 1.0$ MeV was recently reported. In the present study it is concluded that sharp monopole peaks with the width of the order of 1 MeV can hardly be expected for ${}^{68}\mathrm{Ni}$ in that energy region. Instead, a broad shoulder of monopole strength consisting of neutron excitations to nonresonant one-particle states (called “threshold strength”) with relatively low angular momenta $(\ell ,j)$ is obtained in the continuum energy region above the particle threshold, which is considerably lower than that of the isoscalar giant monopole resonance. In the case of monopole excitations of ${}^{68}\mathrm{Ni}$ there are no unperturbed particle-hole states below 20 MeV, in which the particle expresses a neutron (or proton) resonant state. It is emphasized that in the theoretical estimate a proper treatment of the continuum is extremely important.

Figure 1

Monopole strength function (1) of ${}^{68}\mathrm{Ni}$ as a function of excitation energy. The SLy4 interaction is used consistently both in the HF and the RPA calculations. (a) Unperturbed (neutron plus proton) monopole strength and isoscalar monopole RPA strength. The RPA strength denoted by the solid curve includes all possible strengths due to the coupling between bound and unbound states in RPA. On the other hand, in the unperturbed response denoted by the short-dashed curve the p-h strengths, in which both particle and hole orbits are bound, are not included because of the different dimensions of the strength. The energies of those unperturbed p-h excitations are the $1d_{5/2}\to 2d_{5/2}$ excitation at 27.60 MeV for neutrons and the excitations of $1p_{3/2}\to 2p_{3/2}$ at 27.58 MeV and $1p_{1/2}\to 2p_{1/2}$ at 27.46 MeV for protons. In addition, the proton excitation at 27.3 MeV from the bound $1d_{5/2}$ orbit to the one-particle resonant $2d_{5/2}$ orbit has such a narrow width that the strength is not plotted. The narrow peaks at 24.1 and 24.7 MeV in the unperturbed strength curve are the proton $2s_{1/2}\to s_{1/2}$ and $1d_{3/2}\to 2d_{3/2}$ excitations, respectively. ISGMR peak is obtained at slightly less than 21 MeV. (b) Some unperturbed neutron threshold strengths, which contribute appreciably to the total unperturbed strength below the energy of ISGMR in Fig. 1, are shown for respective occupied hole orbits, ${\left(1f_{7/2}\right)}^{-1}$, ${\left(2p_{3/2}\right)}^{-1}$, ${\left(1f_{5/2}\right)}^{-1}$, and ${\left(2p_{1/2}\right)}^{-1}$. Below 15 MeV the sum of the four curves shown is exactly equal to the short-dashed curve in Fig. 1.

Figure 2

Monopole strength function of ${}^{78}\mathrm{Ni}$ as a function of excitation energy. The SLy4 interaction is consistently used both in the HF and the RPA calculations. (a) Unperturbed (neutron plus proton) monopole strength and isoscalar monopole RPA strength. The RPA strength includes all possible strengths. On the other hand, in the unperturbed strength that is denoted by the short-dashed curve the p-h strengths, in which both particle and hole orbits are bound, are not included because of the different dimensions of the strength. (b) Some unperturbed neutron threshold strengths, which contribute appreciably to the total unperturbed strength below the energy of ISGMR in Fig. 2, are shown for respective occupied hole orbits.