Isoscalar and neutron modes in the $E1$ spectra of Ni isotopes and the relevance of shell effects and the continuum

We study theoretically the electric dipole transitions of even Ni isotopes at low energies, using the self-consistent quasiparticle random-phase approximation with the D1S Gogny interaction and a continuum-RPA model with the SLy4 Skyrme force. We analyze isoscalar states, isovector states, and the dipole polarizability. We define a reference value for the polarizability to remove a trivial dependence on the mass number. We compare our results with data and other calculations, with a focus on collective states, shell effects, and threshold transitions. Our results support the presence of a strong isoscalar transition, with little or moderate $E1$ strength, as a universal feature of ordinary nuclei. In moderately neutron-rich Ni isotopes, namely ${}^{68}\mathrm{Ni}$ and neighboring isotopes, this transition is found bimodal due to couplings with surface neutrons. An adequate treatment of the continuum states appears essential for describing suprathreshold $E1$ strength, especially beyond ${}^{68}\mathrm{Ni}$. Very exotic isotopes $\left(N>50\right)$ are found highly polarizable, with practically all their $E1$ strength in the continuum. The dipole polarizability and the neutron-skin thickness are influenced by shell structure in different ways, so they can appear anticorrelated. A comparison with existing results for lighter (Ca) and heavier (Sn) nuclei suggests that the so-called pygmy dipole strength is influenced strongly by shell effects and that, partly for that reason, its isospin structure depends on the mass region.

Figure 1

Calculated photoabsorption cross section for the indicated interactions and isotopes. The calculated cross sections are smoothed with a Lorenzian of width equal to 2 MeV. For ${}^{58}\mathrm{Ni}$ (c) the dashed curve corresponds to a smoothing width of 5 MeV. The data plotted in (c) and (d) for ${}^{58}\mathrm{Ni}$ and ${}^{68}\mathrm{Ni}$ are from Refs. [44] and [38, 45], respectively. The arrows in (d) indicate peaks identified in Refs. [38] (black) and [46] (gray).

Figure 2

Calculated isoscalar strength distribution for the indicated interactions and isotopes. The distributions are smoothed with a Lorenzian of width equal to 2 MeV. For ${}^{58}\mathrm{Ni}$ (c) the dashed curve corresponds to a smoothing width of 5 MeV. The arrows show peaks detected below 10 MeV in $(\alpha ,{\alpha }^{\prime }{\gamma }_0)$ experiments [49] and mean energies of Gaussian fits to the IS strength distribution extracted from $\alpha$ scattering at small angles [50].

Figure 3

For the given isotopes and excitation energies, the IS and $E1$ transition strengths are indicated by green disks and black circles, respectively. The area of a disk or circle is proportional to the strength.

Figure 4

(a) Proton and (b) neutron transition densities of the IS-LED mode in even Ni isotopes, calculated with the QRPA+D1S model. For comparison, the transition densities of the lowest dipole mode in ${}^{80}\mathrm{Ni}$ are also shown.

Figure 5

Calculated occupation probabilities of the neutron $0f1p0g_{9/2}$ states in ${}^{56-78}\mathrm{Ni}$.

Figure 6

Proton and neutron transition densities of the neutron-mode (or pygmy) candidate in ${}^{64,68,72,76}\mathrm{Ni}$, calculated with the QRPA+D1S model. For comparison, the transition densities of the IS-LED mode (from Fig. 4) are also shown, plotted with thinner and paler lines.

Figure 7

The contributions of two-quasiparticle configurations to the transition matrix element of the IS-LED and the 10.6 MeV mode in ${}^{68}\mathrm{Ni}$, calculated in the QRPA+D1S model (a) for the IV operator, Eq. (6), and (b) for the IS operator, Eq. (5).

Figure 8

The IS and $E1$ strength distributions, for ${}^{68}\mathrm{Ni}$ and ${}^{78}\mathrm{Ni}$, calculated with the QRPA+D1S and CRPA+SLy4. The response functions are smoothed with a width of 0.4 MeV. The data in (c) are from Refs. [38, 45].

Figure 9

Sum of $E1$ strength for the indicated energy cutoffs as a function of mass number. For ${}^{58,60}\mathrm{Ni}$, data shown are from Refs. [59, 60]. The data points for $A=68$ are an estimate assuming a single peak with the reported mean energy and energy-weighted strength [38] (lower point) and the actual sum of strength measured between 7.8 and 10 MeV [45] (upper point). An earlier measurement of the energy-weighted summed strength [46, 61] gives the estimate $1.1\pm 0.3e^2{\mathrm{fm}}^2$. The single-neutron unit, $B_{\mathrm{s}.\mathrm{n}.}\left(E1\right)$, is given by Eq. (11).

Figure 10

The electric dipole polarizability $a_D$ of Ni isotopes (left axis) calculated with both models as indicated and the difference of the proton and neutron root-mean-square radii (right axis) calculated with the HFB+D1S model. The polarizability is given relative to the estimate $a_D^{\mathrm{coll}}$ for a single collective mode. The radii difference is divided by $A^{1/3}$. The measurement of $a_D$ in ${}^{68}\mathrm{Ni}$ is from Ref. [38].