Onset of collective low-energy dipole resonances

The next step of investigation starts with RRPA calculations for several nickel isotopes. Results of the isovector dipole strength distributions are shown in the Fig.7, for the $^{48}$Ni,$^{56}$Ni, $^{68}$Ni and $^{78}$Ni nuclei. Low-energy peak for the $^{48}$Ni correspond mainly to the proton particle-hole configuration $(96\%\ 1f_{7/2} \to 2d_{5/2})$. As the number of neutrons increases, the next appearance of the low-lying strength is observed in $^{62}$Ni and heavier isotopes. Apart from the increase of the dipole strength, large relative contribution in low-energy region E$\leq$ 10 MeV is also observed; the ratio of energy weighted moments, $m_{1,low}/m_{1,high}$ varies from 0.9% for $^{62}$Ni, toward 6.1% for the $^{78}$Ni nucleus (see Fig.3). Like in the light nuclei, low-lying excitations correspond mainly to the neutron response. In the case of $^{68}$Ni, single particle-hole configuration dominates in the most of low-lying peaks. The exception of this behavior is peak at 9.0 MeV (4.3% EWSR), with many particle-hole pairs contributing to the corresponding dipole strength. The p-h configurations participating in this peak are shown in the Table . RRPA amplitudes are more evenly distributed among many transitions from all valent shell orbitals, thus describing excitations with increased degree of collectivity similar to GDR. Low-lying dipole strength in $^{78}$Ni has the similar structure of fragmented RRPA amplitudes among several possibilities of excitation. Transition densities of $^{78}$Ni, presented in the Fig.8, have longer tail with strong neutron participation beyond the rms radius for the low-lying peak at 8.9 MeV (4.0% EWSR). The Steinwedel and Jensen hydrodynamic two-fluid model of the core and excess fluid describes the onset of pygmy dipole resonance in which the neutron excess move against the core [12]. In this model, the energy of the low-lying resonance is predicted as

$$E_{PR} = \sqrt{\frac{Z(N-N_c)}{N(Z+N_c)}}E_{GDR},$$ (13)

where $N_c$ refers to the number of neutrons that form the core together with protons, and $E_{GDR}$ is energy of the giant dipole resonance. Comparison of RRPA centroid energies of the low-lying dipole strength and pygmy resonances in hydrodynamical model [12] in the case of nickel isotopes is presented in the lower part of Fig.9. The extreme case for the number of excess neutrons $N_c=Z$ in ( ) is used in calculations within the hydrodynamical model. As an additional check (upper part in Fig.9), the RRPA GDR-energies are compared with the approximate relation describing mass dependence of the isovector giant dipole resonance $E=78A^{-1/3}$ (the best in $A>100$ nuclei), obtained by fitting to the experimental results [1]. The energies of low-lying RRPA modes generally have higher values, and agreement is not so fair as between the GDR and its mass dependence law. In contrast to the RRPA results, the Steinwedel and Jensen hydrodynamical model [12] predicted that the Pygmy dipole resonance (PDR) energy increase with increasing number of excess neutrons. Similar quantitative disagreement with the hydrodynamic model have been observed within the density functional theory [20], with PDR centroid energy of calcium isotopes decreasing in interval (9.5,7.5) MeV as the number of excess neutrons to the proton-neutron core increase from 2-8 neutrons.

We continue this analysis in the region of periodic table from $^{100}$Sn toward the neutron drip line. Isovector dipole response to the RRPA states for the $^{100}$Sn,$^{114}$Sn, $^{120}$Sn and $^{132}$Sn nuclei is displayed in the Fig.10. As the number of neutrons increases, the onset of low-lying strength below 10 MeV is observed. Similar low-lying structure is repeated for tin isotopes: among several peaks with dominant neutron single particle-hole excitation, a peak with more fragmented strength among many p-h excitations, exhausting around 2% of the EWSR is located in the interval 7-9 MeV. In particular, for the case of $^{132}$Sn, neutron p-h pairs contributing to the peak at 8.6 MeV, exhausting 1.4% EWSR, are listed in the Table . Therefore, the corresponding dipole strength is mostly determined by several p-h excitations from the orbitals of excess neutrons. On the other side, protons are contributing only 10.4% to the RRPA amplitude of eigenvalue 8.6 MeV.

In the study of $^{208}$Pb within the Hartree-Fock + RPA model with Skyrme-type interaction  [19], at the energy of the lowest pygmy states the neutron response was found a factor of 10 larger than the proton response. Recent RRPA-analysis [14], results with proton participation of 14% in the pygmy state of $^{208}$Pb. In Fig.11 we plot the transition densities of $^{132}$Sn to the two states at 8.6 MeV and 14.8 MeV. For the GDR state at 14.8 MeV, proton and neutron transition densities display a usual radial dependence of isovector giant dipole resonance, with proton and neutron densities oscillating in opposite phases. In the lower part of Fig.11 contributions of excess neutrons $(50<N\leq82)$ filling the 1g$_{7/2}$, 2d$_{5/2}$, 2d$_{3/2}$, 3s$_{1/2}$ and 1h$_{11/2}$ orbitals are separated from the rest of neutrons which form the core with N=Z=50. Transition densities of the neutron excess correspond approximately to the total neutron transition densities. The contribution from the core neutrons is limited because the p-h configurations which involve core neutrons have higher excitation energies than the energies of the low-energy excitations. The transition densities of neutron excess and proton-neutron core (Fig.11(c)) have the same sign for the state at 8.6 MeV. As the radius increases, the core contribution vanishes, with excess neutrons oscillating out of phase with respect to $^{100}Sn$ core. This is in contrast to the GDR state at 14.8 MeV, where the contributions of protons and neutrons are comparable through the radial coordinate (Fig.11(d)). In the case of many p-h configurations contributing to the low-lying state, this behavior of transition densities can be interpreted as a collective oscillation associated with the pygmy dipole resonance.

In the microscopic density functional theory, the nucleon density variation calculated for the pygmy states, displays similar neutron polarization: surface neutron density oscillating out of phase with a stable core [20]. In order to achieve better understanding of collective nuclear dynamics determined by the microscopic RRPA calculations, the study of transition currents is applied. Transition currents and the corresponding velocity fields are more sensitive to the properties of interaction, than the strength distribution function or the transition densities [29]. The velocity fields are derived from the transition currents using the procedure described in Ref. [29]. Results for the two states at 8.6 MeV and 14.8 MeV are compared in the Figs.12 and 13, respectively. The velocity field of the excess neutrons $(50<N\leq82)$ is separated from the contribution of proton-neutron core $(Z,N\leq50)$, and velocity vectors are normalized to the largest neutron velocity. In the case of peak at 14.8 MeV, the core and excess neutrons mostly contribute to the giant resonance in the similar way. The largest velocities are noticed in the internal region of the core (Fig. 13(c), when compared with the velocities of the neutron excess (Fig.13(d)), and close to the z axis both velocities are in phase. On the other side, for the low energy peak at 8.6 MeV the core velocities (Fig.12(a),(b) are in phase with the velocities of the excess neutrons only in the central part of nuclei. Although the presented velocity fields seem to be more complicated than vibrations in simple macroscopic models, collective dynamics characteristic to the Pygmy resonances have been identified, with the excess neutrons oscillating against the inert core of the rest of nucleons occupying the same shell model orbitals.

In the lower part of Fig.14, the RRPA energies of low-lying modes in tin isotopes are compared with the pygmy resonances calculated in the hydrodynamical model( ) for equal number of protons and neutrons in the core. The RRPA giant resonance energies (Fig.14, upper part) nicely follow the mass dependence law $E=78A^{-1/3}$ describing the GDR energies from experimental study [1]. Evaluated low-lying RRPA modes slightly decrease with the neutron number, with a few MeV higher values when compared to the simplified hydrodynamical prediction.

Finally, let us investigate the isovector dipole response of medium heavy nuclei with extremely large number of neutrons. As an example we take the $^{122}Zr$ (Z=40, N=82) (Fig.15(a)). Low-lying state at 7.7 MeV exhausting 3.3% of EWSR, has the most fragmented RRPA amplitude among 15 neutron p-h pairs, each contributing from 0.1% to 27.5%. In comparison, the GDR state is mainly dominated with 27 neutron p-h pairs, each participating with 0.1%-17.3%. The proton p-h excitations contribute 10.5% to the low-lying RRPA eigenvalue, in contrast to the 29.0% for the GDR state. In the Fig.15(b), (c) transition densities of the low-lying state are shown. As expected, the transition density of excess neutrons $(50<N\leq82)$ is strongly dominating and is out of phase against the proton-neutron core (Z=40, N=50) in the region close to the surface. The analysis of the velocity fields in the low-lying peak (Fig.16) support the characteristic image of the pygmy resonances. Significant, out of phase contributions to the velocities of the neutron excess (Fig.16(b)), when compared to the core (Z=40, N=50)(Fig.16(a)), are obtained in the region close to the nuclear surface.

When moving further to the heavier nuclei, pygmy resonances have been observed both theoretically [19] and experimentally  [22,23] as a well established collective mode even in the stable nuclei close to the line of $\beta$ -stability. Recent RRPA analysis [14] in $^{208}Pb$ has confirmed the existence of the low-lying E1 state corresponding to the dipole pygmy resonance.

Nils Paar, 2001.