Low-lying dipole strength in oxygen and calcium isotopes

The RRPA strength function () for dipole excitation is evaluated for several oxygen isotopes. The calculations have been performed in the large configuration space including both particle-hole pairs, and pairs formed from hole states and negative-energy states. Discrete spectra is averaged with the Lorentzian distribution,

$$R\left( E \right) = \sum_{\nu}B(E1,E_{\nu}) \frac{\Gamma^2}{4\left(E-E_{\nu}\right)^2-\Gamma^2},$$ (8)

with arbitrarily chosen width of distribution $\Gamma$=0.5 MeV, while the strength function ( ) is evaluated for the isovector dipole operator ( ). The mean energy of the resonance is defined as the centroid energy,

$$\bar E = \frac{m_1}{m_0}$$ (9)

with the energy weighted moments for discrete spectra defined in a usual way,

$$m_k=\sum_{\nu}B(E1,E_{\nu})E^k_{\nu}.$$ (10)

In the case of $k=1$, this equation correspond to the energy weighted sum rule (EWSR), which is evaluated in the present analysis within the interval (0,50) MeV.

In Fig.2 we present the dipole strength distributions () for $^{16}$O, $^{22}$O,$^{24}$O and $^{28}$O nuclei. For $^{16}$O the isovector giant resonance is located at $\bar E$=21.8 MeV. The adding of extra neutrons leads to two interesting phenomena: a) increased fragmentation of dipole strength, and b) appearance of the low lying dipole strength below 10 MeV. The first appearance of the low energy peak below 10 MeV is observed in $^{22}$O. When moving further to the neutron drip line, the contribution of the low energy (E$\leq$10 MeV) strength increase relatively more than the high energy response. This can be seen in corresponding curve of Fig.3, where the ratio of separated energy weighted moment m$_1$ from low (E$\leq$10 MeV) and high (E$>$10 MeV) energy region is presented as a function of excess neutrons $N-N_{c}$, with $N_{c}=Z$. In the Hartree-Fock plus random phase approximation, using Skyrme effective interaction [24], in addition to the spreading of the isovector dipole strength, several peaks are obtained for $^{28}$O in the region 6-10 MeV, in contrast to the $^{16}$O, with no low-energy contributions. In the case of $^{28}$O the most collective RRPA peak at 15.2 MeV exhausts 24% of EWSR, while in the non-relativistic RPA approach [24] the most collective state exhausts approximately 15% of total EWSR. Recent experimental investigation [15] of giant resonances in unstable oxygen isotopes $^{18}$O,$^{20}$O and $^{22}$O shows the onset of low-lying dipole strength, exhausting around 5% of the Thomas Reiche Kuhn sum rule for energies up to 5 MeV above the continuum threshold. Further experimental study involving the drip line nucleus $^{24}$O is expected in the near future. In the framework of the relativistic random phase approximation, we observe 2.5%, 7.0% and 8.6% of the energy weighted sum rule (EWSR) contributions from the $E\leq10$ MeV peaks in $^{22}$O,$^{24}$O and $^{28}$O nuclei, respectively. In comparison, dipole strengths below 15 MeV within the large scale shell model calculations exhaust 10% and 8.6% of the classical sum rule in $^{22}$O and $^{24}$O, respectively [21]. Let us investigate whether these soft dipole modes are collective, as a consequence of the coherent superposition of many single particle-hole configurations, like in the giant resonances. The question whether the soft excitation is collective or single-particle, have been already studied in the light nuclei $^{11}$Li and $^{11}$Be, concluding that the soft mode of excitation is a new type of independent particle excitation, characterized by a narrow width and a large transition strength, as a consequence of the large spatial extension of the bound single-particle states [25]. Within the RRPA investigation of oxygen isotopes, we analyze in more details the main peaks contributing to the low-lying isovector dipole strength. In order to estimate the role of each particle-hole excitation, we separate contributions of each proton and neutron p-h configuration by evaluating the fraction parameter

$$\xi_{\tilde{p} h}=\left\vert X^{\nu,JM}_{\tilde{p} h}\right\vert^2- \left\vert Y^{\nu,JM}_{\tilde{p} h}\right\vert^2$$ (11)

for particular components of RRPA-eigenvectors and selected RRPA eigenvalue $\omega_{\nu}$. Since the condition of normalization

$$\sum_{\tilde{p} h}\xi_{\tilde{p} h}=1$$ (12)

is fulfilled, the effect of the most important excitations to the total transition strength can be described by the corresponding percentages for each p-h pair.

Dominant low-energy peak in $^{22}$O which exhausts 2.5% of EWSR, located at 9.3 MeV is mainly due to the neutron particle hole excitations $(93\%\ 1d_{5/2} \to 2p_{3/2})$ and $(3\%\ 1d_{5/2} \to 1f_{7/2})$. In the case of $^{24}$O, three main peaks in the low-energy region appear: 6.9 MeV (3.1% EWSR), 7.4 MeV (1.6% EWSR) and 9.3 MeV (2.3% EWSR). They correspond to the neutron excitations $(93\%\ 2s_{1/2} \to 2p_{3/2})$, $(96\%\ 2s_{1/2} \to 2p_{1/2})$ and $(94\%\ 1d_{5/2} \to 2p_{3/2})$, respectively, which reduce the role of other neutron and proton p-h excitations. For extremely neutron rich nucleus $^{28}$O, larger fragmentation is observed also in the low-energy region. We note here four main low-energy peaks at 4.2 MeV, 4.9 MeV, 7.3 MeV and 8.9 MeV, which are dominated again by the neutron excitations. Dominant particle-hole configurations for low-lying excitations in $^{28}$O are presented in the Table . The main contribution in the low-energy region comes from excitations of the excess neutrons in the 1d$_{3/2}$, 2s$_{1/2}$ and 1d$_{5/2}$ orbitals, but only one p-h excitation is relevant in each low-energy peak, in contrast to large number of p-h excitations contributing in the high-energy giant resonance. The transition densities of characteristic low and high energy isovector dipole modes, are shown in Fig.4 for $^{22}$O and $^{28}$O nuclei. Here, we present the proton and neutron transition densities, and the corresponding isoscalar and isovector transition densities (solid line and dotted line, respectively). According to Ref. [24], it is expected for the isoscalar B(E1) to all states to vanish, but if the neutron and proton transition densities have different shapes, as is the case for the nuclei close to the drip line, the corresponding isoscalar dipole transition density will be non-zero. Transition densities for the main peak at 20.9 MeV and 18.1 MeV for $^{22}$O and $^{28}$O display a radial dependence of isovector giant dipole resonance (GDR), with protons and neutrons oscillating in opposite phases. Similar to the study in the Hartree-Fock plus random phase approximation with Skyrme forces [24], the large neutron component in the surface region contributes to the existence of a node in the isoscalar transition density, moving toward larger value of radial coordinate, as the number of neutrons increase. Proton and neutron transition densities of GDR peak have similar radial dependence at large radii. Transition densities of selected low-energy peaks for $^{22}$O and $^{28}$O show quite different behavior. The proton and neutron densities in the interior region of both nuclei are not out of phase, and the transition densities have characteristic long tail with almost no contribution from the protons. Similar behavior of transition densities have been observed in the neutron-rich light nuclei $^{6}$He, $^{11}$Li and $^{12}$Be where the large extended tails of the loosely-bound neutron wave functions are responsible to cause different radial behavior of the low-energy peaks when compared with those of giant resonances [26].

Next we investigate the low-lying dipole excitations in calcium isotopes. Isovector dipole strength distributions for $^{40}$Ca,$^{48}$Ca, $^{54}$Ca and $^{60}$Ca nuclei are displayed in Fig.5. As the neutron excess increase, the fragmentation of the spectra is increased. In the case of $^{40}$Ca and $^{48}$Ca no contribution for energies less than 10 MeV is observed. The onset of low-lying dipole strength is manifested after $^{54}$Ca. This is in agreement with the non-relativistic RPA calculations of Ref. [24]. It is also noted that the centroid energies move to lower values with increasing neutron number in both models. In addition, no $E\leq10$ MeV contributions in $^{48}$Ca have been recognized when compared with the $^{40}$Ca. In the RRPA investigation of extremely neutron rich nuclei $^{60}$Ca, many peaks are contributing in the E$\leq$10 MeV region, together exhausting 9.9% of the EWSR in contrast to the 39.4% EWSR of the main GDR peak at 15.2 MeV. For the low-lying dipole strength, the most important are excitations of neutrons from orbitals 1f$_{7/2}$, 2p$_{3/2}$, 2p$_{1/2}$ ($^{54}$Ca) and 1f$_{5/2}$ ($^{60}$Ca) in the last shell. Dominant p-h transitions in the highest low-energy peaks for $^{60}$Ca are listed in the Table . In almost all peaks, single particle-hole transition is dominant, only at 7.3 MeV two comparable transitions have important role in the dipole transition strength. The role of other allowed neutron and proton p-h excitations is very small. Therefore the degree of collectivity in the low-energy region of $^{60}$Ca is very limited, single particle-hole nature of these modes dominates. On the other side, GDR peak is characterized by a large number of comparable p-h excitations, where the largest single neutron p-h contribution is 20%, while the ratio of neutron and proton participation 61.8%/36.7%=1.7 is close to the ratio of neutron and proton numbers. In contrast to the density functional theory calculations  [20], collective pygmy oscillations in calcium isotopes have not been observed in the RRPA investigation.

Experimental results on low energy dipole strength in calcium isotopes are still under debate. There was no evidence of low-lying mode when comparing the $^{40}$Ca and $^{48}$Ca in experiments with heavy ion reactions  [27]. On the other side, recent results [18] obtained with the high resolution photon scattering experiments show concentration of low-lying dipole strength in $^{48}$Ca. The sum B(E1) strength between 5 and 10 MeV is about 10 times larger than in $^{40}$Ca.

RRPA transition densities for the case of selected low (7.3 MeV) and GDR (15.2 MeV) states of $^{60}$Ca are displayed in the Fig.6. At the GDR energy, proton and neutron densities oscillate in opposite phases, as it is expected for an isovector resonance. Because of the very large neutron number, the isoscalar transition density does not vanish, similar to the non-relativistic approach [24]. In the interior of the nucleus isoscalar transition density is comparable with the isovector one. However, its contribution decreases close to the surface region, where it changes sign. On the other side, the transition densities of low-energy peak show quite opposite behavior, with a neutron dominated tail beyond the nuclear surface region.

Nils Paar, 2001.