Emergence of low-energy monopole strength in the neutron-rich calcium isotopes

Background: The isoscalar monopole response of neutron-rich nuclei is sensitive to both the incompressibility coefficient of symmetric nuclear matter and the density dependence of the symmetry energy. For exotic nuclei with a large neutron excess, a low-energy component emerges that is driven by transitions into the continuum.

Purpose: While understanding the scaling of the giant monopole resonance with mass number is central to this work, the main goal of this paper is to explore the emergence, evolution, and origin of low-energy monopole strength along the even-even calcium isotopes: from ${}^{40}\mathrm{Ca}$ to ${}^{60}\mathrm{Ca}$.

Methods: The distribution of isoscalar monopole strength is computed in a relativistic random phase approximation (RPA) using three effective interactions that have been calibrated to the properties of finite nuclei and neutron stars. A nonspectral approach is adopted that allows for an exact treatment of the continuum without any reliance on discretization. This is particularly critical in the case of weakly bound nuclei with single-particle orbits near the continuum. The discretization of the continuum is neither required nor admitted.

Results: For the stable calcium isotopes, no evidence of low-energy monopole strength is observed, even as the $1f^{7/2}$ neutron orbital is being filled and the neutron-skin thickness progressively grows. Further, in contrast to experimental findings, a mild softening of the monopole response with increasing mass number is predicted. Beyond ${}^{48}\mathrm{Ca}$, a significant amount of low-energy monopole strength emerges as soon as the weak-binding neutron orbitals ($2p$ and $1f^{5/2}$) become populated. The emergence and evolution of low-energy strength is identified with transitions from these weakly bound states into the continuum—which is treated exactly in the RPA approach. Moreover, given that models with a soft symmetry energy tend to reach the neutron-drip line earlier than their stiffer counterparts, an inverse correlation is identified between the neutron-skin thickness and the inverse energy weighted sum.

Conclusions: Despite experimental claims to the contrary, a mild softening of the giant monopole resonance is observed in going from ${}^{40}\mathrm{Ca}$ to ${}^{48}\mathrm{Ca}$. Measurements for other stable calcium isotopes may be critical in elucidating the nature of the discrepancy. Moreover, given the early success in measuring the distribution of isoscalar monopole strength in the unstable ${}^{68}\mathrm{Ni}$ nucleus, new measurements along the unstable neutron-rich calcium isotopes are advocated in order to explore the critical role of the continuum in the development of a soft monopole mode.

Figure 1

Diagrammatic representation of the RPA equations. The ring diagram with the thick black lines represents the fully correlated RPA polarization while the one depicted with the thin blue lines is the uncorrelated mean-field polarization. The residual interaction denoted with the red wavy line must be identical to the one used to generate the mean-field ground state.

Figure 2

Distribution of isoscalar monopole strength for the neutron-even calcium isotopes from ${}^{40}\mathrm{Ca}$ up to ${}^{48}\mathrm{Ca}$ as predicted by a relativistic RPA calculation using the FSUGarnet parametrization [70]. The numbers in parentheses represent the centroid energies defined as $E_{\mathrm{cen}}=m_1/m_0$. The inset displays the integrated strength, or “running sum,” with the value at large excitation energy equal to the unweighted energy sum $m_0$.

Figure 3

Single-particle spectrum for ${}^{60}\mathrm{Ca}$ as predicted by the relativistic FSUGarnet parametrization. The blue (red) lines denote occupied (empty) orbitals and the arrows indicate discrete transitions into bound states.

Figure 4

Distribution of “uncorrelated” isoscalar monopole strength for ${}^{60}\mathrm{Ca}$ as predicted by the FSUGarnet parametrization [70]. The left-hand panel (a) describes proton particle-hole excitations while the right-hand panel (b) depicts the corresponding neutron excitations. Transitions into bound states are easily identified both in this figure as well as in Fig. 3. See text for further details.

Figure 5

(a) Distribution of isoscalar monopole strength for the neutron-even calcium isotopes from ${}^{48}\mathrm{Ca}$ to ${}^{60}\mathrm{Ca}$ as predicted by a relativistic RPA calculation using the FSUGarnet parametrization [70]. The numbers in parentheses represent the centroid energies defined as $E_{\mathrm{cen}}=m_1/m_0$. The inset displays the integrated strength, or “running sum,” with the value at large excitation energy equal to the unweighted energy sum $m_0$. (b) Contribution from the low-energy (“Pygmy”) and high-energy (“Giant”) resonance region to the total unweighted energy sum $m_0$ for two relativistic parametrizations: FSUGold [69] and FSUGarnet.

Figure 6

Contribution from the low-energy (“Pygmy”) and high-energy (“Giant”) resonance region to the total (a) inverse energy weighted sum $m_{-1}$ and (b) energy weighted sum $m_1$, as predicted by two relativistic parametrizations: FSUGold [69] and FSUGarnet [70].

Figure 7

Total (pygmy plus giant) inverse energy weighted sum as a function of the corresponding neutron-skin thickness for all even-even calcium isotopes from ${}^{50}\mathrm{Ca}$ to ${}^{60}\mathrm{Ca}$. Predictions are displayed from all three relativistic models used in the text.

Figure 8

Centroid monopole energies predicted by all three relativistic models NL3, FSUGold, and FSUGarnet for all even-even calcium isotopes from ${}^{40}\mathrm{Ca}$ to ${}^{60}\mathrm{Ca}$. The various definitions adopted for the centroid energy have been discussed in the text.