Influence of complex configurations on the properties of the pygmy dipole resonance in neutron-rich Ca isotopes

Starting from the quasiparticle random phase approximation based on the Skyrme interaction SLy5, we study the effects of phonon-phonon coupling (PPC) on the low-energy electric dipole response in ${}^{40-58}\mathrm{Ca}$. Using the same set of parameters we describe available experimental data for ${}^{40,44,48}\mathrm{Ca}$ and give the prediction for ${}^{50-58}\mathrm{Ca}$. The inclusion of the PPC results in the formation of low-energy $1^{-}$ states. There is an impact of the PPC effect on low-energy $E1$ strength of ${}^{40,44,48}\mathrm{Ca}$. The PPC effect on the electric dipole polarizability is discussed. We predict a strong increase of the summed $E1$ strength below 10 MeV, with increasing neutron number from ${}^{48}\mathrm{Ca}$ until ${}^{58}\mathrm{Ca}$.

Figure 1

(a) Binding energies of the even-even Ca isotopes as a function of neutron number, compared with experiment and extrapolated energies (filled circles) from the AME2012 atomic mass evaluation [51]. Results of the calculations within the HF-BCS with the SLy5 EDF (open triangles) and with SLy5+T (filled triangles). (b) The neutron skin thickness $\mathrm{\Delta }{R}_{np}$ of the Ca isotopes calculated within the HF-BCS approach with the SLy5 EDF (open triangles) and with SLy5+T (filled triangles). Experimental data of the neutron skin thickness are taken from Ref. [52] (filled circles).

Figure 2

(a) One- and (b) two-neutron separation energies for Ca isotopes as a function of neutron number, calculated with the SLy5 EDF (open triangles) and with SLy5+T (filled triangles). The experimental and extrapolated energies (filled circles) are from the AME2012 atomic mass evaluation [51].

Figure 3

Energies (a) and $B\left(E2\right)$ (b) values for up-transitions to the $2_1^{+}$ states in the neutron-rich Ca isotopes. Results of the calculations within the QRPA (open triangles) with the SLy5 EDF and the QRPA plus PPC (filled triangles) are shown. Experimental data (filled circles) are taken from Refs. [18, 20, 63].

Figure 4

Low-energy $E1$ strength distributions of ${}^{40}\mathrm{Ca}$ (respectively, ${}^{48}\mathrm{Ca})$ are shown in (a) and (b) [respectively, (c) and (d). The dashed and solid lines correspond to the SLy5 calculations within the RPA and taking into account the PPC effects, respectively. (b) and (d) Experimental data are from Ref. [14].

Figure 5

(a) The estimated photoabsorption cross section for ${}^{48}\mathrm{Ca}$ (filled circles) is taken from Ref. [65]. The dotted and solid lines correspond to the calculations within the RPA with the SLy5 EDF and taking into account the PPC, respectively. (b) Running sum of the electric dipole polarizability for ${}^{48}\mathrm{Ca}$ calculated within the RPA with the SLy5 EDF (dotted line) and the RPA plus PPC (solid line) in comparison to experimental determination of ${\alpha }_D$ (the two dashed lines indicate upper and lower limits) [52].

Figure 6

(a) The electric dipole polarizability ${\alpha }_D$ as a function of neutron number, calculated with the SLy5 EDF (open triangles) and with SLy5+T (filled triangles). Experimental values (filled circles) of ${\alpha }_D$ are taken from [52]. (b) The same as (a) but as a function of the neutron skin thickness $\mathrm{\Delta }{R}_{np}$.

Figure 7

Transition proton (dotted line) and neutron (dashed line) densities to selected QRPA $1^{-}$ states of ${}^{50}\mathrm{Ca}$ (respectively, ${}^{56}\mathrm{Ca})$ in (a) and (b) [respectively, (c) and (d)] panels. Panels (a) and (b) correspond to the transition densities for the states at 9.1 and 10.3 MeV in ${}^{50}\mathrm{Ca}$, while (c) and (d) show the transition densities for the QRPA $1^{-}$ states at 8.7 and 10.7 MeV in ${}^{56}\mathrm{Ca}$, respectively. All transition densities are multiplied by $r^2$.

Figure 8

(a) Summed dipole strength below 10 MeV calculated in the QRPA (open triangles) with the SLy5 EDF, PPC (filled triangles), and with RQTBA (filled stars) [17]. Experimental data (filled circles) are taken from Refs. [12]. (b) Ratio of the PDR energy-weighted strength to the TRK sum rule for various Ca isotopes as a function of the neutron skin thickness $\mathrm{\Delta }{R}_{np}$. The open triangles indicate discrete QRPA results, while the filled triangles are PPC results.

Figure 9

Running sums of the low-lying dipole strengths in ${}^{48}\mathrm{Ca}$ (a) and ${}^{50}\mathrm{Ca}$ (b). The dotted and solid lines correspond to the calculations within the QRPA with the SLy5 EDF and taking into account the PPC, respectively. Experimental data (dashed line) are taken from Ref. [12]. The calculated $S_{1n}$ energy is indicated by the solid arrow.

Figure 10

Running sums (solid lines) of the $B(E1;0_{gs}^{+}\to 1_1^{-})$ as a function of the two-quasiparticle energy included in the QRPA calculations for ${}^{48}\mathrm{Ca}$ (a) and ${}^{50}\mathrm{Ca}$ (b). Dashed and dotted lines correspond to the results for the proton and the neutron components of the dipole operator (9), respectively.