Pygmy resonances and radiative nucleon captures for stellar nucleosynthesis

The impact of low-energy multipole excitations and pygmy resonances on radiative neutron and proton-capture cross sections in nuclei close to the $\beta$-stability line is investigated. For this purpose, a microscopic theoretical approach based on self-consistent density functional theory and quasiparticle-random-phase-approximation formalism extended with multiphonon degrees of freedom is implemented in a statistical reaction model. The advantage of the method is the microscopic nuclear structure input for unified description of low-energy multiphonon excitations and pygmy and giant resonances. This is found to be important for the understanding of the fine structure and dynamics of the nuclear response function at low energies, which strongly influences nuclear reaction rates of astrophysical relevance. Calculations of the radiative capture cross sections of the reactions ${}^{85}\mathrm{Kr}\left(n,\gamma \right){}^{86}\mathrm{Kr}, {}^{87}\mathrm{Sr}\left(n,\gamma \right){}^{88}\mathrm{Sr}$, and ${}^{89}\mathrm{Y}\left(p,\gamma \right){}^{90}\mathrm{Zr}$ are discussed in comparison with experimental data. For the reactions ${}^{89}\mathrm{Zr}\left(n,\gamma \right){}^{90}\mathrm{Zr}$ and ${}^{91}\mathrm{Mo}\left(n,\gamma \right){}^{92}\mathrm{Mo}$ theoretical predictions of the reaction cross sections are made.

Figure 1

Systematic $\mathrm{EDF}+\mathrm{QRPA}$ including PDR (blue dashed line), $\mathrm{EDF}+\mathrm{QRPA}$ excluding PDR (green long-dash-dotted line), three-phonon $\mathrm{EDF}+\mathrm{QPM}$ (black solid line), $\mathrm{HFB}+\mathrm{QRPA}$ calculation based on the BSk7 force (red dotted line), and standard Lorentz curves (brown short-dash-dotted line) with parameter sets from Table 2 calculations of dipole photoabsorption cross section below the neutron threshold of (a) ${}^{86}\mathrm{Kr}$, (b) ${}^{88}\mathrm{Sr}$, (c) ${}^{90}\mathrm{Zr}$, (d) ${}^{92}\mathrm{Mo}$ ($N=50$ nuclei) in comparison with experimental data [16].

Figure 2

(a) $\mathrm{EDF}+\mathrm{QRPA}$ calculations of the total $B{\left(E1\right)}_{\text{PDR}}$ strength in stable even-even $N=50$ nuclei (where the summation is taken in the energy region below $E_{\gamma }\le 9$ MeV related to PDR [16, 18]) as a function of the mass number $A$. (b) Neutron skin thickness $\delta r$ as a function of $A$ for the same nuclei.

Figure 3

Ground-state neutron-capture cross sections of (a) ${}^{85}\mathrm{Kr}\left(n,\gamma \right){}^{86}\mathrm{Kr}$, (b) ${}^{87}\mathrm{Sr}\left(n,\gamma \right){}^{88}\mathrm{Sr}$, (c) ${}^{89}\mathrm{Zr}\left(n,\gamma \right){}^{90}\mathrm{Zr}$, and (d) ${}^{91}\mathrm{Mo}\left(n,\gamma \right){}^{92}\mathrm{Mo}$ calculated with talys using EDF plus three-phonon QPM (black solid line), $\mathrm{EDF}+\mathrm{QRPA}$ (PDR included) (blue dashed line), $\mathrm{EDF}+\mathrm{QRPA}$ (PDR excluded) (green dash-dotted line), and $\mathrm{HFB}+\mathrm{QRPA}$ (BSk7) (red dotted line) strength functions. For ${}^{85}\mathrm{Kr}\left(n,\gamma \right){}^{86}\mathrm{Kr}$, the hatched area corresponds to the cross section determined with the experimental strength as derived in Ref. [5]. For ${}^{87}\mathrm{Sr}\left(n,\gamma \right){}^{88}\mathrm{Sr}$, talys cross sections are compared with experimental data [60, 61].

Figure 4

Partial ($n,\gamma$) cross sections to a CN state of energy $E$ for incident neutrons (at an energy $E_n=100$ keV) on (a) ${}^{85}\mathrm{Kr}$, (b) ${}^{87}\mathrm{Sr}$, (c) ${}^{89}\mathrm{Zr}$, and (d) ${}^{91}\mathrm{Mo}$ calculated with talys using EDF plus three-phonon QPM (black solid line), $\mathrm{EDF}+\mathrm{QRPA}$ (PDR included) (blue dashed line), $\mathrm{EDF}+\mathrm{QRPA}$ (PDR excluded) (green dash-dotted line), and $\mathrm{HFB}+\mathrm{QRPA}$ (BSk7) (red dotted line) strength functions. $E$ corresponds to the excited states in the CN, either described by discrete known levels or above the last known level by the HFB plus combinatorial NLD [53]. The upper $x$ axis corresponds to the emitted $\gamma$-ray energy $E_{\gamma }=S_n+E_n-E$.

Figure 5

Neutron-capture cross sections of ${}^{89}\mathrm{Zr}\left(n,\gamma \right){}^{90}\mathrm{Zr}$ and ${}^{91}\mathrm{Mo}\left(n,\gamma \right){}^{92}\mathrm{Mo}$ calculated with talys using the three-phonon QPM strength function and five different NLD models from Refs. [49, 53, 54].

Figure 6

Radiative proton-capture cross section ${}^{89}\mathrm{Y}\left(p,\gamma \right){}^{90}\mathrm{Zr}$ calculated with talys using EDF plus three-phonon QPM (black solid line), $\mathrm{EDF}+\mathrm{QRPA}$ (PDR included) (blue dashed line), $\mathrm{EDF}+\mathrm{QRPA}$ (PDR excluded) (green dash-dotted line), and $\mathrm{HFB}+\mathrm{QRPA}$ (BSk7) (red dotted line) strength functions. Experimental data from [62, 63].