Large-scale deformed quasiparticle random-phase approximation calculations of the $\gamma$-ray strength function using the Gogny force

Valuable theoretical predictions of nuclear dipole excitations in the whole chart are of great interest for different nuclear applications, including in particular nuclear astrophysics. Here we present large-scale calculations of the $E1\gamma$-ray strength function obtained in the framework of the axially symmetric deformed quasiparticle random-phase approximation based on the finite-range Gogny force. This approach is applied to even-even nuclei, the strength function for odd nuclei being derived by interpolation. The convergence with respect to the adopted number of harmonic oscillator shells and the cutoff energy introduced in the 2-quasiparticle $(2-qp)$ excitation space is analyzed. The calculations performed with two different Gogny interactions, namely D1S and D1M, are compared. A systematic energy shift of the $E1$ strength is found for D1M relative to D1S, leading to a lower energy centroid and a smaller energy-weighted sum rule for D1M. When comparing with experimental photoabsorption data, the Gogny-QRPA predictions are found to overestimate the giant dipole energy by typically $\sim 2$ MeV. Despite the microscopic nature of our self-consistent Hartree-Fock-Bogoliubov plus QRPA calculation, some phenomenological corrections need to be included to take into account the effects beyond the standard $2-qp$ QRPA excitations and the coupling between the single-particle and low-lying collective phonon degrees of freedom. For this purpose, three prescriptions of folding procedure are considered and adjusted to reproduce experimental photoabsorption data at best. All of them are shown to lead to somewhat similar predictions of the $E1$ strength, both at low energies and for exotic neutron-rich nuclei. Predictions of $\gamma$-ray strength functions and Maxwellian-averaged neutron capture rates for the whole Sn isotopic chain are also discussed and compared with previous theoretical calculations.

Figure 1

QRPA $E1$ strength $S_{E1}$ calculated with the D1M Gogny interaction as a function of the excitation energy for different numbers of HO shells without any energy cutoff on the $2-qp$ state energies.

Figure 2

QRPA $E1$ strength $S_{E1}$ calculated with the D1M Gogny interaction for different numbers of HO shells for (a) ${}^{92}\mathrm{Zr}$ with an energy cutoff on the $2-qp$ states energies of ${\varepsilon }_c=120$ MeV and (b) for ${}^{238}\mathrm{U}$ with ${\varepsilon }_c=60$ MeV.

Figure 3

QRPA $E1$ strength $S_{E1}$ calculated with the D1M Gogny interaction for different cutoff energies for an arbitrary sample of nuclei. The number of HO shells chosen for each nucleus ensures convergence with respect to the conclusions of Sec. 3a and is summarized in Table 2.

Figure 4

Energy shifts of the two main peaks of the $B\left(E1\right)$ distribution predicted with D1M as a function of the atomic mass for two different values of the cutoff energy (see text for details). Black (respectively, green) squares correspond to a ${\varepsilon }_c=30$ MeV (respectively, 60 MeV). The full squares indicate calculations for which $N_{\mathrm{sh}}$ values are compatible with Table 2.

Figure 5

QRPA $B\left(E1\right)$ distributions in the GDR region for a reduced set of nuclei calculated with the D1S and D1M Gogny interactions for identical energy cutoff and basis size. For deformed nuclei (${}^{76}\mathrm{Ge}$, ${}^{152}\mathrm{Sm}$, ${}^{196}\mathrm{Pt}$, and ${}^{240}\mathrm{Pu}$) $K^{\mathrm{\Pi }}=0^{-}$ and ${\left|K\right|}^{\mathrm{\Pi }}=1^{-}$ contributions are plotted separately. The D1M strengths are multiplied by $-1$ for clarity.

Figure 6

Energies of the centroid of the QRPA $B\left(E1\right)$ distributions calculated with D1S and D1M Gogny interactions for all the nuclei considered in this work. The full line represents the fit of the points.

Figure 7

$E1$ EWSR in TRK units (14.8 $\frac{NZ}{A}{\mathrm{e}}^2{\mathrm{fm}}^2$ MeV) calculated in QRPA with D1S and D1M Gogny interactions for all the nuclei considered in this work.

Figure 8

Energy centroid of the QRPA $B\left(E1\right)$ distributions calculated with D1M as a function of the number of shells using a global 60-MeV energy cutoff for a sample of oblate (${}^{198}\mathrm{Pt}$), spherical (${}^{208}\mathrm{Pb}$), and prolate (${}^{76}\mathrm{Ge}$, ${}^{152}\mathrm{Sm}$, ${}^{240}\mathrm{Pu}$) nuclei.

Figure 9

Centroid energies of the $K^{\pi }=0^{-}$ and $1^{-}$ components for all the deformed nuclei considered in this work. The squares correspond to prolate nuclei and the triangles to oblate ones.

Figure 10

Comparison of the QRPA centroid energies (red) with the peak energies compiled in the RIPL library (black). (a) Nuclei for which a single Lorentzian is recommended. (b) and (c) Lowest (b) and highest (c) peak energies for the nuclei for which two Lorentzians are recommended.

Figure 11

Comparison for 28 nuclei between the experimental photoabsorption cross section [4] (black solid line) and the fits corresponding to Model 0 (dotted blue line), Model 1 (green dashed line), and Model 2 (red solid line). The experimental cross section around the GDR is represented by one or two parametrized Lorentzian functions, as traditionally done [4].

Figure 12

Values of the three parameters ${\mathrm{\Delta }}_0$ (squares), $\mathrm{\Gamma }$ (circles), and ${\delta }_{4qp}$ (diamonds) adjusted on experimental photoabsorption cross sections as a function of the neutron number of the corresponding nucleus for Models 1 (upper panel) and 2 (lower panel).

Figure 13

Comparison between experimental photoabsorption (or photoneutron) cross sections [41, 42, 43, 44] (squares) and the QRPA ones obtained from the interpolated strengths corresponding to Model 0 (dotted line), Model 1 (dashed line), and Model 2 (solid line) for three odd-A nuclei.

Figure 14

Comparison of the QRPA low-energy $E1$-strength functions with the experimental compilation [4] including resolved-resonance and thermal-capture measurements, as well as photonuclear data for nuclei from ${}^{36}\mathrm{Cl}$ up to ${}^{239}\mathrm{U}$ at energies ranging from 4 to 8 MeV.

Figure 15

Comparison between the $E1$ strength functions for Sn isotopes (from $A=115$ to 155 by steps of $\mathrm{\Delta }A=5$) obtained with the three prescriptions used to correct the HFB+QRPA model based on the D1M force.

Figure 16

Comparison between the $E1$ strength functions for Sn isotopes (from $A=115$ to 155 by steps of $\mathrm{\Delta }A=5$) obtained with the Lorentzian model [4] (left panel), the microscopic QRPA calculation based on the Skyrme force [6] (central panel), and the present QRPA calculation with the D1M Gogny force (Model 0) (right panel).

Figure 17

(Left panel) Ratio of the Maxwellian-averaged radiative neutron capture rate (at a temperature of ${10}^9$ K) for the Sn isotopes obtained with the three present D1M+QRPA models. (Right panel) Same where the neutron capture rate with Model 0 of the D1M+QRPA strength is compared with the one obtained with the GLO model (squares) [4] and to the one predicted with the Skyrme-HFB+QRPA model based on the BSk7 force (circles) [6].