Gogny-Hartree-Fock-Bogolyubov plus quasiparticle random-phase approximation predictions of the $M1$ strength function and its impact on radiative neutron capture cross section

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 extend our large-scale calculations of the $E1 \gamma$-ray strength function, obtained in the framework of the axially- symmetric-deformed quasiparticle random phase approximation (QRPA) based on the finite-range D1M Gogny force, to the calculation of the $M1$ strength function. We compare our QRPA prediction of the $M1$ strength with available experimental data and show that a relatively good agreement is obtained provided the strength is shifted globally by about 2 MeV and increased by an empirical factor of 2. Predictions of the $M1$ strength function for spherical and deformed nuclei within the valley of $\beta$ stability as well as in the neutron-rich region are discussed. Its impact on the radiative neutron capture cross section is also analyzed.

Figure 1

(a) $M1$ strength $f_{M1}$ for ${}^{92}\text{Zr}$ and ${}^{240}\text{Pu}$ for a number of HO shells $N_{\mathrm{sh}}=13$ and different cutoff energies $E_c=60$, 120, and 200 MeV. (b) The same as (a) for $E_c=60$ MeV and different shell numbers $N_{\mathrm{sh}}=9$, 11, 13, and 17 (the case of $N_{\mathrm{sh}}=17$ has been calculated in the Pu case only).

Figure 2

Centroid $M1$ energy and total $B\left(M1\right)$ strength estimated for the 412 even-even nuclei around the valley of stability for which QRPA calculations have been performed. The Sn isotopic chain is shown by the red circles.

Figure 3

Total $K=1^{+}$ strength below 4.5 MeV as a function of the quadrupole deformation ${\beta }_2$ for the deformed nuclei in our set of 412 even-even nuclei. The full squares correspond to stable nuclei and long-lived actinides, while open squares depict neutron-rich nuclei. The solid curve is the $18{\beta }_2^2$ function in ${\mu }_N^2$ units.

Figure 4

(a) Comparison between experimental [24, 25] and D1M+QRPA $M1$ strength functions for ${}^{106}\text{Pd}$. Also shown is the SLO approximation recommended in Ref. [27]. (b) The same for ${}^{198}\text{Au}$ with experimental data taken from [24, 26].

Figure 5

Same as Fig. 4 for the $M1$ photoabsorption cross section ${\sigma }_{\gamma }$ of (a) ${}^{128}\text{Xe}$ and (b) ${}^{134}\text{Xe}$. Experimental data are taken from Ref. [28].

Figure 6

Comparison between experimental [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40] (open symbols) and D1M+QRPA (full symbols) values of the integrated strength $\sum B\left(M1\right)\left({\mu }_N^2\right)$ in the well defined energy range given by the measurements. Circles and squares correspond to data below or above 4 MeV, respectively.

Figure 7

Comparison between experimental [41] and D1M+QRPA mean excitation energy of the scissors mode.

Figure 8

Comparison between experimental [31, 42, 43] and D1M+QRPA integrated $M1$ strength in ${}^{208}\text{Pb}$ between 7 and 9 MeV with (solid line) and without (dashed line) energy shift $\mathrm{\Delta }$.

Figure 9

(a) Comparison between experimental [44] and D1M+QRPA strength for ${}^{232}\text{Th}$. The dotted, dashed, and solid lines correspond to the $M1, E1$, and total $E1+M1$ QRPA strengths, respectively. (b) The same for ${}^{238}\text{U}$.

Figure 10

Comparison between experimental (black dots) [3] and D1M+QRPA estimate of the $\langle {\mathrm{\Gamma }}_{\gamma }\rangle$ value (open red diamonds) including both the $E1$ and $M1$ contributions. The theoretical uncertainties stem from the use of different nuclear level density prescriptions [23, 46, 47]. The blue triangles correspond to the predicted $\langle {\mathrm{\Gamma }}_{\gamma }\rangle$ when omitting the $M1$ mode totally.

Figure 11

(a) D1M+QRPA $M1$ strength function in the Ni isotopic chain. (b) The same for the Sn isotopic chain. Spherical nuclei are shown with solid lines and deformed ones with dotted lines.

Figure 12

Contribution (in %) of the D1M+QRPA $M1$ strength function to the total MACS for 1400 nuclei at the energy of $kT=30$ keV.

Figure 13

Ratio of the MACS obtained with the D1M+QRPA $M1$ strength to the one obtained with the RIPL-1 recommendation [Eq. (7)] [27] for the 1400 nuclei at the energy of $kT=30$ keV.