Temperature-dependent combinatorial level densities with the D1M Gogny force

The combinatorial model of nuclear level densities has now reached a level of accuracy comparable to that of the best global analytical expressions without suffering from the limits imposed by the statistical hypothesis on which the latter expressions rely. In particular, it provides naturally, non-Gaussian spin distribution as well as nonequipartition of parities which are known to have a significant impact on cross section predictions at low energies. Our previous global model [S. Goriely, S. Hilaire, and A. J. Koning, Phys. Rev. C 78, 064307 (2008)] suffered from deficiencies, in particular in the way the collective effects—both vibrational and rotational—were treated. We have recently improved the level density calculations using simultaneously the single-particle levels and collective properties predicted by a newly derived Gogny interaction [S. Goriely, S. Hilaire, M. Girod, and S. Péru, Phys. Rev. Lett. 102, 242501 (2009)], therefore enabling a microscopic description of energy-dependent shell, pairing, and deformation effects. In addition, for deformed nuclei, the transition to sphericity is coherently taken into account on the basis of a temperature-dependent Hartree-Fock calculation which provides at each temperature the structure properties needed to build the level densities. This new method is described and shown to give reasonable results with respect to available experimental data.

Figure 1

Comparison between the energy of the experimental quadrupole vibrational $0^{+}$ and $2^{+}$ levels and those predicted using the approach described in Ref. [11] with the D1S and D1M interactions.

Figure 2

(a) Excitation energy as a function of the nuclear temperature $T$. (b) Quadrupole deformation parameter $\beta$ as a function of $T$. (c) Moment of inertia as function of $T$.

Figure 3

Total NLDs for positive parities of ${}^{238}$U calculated for several temperatures. Each level density curve covers an energy interval which starts at the excitation energy corresponding to the chosen temperature.

Figure 4

Total NLDs for positive parities of ${}^{238}$U calculated for several temperatures connected smoothly thanks to the procedure described in the text. The reference points, i.e., those not affected by the smoothing procedure, are identified by circles. The level density obtained using the HFB ingredients determined with $T=0$ at all excitation energies $U$ is shown for comparison (lower grey curve).

Figure 5

Ratio of HFB plus combinatorial ($D_{\mathrm{th}}$) to the experimental ($D_{\mathrm{expt}}$) $s$-wave neutron resonance spacings [20]. The temperature-dependent treatment is shown with circles and the $T=0$ case with squares.

Figure 6

Comparison of the cumulative number of observed levels (colored lines) with the current combinatorial method (thick black line) as a function of the excitation energy for a sample of 24 nuclei. Experimental data for even-even, odd-A, and odd-odd nuclei are shown by the blue, red, and green lines, respectively. Only for ${}^{208}$Pb, both curves have been shifted by 2 MeV, the energy range corresponding consequently to [2–7] MeV instead of [0–5] MeV.

Figure 7

Same as Fig. 6, but for a sample of four nuclei for which a worse agreement with experiment (thin red line) than in Ref. [2] is observed. Theoretical predictions are obtained using the rigid body value for the moment of inertia and either the damping function of Ref. [2] (heavy black solid line) or the current damping function (black dotted line) given by Eq. $\left(6\right)$. See text for more details.

Figure 8

Current $\alpha$ and $\delta$ values (red squares) and those of Ref. [2] (black crosses) plotted as a function of the atomic mass. See text for more details.

Figure 9

Comparison between the total NLDs determined by the Oslo group (grey areas) [23, 24, 25] and the HFB combinatorial predictions (thick lines). The full triangles correspond to the model-dependent normalization point derived by the Oslo group. See text for more details.

Figure 10

Comparison between theoretical and experimental NLDs for given spin and parity in ${}^{40}$Ca, ${}^{58}$Ni, ${}^{90}$Zr, and ${}^{90}$Nb. The experimental data (full squares) are taken from Ref. [26] for ${}^{40}$Ca and from Ref. [27] for ${}^{58}$Ni, ${}^{90}$Zr, and ${}^{90}$Nb. The blue dashed lines correspond to the raw NLDs and the full red lines to the normalized ones.

Figure 11

Ratio of Maxwellian-averaged ($n,\gamma$) rates obtained with the current NLDs and those of Ref. [2] for tin isotopes for a stellar temperature of $T={10}^9$ K.

Figure 12

Ratio of TALYS Maxwellian-averaged ($n,\gamma$) rates ${\langle \sigma v\rangle }_{\mathrm{th}}$ with experimental values [28] at $T=3\times {10}^8$ K. The blue squares correspond to the current normalized NLDs and the red dots to those of Ref. [2].