Comprehensive test of nuclear level density models

For the last two decades, experimental information on nuclear level densities for about 60 different nuclei has been obtained on the basis of the Oslo method. While each of these measurements has been typically compared to one or a few level density models, a global study including all the measurements has been missing. The present study provides a systematic comparison between Oslo data and six global level density models for 42 nuclei for which $s$-wave resonance spacings are also available. We apply a coherent normalization procedure to the Oslo data for each of the six different models, all being treated on the same footing. Our quantitative analysis shows that the constant-temperature model presents the best global description of the Oslo data, closely followed by the mean-field plus combinatorial model and Hartree-Fock plus statistical model. Their accuracies are quite similar, so that it remains difficult to clearly favour one of these models. When considering energies above the threshold where the experimental level scheme is complete, all the six models are shown to lead to rather similar accuracies with respect to Oslo data. The recently proposed shape method can, in principle, improve the situation since it provides an absolute estimate of the excitation-energy dependence of the measured level densities. We show for the specific case of ${}^{112}\mathrm{Cd}$ that the shape method could exclude the Hartree-Fock plus statistical model. Such an analysis remains to be performed for the bulk of data for which the shape method can be applied to the Oslo measurements before drawing conclusions on the general quality of a given nuclear level density model.

Figure 1

Comparison of 6 NLD predictions with Oslo data after renormalization for the four test cases (a) ${}^{106}\mathrm{Pd}$, (b) ${}^{108}\mathrm{Pd}$, (c) ${}^{112}\mathrm{Cd}$, and (d) ${}^{164}\mathrm{Dy}$. The solid black line represents the NLD extracted from known discrete levels using an excitation-energy bin $\mathrm{\Delta }U=0.5$ MeV. The filled circles correspond to the measured NLDs renormalized by Eq. (4) on the corresponding theoretical NLD to minimize $f_{\mathrm{rms}}$. The lines correspond to the six NLD calculations normalized to the experimental $D_0$ value. The full squares give the predicted total NLD at $U=B_n$ for each of the six NLD models.

Figure 2

Theoretical and renormalized Oslo NLD for ${}^{106}\mathrm{Pd}$ for each of the six NLD models considered. The black solid line represents the NLD extracted from known discrete levels using an excitation-energy bin of $\mathrm{\Delta }U=0.5$ MeV. The filled circles correspond to the measured NLD renormalized by Eq. (4) to the theoretical NLD at $E_{exp}^{\max }\simeq 7.4$ MeV. The additional lines correspond to six NLD calculations (Sec. 2a). The full squares at $U=B_n$ give the total NLD extracted for a given NLD model after renormalization to the same experimental $D_0$ value.

Figure 3

Comparison for 30 nuclei between the renormalized experimental Oslo data (filled circles) and NLD curves (solid lines) corresponding to the Cst-T (blue) or HFB+comb (red) models. The black solid lines are the NLDs deduced from the low-lying level schemes.

Figure 4

Energy difference $\mathrm{\Delta }E=B_n-E_{exp}^{\max }$ between the neutron separation energy and the data point at the highest excitation energy as a function of the atomic mass $A$ for all nuclei for which Oslo measurements are available. The nuclei with the largest energy differences ($\mathrm{\Delta }E>3.5$ MeV) are marked specifically.

Figure 5

The first-generation matrix of ${}^{112}\mathrm{Cd}$. The black dashed lines indicate the limits for the decay to the ground state ($D_1$) and the black solid lines the decay to the first $2^{+}$ ($D_2$).

Figure 6

(a) Extracted GSFs for ${}^{112}\mathrm{Cd}$ using the Oslo method (black squares) and the shape method with two different codes (pink squares and light blue diamonds); (b) the original NLD from the Oslo method (black squares), and the renormalized one using the slope from the shape method (light blue diamonds). The error bars on the shape-method renormalized NLD include statistical errors and represent a $2\sigma$ confidence level.

Figure 7

Comparison, for ${}^{112}\mathrm{Cd}$, between the NLD constrained by the shape method (open circles) and the NLD predicted by the six models considered here (solid lines). The full squares at $U=B_n$ give the total NLD extracted for a given NLD model after renormalization to the same experimental $D_0$ value. The black solid line represents the NLD extracted from known discrete levels using an excitation-energy bin of $\mathrm{\Delta }U=0.5$ MeV.