Constraining level densities through quantitative correlations with cross-section data

The adopted level densities (LD) for the nuclei produced through different reaction mechanisms significantly impact the accurate calculation of cross sections for the different reaction channels. Many common LD models make simplified assumptions regarding the overall behavior of the total LD and the intrinsic spin and parity distributions of the excited states. However, very few experimental constraints are taken into account in these models: LD at neutron separation energy coming from average spacings of $s$- and $p$-wave resonances ($D_0$ and $D_1$, respectively) whenever they have been previously measured, and the sometimes subjective extrapolation of discrete levels. These, however, constrain the LD only in very specific regions of excitation energy, and for specific spins and parities. This work aims to establish additional experimental constraints on LD through quantitative correlations between cross sections and LD. This allows for fitting and the determination of detailed structures in LD. For this we use the microscopic Hartree-Fock-Bogoliubov (HFB) LD as a starting point as the HFB LD provide a more realistic spin and parity distributions than phenomenological models such as Gilbert-Cameron (GC). We then associate variations predicted by the HFB model with the structure observed in double-differential cross sections at low outgoing neutron energy, a region that is dominated by the LD input. We also use ($n,p$) on ${}^{56}\mathrm{Fe}$, as an example case where angle-integrated cross sections are extremely sensitive to LD. For comparison purposes we also perform calculations with the GC model. With this approach we are able to perform fits of the LD based on actual experimental data, constraining the model and ensuring its consistency. This approach can be particularly useful in extrapolating the LD to nuclei for which high-excited discrete levels and/or values of $D_0$ or $D_1$ are unknown. It also predicts neutron-induced inelastic $\gamma$ cross sections that in some cases can differ significantly from more standard phenomenological LD models such as GC.

Figure 1

Spin distributions for levels of positive (a) and negative (b) parities, up to the cutoff excitation energy of $E_{\mathrm{cut}}$ = 5.386 MeV. Results are shown for experimental discrete levels (as found in RIPL), and as predicted by the GC and HFB models. Each distribution is normalized by the total number of levels within each formalism. The bottom panels show the difference between models and experiment. (a) Normalized spin distributions for levels with positive parity. (b) Normalized spin distributions for levels with negative parity.

Figure 2

Cumulative number of levels for ${}^{56}\mathrm{Fe}$. The black curve is derived from the cumulative number of levels observed experimentally; the green curve is the LD from the GC model with its parameters fitted according to the ENDF/B-VIII.0 iron evaluation [26]; the red dashed curve is the LD from the HFB model as tabulated in RIPL; the dashed blue curve is the HFB LD rescaled to better agree with calculated neutron double-differential spectra. See text for details on the calculations in each case.

Figure 3

Spin and parity distributions of the ${}^{56}\mathrm{Mn}$ level density at the neutron separation energy ($E_x=S_n$). Black solid curve corresponds to the GC model with the ENDF/B-VIII.0 parametrization for either parity; red and blue curves correspond to the HFB model as defined in RIPL-3 [14] for positive and negative parities, respectively; black dashed corresponds to GC model but with parametrization from RIPL-3. Upside and downside triangles highlight the spins and parities that contribute to $D_0^{-1}$ and $D_1^{-1}$, respectively.

Figure 4

Level densities for ${}^{56}\mathrm{Fe}$. The meaning of the curves is the same as of Fig. 2. The dashed gray line marks the neutron separation energy $S_n$ of ${}^{56}\mathrm{Fe}$.

Figure 5

Cross section for the ${}^{56}\mathrm{Fe}$(n,p)${}^{56}\mathrm{Mn}$ reaction calculated using the different assumptions for the LD, as detailed in text. The green curve is the LD from the GC model with its parameters fitted according to the ENDF/B-VIII.0 iron evaluation [26]; the red dashed curve is the LD from the HFB model as tabulated in RIPL; the red solid curve is the result after fitting ${}^{56}\mathrm{Mn}$ HFB LD parameters to optimize agreement with ($n,p$) data; the blue dashed curve is the same as solid-red but also with HFB LD for ${}^{56}\mathrm{Fe}$ rescaled to better agree with calculated neutron double-differential spectra; the solid blue curve is the same as previous but also with ${}^{56}\mathrm{Mn}$ rescaled and refitted to ($n,p$) data. See text for details on the calculations of each curve. Experimental data retrieved from EXFOR [35, 36].

Figure 6

Level densities for ${}^{56}\mathrm{Mn}$. The black curve is derived from the cumulative number of observed experimentally; the green curve is the LD from the GC model with its parameters fitted according to the ENDF/B-VIII.0 iron evaluation [26]; the red dashed curve is the LD from the HFB model as tabulated in RIPL; the red solid curve is the LD after fitting ${}^{56}\mathrm{Mn}$ HFB LD parameters to optimize agreement with ($n,p$) data; the blue solid curve is the ${}^{56}\mathrm{Mn}$ HFB LD rescaled and refitted to ($n,p$) data after having rescaled HFB LD for ${}^{56}\mathrm{Fe}$ to better agree with calculated neutron double-differential spectra. The dashed gray line marks the neutron separation energy $S_n$ of ${}^{56}\mathrm{Mn}$. See text for details on the calculations of each curve.

Figure 7

Cumulative number of levels for ${}^{56}\mathrm{Mn}$. The meaning of the curves is the same as of Fig. 6.

Figure 8

Example of double-differential spectra for different neutron incident energies and at different scattering angles for the different LD approaches. The green curves are the results from using the LD from the GC model with its parameters fitted according to the ENDF/B-VIII.0 iron evaluation [26]; the solid-red curves are the results from using the HFB model and fitting ${}^{56}\mathrm{Mn}$ HFB LD parameters to optimize agreement with ($n,p$) data; the dashed-blue curve is the same as the solid-red one but also with HFB LD for ${}^{56}\mathrm{Fe}$ rescaled to better agree with calculated neutron double-differential spectra. Data retrieved from EXFOR [35, 36].

Figure 9

Inelastic $\gamma$ cross sections for select transitions, as measured in Ref. [39], with model calculations using Gilbert-Cameron (red curves) and HFB (blue curve) LD models. (a) Transition between states #2 ($E_x=846.8\mathrm{keV}$) and #1 (ground state). (b) Transition between states #5 ($E_x=2.9415\mathrm{MeV}$) and #2 ($E_x=846.8\mathrm{keV}$). (c) Transition between states #7 ($E_x=3.12011\mathrm{MeV}$) and #2 ($E_x=846.8\mathrm{keV}$). (d) Transition between states #31 ($E_x=4.4477\mathrm{MeV}$) and #2 ($E_x=846.8\mathrm{keV}$); no data available.

Figure 10

Assumed parity distribution from the Gilbert-Cameron and HFB LD models for ${}^{56}\mathrm{Fe}$.

Figure 11

Fractional variations of ${}^{56}\mathrm{Fe}(n,p$) cross sections, at a given incident energy, relative to changes in the ${}^{56}\mathrm{Fe}$ (upper panel) and ${}^{56}\mathrm{Mn}$ (bottom panel) LD at specific excitation energies. (a) Fractional variation of ${}^{56}\mathrm{Fe}(n,p)$ relative to changes in ${}^{56}\mathrm{Fe}$ LD. (b) Fractional variation of ${}^{56}\mathrm{Fe}(n,p)$ relative to changes in ${}^{56}\mathrm{Mn}$ LD variations.

Figure 12

Fractional variations of ${}^{56}\mathrm{Fe}(n$, inel) cross sections, at a given incident energy, relative to changes in the ${}^{56}\mathrm{Fe}$ (upper panel) and ${}^{56}\mathrm{Mn}$ (bottom panel) LD at specific excitation energies. (a) Fractional variation of ${}^{56}\mathrm{Fe}(n,\mathrm{inel})$ relative to changes in ${}^{56}\mathrm{Fe}$ LD. (b) Fractional variation of ${}^{56}\mathrm{Fe}(n,\mathrm{inel})$ relative to changes in ${}^{56}\mathrm{Mn}$ LD.