Testing the spin-cutoff parametrization with shell-model calculations

The nuclear level density, an important input to Hauser-Feshbach calculations, depends not only on excitation energy but also on angular momentum $J$. The $J$ dependence of the level density at fixed excitation energy $E_x$ is usually parametrized via the spin-cutoff factor $\sigma$. We carefully test the statistical accuracy of this parametrization for a large number of spectra computed using semirealistic interactions in the interacting shell model, with a nonlinear least-squares fit of $\sigma$ and finding the error bar in $\sigma$. The spin-cutoff parametrization works well as long as there are enough states to be statistical. In turn, the spin-cutoff factor can be related to the average value of $J^2$ at a fixed excitation energy, and we briefly investigate extracting $\langle {\widehat{J}}^2\left(E_x\right)\rangle$ from a thermal calculation such as one might do via a Monte Carlo method.

Figure 1

${}^{26}\mathrm{Al}$ in the $sd$ shell. The error bars are standard $\sqrt{{\rho }_J}$ statistical errors in counting the number of states in a given bin of 1.0 MeV in energy and fixed $J$. The dashed line is the continuous spin distribution using ${\sigma }_{\mathrm{fit}}$ and the gray shading is the one-standard-deviation envelope.

Figure 2

Same as Fig. 1 but for ${}^{26}\mathrm{Mg}$ in the $sd$ shell.

Figure 3

Same as Fig. 1 but for ${}^{33}\mathrm{P}$ in the $sd$ shell.

Figure 4

Evolution with excitation energy of fit spin-cutoff factors for selected $sd$-shell nuclides, using 1-MeV bins.

Figure 5

Evolution with excitation energy of fit spin-cutoff factors for selected $pf$-shell nuclides, using 1-MeV bins.

Figure 6

Evolution with excitation energy of fit spin-cutoff factors for selected $p\text{−}s{d}_{5/2}$-shell nuclides, using 1-MeV bins, for both positive parity (left-hand figures) and negative parity (right-hand figures).

Figure 7

${({\rho }_J/\rho )}_{\mathrm{RMSE}}$ [defined in Eq. (3)] for all analyzed $sd$ and $pf$ shell nuclides organized by structure.

Figure 8

${({\rho }_J/\rho )}_{\mathrm{RMSE}}$ [defined in Eq. (3)] for all analyzed $p\text{−}s{d}_{52}$ cross-shell nuclides.

Figure 9

Dependence of fit spin-cutoff factors on energy bin size for ${}^{28}\mathrm{Al}$ in the $sd$ shell (left) and for ${}^{47}\mathrm{Ti}$ in the $pf$ shell.

Figure 10

Comparison of the spin-cutoff factor extracted by least-squares fit $\left({\sigma }_{\mathrm{fit}}\right)$ against the spin-cutoff factor extracted from the average value of $J(J+1)$ in an energy bin $\left({\sigma }_{\mathrm{approx}}\right)$, for several representative nuclides.

Figure 11

Comparison of the average value of $J(J+1)$ in 1-MeV energy bins, calculated directly from the shell-model diagonalization spectrum for ${}^{22}\mathrm{Na}$ in the $sd$ shell (SM; histogram) and from inverting the thermal average, Eq. (8), using Eq.(13) (Inversion; dashed line).

Figure 12

Same as Fig. 11 but for ${}^{33}\mathrm{S}$ in the $sd$ shell and ${}^{44}\mathrm{Ti}$ in the $pf$ shell (right). The turnover for ${}^{44}\mathrm{Ti}$ is due to the finite model space; the other nuclides also display a turnover, but at higher excitation energies.