Nuclear level density: Shell-model approach

Knowledge of the nuclear level density is necessary for understanding various reactions, including those in the stellar environment. Usually the combinatorics of a Fermi gas plus pairing is used for finding the level density. Recently a practical algorithm avoiding diagonalization of huge matrices was developed for calculating the density of many-body nuclear energy levels with certain quantum numbers for a full shell-model Hamiltonian. The underlying physics is that of quantum chaos and intrinsic thermalization in a closed system of interacting particles. We briefly explain this algorithm and, when possible, demonstrate the agreement of the results with those derived from exact diagonalization. The resulting level density is much smoother than that coming from conventional mean-field combinatorics. We study the role of various components of residual interactions in the process of thermalization, stressing the influence of incoherent collision-like processes. The shell-model results for the traditionally used parameters are also compared with standard phenomenological approaches.

Figure 1

Nuclear level densities for ${}^{24}\mathrm{Mg}, \mathrm{\Pi }=+1$ and various spins, for the $sd$-shell and USDB two-body interaction [45]; full shell-model diagonalization (solid curves) vs moments method (dashed curves); finite-range parameter $\eta =2.7$.

Figure 2

Low-energy level density for ${}^{64}\mathrm{Ge}$, spin $J=0$ and $\mathrm{\Pi }=+1$. The solid curve presents the calculation in the $pf$ shell with the GXPF1A interaction, the dashed curve corresponds to the calculation in the larger model space with the level $g_{9/2}$ added, and the dotted curve presents the results obtained using the HFB single-particle energies and the combinatorial method [47].

Figure 3

Level density for ${}^{64}\mathrm{Ge}$, spin $J=0$ and $\mathrm{\Pi }=+1$. The densities are calculated in $pf$ (solid curve) and in $pf+g_{9/2}$ (dashed curve) shells. The range of excitation energy is increased, compared to Fig. 3, up to 24 MeV.

Figure 4

Level density for ${}^{28}\mathrm{Si}, sd$ model space. Different curves correspond to different scale factors of Eq. (9): $k_1=k_2=\{0.1,0.2,0.3,0.5,1.0\}$ when the pairing and nonpairing parts of the interaction scale similarly. The left graph corresponds to the total density with all $J$ included, while the right graph describes the evolution of the $J=0$ density.

Figure 5

Level density for ${}^{28}\mathrm{Si}, sd$ model space. The nonpairing interaction is always turned off, $k_2=0.1$, while the pairing interaction scales are $k_1=\{0.1,0.2,0.3,0.5,1.0,1.5\}$. The left graph corresponds to the total density with all $J$ included, while the right graph describes the evolution of the $J=0$ density.

Figure 6

Level density for ${}^{28}\mathrm{Si}, J=0, sd$ model space. The black curve presents $k_1=1.0,k_2=0.1$, then the remaining parts of the interaction are increased together with $k_2$ up to the red curve ($k_1=k_2=1.0$) that describes the realistic interaction.

Figure 7

Level density for ${}^{52}\mathrm{Fe}$, all $J, pf$ model space. This figure and its color scheme are similar to the left panel of Fig. 4.

Figure 8

Interpolation of the single-particle level density parameter $a$; see Eq. (15) and the details in the text. The fitting energy range is 5–25 MeV. Left panel: different colors present different isotopes or isotones. Right panel: moments-method calculation with interpolation (black circles), fit using the experimental data on neutron resonances (Ref. [53], red diamonds), and fit using experimental low-lying levels ([54], yellow squares).

Figure 9

Level density parameter (11) fitted in the simple Fermi-gas model for $sd$ and $pf$ shell nuclei. The empty circles (red color) present the fitting range 1-5 MeV and the filled black circles correspond to the fitting range 5–25 MeV. The error bars show the uncertainty of the level density parameter $a$ due to the fitting.

Figure 10

The relative changes in the level density parameter $a$ due to the pairing energy shift $\delta$, Eq. (17), shown for $sd$- and $pf$-shell nuclei. The empty circles present the fit with the energy shift $\delta =-1$ MeV and the filled circles correspond to $\delta =1$ MeV.

Figure 11

Thermodynamic temperature as a function of excitation energy, Eq. (18), calculated in the moments method for ${}^{28}\mathrm{Si}$, top four curves (black color), and for ${}^{56}\mathrm{Fe}$, the bottom four curves (blue color).

Figure 12

The constant temperature $T$ and the constant [see Eq. (19)] fitted for the energy range 5–15 MeV. Different sets of nuclei from the $pf$ shell with complete and almost complete shells are presented by different colors: $Z=20$ (red), $Z=21$ (green), $Z=22$ (blue), and $N=40$ (brown).

Figure 13

Comparison between global level densities for ${}^{28}\mathrm{Si}$ calculated with the moments method (the solid curve) and using the constant temperature for 5–15 MeV (the dotted curve) and for 5–25 MeV (the dashed curve) energy ranges; fits according to Eq. (19). The inset presents the magnified low-energy region.

Figure 14

Constant temperature fitted for $sd$ and $pf$ shell nuclei. The diamonds (red color) present the fitting range 1–5 MeV, the circles (black color) correspond to 5–15 MeV, and the squares (blue color) correspond to 5–25 MeV.

Figure 15

Logarithm of level density, $ln\left[\rho \right(E,M\left)\right]$, versus $M^2$, Eq. (23), calculated for ${}^{28}\mathrm{Si}, {}^{52}\mathrm{Fe}, {}^{44}\mathrm{Ca}$, and ${}^{64}\mathrm{Cr}$. Different colors present different excitation energies: 5 MeV (black), 10 MeV (red), 15 MeV (green), 20 MeV (blue), and 25 MeV (brown). Dots correspond to the moments method calculations, the solid lines are linear interpolations; see the text.

Figure 16

Interpolation of the spin cutoff parameter ${\sigma }^2$ (see the description in the text for details). Parameters $\alpha$ and $\beta$ are shown for different $sd$ and $pf$ nuclei. Left panel: different colors present different isotopes or isotones. Right panel: moments method calculation with interpolation (black circles), statistical calculation (orange diamonds), and rigid-body approximation (yellow squares).

Figure 17

Partial nuclear level densities for ${}^{28}\mathrm{Si}, \mathrm{\Pi }=+1$, and various spin projections $M$. The solid curves present the shell model densities ($sd$ shell and USDB two-body interaction [45]) while the dashed curves show the Fermi-gas model densities (with the fitted parameters, see text for details). The insets show the magnified low-energy area that spans from 5 to 25 MeV, where the densities were actually fitted.