Semiclassical shell-structure micro-macroscopic approach for the level density

Level density $\rho (E,A)$ is derived for a one-component nucleon system with a given energy $E$ and particle number $A$ within the mean-field semiclassical periodic-orbit theory beyond the saddle-point method of the Fermi gas model. We obtain $\rho \propto I_{\nu }\left(S\right)/S^{\nu }$, with $I_{\nu }\left(S\right)$ being the modified Bessel function of the entropy $S$. Within the micro-macro-canonical approximation (MMA), for a small thermal excitation energy $U$, with respect to rotational excitations $E_{\mathrm{rot}}$, one obtains $\nu =3/2$ for $\rho (E,A)$. In the case of excitation energy $U$ larger than $E_{\mathrm{rot}}$ but smaller than the neutron separation energy, one finds a larger value of $\nu =5/2$. A role of the fixed spin variables for rotating nuclei is discussed. The MMA level density $\rho$ reaches the well-known grand-canonical ensemble limit (Fermi gas asymptote) for large $S$ related to large excitation energies, and also reaches the finite micro-canonical limit for small combinatorial entropy $S$ at low excitation energies (the constant “temperature” model). Fitting the $\rho (E,A)$ of the MMA to the experimental data for low excitation energies, taking into account shell and, qualitatively, pairing effects, one obtains for the inverse level density parameter $K$ a value which differs essentially from that parameter derived from data on neutron resonances.

Figure 1

MMA level density $\rho$ [Eq. (42) in units of ${\mathrm{MeV}}^{-1}$] as function of the excitation energy $U$ (in units of MeV) at the inverse level density parameter $K=10$ MeV (a), (b), and at 20 MeV (c), (d) for the relative energy shell corrections ${\mathcal{E}}_{\mathrm{sh}}=1.7$ (a), (c) and 0.6 (b), (d) values. The black solid ($n=0$) and dotted ($n=1$) lines are of MMA2, without [Eqs. (34)] and with [Eq. (31)] the second term, respectively. The magenta dashed line ($n=2$) [numerical, Eq. (15)] with the next leading correction term presents good convergence to the MMA2 results owing to the expansion of the Jacobian factor, ${\mathcal{J}}^{-1/2}$ [see Eq. (18) for the Jacobian $\mathcal{J}$], in the integrand of Eq. (15), over $1/\xi$ (see text). The heavy dashed red line ($n=0$) and blue dotted line ($n=1$), and the dashed cyan line ($n=2$)[see Eqs. (29) (MMA1) and (26), and (15), respectively], show the convergence to the MMA1 results due to the expansion of this Jacobian $\mathcal{J}$, over $\xi$. The particle number $A=200$ was used.

Figure 2

Level density $\rho$ [Eq. (42)], in units of ${\overline{\rho }}_{\nu }$, with the accurate result “1” (solid line), Eq. (29), (a) for $\nu =3/2$ [MMA1 (i)], and (b), Eq. (34), for $\nu =5/2$[(MMA2 (ii)], shown as functions of the entropy $S$ for different approximations: (1) $S\ll 1$ (dashed lines), Eq. (44) at the second order, and (2) $S\gg 1$ (dotted and thin solid lines), Eq. (43); “3” is the main term of the expansion in powers of $1/S$, and “4” is the expansion over $1/S$ up to first [in (a)], and second [in (b)] order terms in square brackets of Eq. (43), respectively.

Figure 3

MMA level density $\rho$ [Eq. (42)] in units of ${\mathrm{MeV}}^{-1}$ as function of the entropy $S$ (a), and excitation energy $U$, in units of MeV (b). The black solid and dotted lines are the MMA2 approach for ${\mathcal{E}}_{\mathrm{sh}}=2.0$ and 0.002, Eq. (34), respectively. Green dashed and blue dotted lines are the general Fermi gas (GFG) approach, Eq. (45), for the same values of ${\mathcal{E}}_{\mathrm{sh}}$, respectively. The red dashed line is the MMA1, Eq. (29); in (b) $K=10$ MeV, of the order of the ETF value of Ref. [20].

Figure 4

Level density, $ln\rho (E,A)$, is obtained for low energy states in ${}^{144}\mathrm{Sm}$ (a), ${}^{166}\mathrm{Ho}$ (b), ${}^{208}\mathrm{Pb}$ (c), and ${}^{230}\mathrm{Th}$ (d) within different approximations: The MMA dashed green line “1”, Eq. (29); the MMA solid black line “2a”, Eq. (34), at the relative realistic shell correction ${\mathcal{E}}_{\mathrm{sh}}$ [58]; the MMA dash-dotted red line “2b”, Eq. (34) at an extremely small ${\mathcal{E}}_{\mathrm{sh}}$, Eq. (34) with (35); and the Fermi gas Bethe3, blue dotted line, Eq. (47). The realistic values of ${\mathcal{E}}_{\mathrm{sh}}=0.37$ (a), 0.50 (b), 1.77 (c), and 0.55 (d) for MMA2 are taken from Ref. [58] (the chemical potential $\lambda =40$ MeV, independent of particle numbers). Heavy dashed red lines test shifts of the excitation energies $U$ for MMA1 and MMA2a by $+1.1$ and $+2.2$ MeV in ${}^{144}\mathrm{Sm}$ and ${}^{208}\mathrm{Pb}$, respectively, which are due, presumably, to the pairing condensation energy shown by arrows in the panels (a) and (c), as explained in the text and Table 1. Experimental dots (with error bars, $\mathrm{\Delta }{\rho }_i/{\rho }_i=1/\sqrt{N_i}$) are obtained directly from the excitation states (with spins and their degeneracies) spectrum [65] in shown nuclei (Table 1) by using the sample method where the sample lengths $U_s=0.45$ (a), 0.15 (b), 0.34 (c), and 0.17 (d) MeV are found on the plateau condition over the inverse level density parameter $K$.