Nuclear level statistics: Extending shell model theory to higher temperatures

The shell model Monte Carlo (SMMC) approach has been applied to calculate level densities and partition functions to temperatures up to $\sim 1.5–2\text{MeV}$, with the maximal temperature limited by the size of the configuration space. Here we develop an extension of the theory that can be used to higher temperatures, taking into account the large configuration space that is needed. We first examine the configuration space limitation using an independent-particle model that includes both bound states and the continuum. The larger configuration space is then combined with the SMMC under the assumption that the effects on the partition function are factorizable. The method is demonstrated for nuclei in the iron region, extending the calculated partition functions and level densities up to $T\sim 4\text{MeV}$. We find that the back-shifted Bethe formula has a much larger range of validity than was previously believed. The present theory also shows more clearly the effects of the pairing phase transition on the heat capacity.

Figure 1

The logarithm of the canonical excitation partition function $Z_{sp}^{\prime }$ (where the energy is measured with respect to the ground-state energy), calculated for ${}^{56}{}_{,}\mathrm{Fe}$ is shown for the following single-particle spaces: all bound states plus continuum as described by Eqs. (6, 8, 12) (solid line); bound states and the narrow resonances listed in Table (dashed-dotted line); and bound states only (dotted line). The corresponding calculation with the space truncated to the $fp{g}_{9/2}$ major shell is shown as the dashed line. The intercept at $T=0$ is $\ln N_{\mathrm{g}.\mathrm{s}.}$, where $N_{{g}.{s}.}=168$ is the number of states in the ${\left(f_{7∕2}^p\right)}^{-2}{\left(p_{3∕2}^n\right)}^2$ shell configuration.

Figure 2

Similar to Fig. 1 but for the nucleus ${}^{66}\mathrm{Cr}$.

Figure 3

The excitation partition function of ${}^{56}\mathrm{Fe}$ using the SMMC method in the truncated space $\left(fp{g}_{9∕2}\right)$. The SMMC partition function ${Z^{\prime }}_{v,tr}$ in the presence of the residual interaction (squares) is compared to the SMMC partition function ${Z^{\prime }}_{sp,tr}$ in the absence of interactions (circles). The dashed line is the partition function of the independent-particle model calculated in the saddle point approximation using Eqs. (6, 8, 12).

Figure 4

Extended partition function for ${}^{56}\mathrm{Fe}$, calculated from Eq. (15), shown as solid squares. The solid line is a fit to Eq. (19) with $a=5.87{\text{MeV}}^{-1}$ and $\Delta =1.03\text{MeV}$. Shown for comparison are the SMMC (open squares) and the full space single-particle (dashed line) results.

Figure 5

The extended entropy of ${}^{56}\mathrm{Fe}$ (solid squares). The solid line is a fit to Eq. (20) with $a=5.82{\text{MeV}}^{-1}$. The open squares describe the SMMC entropy.

Figure 6

The extended partition function (solid squares) for the odd-even nucleus ${}^{57}\mathrm{Fe}$ is compared with the SMMC partition function (open squares). The solid line is a fit to the back-shift formula (19) with $a=6.04{\text{MeV}}^{-1}$ and $\Delta =-1.29\text{MeV}$.

Figure 7

Level density of ${}^{56}\mathrm{Fe}$, calculated with the extended partition function (solid squares). The solid line is a fit to the back-shifted Bethe formula (27) and describes well the extended level density in the full excitation energy range shown in the figure. The SMMC level density in the truncated $fp{g}_{9∕2}$ shell is shown by open squares and the independent-particle model level density is shown by the dashed line.

Figure 8

Extended level density (solid squares) in comparison with the SMMC level density (open squares) for ${}^{57}\mathrm{Fe}$. The solid line is a fit to the back-shifted Bethe formula (27).

Figure 9

The extended heat capacity of ${}^{56}\mathrm{Fe}$ vs temperature (solid squares), showing a bump around the pairing transition temperature. The SMMC heat capacity is shown by the open squares. The dashed line is the heat capacity obtained from the independent-particle model (bound states plus continuum).