Nuclear moment of inertia and spin distribution of nuclear levels

We introduce a simple model to calculate the nuclear moment of inertia at finite temperature. This moment of inertia describes the spin distribution of nuclear levels in the framework of the spin-cutoff model. Our model is based on a deformed single-particle Hamiltonian with pairing interaction and takes into account fluctuations in the pairing gap. We derive a formula for the moment of inertia at finite temperature that generalizes the Belyaev formula for zero temperature. We show that a number-parity projection explains the strong odd-even effects observed in shell model Monte Carlo studies of the nuclear moment of inertia in the iron region.

Figure 1

The pairing gap Δ versus temperature $T$ for protons (left) and neutrons (right). The solid lines are the solution of the BCS equations (32) and (33) for ${}^{56}$Fe. The dotted-dashed lines and the dashed lines are, respectively, the solution to the number-parity projected BCS equations (57) and (58) for $\eta =1$ and $\eta =-1$. The deformation parameter is taken to be ${\beta }_2=0.24$ and the pairing strengths are determined from the zero-temperature BCS to reproduce the experimental values of the gaps (using the second difference formula of Ref. 27).

Figure 2

Moment of inertia [characterizing the spin-cutoff distribution (1)] versus inverse temperature β for an even-even nucleus (${}^{56}$Fe) and an odd-even nucleus (${}^{55}$Fe). The SMMC results calculated from Eq. (68) are shown by the solid circles for ${}^{56}$Fe and the open circles for ${}^{55}$Fe. The results of the model discussed in this work (which includes fluctuations of the pairing field and number-parity projection) are shown by solid line for ${}^{56}$Fe and dashed line for ${}^{55}$Fe. For protons we use even number-parity projection, while for neutrons we use even (odd) number-parity projection for ${}^{56}$Fe (${}^{55}$Fe). In the model we use a deformation of ${\beta }_2=0.14$ and pairing strengths of $G_p=0.42$ and $G_n=0.36$.

Figure 3

Similar to Fig. 2 but the lines correspond to the number-parity projected moments of inertia calculated at the solutions ξ of the corresponding number-parity BCS equations (57). The deformation and pairing parameters are the same as in Fig. 2.

Figure 4

Systematics of the moment of inertia versus β for a series of iron isotopes. The symbols (solid circles for even-mass isotopes and open circles for odd-mass isotopes) are the SMMC results calculated from Eq. (68). The lines describe the results of our model (including fluctuations in the pairing field). The solid (dashed) lines are obtained using the even (odd) number-parity projection for neutrons. The dotted-dashed lines are the rigid-body moment of inertia. We use ${\beta }_2=0.14$ and $G_p=0.42$, $G_n=0.36$.

Figure 5

As in Fig. 4, except that the moments of inertia are shown versus temperature.

Figure 6

The moments of inertia versus temperature for the odd-even nucleus ${}^{55}$Mn and the odd-odd nucleus ${}^{56}$Mn. The notation as in Fig. 4.