Improved treatment of blocking effect at finite temperature

The blocking effect caused by the odd particle on the pairing properties of systems with an odd number of fermions at finite temperature interacting via the monopole pairing force is studied within several approximations. The results are compared with the predictions obtained by using the exact solutions of the pairing Hamiltonian. The comparison favors the approximation with the odd particle occupying the top level, which is the closest to the Fermi surface and whose occupation number decreases with increasing temperature.

Figure 1

Exact occupation numbers of the levels as functions of $T$ for $N=9$, $\mathrm{\Omega }=10$, 14, and 20. (a), (c), and (d) show the results obtained with equidistant levels; (b) those with nonequidistant ones (see text). In each panel, the higher-located line is the occupation number of the lower-located level. The thick line, which starts from 1 at $T=0$, is the occupation number of the level occupied by the odd particle.

Figure 2

Chemical potentials as functions of $T$ for $N=9$. The solid, dotted, and dashed lines are results obtained within the equidistant model with $\mathrm{\Omega }=10$, 14, and 20, respectively. The dot-dashed line stands for the prediction obtained within the nonequidistant model with $\mathrm{\Omega }=10$. The thin lines correspond to the predictions by the blocked BCS, whereas the thick lines are those obtained within the $q_k$-blocked BCS.

Figure 3

Pairing gap $\mathrm{\Delta }$ [(a1)–(a3)], excitation energy $E^{*}$ [(b1)–(b3)], entropy $S$ [(c1)–(c3)], and heat capacity $C$ [(d1)–(d3)] as functions of $T$ for the equidistant model with $N=9$, $\mathrm{\Omega }=10$, 14, and 20. The dotted, dashed, dot-dashed, and solid lines stand for the predictions obtained by using the blocked BCS, average-blocked BCS, Maino's approach, and exact solutions, respectively.

Figure 4

Same thermodynamic quantities as in Fig. 3 for the equidistant model with $N=9$, $\mathrm{\Omega }=10$ and 20. The dotted, dashed, and solid lines denote the predictions obtained by using the $q_k$-blocked BCS, $q_k$-blocked BCS1, and exact solutions, respectively.