Accuracy of BCS-based approximations for pairing in small Fermi systems

We analyze the accuracy of BCS-based approximations for calculating correlation energies and odd-even energy differences in two-component fermionic systems with a small number of pairs. The analysis is focused on comparing BCS and projected BCS treatments with the exact solution of the pairing Hamiltonian, considering parameter ranges appropriate for nuclear pairing energies. We find that the projected BCS is quite accurate over the entire range of coupling strengths in spaces of up to about $~20$ doubly degenerate orbitals. It is also quite accurate for two cases we considered with a more realistic Hamiltonian, representing the nuclei around ${}^{117}\mathrm{Sn}$ and ${}^{207}\mathrm{Pb}$. However, the projected BCS significantly underestimates the energies for much larger spaces when the pairing is weak.

Figure 1

Correlation energies for the Hamiltonian (1), calculated in various approximations: (a) $\Omega =8$, $N_{\mathrm{pair}}=4$; (b) $\Omega =80$, $N_{\mathrm{pair}}=40$.

Figure 2

Errors of the BCS approximation for correlation energies.

Figure 3

Errors for the correlation energies calculated in the PBCS approximation.

Figure 4

Errors for the correlation energies approximated as the sum of the BCS energy and the second-order perturbative energy, Eq. (13).

Figure 5

Pairing gaps [Eq. (14)] calculated in various approximations: (a) $\Omega =8$, $N_{\mathrm{pair}}=4$; (b) $\Omega =80$, $N_{\mathrm{pair}}=40$.

Figure 6

Error in pairing gaps calculated with the PBCS approximation.

Figure 7

Pairing correlation energy in ${}^{116}\mathrm{Sn}$. The pairing interaction is a δ function with a strength scaled from the nominal value $v_0=465$ MeV fm${}^3$ by a factor given on the abscissa. The orbital space is the 16 orbitals around the Fermi energy as described in the text. The solid line shows the result of exact diagonalization.

Figure 8

Neutron pairing gap at ${}^{117}\mathrm{Sn}$. The correlations energies for the three nuclei needed for Eq. (14) were calculated with the same functional and in the same space as in Fig. 7.

Figure 9

Pairing correlation energy in ${}^{206}\mathrm{Pb}$. For these calculations, the space was truncated to $\Omega =16$ by including the 11 highest orbitals below the $N=126$ magic number and the 5 $g_{9/2}$ orbitals above $N=126$. This corresponds to an energy window of about 8 MeV.

Figure 10

Neutron pairing gap at ${}^{207}\mathrm{Pb}$. For these calculations we used the same space as in Fig. 9.

Figure 11

Occupation factors ${\kappa }_i^2=v_i^2(1-v_i^2)$ for $N_{\mathrm{pair}}=8$, $\Omega =16$, and $g=0.32$. The results correspond to the Hamiltonian (1) with ${\varepsilon }_i=i$.

Figure 12

The same as described in the caption to Fig. 11 but for $g=0.42$.

Figure 13

The same as in Fig. 11 but for $g=0.87$.

Figure 14

The Schmidt numbers [Eq. (24)] for a system with $N_{\mathrm{pair}}=8$ and $\Omega =16$.