Test of the modified BCS model at finite temperature

A recently suggested modified BCS (MBCS) model has been studied at finite temperatures. We show that this approach does not allow the existence of the normal (nonsuperfluid) phase at any finite temperature (FT). Other MBCS predictions, such as a negative pairing gap, pairing induced by heating in closed-shell nuclei, and superfluid to super-superfluid phase transition are discussed also. The MBCS model is tested by comparing it with exact solutions for the picket fence model. Severe violation of the internal symmetry of the problem is detected. The MBCS equations are found to be inconsistent. The limit of the MBCS applicability has been determined to be far below the superfluid–normal phase transition of the conventional FT-BCS, where the model performs worse than the FT-BCS.

Figure 1

(Color online) MBCS (a) pairing gap $\overline{\Delta }$ and (b) ${\overline{v}}_j$ coefficients for particle levels near the Fermi surface in ${}^{76}\mathrm{Ni}$ (neutrons). The conventional critical temperature $T_c$ is shown by the vertical arrow, and the single-particle spectrum used is from Ref. 4.

Figure 2

MBCS (a) pairing gap $\overline{\Delta }$, (b) chemical potential $\overline{\lambda }$, and (c) excitation energy $E^{*}$ in ${}^{84}\mathrm{Ni}$ (neutrons). The single-particle spectrum from Ref. 4 is extended by including levels from $N=0-28$ shells.

Figure 3

MBCS (a) pairing gap $\overline{\Delta }$ and (b) chemical potential $\overline{\lambda }$ for neutrons (solid curves) and protons (dashed curves) in ${}^{120}\mathrm{Sn}$. ${\overline{\Delta }}_{\mathrm{CS}}$ and ${\overline{\lambda }}_{\mathrm{CS}}$ (dotted curves) correspond to closed-shell calculations in Ref. 5. See text for details.

Figure 4

(Color online) (a) Pairing gap $\overline{\Delta }$, (b) excitation energy $E^{*}$, and (c) specific heat $C_{\nu }$ as predicted by the MBCS (solid curves) and FT-BCS (dashed curves) for the PFM ($N=10$ and $G=0.4$ MeV). The exact results in (b) and (c) are plotted as dotted curves.

Figure 5

(Color online) Spectroscopic factors of the two lowest particle and hole levels in the PFM ($N=10$ and $G=0.4$ MeV) calculated within the MBCS (solid curves) and FT-BCS (dashed curves). Dotted curves represent the exact results.

Figure 6

Quasiparticle spectrum in the PFM ($N=10$ and $G=0.4$ MeV) within (a) the FT-BCS and (b) the MBCS. Particle (hole) levels are plotted as thick (thin) curves.

Figure 7

(Color online) The $(u_1^2+u_{-1}^2)$ quantity in the MBCS-PFM calculations with different N. $T_c=0.5$ MeV is the same in all calculations. Analytical identity ($y=1$) is plotted as a dotted curve.

Figure 8

(Color online) MBCS ${\overline{b}}_j$ and ${\overline{c}}_j$ quantities as functions of temperature for some neutron levels in ${}^{120}\mathrm{Sn}$. See text for details.

Figure 9

(Color online) Excitation energy of the PFM calculated as $E^{*}$ (thick curves) and $\langle H\rangle {}^{*}$ (thin curves) in the MHFB (solid curves) and the FT-HFB (dashed curves) with $T_c=0.42$ MeV for (a) $N=10$ and (b) $N=2$.