Pairing properties from random distributions of single-particle energy levels

Exploiting the similarity between the bunched single-particle energy levels of nuclei and of random distributions around the Fermi surface, pairing properties of the latter are calculated to establish statistically based bounds on the basic characteristics of the pairing phenomenon. When the most probable values for the pairing gaps germane to the BCS formalism are used to calculate thermodynamic quantities, we find that while the ratio of the critical temperature $T_c$ to the zero-temperature pairing gap is close to its BCS Fermi gas value, the ratio of the superfluid to the normal phase specific heats at $T_c$ differs significantly from its Fermi gas counterpart. The largest deviations occur when a few levels lie closely on either side of the Fermi energy but other levels are far away from it. The influence of thermal fluctuations, expected to be large for systems of finite number of particles, were also investigated using a semiclassical treatment of fluctuations. When the average pairing gaps along with those differing by one standard deviation are used, the characteristic discontinuity of the specific heat at $T_c$ in the BCS formalism was transformed to a shoulderlike structure indicating the suppression of a second-order phase transition as experimentally observed in nanoparticles and several nuclei. Contrasting semiclassical and quantum treatments of fluctuations for the random spacing model is currently underway.

Figure 1

Neutron single-particle energy levels in the indicated nuclei from HFB calculations [22, 23] using the SkO${}^{\prime }$ energy density functional with full pairing. The dotted lines indicate the location of the Fermi energies in each case.

Figure 2

Neutron single-particle energy levels in ${}^{126}\mathrm{Sn}$ from HF+BCS calculations [22, 23] using the ${\mathrm{SkO}}^{\prime }$ energy density functional with constant pairing force (leftmost set) and three randomly generated single-particle energy levels.

Figure 3

(a) Pairing gap vs temperature for 500 sets of randomly generated sp levels. (b) Pairing gap normalized to its zero-temperature value vs temperature normalized to the critical temperature for the results in (a).

Figure 4

Ratio of the superfluid to normal phase specific heats at constant volume at $T_c$.

Figure 5

Pairing gap $\mathrm{\Delta }$ vs temperature, $T$, and the projection of the total angular momentum, $M$.

Figure 6

Examples of random sp energy spectra illustrating the origin of large deviations in $C_V^s/C_V^n$. The long horizontal lines show the locations of the corresponding Fermi energies.

Figure 7

Probability distributions $P\left(\mathrm{\Delta }\right)$ vs $\mathrm{\Delta }$ at different $T$'s. The maximum for each $P\left(\mathrm{\Delta }\right)$ occurs at the corresponding ${\mathrm{\Delta }}_{\mathrm{mp}}$ obtained from Eq. (2).

Figure 8

The most probable and average chemical potentials for pairing gaps, ${\mathrm{\Delta }}_{\mathrm{mp}}$ and ${\mathrm{\Delta }}_{\mathrm{av}}$.

Figure 9

The most probable and average pairing gaps, ${\mathrm{\Delta }}_{\mathrm{mp}}$ and ${\mathrm{\Delta }}_{\mathrm{av}}$, along with those differing by one standard deviation, $\sigma$.

Figure 10

(a) Excitation energies with the gaps shown in Fig. 9. (b) Specific heats at constant volume with the gaps shown in Fig. 9.

Figure 11

The most probable and average pairing gaps, ${\mathrm{\Delta }}_{\mathrm{mp}}$ and ${\mathrm{\Delta }}_{\mathrm{av}}$, for 50 independent random realizations of sp energies.

Figure 12

Excitation energies with (a) ${\mathrm{\Delta }}_{\mathrm{mp}}$ and (b) ${\mathrm{\Delta }}_{\mathrm{av}}$ shown in Fig. 11.

Figure 13

Specific heats at constant volume with (a) ${\mathrm{\Delta }}_{\mathrm{mp}}$ and (b) ${\mathrm{\Delta }}_{\mathrm{av}}$ shown in Fig. 11.

Figure 14

Same as Fig. 11, but for the RS model with degeneracy $d=2j+1$.

Figure 15

Same as Fig. 10, but for the RS model with degeneracy $d=2j+1$.

Figure 16

Same as Fig. 9, but for protons (a) and neutrons (b) in ${}^{197}\mathrm{Pt}$.

Figure 17

Same as Fig. 10, but for protons and neutrons in ${}^{197}\mathrm{Pt}$.