Ⅱ-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 53

Pairing in excited nuclei: a review†N. Quang Hung,∗1 N. Dinh Dang,∗2 and L. G. Moretto∗31012

The present review summarizes the recent progressin the study of pairing properties in excited nuclei andtheir analogy to those appeared in other ﬁnite systems including superconductors, metallic nano sizedclusters/grains, and solid-state materials such as ferromagnets. The review consists of four parts.
The ﬁrst part discusses the treatment of pairing
within a uniform model based on the BCS Hamiltonian within the grand canonical ensemble.
The second part focuses on the grandcanonical treatment of pairing within the Hartree-Fock-Bogoliubov
theory and ﬁnite-temperature pairing reentrance phenomenon in superconducting ultrasmall metallic grains
as well as in even-even and odd nuclei.
The third part presents the results obtained fromthe treatment of nuclear pairing within the canonicaland microcanonical ensembles, from which the eﬀectcaused by the ﬁnite size of the systems is highlighted.
Diﬀerent approaches to canonical and microcanonical ensembles including particle-number projection,
particle-number projection plus static path approximation, solutions of BCS with Lipkin-Nogami particlenumber projection incorporating the self-consistentquasiparticle random-phase approximation embedded
into the canonical and microcanonical ensembles, shellmodel Monte Carlo method at ﬁnite temperature, andexact solutions of the pairing problems, which are embedded into the canonical and microcanonial ensembles, are introduced.
In the fourth part, the ﬁrst experimental evidenceof the pairing reentrance phenomenon in a warm rotating 104 Pd nucleus, which was observed via the local enhancement of nuclear level density at low temperature and high angular momentum extracted fromthe reaction 12 C + 93 Nb at the incident energy of 40–
50 MeV, is analyzed and discussed within the framework of the ﬁnite-temperature BCS (FTBCS), which
includes the quasiparticle-number ﬂuctuations (FTBCS1) at angular momentum J (Fig. 1). The similar behavior of pairing reentrance observed in thecondensed-matter counterpart such as metallic compound of Eu0.75 Sn0.25 Mo6 S7.2 Se0.8 and heavy-fermion
cubic system of CePb3 , quasi-2D organic conductor κ(BETS)2 FeCl4 , and ferromagnetic superconductor ofURhGe under the strong magnetic ﬁeld is also summarized.
It has been found that, by applying the BCS theory
†
∗1
∗2
∗3

ρ (MeV -1 )1010
810ExpFTBCSFTBCS1

10 6104__102(b) J = 10 h(a) J = 0 h

100

ρ (MeV -1 )

1010
108
10 6104_10_(c) J = 20 h

102(d)J = 30 h

005101520

E* (MeV)

250510152025

E* (MeV)Fig. 1. Level densities for 104 Pd as function of excitationenergy E ∗ at the quadrupole deformation parameterβ2 = 0.276 and diﬀerent values of total angular momentum J. The dotted and dashed lines stand for theFTBCS and FTBCS1 results, respectively. The solidlines are the experimentally extracted level densities.

at ﬁnite temperature and angular momentum to theuniform nuclear model, for which the analytic solutions of the BCS equation can be obtained, one is able
to study the appearance of the ﬁrst-order and secondorder phase transitions in ﬁnite nuclear systems wheneither the total angular momentum or the number ofquasiparticles or the total energy of the system is ﬁxed.
However, it has also been pointed out that these shapephase transitions are simply an artifact caused by theapplication of the BCS theory to ﬁnite nuclei, neglecting all the thermal ﬂuctuations. In fact, strong thermal
ﬂuctuations in ﬁnite systems smooth out all the phasetransitions, leading to diﬀerent behaviors of nuclear
thermodynamic quantities. For example, the ﬂuctuations of the quasiparticle number are one of the microscopic origins caused the nonvanishing nuclear pairing
gaps at ﬁnite temperature, resulting in the smoothingof the superﬂuid-normal (second-order) phase transition. This ﬁnding is important in the sense that thermal ﬂuctuations should always be considered whenever
the statistical methods are applied to ﬁnite systems.

Condensed from the article in Rep. Prog. Phys. 82, 056301(2019)
Duy Tan University
RIKEN Nishina Center
University of California at Berkeley and Lawrence Berkeley
National Laboratory- 70 -30