RIKEN Accel. Prog. Rep. 48 (2015)

Ⅱ-2. Nuclear Physics (Theory)

Reentrance phenomenon of superfluid pairing in hot rotating nuclei
N. Quang Hung,∗1 N. Dinh Dang,∗2 B.K. Agrawal,∗3 V.M. Datar,∗4 A. Mitra,∗4 and D.R. Chakrabarty∗4∗1
∗2
∗3
∗4

School of engineering, Tan Tao University, Long An
RIKEN Nishina Center
Theory Division, Saha Institute of Nuclear Physics, Kolkata
Nuclear Physics Division, Bhabha Atomic Research Center,
Mumbai1012

ρ (MeV -1 )1010
810ExpFTBCSFTBCS110 6104__102(b) J = 10 h(a) J = 0 h1001010

ρ (MeV -1 )When a nucleus rotates (total angular momentum J
and/or rotational frequency ω are not zero), the nucleon (proton and neutron) pairs located around theFermi surface will scatter to the empty levels nearby
and lead to the decreasing of pairing correlation. When
the J or ω is sufficiently high, i.e., equal to the criticalvalue Jc or ωc , the scattered nucleons completely block
the single-particle levels around the Fermi surface.
Consequently, pairing correlation disappears. However, when J is slightly higher than Jc (or ω ≥ ωc ), theincrease of temperature T will relax the particles scattered around the Fermi surface and causes some levels
become partially unoccupied, making them available
for scattered pairs. As a result, the pairing correlation reappears at some critical value T1 . As T goeshigher, e.g., at T2 > T1 , the newly created pairs will
be eventually broken down again. This phenomenon
is called the pairing reentrance. The recently developed FTBCS1 theory that includes the effect due toquasiparticle-number fluctuations in the pairing field
and angular momentum z projection at T �= 0 has
predicted the pairing reentrance effect in some realistic nuclei1) . The shell-model Monte Carlo calculations
have suggested that the pairing reentrance effect canbe observed in the nuclear level density in a form of
a local maximum at low T (or excitation energy E ∗ )and high J (or ω). Recently, an enhancement of level
density of 104 Pd at low E ∗ and high J has been experimentally reported2) . In this work we try to see
whether the enhancement observed in the extracted
level density of 104 Pd is an experimental evidence ofpairing reentrance phenomenon in atomic nuclei.
The FTBCS1 equations at finite temperature andangular momentum are derived based on the variational method to minimizethe expectation value of thepairing Hamiltonian H = k ǫk (a†+k a+k + a†−k a−k ) −G kk′ a†k a†−k a−k′ ak′ − λN̂ − ω M̂ , in the grandcanonical ensemble. This Hamiltonian describes a system rotating about the symmetry axis, which is chosen
to coincide with its z component. The particle-numberoperator N̂ and the z projection M̂ of the total angular momentum Jˆ (which coincides with M̂ for spherical nuclei) are defined as N̂ = k (a†+k a+k + a†−k a−k ) ,M̂ = k mk (a†+k a+k − a†−k a−k ) , where a†±k (a±k ) are
the creation (annihilation) operators of a particle in
the k-th deformed state, whereas ǫk , λ, and ω are respectively the single-particle energies, chemical poten-

108
10 6104_10_(c) J = 20 h102(d)J = 30 h

005101520

E* (MeV)

25051015202530

E* (MeV)Fig. 1. (Color online) Total level densities for 104 Pd asfunction of E ∗ obtained within at the quadrupole deformation parameter β2 = 0.276 at several J. The dottedand dashed lines stand for the FTBCS and FTBCS1results, respectively. The solid lines are the experimentally extracted level densities

tial, and rotational frequency. The FTBCS1 equation for the pairing gap has the final form as ∆−k =∆ + δ∆k , where ∆ = G k′ uk′ vk′ (1 − n+k ′ − nk ′ ) ,−δ∆k = GδNk2 uk vk /(1 − n+−n),withu,vkk , and nkkk
being the Bogolyubov’ u, v coefficients and quasiparticle occupation numbers, respectively. The total leveldensity ρ(E, J) is calculated as ρ(E, J) = ρ(E, M =J) − ρ(E, M = J + 1) , where ρ(E, M ) is obtained byusing the inverse Laplace transformation of the grandpartition function.
Because of quasiparticle number fluctuations, theFTBCS1 gaps decrease monotonically with increasing
excitation energy E ∗ and do not collapse at E ∗ = Ec∗
as in the case of the FTBCS. Within the FTBCS1, thepairing reentrance is seen very clearly at J = 20h̄ forneutrons and at J = 30h̄ for protons. Consequently,
there appear local enhancements in the FTBCS1 leveldensities at around 2 < E ∗ < 5 MeV at these two values of J(Fig. 1). The FTBCS1 level densities agree
fairly well with the experimental data at all J values
considered in present work.
References
1) N.D. Dang and N.Q. Hung, Phys. Rev. C 77 ), 064315(2008); N.Q. Hung and N.D. Dang, Phys. Rev. C 78,064315 (2008), Phys. Rev. C 84, 054324 (2011).
2) A. Mitra et al., J. Phys. G 36, 095103 (2009); A. Mitraet al., EPJ Web of Converences 2, 04004 (2010); V.
Datar, A. Mitra, and D.R. Chakrabarty, unpublished.

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