Ⅱ-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 50 (2017)

Improved treatment of blocking eﬀect at ﬁnite temperature†

In odd nuclei with odd numbers of nucleon (neutronor proton), due to the Pauli principle, the odd particle at zero temperature (T = 0) completely blocks thelevel k which it occupies. Consequently, this level k remains unpaired with the occupation number equal to
1. Within the approximation such as the BCS theory,
this level is excluded from the BCS equation for thepairing gap, whereas the particle-number equation issolved only for the even core. The resulting approximation is called the “blocked BCS”1) . At ﬁnite temperature T = 0, the odd particle has the probabilities
of occupying all the levels above the Fermi surface. As
the result, any observable is then obtained by taking
the average of all the above probabilities over the statistical ensembles such as the grand canonical ensemble(GCE) or canonical ensemble (CE). Several approximations have been proposed to describe the propertiesof odd nuclei at T = 0. One of the approximations,which is a direct extension of the blocked BCS and is
widely used in nuclear physics, was proposed by Mainoet al.2) . This approach assumes that at T = 0, the odd
particle can occupy the level k with the unity occupation number (qk = 1) as at T = 0, but k can be anylevel above the Fermi surface, not only k = k0 with
k0 being the lowest level located just above the Fermi
level. The approach, called “Maino’s” hereafter, thentakes the average of any observable as the sum of all
its average values taken for each blocked level k withthe statistical weight as the ratio of the grand partition
function for the blocked level k to the total grand partition function. The assumption of the unity occupation
probability of the odd level in the Maino’s approachmeans that temperatures have no eﬀect on it, whereas
the occupation numbers of all the levels in the pairing core are obtained by averaging over the GCE, and,they are therefore temperature-dependent. An obvious
inconsistency of such assumption is that it fails to reproduce the zero-pairing limit of the equation for theparticle number at ﬁnite temperature, where all thesingle particle occupation numbers should follow theFermi-Dirac distribution of noninteracting fermions.
Present paper proposes an improved treatment of
the blocking eﬀect in odd nuclear systems interacting
via the constant pairing force. Based on the analysisof the exact solutions of the doubly-degenerate equidistant multilevel pairing model for N = 9 particles andΩ = 10, 14, and 20 levels whose the single-particle en†
∗1
∗2
∗3

Occupation numbers

N. Quang Hung,∗1 N. Dinh Dang,∗2 and L. T. Quynh Huong∗32.0

Ω = 14 (b)Ω = 10 (a)Ω = 20 (c)

1.5
1.0
0.50

1

2 3T (MeV)40

12 3 4T (MeV)01

2 3 4T (MeV)

5Fig. 1. Exact occupation numbers of the levels as functionsof T obtained within the doubly-degenerate equidistant multilevel pairing model for N = 9 particles andΩ = 10 (a), 14 (b), and 20 (c) levels. In each panel,
the higher-located line is the occupation number of thelower-located level. The thick line, which starts from 1at T = 0, is the occupation number of the level occupiedby the odd particle.

ergies are k = k (k = 1, 2, ..., Ω), we show that theexact occupation number of the blocked level occupiedby the odd particle is equal to 1 at T = 0 and thendecreases with increasing T (See e.g., thick solid lines
in Fig. 1). As the result, the conventional assumption of the unity occupation number for this blocked
level is no longer valid at T = 0. Instead, we introduce the temperature-dependent occupation number
qk for the blocked level, which is directly derived fromthe standard equation for the particle number within
the standard ﬁnite-temperature BCS [Eqs. (13), (23),and (24) of Ref.3) ]. We also construct the average
qk -blocked BCS [Eqs. (13), (23), and (25) of Ref.3) ],which follows Maino’s idea and the results obtained
agree well with those obtained within the Mainos approach. However, again by using the exact solutionsof the pairing Hamiltonian, we demonstrate that theaverage procedure over all the levels above the Fermi
surface as proposed in the Mainos approach does notcorrespond to the real situation observed in the exact
solutions, where the odd particle actually always remains on the top level k = k0 . Based on the analysisof the results obtained in the present paper, we believe
that, in the study of the odd systems with pairing, suchas atomic nuclei, at ﬁnite temperature, if the BCS approach with blocking is ever applied, the qk -blocked
BCS with k = k0 proposed here should be used instead of the blocked BCS, whereas neither the Maino’s
approach nor the average qk -blocked BCS is necessary.
References1) P. Ring and P. Schuck, The Nuclear Many-Problem(Springer, Berlin, 2004).
2) G. Maino, E. Menapace, and A. Ventura, Il Nuovo Cimento A 57, 427 (1980).
3) N. Quang Hung, N. Dinh Dang, and L. T. Quynh
Huong, Phys. Rev. C 94, 024341 (2016).

Condensed from the article in Phys. Rev. C 94, 024341(2016)
Institute of Research and Development, Duy Tan University
RIKEN Nishina Center
Department of Natural Science and Technology, Universityof Khanh Hoa- 93 -