II-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 54 (2021)

RIKEN Accel. Prog. Rep. 54 (2021)

Parity-conserved self-consistent CHFB solution
K. Sugawara-Tanabe∗1,∗2 and K. Tanabe∗3
We developed a new program for solving the constrained Hartree-Fock-Bogoliubov (CHFB) equation
without parity mixing. In this scheme (CHFB5), we
require ﬁve constraints, one each on the total angularmomentum I, proton number Z+ in the + parity shell(p+ ), proton number Z− in the − parity shell (p− ),neutron number N+ in the + parity shell (n+ ), andneutron number N− in the − parity shell (n− ). As
an example, we choose 134 Nd with the same parameter set as that adopted in Ref. 1). Here, we solved
the full CHFB equation2) including all exchange terms(Fock terms), while Ref. 1). adopts only the Hartreeterms. The values of (Z+ , Z− , N+ , N− ) are selected inreference to the usual CHFB solutions with three constraints (CHFB3). The usual CHFB3 solutions show
(Z+ , Z− , N+ , N− ) = (14.59, 17.41, 13.87, 10.13) at I =0, while (14.04, 17.96, 14.0, 10.0) at I = 26. Here,(Z, N ) = (32, 24) are numbers outside the closed core
(28, 50). Thus, we select (14, 18, 14, 10) for the CHFB5equation. The intrinsic diﬀerence between CHFB3 andCHFB5 solutions is in the quasi-particle (QP) energies. In Fig. 1, we compare the behavior of the lowest
QP energies of Λ with its time-reversed energy Λ̃ vs.
I. The equations for Λ and Λ̃ have been provided inRef. 2). The degeneracy is lifted by the Coriolis antipairing eﬀect with increasing I. Figure 1(A) shows theneutron shell, and (B) the proton shell. In both panels, ± speciﬁes the ± shell; the ﬁlled symbols expressΛ̃ and the open symbols Λ. Those in the abbreviation “with" denote CHFB5 solutions, while the others
denote CHFB3 solutions. At low I, QP energies byCHFB3 and CHFB5 solutions coincide in the neutronshells (A); however, there is a considerable diﬀerence
among the p+ shell (B). The negative value of Λ in then+ shell is observed at I = 10 in both CHFB3 andCHFB5 solutions; this indicates the ﬁrst backbendingis caused by the i13/2 level in the n+ shell. There occur
negative values of Λ in the n+ and n− shells aroundI = 20 to 26, and they correspond to decreasing ∆n ,i.e., 0.00021 (CHFB3) and 0.00035 (CHFB5).

lowest QP energies  &  MeV3∗3

A Neutron shells in the Yrast band1

5


10
0

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with 10n 
 n  with 14n



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15


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20

n with 14



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25

30

35

I 

3.0

lowest QP energies  &  MeV

134

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1.5


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Nd

B Proton shells in the Yrast band


1.0p with 14




0.50.00pp  with 18
p with 18


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p

p





p





p with 14



51015

20

253035I Fig. 1. (A) Lowest QP energies Λ and Λ̃ in the neutron shell
as functions of angular momentum I. The red opencircles represent Λ, and the red filled-circles representΛ̃ in the n+ shell by the CHFB3 solutions; the orange
open-triangles denote Λ, and the orange filled-triangles
represent Λ̃ in the n+ shell by the CHFB5 solutions.
The open squares represent Λ, and the filled squares
represent Λ̃ in the n− shell by the CHFB3 solutions,while the blue open-triangles-down represent Λ, and theblue filled-triangles-down represent Λ̃ in the n− shell by
the CHFB5 solutions. (B) The lowest QP energies ofΛ and Λ̃ in the proton shell as functions of I. The
red open-circles represent Λ, and the red filled-circlesrepresent Λ̃ in the p+ shell by the CHFB3 solutions,while the orange open-triangles represent Λ, and theorange filled-triangles represent Λ̃ in the p+ shell by
the CHFB5 solutions. The open squares represent Λ,and the filled squares represent Λ̃ in the p− shell by theCHFB3 solutions, while the blue open-triangles-downrepresent Λ and the blue filled-triangles-down representΛ̃ in the p− shell by the CHFB5 solutions.

References∗22Nd101) K. Tanabe, K. Sugawara-Tanabe, Phys. Lett. 259B, 12(1991).
2) K. Sugawara-Tanabe, K. Tanabe, RIKEN Accel. Prog.
Rep. 52, 1148 (2019).

∗1134

RIKEN Nishina Center
Department of Information Design, Otsuma Women’s University
Department of Physics, Saitama University- 48 -1