Ⅱ-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 53

Canonical base in self-consistent constrained HFB in odd-A nuclei†
K. Sugawara-Tanabe∗1,∗2 and K. Tanabe∗3
We developed a program for solving the selfconsistent Hartree-Fock-Bogoliubov (CHFB) equation
under four constraints, one each on the total angularmomentum I, proton number Z+ in the + parity proton shell (p+ ), proton number Z− in the − parity
proton shell (p− ), and neutron number N , so thatit reproduces the 11/2− band in 135 Pr. We choose
the signature-invariant base Ck and Ck̂ , where k̂ isthe time-reversed level of k, and adopt a Hamiltonian
with spherical single-particle energies plus the residual quadrupole-quadrupole, monopole-pairing, andquadrupole-pairing interactions. In this signatureinvariant base, the generalized density matrix K(=K 2 ) is expressed in terms of ρ1kl = ⟨Cl† Ck ⟩, ρ2k̂l̂ =Ρ1 C � Ρ2 C in canonical base

1.4ii

∗1∗2
∗3      

0.8odd nucleon in p�

0.6�p p�
 p� 14� p� 17

0.40.015 10

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20

Ξ1 � Ξ2 2 MeVii

�c� in canonical base MeV

1.4

B Gap in the canonical base

1.2

 p�
 p�
 p� 14
� p� 17

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0.4  0.2    
 


 

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15 10

505101520Ξ1 � Ξ2 2 MeVîîFig. 1. (A) ρ1c = ρ2c in the canonical base as a functionof (ξ 1 + ξ 2 )/2 at the I = 31/2− state. (B) ∆c in thecanonical base as a function of (ξ 1 + ξ 2 )/2 at the I =31/2− state. In both figures, filled circles represent theproton levels in the p+ shell with Z+ = 14, and filledsquares represent the proton levels in the p− shell with
Z− = 17. Open circles and open squares indicate theCHFB solution under Z = 14+17 = 31. The red filledsquare level with Z− = 17 and the red open-squarelevel with Z = 31 are mainly from h11/2 . See the text
for further details.

where ξ 1 = (F 1 )−1 (h1 − ωjx )F 1 , ξ 2 = (F 2 )−1 (h2 +ωjx )F 2 , and ∆c = (F 2 )−1 ∆F 1 . Here h1 , h2 , and ∆are deﬁned in Eq. (1) in Ref. 1). In Fig. 1, we compare the CHFB solution under three constraints for
I = 31/2, Z = Z+ + Z− , and N (open squares andopen circles) with the CHFB solution under four constraints (ﬁlled squares and ﬁlled circles). Here, thecircles correspond to the p+ shell, and squares correspond to the p− shells. The quantities represented by
the red ﬁlled square under the constraint Z− = 17 andthe red open square under Z = 31 are composed of thecontribution mainly from the h11/2 level with jz = 5/2.
The number Z = 31 corresponds to the proton number
outside core 28 for 135 Pr. As shown in (A) of Fig. 1,
ρ1c = ρ2c ∼ 1/2, i.e., 1/2 × 2 = 1 in the p− shell, weconﬁrm that the solution certainly describes the negative parity band in the odd-Z nucleus (11/2− band in†

1.00.2⟨Cl̂† Ck̂ ⟩ and κk̂l = ⟨Cl Ck̂ ⟩, where the state | ⟩ is thequasiparticle vacuum. The CHFB equation is given as
Eq. (1) in Ref. 1). After rendering the dangerous term
zero through the iteration procedure, we can transform the elements in K to the canonical forms, i.e.,
111c2 −1 2
22c(F 1 )−1km ρmn Fnk ≡ ρkk and (F )k̂m̂ ρm̂n̂ Fn̂k̂ ≡ ρk̂k̂ ; thus(F 2 )−1κ F 1 ≡ κck̂k . In other words, K is transk̂m̂ m̂n nk
formed to Kc , and the relation Kc2 = Kc follows. Subsequently, we obtain ρ1c = ρ2c . If we apply the sametransformation to the CHFB matrix (see Eq. (1) inRef. 1)), we obtain
12(ξ+ξ)/21
ii
2cîî
 , (1)1−√ρ1cii = ρîî =2c2(∆ ) +((ξ 1 +ξ 2 )/2)2îi

A Density in the canonical base1.2135Pr). In (B) of Fig. 1, we compare ∆ciî between theCHFB solution under three constraints and the CHFB
solution under four constraints The blocking eﬀect isclearly indicated by the red ﬁlled or open squares because these gap values are much smaller than those forthe other levels.

Condensed from the talk at the autumn meeting of Japan
Physical Society, Yamagata University, Sept. 18th (2019)
RIKEN Nishina Center
Department of Information Design, Otsuma Women’s University
Department of Physics, Saitama University

Reference
1) K. Sugawara-Tanabe, K. Tanabe, RIKEN Accel. Prog.
Rep. 52, 1148 (2019).

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