Ⅱ-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 51 (2018)

New interpretation of pairing anti-halo eﬀect†)

K. Hagino∗1,∗2,∗3 and H. Sagawa∗4,∗5

H = ĥ(1) + ĥ(2) + vpair (⃗r1 , ⃗r2 ) +p⃗1 · p⃗2,
mc

quasi-particle wf (fm-1/2The pairing anti-halo eﬀect is a phenomenon bywhich a pairing correlation suppresses the divergenceof nuclear radius, which happens for single-particle
states with orbital angular momenta of l = 0 and
1 in the limit of vanishing binding energy. Thisphenomenon was originally proposed based on theHartree-Fock-Bogoliubov (HFB) theory. Although theHFB method provides a clear mathematical interpretation of the pairing anti-halo eﬀect, its physical mechanism is less transparent. The aim of this paper is topropose a more intuitive idea on the pairing anti-haloeﬀect, using a three-body model. This model is formulated to include many-body correlations beyond theHFB model, providing a complementary opportunity
to clarify the concept based on the HFB method. It
can be used to test whether the pairing anti-halo eﬀect
is speciﬁc only to the mean-ﬁeld treatment or not.
The Hamiltonian for the three-body model reads1)

unlj (r)Yjlm (r̂),
r(1)

(2)

∗2
∗3
∗4
∗5-0.4010

n=2s1/2r (fm)2030(c)(5)n=cont.

The main feature of this quasi-particle wave functionis that the bound state and continuum state contributions largely cancel each other outside the potential
while the two components contribute coherently in theinner region. We recognise that the localization due
to a coherent superposition of continuum states is thesame mechanism as the formation of a localized wave
packet. This is an essential ingredient of the pairing
anti-halo eﬀect, that is, the formation of a localized
wave packet induced by a pairing interaction.

k,l,j

∗1

total
bound statecontinuum(b)where the coeﬃcients Cnn′ lj are calculated by diagonalizing the three-body Hamiltonian (1). The
one-particle density constructed with this two-particle
wave function is given by�2�∫
1 ∑ �� ũklj (r) ��ρ(⃗r) = d⃗r′ |Ψ(⃗r, ⃗r′ )|2 =(4)� r � ,4π†-0.2ũklj (r) = ũklj (r) + ũklj (r)∑∑Cnklj unlj (r)+Cnklj unlj (r).
=n,n′ ,l,j∑0Note that this is in a similar form as the oneparticle density in the HFB approximation, especially
if the quasi-particle wave function is expended on theHartree-Fock basis, unlj .2)
The solid line in Fig. 1 shows the radial dependenceof the quasi-particle wave function for the weaklybound 2s1/2 state; that is, ũklj (r) with (klj) = 2s1/2
in 24 O. The dashed and dot-dashed lines show its decomposition into the bound state and the continuum
state contributions, respectively. They are deﬁned aswhere ϕnlj (r) and Yjlm (r̂) are the radial and spin angular parts of the s.p. wave function, respectively. Using
these eigen-functions, the two-particle wave function
for the ground state of the three-body system with
spin-parity of J π = 0+ is given as∑Cnn′ lj [ψnlj (⃗r1 )ψn′ lj (⃗r2 )]J=0 , (3)Ψ(⃗r1 , ⃗r2 ) =where ũklj (r) is deﬁned as ũklj (r) ≡0.2

Fig. 1. The radial part of the quasi-particle wave function,ũ2s1/2 (r), defined in Eq. (4), for the weakly bound 2s1/2
state in 24 O. In the uncorrelated case, the two valence
neutrons occupy the 2s1/2 state at ϵ = −0.275 MeV,
while the continuum states are also taken into accountin the three-body model calculations. A zero-range
pairing interaction is employed, which yields a groundstate energy of Eg.s. = −2.46 MeV. The solid line
shows the total wave function, while the dashed anddot-dashed lines denote its bound state and continuum
state contributions, respectively.

where ĥ is a single-particle (s.p.) Hamiltonian andvpair (⃗r1 , ⃗r2 ) is the pairing interaction between the twovalence neutrons. The last term is the two-body partof the recoil kinetic energy of the core nucleus.
The eigen-functions ψnljm (⃗r) of ĥ is given byψnljm (⃗r) = ϕnlj (r)Yjlm (r̂) =0.4

n Cnklj unlj (r).

Condensed from the article in Phys. Rev. C 95, 024304(2017)
Department of Physics, Tohoku University
Research Center for Electron Photon Science, Tohoku University
National Astronomical Observatory of JapanRIKEN Nishina Center
Center for Mathematics and Physics, University of Aizu

References
1) K. Hagino, H. Sagawa, Phys. Rev. C 72, 044321 (2005).
2) K. Hagino, H. Sagawa, Phys. Rev. C 71, 044302 (2005).

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