Ⅰ. HIGHLIGHTS OF THE YEAR
RIKEN
Accel. Prog. Rep. 56 (2023)

RIKEN Accel. Prog. Rep. 56 (2023)

Isovector density and isospin impurity in 40 Ca†
H. Sagawa,∗1,∗2 S. Yoshida,∗3 T. Naito,∗4,∗1 T. Uesaka,∗5 J. Zenihiro,∗6 J. Tanaka,∗1 and T. Suzuki∗7

ε2 = (0.928 ± 0.586) %.

(1)

This central value is about 50% larger than thevalue of RPA calculations without the ISB forces.
From this value of the isospin impurity, the strengthof CSB interaction s0 is further obtained as s0 =− (8.80 ± 16.0) MeV fm3 .
In summary, we studied the IV density of 40 Ca using†∗1
∗2
∗3
∗4
∗5
∗6
∗7

Condensed from the article in Phys. Lett. B 829, 137072(2022)RIKEN Nishina Center
Center for Mathematical Sciences, University of Aizu
Science Research Center, Hosei University
Department of Physics, University of Tokyo
RIKEN Cluster for Pioneering Research
Department of Physics, Kyoto University
Department of Physics, Nihon University

0.3
0.2

4r2IV ( fm-1 )Experimental charge densities have been studied byelectron scattering experiments since the 1950s. Protondensities can be extracted from charge densities removing the proton ﬁnite-size eﬀect. It has been shown in
Ref. 1) that proton elastic scattering is quite useful toextract the neutron density from the angular distributions of cross sections and analyzing powers of polarized
proton beams.
Charge symmetry breaking (CSB) and charge independence breaking (CIB) forces have been discussed
in the context of isospin impurity eﬀect on the superallowed Fermi decays. The quantitative information ofisospin symmetry breaking (ISB) forces has been recently examined to calculate the binding energies of
isodoublet and isotriplet nuclei and also the excitationenergies of isobaric analogue states (IAS). This study
introduces the Skyrme-type ISB interactions and study
their eﬀect on the IV density in 40 Ca.
The isovector density deﬁned as the diﬀerence between neutron and proton densities as ρIV = ρn − ρpis plotted with a factor of 4πr2 (Fig. 1). The theoretical predictions of the isovector density are qualitatively
diﬀerent from the experimental results: the experimental result of the isovector density exhibits a peak atr ∼ 3.2 fm, whereas the Hatree-Fock (HF) model withSAMi-J27 interaction predicts a peak at r ∼ 2.5 fm
with positive values (neutron excess) in the interior andnegative values on the surface.
A strong correlation between the maximum of IV
density and isospin impurity was reported in Ref. 2)
with the correlation coeﬃcient r = 0.992. The isospin
impurity can be extracted from the peak height of theIV density when the proton and neutron densities areavailable experimentally. The experimental peak heightof the IV density is 0.208 ± 0.066 fm−1 in Fig. 1. Then,the isospin impurity is then extracted from the correlation plot as40

SAMi-J27
PVC(J27)
SAMi-J27modExp.

Ca

0.1
0.0
-0.1
-0.2
-0.3
0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0r ( fm )Fig. 1. IV densities multiplied by a factor of 4πr 2 of 40 Ca
calculated by the HF, modiﬁed HF and PVC models as well as the experimental IV density. The red(blue) solid and dashed curves denote the HF (modiﬁedHF) and PVC densities with SAMi-J27 (SAMi-J27mod)model, respectively. The occupation probabilities near
the Fermi surface are optimized to ﬁt the IS density inthe case of modiﬁed HF. The experimental data are denoted with a black solid curve considering the experimental uncertainty depicted by the shaded area.

the mean-ﬁeld and particle-vibration coupling (PVC)
models. We adopted the Skyrme SAMi-J model as themean-ﬁeld model. We observed a signiﬁcant diﬀerence
between the experimental and calculated IS densities in
the interior part and also dilute density region of 40 Ca.
This diﬀerence implies the modiﬁcation of density distribution owing to the reduced occupation probabilities
of single-particle states near the Fermi surface, which
may be caused by many-body correlations beyond themean-ﬁeld model.
It was shown in Ref. 2) that the magnitude of theIV density signiﬁcantly changes owing to the CSB interaction, whereas the CIB interaction exhibits no appreciable eﬀect. It is found that the CSB interaction
exhibits a strong linear correlation with the maximumof IV density and isospin impurity. Thus, the magnitude of IV density helps experimentally to determine
the isospin impurity and the magnitude of CSB interaction. This characteristic of the IV density appears ingeneral in N = Z nuclei, 40 Ca, 80 Zr, and 100 Sn. The
measurements of the IV density is desired to obtain theexperimental information of the isospin impurity andthe CSB interaction.
References
1) H. Sakaguchi et al., Prog. Part. Nucl. Phys. 97, 1 (2017).
2) H. Sagawa et al., Phys. Lett. B 829, 137072 (2022).

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