Ⅱ-2. Nuclear Physics (Theory)
RIKEN
Accel. Prog. Rep. 56 (2023)

RIKEN Accel. Prog. Rep. 56 (2023)

He cluster formation in light neutron-rich nuclei†
H. Motoki,∗1 S. Shin,∗1 Y. Suzuki,∗2 Q. Zhao,∗3 B. Zhou,∗4 and M. Kimura∗5
Recently, a negative correlation between the neutron
number and 4 He cluster formation probability on thenuclear surface was experimentally reported for Sn isotopes.1) A similar trend was theoretically observed forC isotopes.2) However, this trend contradicts to the previous expectation; 4 He clustering develops toward theneutron drip-line of Be and B isotopes.3) To understandthese ambiguous results, we have quantitatively evaluated the 4 He cluster formation probability based on theantisymmetrized molecular dynamics (AMD).
Figure 1 shows the calculated intrinsic proton density
distributions of the ground states.4) Except for 11 B, thedistributions exhibit two peaks, indicating the formation of He + He and He + He clusters in Be and B isotopes, respectively. The development of clusters can be
estimated from the distance between the clusters (distance between two peaks), which is shown in the lower
panel of Fig. 1. It suggests a gradual growth of clusters
toward the drip line, which is qualitatively consistent
with the previous expectation.3)

A′⟨δ(x − r)ΦHe ΦA′ Yl=0 (x̂)]0 |ΦA ⟩ ,where ΦHe , ΦA′ and ΦA denote the wave functions of
the He cluster, daughter, and target nuclei. RWA is
the probability amplitude to determine the He cluster
at distance r from the daughter nucleus. Hence, itssquare integral (herein referred to as S-factor) deﬁnes
the cluster formation probability.
The results are denoted by solid lines in Fig. 2. Unexpectedly, the S-factor decreases as neutron number increases. This contradicts the results illustrated in Fig. 1and previous expectation;3) however, this trend remains
consistent in Sn isotopes.1) To resolve this issue, we investigated the formation probabilities of 6 He and 8 Heclusters (Fig. 2). The 6 He and 8 He clusters dominate
over the 4 He cluster as neutron number increases. Considering the sum of the formation probabilities of He
isotopes, we observe an increasing trend, as shown in
Fig. 1.

S-factor

0.3

12Be

10Be

14Be

0.2

0.2

S(4He)11B

S(8He)

S(6He)
13B

15B

17B

0.1

19B

0.1

15B

13B

11B

S(4He)

17B

0.0

0.3

cluster distance (fm)

3.0

68
neutron number

13B 4He 4He

0.0

Sum
0.0

10

84He

15B

6He

610Be

12Be

14Be

0.0

10
12
neutron number

14

17B

6He

6He

8He

8He

0.1
2.0

4.0

6.0

2.0

4.0

6.0

2.0

4.0

6.0

-0.1

2.0
1.0

Sum

S(6He)

0.2

4.0

(1)

0.4

14Be

12Be√( )A

S-factor

10Be

rY(r) =

11B

613B

815B

17BFig. 2. Upper panels show the 4 He, 6 He and 8 He cluster
formation probabilities. The lower panels show the RWAof the 4 He, 6 He and 8 He clusters in B isotopes.

BeB

1012Neutron number

14

To quantitatively evaluate He cluster formation andrelate it to 4 He knockout reaction, we calculated thereduced width amplitude (RWA).

The aforementioned results may be summarized asfollows: In neutron-rich Be and B isotopes, 4 He isformed as a cluster core; however, but it is always accompanied by excess neutrons. As a result, we should
sum up the 4 He clusters and 6, 8 He clusters to accurately
evaluate the cluster formation probability. We believethat this new insight is important for future studies tounderstand the relationship between clustering and excess neutrons.

†References

Fig. 1. Upper panels show the intrinsic proton density distribution where peaks of the density distribution areplotted by ﬁlled circles. Lower panel shows the distance
between cluster centroids, i.e. inter-cluster distance.

∗1
∗2
∗3
∗4
∗5

Condensed from the articles in Eur. Phys. J. A 57, 1 (2021)
and Prog. Theor. Exp. Phys. 2022, 113D01 (2022)
Department of Physics, Hokkaido University
Research Center for Nuclear Physics, Osaka University
School of Science, Huzhou University
Institute of Modern Physics, Fudan University
RIKEN Nishina Center- 35 -1) J. Tanaka et al., Science 371, 260 (2021).
2) Q. Zhao et al., Eur. Phys. J. A 57, 1 (2021).
3) Y. Kanada-En’yo et al., Phys. Rev. C 52, 647 (1995).
4) H. Motoki et al., Prog. Theor. Exp. Phys. 2022, 113D01(2022).