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

RIKEN Accel. Prog. Rep. 56 (2023)

Indication of α clustering in the density proﬁles of 44, 52 Ti†
W. Horiuchi∗1,∗2,∗3,∗4 and N. Itagaki∗1,∗2
The standard nuclear structure is shell structure; however, α-cluster structure often appears in the ground andexcited states of light nuclei. Here, we discuss the αcluster structure in medium mass nuclei, 44 Ti and 52 Ti,
and show that the development of core+α structure canbe quantiﬁed in the density proﬁles near the nuclear surface using proton-nucleus elastic scattering.
In the present study, we examine two diﬀerent density
proﬁles. Shell- and α-cluster conﬁgurations are generated by utilizing the basis states of the antisymmetrized
quasi-cluster model (AQCM),1) which can describe boththe j-j coupling shell and α-cluster conﬁgurations in asingle scheme. The AQCM basis for the core (40 Ca or48Ca) plus α cluster is expressed by the antisymmetrized
product of the core and α-cluster wave functions, with R
being their distance. The wave function of the core nucleus is constructed based on the multi-α cluster modelat the zero-distance limit, corresponding to the N = 20closure for 40 Ca and eight neutrons are additionally putfor the N = 28 closure of 48 Ca. The size parameters of
the core and α-cluster wave functions are taken as thesame for simplicity.
The shell-model wave functions for 44, 52 Ti are obtained by taking R → 0. The size parameter is ﬁxed toreproduce the measured charge radius of 44 Ti or 52 Ti.
We call this shell-model wave function as S-type.
For the α-cluster wave function, we determine the size
parameter to reproduce the measured charge radius ofthe core nucleus and set the R value to reproduce thecharge radius of 44 Ti or 52 Ti. We call this α-cluster wave
function as C-type. The determined R value is large(2.85 fm) for 44 Ti, indicating a well-developed α-cluster
structure. The charge radius of 52 Ti is not known. However, to understand the role of excess neutrons, we assume R = 3 fm for 52 Ti, which is comparable to that of44Ti.
Although both S- and C-types reproduce the charge
radius data, we discover that α clustering signiﬁcantly
changes the density proﬁles near the surface. To
quantify these changes, we evaluate the nuclear “diffuseness,”2) which is practically obtained by a twoparameter Fermi function, ρ0 /{1+exp[(r−R̄)/a]}, where
the radius (R̄) and diﬀuseness (a) parameters are determined by the least-square ﬁtting with the obtained
one-body density. The a values for S- and C-types are
0.56 fm and 0.63 fm for 44 Ti and 0.58 fm and 0.61 fm for52Ti, respectively. The α clustering signiﬁcantly changes
†
∗1
∗2
∗3
∗4

Condensed from the article in Phys. Rev. C 106, 044330(2022)
Department of Physics, Osaka Metropolitan University
Nambu Yoichiro Institute of Theoretial and Experimental
Physics, Osaka Metropolitan University
RIKEN Nishina Center
Department of Physics, Hokkaido University- 39 -

the nuclear surface diﬀuseness, especially for 44 Ti, owing
to the occupation of the diﬀused low-angular momentum
orbit, i.e., 1p orbit. For 52 Ti, the diﬀerence between Sand C-types is less signiﬁcant that of 44 Ti because theshell-model conﬁguration (1p3/2 )4 already has a diﬀused
nuclear surface.
We show that this diﬀerence can be detected by measuring proton-nucleus elastic scattering at intermediate
energies. The proton-nucleus diﬀerential elastic scattering cross sections are calculated by the optical-limit
approximation in the Glauber model. Inputs to the reaction model are one-body density distributions and proﬁlefunction. By using the standard proﬁle function, we conﬁrm that the theory reproduces the experimental crosssections for 40 Ca and 48 Ca using the wave functions obtained by the AQCM approach without introducing any
adjustable parameter.
Figure 1 plots the proton-nucleus diﬀerential elastic
scattering cross sections at 320 MeV/nucleon at aroundthe ﬁrst peak position, where the diﬀerence of the nuclear diﬀuseness is well reﬂected.2) For 44 Ti, we see thatthe diﬀerence between the S- and C-type cross sections
is large, considering that the uncertainties of the experimental proton-40 Ca cross sections at around the ﬁrst
peak position is only ≈4%.3) The diﬀerence is smallerfor 52 Ti, but it is signiﬁcant.

Fig. 1. Proton-44, 52 Ti diﬀerential elastic scattering cross sections at 320 MeV/nucleon for scattering angles near theﬁrst peak position.

In summary, we have investigated the density proﬁlesof 44 Ti and 52 Ti with the shell and α-cluster conﬁgurations and found that these two aspects can be distinguished by measuring the proton-nucleus diﬀerential
elastic scattering cross sections up to the ﬁrst peak position.
References
1) N. Itagaki et al., Phys. Rev. C 71, 064307 (2005).
2) S. Hatakeyama et al., Phys. Rev. C 97, 054607 (2018).
3) J. J. Kelly et al., Phys. Rev. C 44, 2602 (1991).