Ⅰ. HIGHLIGHTS OF THE YEARRIKEN Accel. Prog. Rep. 52 (2019)

Nuclear surface diﬀuseness revealed in nucleon-nucleus diﬀraction†
S. Hatakeyama,∗1 W. Horiuchi,∗1 and A. Kohama∗2A systematic study of the nuclear surface density ofneutron-rich nuclei is interesting, as it reveals the neutron number dependence of the nuclear structure dominated by nuclear dynamics at around the nuclear surface. The study can also be extended to understand theproperties of the equation of state of asymmetric nuclear
matter. However, it is diﬃcult to probe the neutron
density by using traditional electron scattering. In thispaper, we use a proton target as a probe of the surface
density. Extended systematic studies have shown that
it is advantageous to use proton to study the neutron
and proton radii because it has more sensitivity to theneutrons at low-incident energy.1,2) To know more than
the nuclear radii, we study the proton-nucleus elastic
scattering, focusing on the reactions of small scattering
angles up to a few diﬀraction peaks.
Here, we employ the optical-limit approximation of
the Glauber model. As the input to the theory, a twoparameter Fermi (2 pF) density, ρ(r) = 1+expρ0 r−R , is( a )assumed. The parameter R is chosen to give the sameroot-mean-square (rms) radius with several choices ofa. By calculating diﬀerential elastic scattering cross sections (DECS), it is found that the ﬁrst peak position
does not change and that its magnitude increases withsmaller diﬀuseness (sharper surface). The magnitude of
the DECS at the ﬁrst peak position must have information on the nuclear surface.
We perform a “numerical experiment” to deduce thenuclear “diﬀuseness” from the reaction data. We havethe DECS obtained from the realistic densities, theSkyrme-Hartree-Fock (HF) + BCS method used inRefs. 2) and 3). Regarding the DECS used as experimental data, we determine R and a in such a way thatthe DECS with the 2 pF density matches the ﬁrst peak
position and its magnitude of the DECS obtained by
the HF+BCS density. The extracted a values are displayed in Fig. 1. We can clearly see the isotope dependence of the nuclear surface, indicating shell evolution,
weak binding, and nuclear deformation, which cruciallychange the density proﬁle at around the nuclear surface.3) We ﬁnd that the extracted a values have a robust
structure information that is independent of the choiceof the incident energy.
To extract detailed structure information of unstable
nuclei, separation of proton and neutron diﬀuseness isimportant because neutron diﬀuseness is expected to be
more sensitive to the ground-state structure of neutronrich isotopes as it is dominated by the neutron motionat the nuclear surface. We investigate the possibility ofutilizing the incident energy dependence of the nucleonnucleon total cross sections.1,2) Again, we perform a nu†∗1∗20.65a (fm)

0.6
0.55

CaNiZr
SnYb
Pb

0.5
0.45

204060 80 100 120 140 160
Neutron numberFig. 1. Nuclear “diﬀuseness” extracted from the “numericalexperiment” at an incident energy of 550 MeV.

merical experiment. We respectively assume 2 pF densities for protons and neutrons and determine these fourparameters to reproduce the ﬁrst peak positions andtheir DECS at low and high incident energies. We calculate the diﬀuseness parameters and rms radii of theprotons and neutrons for 120 Sn, 208 Pb, and neutron-rich132
Sn with several sets of two incident energies among
200, 300, 550, and 800 MeV. For 120 Sn and 208 Pb, theextracted diﬀuseness parameters are scattered around
0.4–0.6 fm depending on the incident energies chosen,though the rms radii are converged within ∼0.5%. In
such cases where the proton and neutron surfaces are located at almost the same position, the separation of theproton and neutron surface proﬁles might be diﬃcult,whereas in the case of 132 Sn, all the extracted values are
consistent with each other within 0.003 fm for protons
and neutrons.
In summary, we have performed a numerical experiment using the theoretically obtained DECS of highenergy nucleon-nucleus scattering based on the Glaubermodel starting from the nucleon-nucleon total cross sections. We have shown that ﬁrst diﬀraction peak reﬂects
the nuclear density proﬁle at around the nuclear surface, and have proposed a way to quantify it with thediﬀuseness parameter of the 2 pF function, which can bedetermined uniquely. This method is advantageous forapplication to studies of unstable nuclei since the experimental data is limited to forward angles in the inverse
kinematics. This can also be useful for extracting the information of the proton and neutron surfaces, although
the separation of the proton and neutron surfaces may
be possible only for such neutron (proton)-rich systems
in which the surfaces of the proton and neutron surfaces
are well separated.
References1) W. Horiuchi et al., Phys. Rev. C 89, 011601(R) (2014).
2) W. Horiuchi et al., Phys. Rev. C 93, 044611 (2016).
3) W. Horiuchi et al., Phys. Rev. C 96, 035804 (2017).

Condensed from the article in Phys. Rev. C 97, 054607 (2018)
Department of Physics, Hokkaido University
RIKEN Nishina Center-7-