II-2. Nuclear Physics (Theory)

RIKENAccel.
Accel.Prog.
Prog. Rep.
Rep. 54
RIKEN
54(2021)(2021)

Trajectory in 2D plot of IS and IV densities of 48 Ca and 208 Pb†
S. Yoshida,∗1 H. Sagawa,∗2,∗3 J. Zenihiro,∗2,∗4 and T. Uesaka∗2†∗1
∗2
∗3
∗4

Condensed from the article in Phys. Rev. C 102, 064307(2020)
Science Research Center, Hosei University
RIKEN Nishina Center
Center for Mathematical Sciences, the University of Aizu
Department of Physics, Kyoto University

0.025

48Ca0.020

IV ( fm-3 )Recently, the neutron skin thicknesses of doubly
48closed shell nuclei 208
82 Pb126 and 20 Ca28 have been investigated intensively. Theoretical studies with modern energy density functionals (EDFs) indicate that thethickness of the neutron skin, ∆rnp ≡ rn − rp , embodies
the stability of pure neutron matter and provides important information on the symmetry energy of neutron
matter equation of states (EoS), which is a sum of thewell-known EoS of the symmetric nuclear matter andthe symmetry energy. The two EoSs govern the formation of not only nuclei but also astrophysical phenomena
such as neutron stars and supernova explosions.
In this paper, we propose a model to examine the details of the symmetry energy by using the isoscalar (IS)and isovector (IV) density distributions. The neutron
skin is an integrated quantity extracted from neutron
and proton density distributions. However, the radial
density distributions will provide more information toelucidate quantitatively the EoS of both nuclear matterand neutron matter. To this end, it is essential to study
the radial dependence of IS and IV densities, which canbe extracted from neutron and proton densities.
The IS and IV densities are deﬁned as ρIS (r) =ρn (r) + ρp (r) and ρIV (r) = ρn (r) − ρp (r), respectively.
Experimental IS and IV densities of 48 Ca1) are plottedin Fig. 1. The solid black curve sandwiched by the dotted lines is the experimental trajectory of IS versus IVdensities at diﬀerent radii r. The equi-energy contour
lines of HF EDF, ε(ρn , ρp ) = ε(ρIS , ρIV ) = constant, arecalculated using SAMi-J27 and plotted for the values
from 3 MeV at ρIS = ρIV ≈ 0 to −15 or −16 MeV at thesaturation density ρIS ≈ 0.16 fm−3 with an energy stepof 1 MeV.
The dashed lines correspond to the constant IV density line ρIV = ρIS (N − Z)/A of the asymmetric nuclear matter. The experimental trajectory is above theconstant density limit in the surface region. This feature suggests a strong IV pressure to push the IV density towards the surface region rather than the interiorof nucleus. The Fermi liquid of a ﬁnite nucleus mayﬁll the valley determined by EDF from the bottom atapproximately ρIS = 0.16 fm−3 to the top of valley atρIS = ρIV = 0.0 fm−3 . The trajectory plots are started
from the right-hand side at r = 0 fm and ends at theleft corner at r = 6 fm. The dots on the solid line indicate diﬀerent radius points with r = 0, 1, 2, 3, . . . fm
from the right to the left. The positive IV density atr = 0 fm reﬂects a larger neutron density than that of

r=40.015
r=3

r=00.010 r = 5
r=20.005

0.000
0.00

SAMi-J27
SAMi-J29
SAMi-J31

r=60.05

0.10r=1

SAMi-J33
SAMi-J35
Exp.

0.15

0.20

IS ( fm-3 )Fig. 1. Trajectories of isoscalar (IS) and isovector (IV) densities of 48 Ca. The experimental curve (black solid line)
is compared with theoretical ones with Skyrme interactions. The equi-energy contour lines are calculated using
the SAMi-J family. The equi-energy contour lines areplotted for SAMi-J27. See the text for details.

protons at the central region corresponding to r = 0 fm.
Keeping the ρIS almost constant, the experimental curve
rises to the surface region at r = 3 fm, where the dots
are separated largely because of a rapid change in theIV density. This can be understood qualitatively froma ﬂat contour map around the IS density ρ = 0.16 fm−3
so that the IV density can have large freedom within thelimit of the equi-energy contour line.
We can see a clear J-dependence for the enhancementof IV density at r = 4 fm; a larger J value yields agreater neutron density at the surface. Compared withthe experimental data, SAMi-J27 shows the best agreement in the entire region of this plot.
In summary, we propose a new 2D plotting methodof IS-IV densities to extract not only the empirical symmetry energy coeﬃcients J, L, and Ksym , but also theasymmetric isospin term Kτ in the nuclear incompressibility. We found strong correlations between the curvature of the 2D density at the density ρIS = 0.1 fm−3 andthe symmetry energy coeﬃcients J, L, and Kτ . The
optimal values are found to be J = 27.2 MeV, L =31.6 MeV, Ksym = −154.7 MeV, and Kτ = −300.6 MeVfor the SAMi-J EDF from the IS-IV density plots of 48 Ca
and 208 Pb.
Reference1) J. Zenihiro et al., arXiv:1810.11796 (2018).

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