pISSN 0374-4914 eISSN 2289-0041

## Research Paper

New Phys.: Sae Mulli 2022; 72: 371-375

Published online May 31, 2022 https://doi.org/10.3938/NPSM.72.371

## Neutron Skin Thickness of 48Ca, 132Sn, and 208Pb with KIDS Density Functional

Chang Ho Hyun*

Department of Physics Education, Daegu University, Gyeongsan 38453, Korea

Correspondence to:*E-mail: hch@daegu.ac.kr

Received: February 21, 2022; Revised: April 1, 2022; Accepted: April 5, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License(http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

Neutron skin thicknesses (Rnp) of atomic nuclei 48Ca, 132Sn, and 208Pb are considered in the Korea-IBS-Daegu-SKKU (KIDS) density functional formalism. The most relevant model parameters to Rnp are constrained from standard nuclear data and neutron star properties determined from modern observations. Rnp mean values for the models satisfying both nuclear and astronomical data are obtained as 0.161, 0.220, and 0.160~fm for 48Ca, 132Sn, and 208Pb, respectively, and the corresponding standard deviations are 0.011, 0.020 and 0.021~fm. Correlations between the Rnp of the three nuclei are estimated. We obtain a correlation coefficient of 0.978 between Rnp of 48Ca and 208Pb, and 0.996 between Rnp of 132Sn and 208Pb.

Keywords: Density functional theory, Neutron skin thickness

### I. INTRODUCTION

The equation of state (EoS) of infinite nuclear matter is commonly parametrized in terms of parameters K0, J, L, and Ksym over a density range accessible in laboratory experiments and observations of compact objects in the universe. These parameters are defined in the energy per nucleon in the nuclear matter as follows:

$E(ρ,δ)=E(ρ)+S(ρ)δ2+O(δ4),E(ρ)=E0+12K0x2+⋯,S(ρ)=J+Lx+12Ksymx2+⋯,$

where $x=(ρ−ρ0)/3ρ0$ and $δ=(ρn−ρp)/ρ$. ρn and ρp are the neutron and proton densities, respectively, ρ is the baryon density ($=ρn+ρp$ in the nucleon matter), and ρ0 is the nuclear saturation density. In nature, it is frequent to have $δ≠0$ at densities below and above ρ0, where the roles of K0, L and Ksym become important. Neutron skin thickness of neutron-rich nuclei, and the neutron star EoS are representative examples that highlight the roles of L and Ksym. Precise determination of the EoS parameters K0, J, L, Ksym is a prerequisite requirement for the correct description of nuclear structure at $ρ<ρ0$ and the EoS of nuclear matter at $ρ>ρ0$ where δ is substantially away from 0,

Recently[1], to minimize the uncertainties of the EoS parameters, the parameter space (K0, J, L, Ksym) was surveyed over the ranges presented in the literature $230≤K0≤260$, $30≤J≤34$, $40≤L≤70$, and $−420≤Kτ≤−240$ in units of MeV. $Kτ$ is defined as $Kτ=Ksym−(6+Q0/K0)L$. Dividing the interval of each parameter by 5, 0.5, 5 and 20 MeV for K0, J, L, and $Kτ$ respectively, the number of EoS considered initially is 4410. Each energy density functional (EDF) is transformed into a Skyrme-type force. Skyrme force has two constants absent in the nuclear matter EDF, the coefficients of $∇2$ and spin-orbit terms. The two constants are fit to 13 nuclear data using a standard Skyrme-Hartree-Fock code with BCS approximation, the energy and radius of 16O, 40Ca, 48Ca, 90Zr, 132Sn, and 208Pb, and the energy of 218U. With the 4410 Skyrme force models fit to 13 nuclear data, the accuracy of a model is measured in terms of the average deviation per datum (ADPD)

$ADPD(N)=1N∑ i=1 N Oi exp−Oi cal Oi exp,$

where $Oiexp$ ($Oical$) denotes an observable from the experiment (theory). To reduce the model space, we assume a condition ADPD < 0.3 % and obtain 358 models satisfying the condition among the original 4410 models. We call the set of 358 models ADPD03. The model space is further reduced by imposing a constraint on the radius of $1.4M⊙$ mass neutron star $R1.4=11.8$ — 12.5 km. 158 models among the 358 models in ADPD03 satisfy the constraint of R1.4. We call the set of 158 models R14.

This study investigates the neutron skin thickness, Rnp, of 48Ca, 132Sn, and 208Pb with ADPD03 and R14 sets. For 48Ca and 208Pb, accurate measurements were performed in recent experiments. PREX Collaboration reported the first stage measurement $Rnp208$(PREX-I) = 0.15 — 0.49 fm[2]. The accuracy was improved in an upgraded second stage experiment, $Rnp208$(PREX-II) = 0.212 — 0.354 fm[3]. Indirect determination is also accessible by measuring the E1 polarizability, $αD$, and pygmy dipole resonance of the nuclei. Measurement of $αD$ at RCNP reports $Rnp208(αD)=$ 0.135 — 0.181 fm[4], and the pygmy resonance experiment gives $Rnp208(pygmy)=$ 0.20 — 0.28 fm[5]. The measurement of $αD$ was also applied to 48Ca, and the result is obtained as $Rnp48(αD)=0.14$ — 0.20 fm[6]. A recent experiment at RIKEN performed the scattering of 48Ca on the C target, and extracted a result $Rnp48(RIKEN)=0.146±0.048$ fm[7]. Measurement of $Rnp48$ in the parity-violating electron scattering was performed at JLab by the CREX Collaboration. They announced “thin skin” of 48Ca in the press release, and the numerical results are expected to be available soon. It was shown that $Rnp132$ is correlated linearly with $Rnp208$[8]. Correlations between Rnp's of different nuclei provide useful criteria for investigating the consistency of theoretical predictions and measurements from experiments. The proposal for measuring $Rnp132$ by the R3B Collaboration at FAIR motivates the calculation of $Rnp132$.

### II. RESULT

Table 1 summarizes the mean and standard deviation of the EoS parameters K0, J, L, and Ksym for the ADPD03 and R14 sets. Isoscalar and isovector effective masses at the saturation density are shown in terms of the ratio to free mass $μS=mS*/mN$ and $μV= mV*/mN$. Reference[1] shows that the correlation of K0 and J with R1.4 is 0.24 and 0.22, respectively, in the extended R14 (eR14) set.1 The slight differences in the mean and standard deviation of K0 and J between the ADPD03 and R14 sets could be due to this weak correlation. The result demonstrates that the data of extremely neutron-rich systems or phenomena are not necessary to accurately determine K0 and J.

Mean and standard deviation (s.d.) of EoS parameters K0, J, L, and Ksym in the unit of MeV, and the isoscalar and isovector effective masses to the nucleon mass in the free state μS and μV from the 358 models in the ADPD03 set and the 158 models in the R14 set.

means.d.means.d.
K0252.05.6251.15.3
J30.90.730.70.6
L54.79.749.85.2
Ksym-52.472.0-82.433.7
μS0.980.090.970.09
μV0.810.070.800.07

Parameters L and Ksym, unlike K0 and J, exhibit substantial relocation of the mean value and reduction in the standard deviation when the R1.4 constraint is applied. This dramatic change in L and Ksym from the ADPD03 set to the R14 set could also be understood in terms of the correlation with R1.4. Because correlation coefficients of L and Ksym with R1.4 are 0.79 and 0.90 in the eR14 sets, respectively, significant changes occur in L and Ksym when the constraint of R1.4 is considered.

The mean values and ranges of J and L of the R14 set are consistent with the result in Ref.[9]; $J=29.0$ — 32.7 MeV, and $L=40.5$ — 61.9 MeV. The result of Ksym in the R14 set is also consistent with the results in the literature (see Ref.[1] for detail), and suggests a reduced range of the uncertainty.

$μS$ and $μV$ can be assumed to take specific values within the KIDS formalism without affecting the nuclear matter properties at saturation density and the basic properties of the nuclei[10]. However, because we do not assume specific values for $μS$ and $μV$ in this work, they are obtained as results of fitting to 13 nuclear data. The results in Table 1 show that $μS$ and $μV$ are weakly dependent on the model set. In recent studies[11, 12], $μS$ and $μV$ values are probed in the electron-nucleus scattering in the quasielastic region. Values considered in Ref.[11, 12] are $(μS,μV)=$ (1.0, 0.8), (0.7, 0.7), (0.9, 0.9). It is shown that the dependence on the effective mass appears evident for the response function in the exclusive scattering on 16O[11] and the cross section in the inclusive scattering on 12C and 40Ca[12]. Theoretical investigation shows better agreement with data with $μS$ close to 1. Models in the ADPD03 and R14 sets show that the effective mass is consistent with the quasielastic electron scattering data. Table 2 summarizes the results of the neutron skin thickness of 48Ca, 132Sn, and 208Pb from the ADPD03 and R14 sets. The mean values and standard deviations of the R14 set are similar to those of the ADPD03 set. The correlation of $Rnp208$ with R1.4 is -0.02 in the ADPD03 set and 0.09 in the eR14 set[1], implying that the determination of Rnp is seldom influenced by the inclusion or omission of the R1.4 constraint. The result demonstrates that such insensitivity is not limited to 208Pb, but is valid globally from light to heavy nuclei.

Mean, standard deviation, minimum and maximum values of the neutron thickness of 48Ca, 132Sn, and 208Pb obtained from the ADPD03 and R14 sets.

means.d.minmaxmeans.d.minmax
Rnp480.1620.0110.1340.1870.1610.0110.1340.182
Rnp1320.2240.0180.1750.2670.2200.0200.1750.259
Rnp2080.1640.0200.1120.2120.1600.0210.1120.203

Compared with recent measurements, the result of $Rnp48$ is compatible with the data of RCNP (0.14 — 0.20 fm), and RIKEN (0.098 — 0.194 fm). In Ref.[13], $Rnp48$ is evaluated using 18 experimental data, and the result is obtained as $Rnp48=0.178$ — 0.204 fm. This result is within the RCNP and RIKEN range, but it is significantly larger than the result of this work. As for $Rnp208$, the mean values of ADPD03 and R14 sets are compatible with PREX-I (0.15 — 0.48 fm) and $αD$ (0.135 — 0.181 fm). However, our result is sizably small compared with PREX-II (0.212 — 0.354 fm) and pygmy resonance (0.20 — 0.28 fm). Reference[13] also evaluated $Rnp208$ with 20 experimental data, and obtained a range $Rnp208=0.156$ — 0.178 fm.

Because the results of the CREX experiment are expected to be reported in near future, it is worthwhile to refer to theoretical predictions of $Rnp48$ in the literature. We summarize the result of three studies. In Ref.[8], Rnp of 48Ca, 132Sn, and 208Pb are calculated from the correlation between $αD$ and $Rnp208$ using 48 nuclear energy density functionals. 25 models that satisfy the data of $αD$ at RCNP were sorted out among the 48 mdoels. The resulting 25 models give $0.176±0.018$, $0.232±0.022$, and $0.168±0.022$ fm for $Rnp48$, $Rnp132$, and $Rnp208$, respectively. The center value and uncertainty of $Rnp48$ are noticeably larger than that of our result. Alternatively, our results for the mean values of $Rnp132$ and $Rnp208$ are smaller than the results in Ref.[8], but a major portion of the ranges is shared by our result and that of Ref.[8].

Another interesting theoretical study is the coupled-cluster calculation with the nuclear potential obtained from low-energy effective field theory (EFT)[14]. The EFT is a qualified tool for describing few-nucleon systems. The method is extended to the calculation of $Rnp48$, and the result is obtained as $Rnp48=0.12$ — 0.15 fm. The uppermost part of the range is consistent with our result, but there is no overalp with the result of Ref.[8].

The third reference for the theoretical study is Ref.[15], in which ranges of J and L are inferred from the Bayesian method using the data of Rnp of Sn isotopes and the neutron star properties. It is concluded that $Rnp48$ must be larger than 0.15 fm and smaller than 0.25 fm to be compatible with the Sn and neutron star data. Uncertainty is broad, but the work provides a lower limit on $Rnp48$. Correlations between the EoS parameters, EoS parameters and observables, and observables have been studied extensively in many works. The left section of Table 3 shows the correlation of Rnp with the EoS parameters and $Rnp208$ for the R14 set. The slope of the symmetry energy, L, shows the strongest correlation with Rnp among the EoS parameters. This result is consistent with the previous findings that L is most strongly correlated with Rnp. The EoS parameter most uncorrelated with Rnp is Ksym. It has been discussed that K0 and J are relevant with to nuclear properties such as mass and radius. Therefore, accurate measurements of Rnp will be a promising way to accurately determine L.

Left section: Correlation coefficients Ci for i=K0, J, L, Ksym and Rnp208 from the R14 set. Right section: Linear approximation (LA) of Rnp48 and Rnp132 as a function of x=Rnp208, and their ranges obtained by subsitituing the data of Rnp208 in the linear function.

CK0CJCLCKsymC208LAPREX-IPREX-IIαD
Rnp208-0.5900.6110.698-0.4581x0.15 — 0.490.212 — 0.3540.135 — 0.181
Rnp48-0.6260.5770.645-0.5510.9780.524x + 0.0770.16 — 0.330.188 — 0.2620.148 — 0.172
Rnp132-0.6050.6270.712-0.4680.9960.976x + 0.0640.21 — 0.540.271 — 0.4100.196 — 0.241

In Ref.[8], the correlations of $Rnp48$ and $Rnp132$ with $Rnp208$ are considered using 48 EDFs. Results are 0.852 for $Rnp48$ and 0.997 for $Rnp132$. Extended model space was surveyed in Ref.[16], and the correlation between $Rnp48$ and $Rnp208$ is determined to be 0.99 using 206 EoSs. To visualize the correlation of this work, we plot $Rnp48$ and $Rnp132$ as functions of $Rnp208$ for the R14 set in Fig. 1. Distribution of $Rnp132$ is almost a straight line. The correlation coefficient, similar to the value of Ref.[8], is 0.996, indicating that the distribution is extremely close to a straight line. Distribution of $Rnp48$ is scattered compared with $Rnp132$, but the correlation coefficient is 0.978, thus it is also close to a straight line. The correlation coefficients, C208, obtained in Refs.[8, 16] are similar to the results of this work. Strong correlations between Rnps are not limited to specific nuclear models, but appear to be a global feature of the nuclear structure.

Figure 1. (Color online) Distribution of $Rnp48$ and $Rnp132$ with respect to $Rnp208$; the correlations are 0.978 and 0.996, respectively, as shown in Table 3.

In the right section of Table 3, linear approximations of $Rnp48$ and $Rnp132$ are given as functions of $Rnp208$ in the column denoted by “LA”. The next column titled “PREX-I" shows the range of $Rnp48$ and $Rnp132$ by substituting the PREX-I data of $Rnp208$ in the linearly approximated functions of $Rnp48$ and $Rnp132$. Columns for “PREX-II” and “$αD$" are obtained in the same way. The ranges of neutron skin thickness of 48Ca and 132Sn can be predicted if a measurement of $Rnp208$ is correct. Since the PREX-II data of $Rnp208$ are incompatible with those of $αD$, there is no overlap in the ranges of $Rnp48$ (and $Rnp132$) obtained from PREX-II and $αD$.

### III. SUMMARY

We calculated the neutron skin thicknesses of 48Ca, 132Sn and 208Pb with the models constrained by accurate reproduction of nuclear data and the radius of $1.4M⊙$ mass neutron stars determined from astronomical observations. The set of models determined from the nuclear data constitute 385 combinations of the EoS parameters (K0, J, L, Ksym), and the neutron star radius constraints select 158 models from the 358 models. The mean and standard deviation of $Rnp48$, $Rnp132$, and $Rnp208$ are not significantly different between the two sets, and are consistent with the results of other theories. Result of $Rnp208$ is compatible with the range determined from an electric dipole polarizability experiment; however, there is no value common with the result of the PREX-II experiment.

Correlations between $Rnp48$, $Rnp132$ and $Rnp208$ are calculated using the 158 models, and the results are extremely close to 1. From the linear approximation of $Rnp48$ and $Rnp132$ as functions of $Rnp208$, we obtained the ranges of $Rnp48$ and $Rnp132$ corresponding to the data of $Rnp208$ from PREX-I, PREX-II, and electric dipole polarizability. The ranges corresponding to PREX-II are incompatible with those obtained from the electric dipole polarizability data. $Rnp48$ result from the CREX Collaboration, and measurement of $Rnp132$ in the R3B Collaboration are expected to shed light on resolving the uncertainty and inconsistency

### Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea govenment (No. 2020R1F1A1052495).

### Footnote

1In Ref.[1], round off is applied to R1.4, so the actual range is $11.75≤R1.4≤12.54$ km. In this work, we do not use round off, so the models are selected for $11.80≤R1.4≤12.50$ km. Therefore, the number of models in the eR14 set is slightly larger than the R14 set.

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