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 and . Reference 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.
Table 1 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.
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.; — 32.7 MeV, and — 61.9 MeV. The result of Ksym in the R14 set is also consistent with the results in the literature (see Ref. for detail), and suggests a reduced range of the uncertainty.
and 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. However, because we do not assume specific values for and in this work, they are obtained as results of fitting to 13 nuclear data. The results in Table 1 show that and are weakly dependent on the model set. In recent studies[11, 12], and values are probed in the electron-nucleus scattering in the quasielastic region. Values considered in Ref.[11, 12] are (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 and the cross section in the inclusive scattering on 12C and 40Ca. Theoretical investigation shows better agreement with data with 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 with R1.4 is -0.02 in the ADPD03 set and 0.09 in the eR14 set, 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.
Table 2 Mean, standard deviation, minimum and maximum values of the neutron thickness of 48Ca, 132Sn, and 208Pb obtained from the ADPD03 and R14 sets.
Compared with recent measurements, the result of is compatible with the data of RCNP (0.14 — 0.20 fm), and RIKEN (0.098 — 0.194 fm). In Ref., is evaluated using 18 experimental data, and the result is obtained as — 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 , the mean values of ADPD03 and R14 sets are compatible with PREX-I (0.15 — 0.48 fm) and (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 also evaluated with 20 experimental data, and obtained a range — 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 in the literature. We summarize the result of three studies. In Ref., Rnp of 48Ca, 132Sn, and 208Pb are calculated from the correlation between and using 48 nuclear energy density functionals. 25 models that satisfy the data of at RCNP were sorted out among the 48 mdoels. The resulting 25 models give , , and fm for , , and , respectively. The center value and uncertainty of are noticeably larger than that of our result. Alternatively, our results for the mean values of and are smaller than the results in Ref., but a major portion of the ranges is shared by our result and that of Ref..
Another interesting theoretical study is the coupled-cluster calculation with the nuclear potential obtained from low-energy effective field theory (EFT). The EFT is a qualified tool for describing few-nucleon systems. The method is extended to the calculation of , and the result is obtained as — 0.15 fm. The uppermost part of the range is consistent with our result, but there is no overalp with the result of Ref..
The third reference for the theoretical study is Ref., 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 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 . 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 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.
Table 3 Left section: Correlation coefficients for , J, L, Ksym and from the R14 set. Right section: Linear approximation (LA) of and as a function of , and their ranges obtained by subsitituing the data of in the linear function.
|-0.590||0.611||0.698||-0.458||1||x||0.15 — 0.49||0.212 — 0.354||0.135 — 0.181|
|-0.626||0.577||0.645||-0.551||0.978||0.524x + 0.077||0.16 — 0.33||0.188 — 0.262||0.148 — 0.172|
|-0.605||0.627||0.712||-0.468||0.996||0.976x + 0.064||0.21 — 0.54||0.271 — 0.410||0.196 — 0.241|
In Ref., the correlations of and with are considered using 48 EDFs. Results are 0.852 for and 0.997 for . Extended model space was surveyed in Ref., and the correlation between and is determined to be 0.99 using 206 EoSs. To visualize the correlation of this work, we plot and as functions of for the R14 set in Fig. 1. Distribution of is almost a straight line. The correlation coefficient, similar to the value of Ref., is 0.996, indicating that the distribution is extremely close to a straight line. Distribution of is scattered compared with , 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 and with respect to ; the correlations are 0.978 and 0.996, respectively, as shown in Table 3.
In the right section of Table 3, linear approximations of and are given as functions of in the column denoted by “LA”. The next column titled “PREX-I" shows the range of and by substituting the PREX-I data of in the linearly approximated functions of and . Columns for “PREX-II” and “" are obtained in the same way. The ranges of neutron skin thickness of 48Ca and 132Sn can be predicted if a measurement of is correct. Since the PREX-II data of are incompatible with those of , there is no overlap in the ranges of (and ) obtained from PREX-II and .