Nuclear dipole polarizability from mean-field modeling constrained by chiral effective field theory

TL;DR

This paper develops a new Skyrme interaction constrained by chiral effective field theory to accurately predict nuclear electric dipole polarizabilities and neutron skin thicknesses, aligning well with experimental data.

Contribution

A novel Skyrme interaction, Skχm*, was constructed using chiral EFT constraints, improving predictions of nuclear properties without fine tuning.

Findings

Predicted electric dipole polarizabilities match experimental data.

Neutron skin thicknesses are consistent with measurements.

The new interaction effectively models nuclear responses.

Abstract

We construct a new Skyrme interaction Sk$\chi$m$^*$ by fitting the equation of state and nucleon effective masses in asymmetric nuclear matter from chiral two- and three-body forces as well as the binding energies of finite nuclei. Employing this interaction to study the electric dipole polarizabilities of $^{48}$Ca, $^{68}$Ni, $^{120}$Sn, and $^{208}$Pb in the random-phase approximation, we find that the theoretical predictions are in good agreement with experimentally measured values without additional fine tuning of the Skyrme interaction, thus confirming the usefulness of the new Skyrme interaction in studying the properties of nuclei. We further use this interaction to study the neutron skin thicknesses of $^{48}$Ca and $^{208}$Pb, and they are found to be consistent with the experimental data.

Figures

Tables

Table 1. Table 1: Parameters of the Skyrme interaction Sk χ ​ m ∗ 𝜒 superscript 𝑚 \chi m^{*} and their errors σ 𝜎 \sigma : lines 1-9 show the chosen 9 macroscopic quantities used to express the Skyrme parameters (see text for details); lines 10-19 show the Skyrme parameters.

Quantity	Sk$\chi m^{\ast }$	$\sigma$
${\rho }_0\left({\mathrm{fm}}^{-3}\right)$	0.1651	0.0025
$E_0\left(\mathrm{MeV}\right)$	-16.07	0.06
$K_0\left(\mathrm{MeV}\right)$	230.4	6.6
$E_{\mathrm{sym}}\left({\rho }_0\right)\left(\mathrm{MeV}\right)$	30.94	0.46
$L\left(\mathrm{MeV}\right)$	45.6	2.5
$G_S\left(\mathrm{MeV}\cdot {\mathrm{fm}}^5\right)$	141.5	7.3
$G_V\left(\mathrm{MeV}\cdot {\mathrm{fm}}^5\right)$	-70.5	32.6
$m_{s,0}^{\ast }/m$	0.750	0.040
$m_{v,0}^{\ast }/m$	0.694	0.020
$W_0\left(\mathrm{MeV}\cdot {\mathrm{fm}}^5\right)$	119.8	4.9
$t_0\left(\mathrm{MeV}\cdot {\mathrm{fm}}^3\right)$	-2260.7	365.8
$t_1\left(\mathrm{MeV}\cdot {\mathrm{fm}}^5\right)$	433.189	30.661
$t_2\left(\mathrm{MeV}\cdot {\mathrm{fm}}^5\right)$	274.553	173.611
$t_3\left(\mathrm{MeV}\cdot {\mathrm{fm}}^{3+3\alpha }\right)$	12984.4	1762.4
$x_0$	0.327488	0.110201
$x_1$	-1.088968	0.303910
$x_2$	-1.822404	0.366035
$x_3$	0.442900	0.215646
$\alpha$	0.198029	0.054443

Table 2. Table 2: Comparisons of binding energies B 𝐵 B (in MeV) and charge radii r c subscript 𝑟 𝑐 r_{c} (in fm) of 208 Pb, 132 Sn, 100 Sn, 56 Ni, 48 Ca, 40 Ca and 16 O from Hartree-Fock calculations with the Sk χ ​ m ∗ 𝜒 superscript 𝑚 ∗ \chi m^{\ast} interaction to experimental data.

	$B^{\exp }$	$B$	$r_c^{\exp }$	$r_c$
208Pb	-1636.44	-1635.71	5.50	5.46
132Sn	-1102.90	-1103.47	4.71	4.68
100Sn	-825.78	-826.77	-	4.45
56Ni	-483.99	-482.83	-	3.74
48Ca	-415.99	-416.55	3.48	3.47
40Ca	-342.05	-342.08	3.48	3.46
16O	-127.62	-127.24	2.70	2.74

Table 3. Table 3: Experimental data for the electric dipole polarizabilities α D subscript 𝛼 D \alpha_{\mathrm{D}} (in units of fm 3 ) for Ca 48 superscript Ca 48 {}^{48}\mathrm{Ca}   Bir16 , Ni 68 superscript Ni 68 {}^{68}\mathrm{Ni}   Ros13 , Sn 120 superscript Sn 120 {}^{120}\mathrm{Sn}   Has15 , and Pb 208 superscript Pb 208 {}^{208}\mathrm{Pb}   Tam11 together with the results of RPA calculations using the Sk χ ​ m ∗ 𝜒 superscript 𝑚 ∗ \chi m^{\ast} Skyrme parametrization .

	Expt.	Sk$\chi m^{\ast }$
${}_{}{}^{48}\mathrm{Ca}$	2.07(22)	2.27(5)
${}_{}{}^{68}\mathrm{Ni}$	3.88(31)	4.06(7)
${}_{}{}^{120}\mathrm{Sn}$	8.59(36)	9.28(14)
${}_{}{}^{208}\mathrm{Pb}$	19.6(6)	19.87(29)

Equations

$$\frac{1}{m_{s,0}^{\ast }}$$

$$\frac{1}{m_{v,0}^{\ast }}$$

$$\frac{1}{m_n^{\ast }}$$

$$\frac{1}{m_p^{\ast }}$$

$$\frac{p}{m^{\ast }}=\frac{de}{dp}.$$

$$v({\mathbf{r}}_1,{\mathbf{r}}_2)$$

$${\mathcal{H}}_{\mathrm{fin}}=\frac{G_S}{2}(\nabla \rho {)}^2-\frac{G_V}{2}(\nabla {\rho }_n-\nabla {\rho }_p{)}^2$$

$$G_S$$

$$G_V$$

$$\mathbf{p}$$

$${\chi }^2$$

$${\chi }^2(\mathbf{p})\approx {\chi }^2({\mathbf{p}}_0)+\frac{1}{2}\sum _{i,j}^N(p_i-p_{0i}){\mathcal{M}}_{ij}(p_j-p_{0j}),$$

$$e_i=\sqrt{({\mathcal{M}}^{-1}{)}_{ii}},$$

$${\mathcal{C}}_{ij}=\frac{{\Sigma }_{ij}}{\sqrt{{\Sigma }_{ii}{\Sigma }_{jj}}},$$

$$\hat{F}=\frac{N}{A}\sum _{i=1}^Z{r}_i{Y}_{\text{1M}}({\hat{r}}_i)-\frac{Z}{A}\sum _{i=1}^N{r}_i{Y}_{\text{1M}}({\hat{r}}_i),$$

$$S(E)=\sum _{\nu }|⟨\nu \Vert \hat{F}\Vert \tilde{0}⟩{|}^2\delta (E-E_{\nu }),$$

$$m_k=\int dE{E}^{k}S(E)=\sum _{\nu }|⟨\nu \Vert \hat{F}\Vert \tilde{0}⟩{|}^2{E}_{\nu }^k,$$

$${\alpha }_D=\frac{8\pi }{9}{e}^2\int dE{E}^{-1}S(E)=\frac{8\pi }{9}{e}^2{m}_{-1}.$$

$$E_x=\sqrt{\frac{m_1}{m_{-1}}}.$$

$$m_1=\frac{9}{4\pi }\frac{1}{2m}\frac{NZ}{A}(1+\kappa ),$$

$$\Delta {\alpha }_{\mathrm{D}}\sim {\alpha }_{\mathrm{D}}\frac{{\Gamma }^2}{4E_x^2}.$$

$$\Delta r_{np}({}^{48}\mathrm{Ca}):$$

$$\Delta r_{np}({}^{208}\mathrm{Pb}):$$

Full text

I Introduction

The nuclear electric dipole polarizability $\alpha_{\mathrm{D}}$, which is proportional to the inverse energy-weighted sum of the dipole response of a nucleus to an external electric field, has recently attracted much attention due to its strong correlation with neutron skin thicknesses, the density dependence of the nuclear symmetry energy Rei10 ; Pie12 ; Zha14 ; Zha15 ; Pie14 ; Col14 ; Lat14 ; Roc13 ; Roc15 , and the properties of neutron stars Horowitz01 ; Fattoyev12 ; Erler13 . The electric dipole response is dominated by the nuclear isovector giant dipole resonance (IVGDR), which is the oldest known nuclear collective excitation and has been extensively studied both theoretically and experimentally Liu75 ; Speth81 ; Woude87 ; Chak87 . It is well known from theoretical studies based on various models that the properties of the IVGDR, in which neutrons oscillate against protons in a nucleus, is affected by the density dependence of the nuclear symmetry energy Mey82 ; Kri84 ; Lip89 ; Dan03 ; Tri08 .

Experimentally, the electric dipole strength distributions in 48Ca Bir16 , ${}^{120}\mathrm{Sn}$ Has15 , and ${}^{208}\mathrm{Pb}$ Tam11 have recently been measured accurately at the Research Center for Nuclear Physics (RCNP) from proton inelastic scattering experiments at forward angles, while that of 68Ni has been investigated at GSI using Coulomb excitations in inverse kinematics and measuring the invariant mass in the one- and two-neutron decay channels Ros13 . Recently, ab initio calculations based on chiral effective field theory interactions have been successful at describing $\alpha_{\mathrm{D}}$ in medium-mass nuclei Hag16 ; Mio16 , but the description of the dipole response of heavy nuclei remains a challenge. Therefore, nuclear energy density functionals are still the most widely-used approach to explore the nuclear equation of state (EOS) from the electric dipole response.

In the present work, we construct a new Skyrme interaction Sk$\chi$m∗ by fitting the equation of state and nucleon effective masses in asymmetric nuclear matter predicted by chiral two- and three-body forces as well as the binding energies of finite nuclei. We then employ it to study the dipole response of 48Ca, 68Ni, 120Sn, and 208Pb within the random-phase approximation (RPA). The calculated electric dipole polarizabilities are in good agreement with existing experimental values. In addition, the neutron skin thickness in 48Ca and 208Pb is discussed.

The paper is organized as follows. In Section II, we explain in detail the new Skyrme interaction Sk$\chi$m∗ and how it is obtained. In Section III, we describe the random-phase approximation employed in our calculations, and in Section IV we present results and discussions for the isovector dipole response, electric dipole polarizability, and neutron skin thicknesses of 48Ca and 208Pb from the new Skyrme mean-field model. We end with a summary in Section V.

II A Skyrme interaction from chiral effective field theory

Recently, extended Skyrme interactions have been constructed in Ref. Lim17 by fitting to the asymmetric nuclear matter equation of state from chiral two- and three-nucleon forces Wellenhofer15 ; Wellenhofer16 . In contrast to previous works Brown14 ; Rrapaj16 imposing constraints on mean-field models from low-density neutron matter, the full density dependence of the asymmetric matter equation of state was used in Ref. Lim17 for the $\chi^{2}$ minimization function. In these calculations, the chiral two-body force was treated at next-to-next-to-next-to-leading order (N3LO) in the chiral power counting, while the three-body force was treated at N2LO. Three different choices of the momentum-space cutoff were considered {$\Lambda=414,~{}450~{}\mathrm{and}~{}500~{}\mathrm{MeV}$}, and the unknown low-energy constants associated with short-distance dynamics were fitted Entem03 ; Coraggio07 ; Coraggio13 ; Coraggio14 in each case to nucleon-nucleon scattering phase shifts and deuteron properties (in the case of the nucleon-nucleon interaction) as well as the binding energy and beta-decay lifetime of 3H (in the case of the three-nucleon force). To fix the gradient contributions to the energy density functional, the extended Skyrme interactions were also fitted to reproduce the binding energies of 7 doubly-closed-shell nuclei: ${}^{16}\mathrm{O}$, ${}^{40}\mathrm{Ca}$, ${}^{48}\mathrm{Ca}$, ${}^{56}\mathrm{Ni}$, ${}^{100}\mathrm{Sn}$, ${}^{132}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$. The conventional Skyrme interaction Cha97 was found to be inadequate to describe the density dependence of the equation of state from chiral effective field theory, leading the authors of Ref. Lim17 to include an additional density-dependent and momentum-independent term. The three Skyrme interactions (Sk$\chi 414$, Sk$\chi 450$ and Sk$\chi 500$) reported in Ref. Lim17 predict, however, too large (even larger than the nucleon bare mass $m$) isoscalar and isovector effective masses $m_{s,0}^{\ast}$ and $m_{v,0}^{\ast}$ at saturation density, which are given by Cha97

[TABLE]

This fact makes Sk$\chi 414$, Sk$\chi 450$ and Sk$\chi 500$ unsuitable for studying the nuclear giant dipole resonances. It is well known that the $m_{v,0}^{\ast}$ is related to the enhancement factor $\kappa$ in the Thomas-Reiche-Kuhn sum rule Ring , given by the relation $1+\kappa=m/m_{v,0}^{\ast}$, and thus affects the nuclear dipole response function. We note that in Skyrme functionals the neutron and proton effective masses $m_{n}^{*}$ and $m_{p}^{*}$ at total baryon number density $\rho$ are related to the isoscalar and isovector effective masses $m_{s}^{*}(\rho)$ and $m_{v}^{*}(\rho)$ by Farine01

[TABLE]

where $\delta=(\rho_{n}-\rho_{p})/(\rho_{n}+\rho_{p})$ is the isospin asymmetry with $\rho_{n}~{}(\rho_{p})$ being the neutron (proton) density.

In the present work, instead of directly fitting the three EOSs predicted by chiral effective field theory as in Ref Lim17 , we treat them as theoretical uncertainties on the equation of state and construct a Skyrme interaction by fitting the central values. As shown later, taking into account the theoretical uncertainties makes it possible to use the conventional Skyrme interaction to reproduce the predicted EOS from chiral effective field theory. To improve the description of nucleon effective masses, we further include in our fit the isoscalar and isovector effective masses $m_{s,0}^{\ast}/m=0.82\pm 0.08$ and $m_{v,0}^{\ast}/m=0.69\pm 0.02$ at saturation density that are extracted from the single-particle energies of protons and neutrons in asymmetric nuclear matter, computed recently in Refs. holt13prc ; holt16prc from chiral two- and three-body forces. From the momentum- and energy-dependent nucleon single-particle energy $e(p)=\frac{p^{2}}{2m}+\mathrm{Re}\Sigma(p,e(p))$, where $p$ is the nucleon momentum and the self-energy $\Sigma$ is computed at second order in perturbation theory, the nucleon effective mass $m^{*}$ can be extracted by the definition

[TABLE]

The $m_{s,0}^{\ast}$ and $m_{v,0}^{\ast}$ then can be obtained by invoking Eqs. (3) and (4). Since the effective masses in the Skyrme-Hartree-Fock model are momentum-independent, we average the momentum-dependent effective masses from chrial effective theory over a momentum range around the Fermi momentum, which accounts for most of the theoretical uncertainties. It is interesting to see that the $m^{*}_{v,0}$ from chiral effective field theory is consistent with that from an analysis of dipole resonances in 208Pb based on conventional Skyrme interactions Klu09 .

Our new Skyrme interaction therefore has the usual form:

[TABLE]

where $\bm{\sigma}_{i}$ is the Pauli spin operator, $P_{\sigma}=(1+\bm{\sigma}_{1}\cdot\bm{\sigma}_{2})/2$ is the spin-exchange operator, $\bm{k}=-i(\bm{\nabla}_{1}-\bm{\nabla}_{2})/2$ is the relative momentum operator, and $\bm{k}^{\prime}$ is the conjugate operator of $\bm{k}$ acting on the left.

Following Refs. LWC10 ; Kor10 , we express the 9 parameters $t_{0}-t_{3}$, $x_{0}-x_{3}$ and $\alpha$ of the Skyrme interaction in terms of 9 macroscopic quantities: $m_{s,0}^{\ast}$, $m_{v,0}^{\ast}$, the nuclear matter saturation density $\rho_{0}$, the energy per particle of symmetric nuclear matter $E_{0}(\rho_{0})$, the incompressibility $K_{0}$, the gradient coefficient $G_{S}$, the symmetry-gradient coefficient $G_{V}$, and the magnitude $E_{\mathrm{sym}}(\rho_{0})$ and density slope $L$ of the nuclear symmetry energy at $\rho_{0}$ LWC10 ; Kor10 . Here the $G_{S}$ and $G_{V}$ are defined by expressing the momentum-dependent or finite-range terms of the Skyrme energy density functional $\mathcal{H}_{\mathrm{fin}}$ as LWC10

[TABLE]

with

[TABLE]

Consequently, the Skyrme mean-field models have the following 10 parameters:

[TABLE]

These parameters have clear physical meanings and their ranges are empirically known, which increases significantly the efficiency of the optimization algorithm. The parameters $G_{S}$, $G_{V}$ and $W_{0}$, which are not constrained by the nuclear equation of state, have been studied Kai11 ; Kai12 ; Holt16 consistently in chiral effective field theory via the density matrix expansion in the Hartree-Fock approximation. However, higher-order terms in the expansion are needed to produce reliable constraints on the associated parameters in Skyrme functionals. In the present work we constrain the values of $G_{S}$, $G_{V}$ and $W_{0}$ by fitting to the binding energies of double-closed-shell nuclei as in Ref. Lim17 .

As in the usual fitting procedure, we optimize the parameters $\bm{p}(p_{1},p_{2},\cdots,p_{N})$ by minimizing the weighted sum of the squared deviations from the data as

[TABLE]

where $E_{i}$ is the binding energy per nucleon in nuclear matter with the nucleon density $\rho_{i}=\{0.01,~{}0.04,~{}0.08,~{}0.16,~{}0.20,~{}0.24,~{}0.28,~{}0.32\}~{}\mathrm{fm}^{-3}$ and the isospin asymmetry $\delta_{i}^{2}=\{0,~{}0.25,~{}0.5,~{}0.75,~{}1\}$; $B_{i}$ is the total binding energy of the $i$th nucleus; ‘Sky’ and ‘EFT’ indicate predictions of the Skyrme-Hartree-Fock model and chiral effective field theory, while ‘exp’ indicates the experimental data. The adopted error $\sigma_{B}$ is set to be $1$ MeV, while $\sigma_{E}$ is taken to be max($0.25,\sigma_{\mathrm{EFT}}(\rho)$) MeV where $\sigma_{\mathrm{EFT}}(\rho)$ is the theoretical error of the EOS due to the variation of the momentum cutoff used in chiral effective field theory. Since the predictions of chiral effective field theory are very precise at low densities (e.g., the error is $0.02$ MeV in neutron matter at $\rho=0.01$ fm*-3*), a minimum error of $0.25$ MeV for the EOS is introduced to take account of the deficiency of the Skyrme parametrization. The optimization is carried out by employing the POUNDERS algorithm in the distribution of PETSc/Tao Tao , which has been successfully used previously to calibrate nuclear energy density functionals Kor10 ; Kor12 ; Kor14 .

Once the optimized parameter set $\bm{p}_{0}$ is obtained, the $\chi^{2}$ near $\bm{p}_{0}$ can be approximately written as

[TABLE]

with $\mathcal{M}_{ij}=\left.\frac{\partial^{2}\chi^{2}}{\partial{p_{i}}\partial{p_{j}}}\right|_{\bm{p}_{0}}$. The errors of parameters then can be estimated as

[TABLE]

which is the allowed variation of the parameter $p_{i}$ within the ellipsoid of $\chi^{2}-\chi^{2}(\bm{p}_{0})\leq 1$. Using the covariance matrix $\Sigma=\mathcal{M}^{-1}$, the correlation matrix can be estimated as

[TABLE]

where $\mathcal{C}_{ij}$ is the correlation coefficient between parameters $p_{i}$ and $p_{j}$. The extrapolation error of any observables and the correlation coefficient between them can also be determined (see e.g., Dob14 ; Pie15 ).

The obtained parameter set $\bm{p}_{0}$, named Sk$\chi m^{*}$, including its uncertainties, together with the corresponding Skyrme parameters are listed in Tab. 1. It is seen that the parameters $\rho_{0},~{}E_{0}(\rho_{0}),~{}K_{0},~{}E_{\mathrm{sym}}(\rho_{0})$ and $L$, which characterize the density- and isospin-dependence of the nuclear EOS, are well constrained by the predicted EOS from chiral effective field theory. If we had removed from the fit all data points shown in Fig. 1 for the equation of state of asymmetric matter, the uncertainty in $L$ would be roughly 60 MeV. Also, the isoscalar gradient coefficient $G_{S}=141.5\pm 7.3~{}\mathrm{MeV}\cdot\mathrm{fm}^{5}$ and the spin-orbit coupling strength $W_{0}=119.8\pm 4.9(\mathrm{MeV}\cdot\mathrm{fm}^{5})$ are consistent with the values at sub-saturation densities of about $0.03$ fm*-3* in the energy density functional derived from chiral two- and three-body forces Holt16 . We show in Fig. 1 the EOSs of asymmetric nuclear matter predicted by the Sk$\chi m^{*}$ interaction together with predictions from chiral effective field theory. One can see that the new interaction can well reproduce the EOSs from chiral effective field theory calculations. In Tab. 2, we list the binding energies and charge radii of ${}^{16}\mathrm{O}$, ${}^{40}\mathrm{Ca}$,${}^{48}\mathrm{Ca}$, ${}^{56}\mathrm{Ni}$, ${}^{100}\mathrm{Sn}$, ${}^{132}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$ obtained by Hartree-Fock calculations using Sk$\chi m^{*}$, together with their experimental values. It is seen that our results are overall in very good agreement with the experimental data for the 7 doubly-closed-shell nuclei.

Fig. 2 shows the absolute values of correlation coefficients calculated among the 10 parameters $\rho_{0}$, $E_{0}(\rho_{0})$, $K_{0}$, $E_{\mathrm{sym}}(\rho_{0})$, $L$, $G_{S}$, $G_{V}$, $W_{0}$, $m_{s,0}^{\ast}$ and $m_{v,0}^{\ast}$. We find that $m_{v,0}^{\ast}$ is almost independent of all other parameters, which reflects the fact that it is poorly constrained by the data from the EOSs and nuclear binding energies used in our fit. Except $m_{v,0}^{\ast}$, most parameters are correlated with each other, but generally the correlations are weak with the absolute value of the Pearson coefficient $R$ less than 0.8. The spin-orbit coupling strength $W_{0}$ is strongly correlated with both the isoscalar effective mass $m_{s,0}$ and the isoscalar gradient coefficient $G_{S}$, which are related to the momentum-dependent terms in the Skyrme interaction.

III Isovector dipole response and random-phase approximation

The dipole response of a nucleus can be calculated in the random-phase approximation using the dipole operator

[TABLE]

where $r_{i}$ is the nucleon’s radial coordinate and $Y_{\text{1M}}(\hat{r_{i}})$ is the corresponding spherical harmonic function. In the RPA method, the isovector dipole strength function is evaluated as

[TABLE]

where $|\tilde{0}\rangle$ is the RPA ground state, and $|\nu\rangle$ is the RPA excited state with $E_{\nu}$ being its energy. Defining the moments of the strength function as

[TABLE]

the electric dipole polarizability $\alpha_{D}$ can then be calculated according to

[TABLE]

In the present work, we employ the Skyrme-RPA code by Colo et al. Col13 to calculate the electric dipole polarizabilities of ${}^{48}\mathrm{Ca}$, ${}^{68}\mathrm{Ni}$, ${}^{120}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$. As pointed out in Ref. Roc15 , to compare the RPA results with experiment, the experimentally extracted values of $\alpha_{\mathrm{D}}$ for ${}^{120}\mathrm{Sn}$ Has15 and ${}^{208}\mathrm{Pb}$ Tam11 need to be corrected by subtracting the contributions from quasi-deuteron excitations Lep81 ; Sch88 , while that of ${}^{68}\mathrm{Ni}$ Hag16 should be modified by including the corrections from the extrapolated low-energy and high-energy regions. The corrected experimental data are shown in Table 3.

IV Results and discussions

In Fig. 3, we show the isovector dipole transition strength functions for ${}^{48}\mathrm{Ca}$, ${}^{68}\mathrm{Ni}$, ${}^{120}\mathrm{Sn}$, and 208Pb obtained from the RPA calculations using the Skyrme interaction Sk$\chi m^{*}$. Here the curves are obtained by smearing each of the discrete RPA peaks by a Lorentzian function of 2 MeV width, which is taken to reproduce the experimental width of the IVGDR. For comparison, the measured strength functions for these nuclei are shown as black solid circles, and the experimental peak energies $E_{x}=18.9,~{}17.1,~{}15$ and $13.43~{}\mathrm{MeV}$ Tam11 ; Ros13 ; Bir16 ; Has15 ; Die88 of the IVGDR in ${}^{48}\mathrm{Ca}$, ${}^{68}\mathrm{Ni}$, ${}^{120}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$ are indicated by the arrows. It is seen that the Sk$\chi m^{*}$ interaction gives dipole strength functions which are consistent with experimental measurements, except the predicted second peak at higher energy that is not found in experimental data, especially for 120Sn. Such an unphysical high-energy peak is expected to become a long high-energy tail if more complex configurations beyond the $1p-1h$ states are included in the RPA calculation Lyu16 . For 208Pb, the predicted peak energy $E_{x}$ from Sk$\chi m^{*}$ at 13.6 $\mathrm{MeV}$ is in particularly good agreement with the experimental value. This is due to the $m_{v,0}^{\ast}=0.694m$, or equivalently $\kappa=0.44$, predicted by chiral effective field theory, which is roughly consistent with the value $\kappa=0.4$ determined by fitting the IVGDR peak energy in 208Pb using conventional Skyrme interactions Klu09 .

The relation between $m_{v,0}^{\ast}$ or $\kappa$ and $E_{x}$ can be understood from the approximate relation Lip89 ; Tri08

[TABLE]

While the inverse energy weighted sum rule $m_{-1}$ [Eq. (17)] is related to the electric dipole polarizability via Eq. (18), the energy weighted sum rule $m_{1}$ depends inversely on the isovector effective mass $m_{v,0}^{\ast}$ at saturation density in the conventional Skyrme mean-field model Zha15a . The latter can be understood by the fact that the $m_{1}$ can be expressed as Col13

[TABLE]

where the factor $\kappa$ is inversely proportional to $m_{v,0}^{\ast}$.

For 68Ni, panel (b) of Fig. 3 shows that there is a so-called pygmy dipole resonance around 9.55 MeV in the experimental strength function that carries 2.8(5)% of the energy weighted sum rule $m_{1}$ Ros13 . Due to the smearing of the RPA states in the calculation, the pygmy resonance is not seen and appears instead as a low-energy tail in the theoretical strength function. The RPA strength in the energy region of 0-11 MeV carries 2.7% of the total energy weighted sum rule and is similar to that carried by the pygmy dipole resonance. Its contribution to the dipole polarizability is 0.38 fm3 and thus underestimates the experimental value of 1.13 fm3 Ros13 ; Roc15 as for most Skyrme-type interactions.

Fig. 4 exhibits the electric dipole polarizabilities of ${}^{48}\mathrm{Ca}$, ${}^{68}\mathrm{Ni}$, ${}^{120}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$ predicted by the Sk$\chi m^{\ast}$ mean-field model together with the experimental results Tam11 ; Ros13 ; Has15 ; Bir16 (detailed numerical values are listed in Table 3). We find that RPA calculations using the Sk$\chi m^{*}$ interaction can overall well reproduce the experimental data. For 120Sn the prediction of Sk$\chi m^{*}$ is slightly larger than the experimental value. This could be due to the fact that, in the present work, we do not include pairing correlations, which reduce the electric dipole polarizability in the Sn isotope chain Roc15 . In the case of ${}^{48}\mathrm{Ca}$, we observe that the results are consistent with recent ab initio calculations Hag16 of the dipole polarizability from chiral effective field theory. We would like to point out that we have calculated the electric dipole polarizability by using discrete RPA peaks because the $1p-1h$ RPA cannot give rise to the spreading width of nuclear giant resonances observed in experiments. Including the spreading width, which can be done by taking into account the coupling of $1p-1h$ states with more complicated multi-particle–multi-hole configurations, is expected to reduce the calculated $\alpha_{D}$ slightly. This can be seen by smearing the RPA peaks using the Lorentzian function with a certain width. In the case of only one Lorentzian function of width $\Gamma$, the electric dipole polarizability is at most reduced by Roc15

[TABLE]

According to this formula, using the experimental value of the peak energy (13.43 MeV) and width (4.07 MeV) of the IVGDR Die88 , we estimate that the reduction should be less than about $0.45~{}\mathrm{fm^{3}}$ $(\sim 2\%)$ for 208Pb. Such a small correction does not influence our conclusion.

The neutron skin thickness, which has been demonstrated to be strongly correlated with the electric dipole polarizability, is an important probe of the density slope of the nuclear symmetry energy as well as the EOS of pure neutron matter. Therefore, we also employ the Sk$\chi m^{\ast}$ mean-field model to predict the neutron skin thicknesses of 48Ca and 208Pb, and the obtained results are given by

[TABLE]

We find that the predicted results for 208Pb are in very good agreement with the constraint ${\Delta}r_{np}=0.15\pm 0.03~{}({\text{stat.}})^{+0.01}_{-0.03}~{}({\text{sys.}})$ fm extracted from coherent pion photoproduction cross sections Tar14 and are also consistent with the constraint $\Delta r_{np}=0.302\pm 0.175~{}(\mathrm{exp})\pm{0.026}~{}(\mathrm{model})\pm 0.005~{}(\mathrm{strange})$ fm Hor12 extracted from the parity-violating asymmetry measurement in the Lead Radius Experiment (PREX) Abr12 .

V Summary

We have constructed a new Skyrme interaction Sk$\chi$m∗ by fitting the EOSs and nucleon effective masses in asymmetric nuclear matter predicted by chiral effective field theory together with the binding energies of selected closed shell doubly-magic nuclei. The new model was employed to study the isovector dipole response of 48Ca, 68Ni, 120Sn and 208Pb. We have found that the new interaction can well reproduce the experimental data on the peak energy of the giant dipole resonance and the electric dipole polarizability. We have further calculated the neutron skin thicknesses of 48Ca and 208Pb. The predicted neutron skin thickness of 208Pb from our study is also consistent with the experimental values Tar14 ; Hor12 ; Abr12 . Our results thus confirm the usefulness of the new Skyrme interaction Sk$\chi$m∗ in studying the isovector properties of nuclei in regimes where ab initio calculations with chiral nuclear forces may not be feasible.

Acknowledgements

This work was supported by the US Department of Energy under Contract No. DE-SC0015266, the National Science Foundation under Grant No. PHY1652199, and the Welch Foundation under Grant No. A-1358.