Optimizing the relativistic energy density functional with nuclear ground state and collective excitation properties

TL;DR

This paper develops a new relativistic energy density functional, DD-PCX, that accurately models nuclear ground states and collective excitations, using experimental data to better constrain nuclear matter properties.

Contribution

It introduces the DD-PCX functional, combining ground state and excitation data in a unified framework with covariance analysis for uncertainty estimation.

Findings

Accurately describes nuclear ground state properties and excitations.

Constrains symmetry energy near saturation density.

Provides statistical uncertainties for model parameters.

Abstract

We introduce a new relativistic energy density functional constrained by the ground state properties of atomic nuclei along with the isoscalar giant monopole resonance energy and dipole polarizability in $^{208}$Pb. A unified framework of the relativistic Hartree-Bogoliubov model and random phase approximation based on the relativistic density-dependent point coupling interaction is established in order to determine the DD-PCX parameterization by $\chi^2$ minimization. This procedure is supplemented with the co-variance analysis in order to estimate statistical uncertainties in the model parameters and observables. The effective interaction DD-PCX accurately describes the nuclear ground state properties including the neutron-skin thickness, as well as the isoscalar giant monopole resonance excitation energies and dipole polarizabilities. The implementation of the experimental data on nuclear excitations allows constraining the symmetry energy close to the saturation density, and the incompressibility of nuclear matter by using genuine observables on finite nuclei in the $\chi^2$ minimization protocol, rather than using pseudo-observables on the nuclear matter, or by relying on the ground state properties only, as it has been customary in the previous studies.

Figures

Tables

Table 1. Table 1: Parameters of the DD-PCX interaction with the corresponding statistical uncertainties. The value of the nucleon mass is 939.0 MeV and the saturation density is set to 0.152 fm -3 .

parameters	DD-PCX	$\sigma$
$a_s$ (fm2)	-10.979243836	0.010808546
$b_s$ (fm2)	-9.038250910	0.023987420
$c_s$ (fm2)	-5.313008820	0.047152813
$d_s$	1.379087070	0.003900800
$a_v$ (fm2)	6.430144908	0.024888709
$b_v$ (fm2)	8.870626019	0.019549460
$d_v$	0.655310525	0.003073028
$b_{tv}$ (fm2)	2.963206854	0.092150525
$d_{tv}$	1.309801417	0.053360277
${\delta }_s$ (fm4)	-0.878850922	0.004512226
$G_n$ (MeV.fm3)	-800.663126037	6.279054350
$G_p$ (MeV.fm3)	-773.776776597	4.003044910

Table 2. Table 2: The saturation properties of nuclear matter for the DD-PCX, DD-PC1 nik08 , DD-ME2 lal05 , and PC-PK1 zha10 interactions.

	DD-PCX	DD-PC1	DD-ME2	PC-PK1
E/A (MeV)	-16.026 $\pm$ 0.018	-16.06	-16.14	-16.12
$m_D^{\ast }/m$	0.5598 $\pm$ 0.0008	0.580	0.572	0.590
$K_0$ (MeV)	213.03 $\pm$ 3.54	230.0	250.89	238.0
$J$ (MeV)	31.12 $\pm$ 0.32	33.0	32.30	35.60
$L$ (MeV)	46.32 $\pm$ 1.68	70.0	51.26	113.0

Table 3. Table 3: The root mean square error ( Δ Δ \Delta ) and root mean square relative error ( δ 𝛿 \delta ) in percentage for the binding energies (B.E.) (MeV), charge radii ( r c subscript 𝑟 𝑐 r_{c} ) (fm) and mean gaps (MeV) for a set of spherical nuclei, using the relativistic DD-PCX, DD-PC1 and DD-ME2 interactions. The numbers of the nuclei considered in the calculations are given in parentheses.

		B.E.	$r_c$	Mean Gap
Interaction		(65)	(46)	(56)
DD-PCX	$\Delta$	1.38 MeV	0.016 fm	0.18 MeV
	$\delta$	0.21%	0.47%	15.40%
DD-PC1	$\Delta$	3.05 MeV	0.017 fm	0.29 MeV
	$\delta$	0.48%	0.49%	21.73%
DD-ME2	$\Delta$	2.08 MeV	0.016 fm	0.35 MeV
	$\delta$	0.27%	0.44%	26.13%

Table 4. Table 4: The constrained ISGMR energies (in MeV) for 90 Zr, 120 Sn and 208 Pb in comparison with the experimental data.

	Exp.	DD-PCX	DD-PC1	DD-ME2
90Zr	17.58${}_{}{}^{+0.06}{}_{-0.04}{}^{}$kr15	18.00 $\pm$0.10	18.83	17.80
	17.66$\pm$0.07 gup16
120Sn	15.5$\pm$0.1 li10	16.18 $\pm$0.09	16.92	16.07
208Pb	13.5$\pm$0.1 patel13	13.66 $\pm$0.08	14.22	13.49
	14.18$\pm$0.11 you99

Equations

$$\begin{array}{rl}\mathcal{L}& =\bar{\psi }(i\gamma \cdot \partial -m)\psi \\ & -\frac{1}{2}{\alpha }_S(\rho )(\bar{\psi }\psi )(\bar{\psi }\psi )-\frac{1}{2}{\alpha }_V(\rho )(\bar{\psi }{\gamma }^{\mu }\psi )(\bar{\psi }{\gamma }_{\mu }\psi )\\ & -\frac{1}{2}{\alpha }_{TV}(\rho )(\bar{\psi }\vec{\tau }{\gamma }^{\mu }\psi )(\bar{\psi }\vec{\tau }{\gamma }_{\mu }\psi )\\ & -\frac{1}{2}{\delta }_S({\partial }_{\nu }\bar{\psi }\psi )({\partial }^{\nu }\bar{\psi }\psi )-e\bar{\psi }\gamma \cdot A\frac{1-{\tau }_3}{2}\psi .\end{array}$$

$${\alpha }_i(\rho )=a_i+(b_i+c_{i}x)e^{-d_{i}x},(i=S,V,TV),$$

$$cov(A,B)=cov(B,A)=\sum _{i,j=1}^N\left(\frac{\partial A}{\partial p_i}\right){\mathcal{M}}_{ij}^{-1}\left(\frac{\partial B}{\partial p_j}\right),$$

Full text

Solving the quantum many-body problem of strongly interacting nucleons represents one of the fundamental challenges not only for understanding the phenomena of nuclear structure and dynamics, but also for various applications of astrophysical relevance, e.g., modeling the stellar evolution, supernova explosion, the properties of compact stars, the synthesis of chemical elements in the universe, etc. Among a variety of theoretical frameworks to address this problem, the nuclear energy density functional (EDF) represents a unified approach to study quantitatively static and dynamic properties of finite nuclei along the nuclide map nik11 ; paar07 as well as the equation of state of nuclear matter roc18 . A considerable progress has been achieved in constructing and optimizing the phenomenological EDFs, both in non-relativistic  klup09 ; gog09 ; kor10 and relativistic nik11 ; lal05 ; nik08 ; zha10 ; pie14 frameworks. Recently, the construction of the EDFs was also inspired by ab initio calculations dobaczewski16 and effective field theories bonnard18 . As pointed out in Ref. shen18 , strengths of the tensor forces guided by ab initio relativistic Brueckner-Hartree-Fock calculations can also be used as a guide for the future ab initio derivations of the EDFs. At present, only the phenomenological EDFs provide a level of accuracy required to quantitatively describe the nuclear properties across the whole nuclide map.

So far, the EDFs have mainly been parametrized with the experimental data on the ground state properties of nuclei. These observables alone are often not enough to constrain the effective interaction completely, especially its isovector channel, thus the protocols to determine the EDF’s parameters often included constraints on the pseudo-observables on the nuclear matter properties. The neutron skin thickness $r_{np}$, isovector dipole excitations in nuclei and neutron star mass and radii represent some of the possible observables that could be used to probe the isovector channel of the EDFs roc18 ; bal16 ; piek14 . However, the data on $r_{np}$ are often model dependent, and the most recent data from parity violating electron scattering experiment (PREX) on 208Pb prex12 have large uncertainties, while the neutron star mass-radius data are still rather limited. An alternative observable to improve the performance of the EDF’s isovector sector is the dipole polarizability ($\alpha_{D}$), which is proportional to the inversely energy-weighted sum of the isovector dipole excitation in a nucleus. Recently, it has been represented as an observable strongly correlated with the symmetry energy parameters of the nuclear equation of state roc15 ; rein10 . The dipole polarizability was also measured in several nuclei, including 48Ca, 68Ni, 120Sn and 208Pb bir17 ; ros13 ; has15 ; tami11 . So far, mainly due to computational difficulties, the dipole polarizability was not employed to constrain the EDFs directly; instead, already established EDFs were tested for their performance in reproducing the experimental data on $\alpha_{D}$ roc18 . On the other side, the isoscalar giant monopole resonance (ISGMR) energy in nuclei also represents an important observable to probe the isoscalar channel of the EDFs and incompressibility of nuclear matter $K_{0}$ gar18 . To date, only in Ref. pie14 the ISGMR was used directly in the optimization of an EDF (relativistic FSUGold2 interaction) along with the properties of finite nuclei and neutron stars.

The purpose of this work is to establish the first unified framework based on the EDF, that allows to constrain an effective interaction not only by the experimental data on nuclear ground state properties, but also by a direct implementation of the measured properties of collective nuclear phenomena: the ISGMR excitation energy and the dipole polarizability. In this way, the properties of nuclei and the nuclear equation of state; incompressibility of nuclear matter and the symmetry energy around the saturation density will be constrained directly by the experimental data on finite nuclei.

In the framework of the relativistic nuclear energy density functional, the nuclear ground-state density and energy are determined by the self-consistent solution of relativistic single-nucleon Kohn-Sham equations koh1 ; koh2 . In the present study these equations are implemented from an interaction Lagrangian density with four fermion contact interaction terms including the isoscalar-scalar, isoscalar-vector, isovector-vector and isospace-space channels (for more details see Refs. nik08 ; nik14 ),

[TABLE]

In addition to the free-nucleon terms, the effective Lagrangian includes point coupling interaction terms, coupling of protons to the electromagnetic field, and the derivative term accounting for the leading effects of finite-range interactions necessary for a quantitative description of nuclear density distribution and radii. Starting from the microscopic density dependence of the scalar and vector self-energies, the following functional form of the couplings is employed nik08 ; Entem.03 ,

[TABLE]

where $x=\rho/\rho_{0}$, and $\rho_{0}$ represents the nucleon density in symmetric nuclear matter at saturation point. The point-coupling model includes 10 parameters ($a_{S}$, $b_{S}$, $c_{S}$, $d_{S}$, $a_{V}$, $b_{V}$, $d_{V}$ $b_{TV}$, $d_{TV}$ and $\delta_{S}$). For the description of open-shell nuclei, the relativistic Hartree-Bogoliubov (RHB) model nik14 is used and the pairing field is formulated using separable pairing force, which also includes two parameters for the pairing strength ($G_{p}$ and $G_{n}$) Tian . More details and various implementations of the RHB model can be found in Refs. vre05 ; meng06 . In the small amplitude limit, the collective excitations are described by the relativistic (quasiparticle) random phase approximation (Q)RPA paar07 . In the present study, a unified computational framework of the RHB model nik14 and self-consistent relativistic (Q)RPA paar07 is established to constrain the 12 model parameters by minimizing the $\chi^{2}$ objective function rocc15 . In order to constrain the model parameters, the binding energies (34 nuclei), charge radii (26 nuclei) and mean pairing gaps (15 nuclei) of the selected open-shell nuclei are used along with the two observables on collective excitations: the constrained ISGMR energy ($E_{ISGMR}=\sqrt{m_{1}/m_{-1}}$) and dipole polarizability for 208Pb (see the supplementary material for details on the data set). Here, $m_{1}$ and $m_{-1}$ represent the energy-weighted moment and inverse energy-weighted moment of the strength distribution paar07 , respectively. The mean gap values for protons and neutrons are calculated using the five-point formula ben00 . The adopted errors for the binding energies, charge radii and pairing gaps are taken as 1.0 MeV, 0.02 fm and 0.05 MeV, respectively. Recently, the dipole polarizability was measured in 208Pb using polarized proton inelastic scattering at extreme forward angles tami11 . After the subtraction of the quasi-deuteron effect from the experimental data, the dipole polarizability was obtained as 19.6$\pm$0.6 fm3 in 208Pb roc15 . Several experimental studies were also performed to explore the ISGMR in 208Pb you04 ; you99 ; uc04 ; pat15 ; patel13 . Although the uncertainties for the measured ISGMR energies are small, there are differences in the excitation energies from different studies. Recently, the ISGMR energies were measured for 204,206,208Pb using inelastic $\alpha$-scattering at extremely forward angles, and the constrained ISGMR energy was found as 13.5$\pm$0.1 in 208Pb patel13 , whereas this energy was previously obtained as 14.18$\pm$0.11 MeV in the Texas A$\&$M experiment  you99 . In optimizing the EDF, the constrained ISGMR energy is taken as 13.5 MeV patel13 and due to experimental uncertainties we adopted slightly large error (1.0%) in the fitting protocol. The dipole polarizability is also taken as 19.6 fm3 roc15 and the adopted error used is 0.5%. As mentioned above, the implementation of the collective nuclear excitations in the fitting protocol is crucial in constraining both the isoscalar and isovector channels of the EDF’s gar18 ; roc18 .

Using the observables introduced above, the ${\chi}^{2}$ minimization for the relativistic point coupling interaction is performed. This procedure is supplemented with the co-variance analysis that allows to determine statistical uncertainties of the model parameters and other quantities, as well as relevant correlations between various properties rein10 ; rocc15 . Accordingly, the curvature matrix is determined at the ${\chi}^{2}$ minimum, $\mathcal{M}\equiv\partial_{p_{i}}\partial_{p_{j}}\chi^{2}$, where $p_{i}$ and $p_{j}$ denote the interaction parameters ($i,j=1,..,12$). Then, it is used to estimate the uncertainties of the model parameters, using $\sigma(p_{i})\equiv\sqrt{\left(\mathcal{M}^{-1}\right)_{ii}}$ relation rocc15 . The covariance between the two observables ($A,B$) is defined as  rocc15

[TABLE]

where the derivatives of the observables and the inverse of the curvature matrix are calculated at the $\chi^{2}$ minimum. The statistical uncertainty of any quantity of interest $A$ is calculated using $\sigma(A)=\sqrt{cov(A,A)}.$

The resulting DD-PCX parameterization with the respective statistical uncertainties are given in Table LABEL:p. The statistical uncertainties of the parameters are found to be small, indicating that the parameters of the interaction are well constrained.

Table LABEL:nmp shows the nuclear matter properties at the saturation for density-dependent point coupling interactions, DD-PCX (with uncertainties) and DD-PC1 nik08 , density dependent meson-exchange interaction DD-ME2 lal05 , and non-linear point coupling interaction PC-PK1 zha10 . These include the energy per nucleon $E/A$, the Dirac effective nucleon mass $m_{D}^{*}$ nik08 , the nuclear matter compression modulus $K_{0}$, the symmetry energy at saturation density $J$ and the slope of the symmetry energy at saturation $L$ roc18 . Compared to the DD-PCX, the DD-PC1, DD-ME2, and PC-PK1 interactions were established using different protocols. The DD-ME2 parameterization is based on the density-dependent meson-exchange interaction, constrained using the ground state properties of spherical nuclei lal05 , whereas the DD-PC1 interaction is based on the point-coupling model, and optimized using the binding energies of deformed nuclei nik08 . In addition, selected nuclear matter properties were fixed in both interactions, and no data on excitations have been used in the ${\chi}^{2}$ minimization. In Ref. zha10 the non-linear point coupling interaction PC-PK1 has been constrained by fitting to observables of 60 selected spherical nuclei, including the binding energies, charge radii, and empirical pairing gaps, and no constraints on the nuclear matter properties have been used in the fitting protocol. Due to the implementation of the nuclear excitations for 208Pb in constraining the DD-PCX interaction, we find that the incompressibility ($K_{0}$) and symmetry energy parameters ($J$ and $L$) at saturation density are lower than for the DD-PC1, DD-ME2, and PC-PK1 effective interactions. The respective uncertainties are also found to be small, indicating that within the fitting protocol employed both the isoscalar and isovector channels of the interaction are tightly constrained. The DD-PCX interaction values for $J$ and $L$ are in agreement with the suggested values from previous studies roc18 . The compression modulus $K_{0}$ is also found at around 213 MeV, which is lower in comparison to other relativistic interactions nik08 ; lal05 . We realize that small $K_{0}$ is a direct consequence of the more recent experimental data ($E_{ISGMR}$=13.5$\pm$0.1 MeV) patel13 used in the fitting protocol, and the implementation of the data from Texas A$\&$M experiment ($E_{ISGMR}$=14.18$\pm$0.11 MeV) you99 would lead toward the higher value of $K_{0}$. Clearly, resolving the ambiguities between different experimental studies on the ISGMR data for 208Pb is essential for constraining the nuclear matter incompressibility.

In the following, the performance of the DD-PCX interaction in description of various nuclear ground state and excitation properties is presented. Figure 1 shows the differences between the experimental mass and calculated binding energies (upper panel) and the charge radii for Ca, Ni, Sn and Pb isotopes (lower panel) for the DD-PCX interaction. For comparison, the results are also shown for the density dependent point coupling interaction DD-PC1 nik08 , density dependent meson-exchange interaction DD-ME2 lal05 , and non-linear point coupling interaction PC-PK1 zha10 ; zha10b . In this work, the calculations are performed using the RHB model with separable pairing force and spherical symmetry is assumed. It is seen that all interactions under consideration provide a reasonable description of the binding energies and produce similar isotopic dependencies. Compared to the DD-PC1 and DD-ME2, the newly parametrized DD-PCX interaction seems to be more successful in the predictions of the experimental binding energies of spherical nuclei. Since the pairing parameters ($G_{n}$, $G_{p}$) are also included in the fitting protocol, the success of the DD-PCX can also be related to the better reproduction of pairing properties of nuclei as discussed below. In comparison to the experimental data charge , the charge radii are also accurately reproduced, with a few exceptions for Ca isotopes and 60Ni. In order to assess a more general overview about the performance of the DD-PCX interaction, we calculate the root mean square error ($\Delta$) and root mean square relative error ($\delta$) in percentage for binding energies, charge radii, and mean pairing gaps for a set of nuclei (see supplementary material for the list of nuclei), and the results are given in Table 3. It is seen that the resulting binding energies and mean gap values are better reproduced using the DD-PCX interaction, while the deviations for the charge radii are similar in all cases. An extended list with the calculated binding energies and charge radii for the selected nuclei using the same interactions is also given in the supplemental material with the corresponding experimental data.

The predictive power of the DD-PCX interaction is also tested on the nuclear excitations of nuclei. Table LABEL:gmr shows the constrained ISGMR energies for 90Zr, 120Sn, 208Pb calculated using various relativistic interactions, and compared to the experimental results  gup16 ; li10 ; kr15 ; patel13 ; you99 . In this part, we should mention that the DD-ME2 calculations are performed using the finite range Gogny interaction D1S in the particle-particle channel for open-shell nuclei paar03 , while separable pairing is used with the DD-PCX and DD-PC1 interactions. Among the relativistic interactions, the DD-ME2 (DD-PC1) interaction predicts the lowest (highest) values for the constrained ISGMR energies. Considering the point coupling interactions, the table clearly demonstrates the relevance of including the constraint on the ISGMR excitation energy for 208Pb in order to provide a reasonable description of the ISGMR energies in all nuclei under consideration. Using the DD-PCX interaction, the calculated values for 90Zr and 120Sn are slightly above the experimental values and further fine-tuning of the interaction may be achieved by considering the ISGMR energies of additional nuclei and using smaller adopted error within the $\chi^{2}$ minimization of the interaction. However, the softness of the Sn nuclei still represents an open question, and their inclusion in constraining the interaction may cause difficulties.

Due to the empirical constraint imposed on the dipole polarizability for 208Pb, it is expected that the isovector channel of the DD-PCX interaction is improved in comparison to other effective interactions, as illustrated in a few following examples. Figure 2 shows the dipole polarizabilities for 48Ca, 68Ni and 208Pb (left panel) and Sn isotopic chain (right panel). The calculations are performed using the DD-PCX, DD-PC1 and DD-ME2 interactions, and the available experimental data are also provided bir17 ; ros13 ; has15 ; tami11 ; roc15 . As shown in Fig. 2, only the DD-PCX interaction systematically reproduces the experimental results on $\alpha_{D}$ for all nuclei under consideration, while the DD-ME2 and DD-PC1 interactions mainly overestimate the measured values. Although in constraining the functional we only use the dipole polarizability for 208Pb, the DD-PCX interaction is also successful in the prediction of the dipole polarizabilities for other nuclei, as expected due to the strong correlation between the dipole polarizabilities in different nuclei roc15 .

Another important isovector property to be considered is the neutron skin thickness $r_{np}$. In Fig. 3, the neutron skin thickness for 208Pb is shown for a set of relativistic rocc15 ; zha10 ; todd05 ; lal05 ; nik08 and non-relativistic calculations klup09 ; rocc15 ; zhen18 along with the experimental data from Refs. tar14 ; prex12 ; tami11 ; pp11 ; antip07 ; klim07 . Using the DD-PCX interaction, the neutron skin thickness for 208Pb is predicted as $r_{np}$=0.159$\pm$0.005 fm, and the (yellow) band denotes the calculated statistical uncertainty. Considering the experimental results, it is clear that there are discrepancies in the predictions for $r_{np}$ values in 208Pb. The non-relativistic functionals predict lower values for the neutron skin thickness compared to the relativistic ones, with the exception of the new effective interaction DD-PCX. Among the relativistic interactions, the DD-PCX prediction provides the lowest neutron skin thickness for 208Pb, which is also in a good agreement with the majority of the experimental data. The neutron-skin thicknesses of neutron-rich 48Ca and 132Sn nuclei are also calculated as 0.172$\pm$0.003 fm and 0.218$\pm$0.005 fm, respectively. We find that the calculated neutron skin thicknesses are also in a good agreement with the “model-averaged” results in Ref. pie12 . The results demonstrate that the DD-PCX interaction can also be used to make reliable predictions for the neutron skin thicknesses of other nuclei.

In conclusion, we have established a unified theoretical framework to constrain the relativistic nuclear EDF, based on the RHB plus (Q)RPA, supplemented with the co-variance analysis, using not only the nuclear ground state properties, but also relevant properties on collective nuclear excitations. The relativistic point-coupling interaction DD-PCX introduced in this work, represents the first effective interaction that is constrained using the binding energies, charge radii and pairing gaps, together with a direct implementation of the ISGMR energy and dipole polarizability in the $\chi^{2}$ minimization. In comparison to the previous studies, where the functionals have the properties of the nuclear symmetry energy either unconstrained, constrained by the pseudo-observables on nuclear matter that are often arbitrary, or validated by the data after the parameters have been determined, the present study implements directly genuine observables on collective excitations to optimize the effective nuclear interaction. The success of the DD-PCX interaction in the predictions of the dipole polarizabilities and neutron skin thicknesses in other nuclei not used in optimizing the model parameters validates the isovector channel of the functional and the respective symmetry energy properties. The present analysis clearly demonstrates the relevance of accurate measurements of the nuclear collective phenomena, as well as the necessity to resolve the ambiguities in the existing data from different experiments, both in the isoscalar (e.g., the ISGMR energy in 208Pb you99 ; patel13 and isovector sectors for constraining modern nuclear energy density functionals.

We thank T. Nikšić for stimulating discussions while preparing this paper. This work is supported by the Croatian Science Foundation under the project Structure and Dynamics of Exotic Femtosystems (IP-2014-09-9159) and by the QuantiXLie Centre of Excellence, a project co financed by the Croatian Government and European Union through the European Regional Development Fund, the Competitiveness and Cohesion Operational Programme (KK.01.1.1.01). E. Y. also acknowledges financial support from the Scientific and Technological Research Council of Turkey (TÜBİTAK) BIDEB-2219 Postdoctoral Research program.