From homogeneous matter to finite nuclei: Role of the effective mass
We present a method for extracting an energy density functional (EDF) for nuclei starting from a given immutable equation of state (EoS) of homogeneous matter. The scheme takes advantage of a natural Ansatz for homogeneous nuclear matter and basic EDF theory tenets. We apply it to closed-(sub)shell nuclei to show that a good-quality Skyrme force model can easily be reverse-engineered from a good-quality EoS without refinement or extensive fitting. Then the bulk and static nuclear properties are found practically to be independent of the assumed value for the effective mass, which is a unique result in bridging EDF for finite and homogeneous systems. Our methodology can be applied to study the effect of the EoS parameters and of the effective mass on nuclear observables including excitations without inducing artificial correlations amongst the EoS parameters and the effective mass, thanks to the freedom to vary them independently.
Atomic nuclei can be viewed as charged, self-bound droplets of nuclear matter. Unlike other quantum liquids such as atomic Helium, nuclear matter’s thermodynamic limit is hardly realized in nature but represents a fictitious system which is a very useful theoretical playground ST2 (); FW (). With certain qualifications such as the stabilizing and binding effects of charged leptons and gravity, “infinite” nuclear matter is found in compact stars so that many astronomical observations serve as nuclear physics observations as well Piekarewicz18 ().
Energy density functional (EDF) theory provides a unified framework for both finite, self-bound nuclei and the equation of state (EoS) of infinite nuclear matter. The founding theorems HK64 (); KS65 (), originally for an externally bound, non-degenerate, unpolarized electron gas in its ground state, were followed over the years by numerous generalizations to situations including degenerate, polarized, and finite systems, and to excited states, as well as by practical justifications and refinements in a hierarchy of approximations PS01 (); Bechstedt (). Attempts exist to formalize intrinsic EDFs for self-bound systems Engel06 (); Messud11 (); Messud12 (). Formal justifications aside, the Kohn-Sham framework is widely used in nuclear physics BHR03 (); NMMY16 (), as exemplified by (though not restricted to) Skyrme models. There are various approaches based on the Hartree-Fock (HF) approximation, with or without explicit correlations beyond mean field or pairing.
Current major goals of nuclear EDF research include 1) establishing connections between EoS parameters and nuclear observables and 2) bridging phenomenological EDF approaches with ab initio approaches, which contain the use of precise phenomenological two- and three-nucleon potentials in Monte Carlo simulations FP81 (); APR98 (); GC09 () as well as the recent intense efforts devoted to nuclear potentials obtained from effective field theories EKLM08 (); DSS13 (); DHS15 (). After decades of work and hundreds of EDF models, the search for a universal functional is ongoing and stumbling blocks remain in reaching the above goals. Traditional EDF models demonstrate spurious correlations amongst parameters, in particular, involving the in-medium effective mass BDMNP17 (); DNMBP17 (). It is also commonly said that certain observables “prefer” a low or high effective mass. (Respective examples are radii or energies BHR03 ().) It turns out that most available models fail to reproduce reasonably constrained EoS properties and nuclear observables simultaneously DLSD12 (); SGSD12 (). The situation is certainly unsatisfactory: if an EoS is “realistic,” then by definition it should be able to reproduce nuclear properties.
This situation may give an impression that the true density dependence of nuclear EDF is likely so complex that in the construction of EDF guesswork prevails. Starting with the analysis of Ref. PPLH16 () we are trying to make a systematic construction of EDF. In that work a power expansion of a homogeneous-matter EDF in Fermi momentum was investigated. The Ansatz and related strategy are dubbed KIDS (Korea: IBS-Daegu-SKKU) after the locale or institute of the original developers GPHPO16 (); GOHP17 (). A statistical analysis showed that three terms suffice for isospin-symmetric nuclear matter (SNM) (the same is concluded in Ref. BFJPS17 () for nuclei) in a converging hierarchy, and that four terms suffice for pure neutron matter (PNM) in a broad regime of densities. An EDF set of parameters could be selected for applications.
In the present work, we address the question: Given such a realistic parameterization for homogeneous matter, can we apply it to nuclei with no refitting? This is the major motivation of the present work and we find that key to a positive answer is the treatment of the in-medium effective mass. In the following we present our method for extracting an EDF for nuclei from a given, immutable EoS, and its applications to nuclei.
The basic idea of the KIDS model for nuclear EDF is the expansion of the EDF in terms of the Fermi momentum or the cubic root of density in homogeneous infinite nuclear matter PPLH16 ():
where is the baryon density ( and are neutron and proton densities, respectively), and asymmetry is defined as . Kinetic energy per particle is written as
where () is the proton (neutron) mass. In Ref. PPLH16 (), a set of SNM parameters was determined by using the established properties at saturation: the saturation density , the binding energy per particle at saturation , and the compression modulus . This information can uniquely fix three unknowns . (Because it gives a marginal contribution, we set PPLH16 (); GPHPO16 (); GOHP17 ().) The four PNM parameters were fitted PPLH16 () to a microscopic EoS of PNM APR98 (). Thus was generated the set of parameters called “KIDS-ad2” in Ref. PPLH16 ().
Focusing on high densities, in Ref. PPLH16 (), the efficiency of the scheme was demonstrated in the regime of neutron stars, while the convergence of the expansion was explored in Ref. GOHP17 (). The nuclear symmetry energy is calculated as shown in Fig. 1. The results show that the symmetry energy may not be soft or stiff but may have a nontrivial density dependence. It is interesting that a similar behavior is seen in Refs. LPR10 (); PKLMR17 () where the cusp originates from skyrmion–half-skyrmion phase transition.
Very relevant for nuclei, on the other hand, is the behavior at low densities. Having used the PNM EoS of Ref. APR98 () in the fitting, with pseudodata down to fm, we test the validity of extrapolation to the region of fm. The results are shown in Fig. 2 which are compared with the EoS’s obtained from ab initio methods (QMC AV4 GC09 () and EFT DSS13 ()) and from a resummation formula YGLO YGL16 () fitted to ab initio pseudodata in fm GC09 () and to the PNM EoS of Ref. APR98 () at fm and representative Skyrme models DLSD12 (). This shows that the KIDS model reproduces the low-density curvature best with respect to EFT.
We now take this EoS, namely the KIDS-ad2 parameterization of Ref. PPLH16 (), which has a wide range of applicability in density, to investigate nuclei without altering its parameter values. Additionally, for the present “proof of principle” we use a similarly obtained EoS but with MeV. For the application to nuclei we reverse-engineer a Skyrme-type force for HF calculations. The HF equations can then be solved with a straightforward extension of a standard numerical code Reinhard91 (). Minimally, and to reproduce the EDF of Eq. (1), we adopt the form of generalized Skyrme force given as
where and , is the relative coordinate, and is the spin-exchange operator. The strength of the spin-orbit coupling, which is absent in the EoS of Eq. (1), is introduced by the term. It should be noted that the above “force” is an auxiliary entity, with no direct relation to a true Hamiltonian, being used as a stepping stone to the equivalent independent-particle “external” potential and resulting Kohn-Sham orbitals.
The Skyrme force of Eq. (3) resembles other generalized Skyrme models with multiple density-dependent couplings CBBDM04 (); ADK06 (); XPC15 (). However, our strategy for determining both the precise form and the strength of the parameters is completely different, as elaborated in Ref. PPLH16 () and below. By comparing the EDF of Eq. (1) and the EDF corresponding to the above Skyrme force BHR03 (); DLSD12 (), relations among parameters can be straightforwardly obtained as
This reveals that the above EDF for nuclei has two sources for the term (): one from the density-dependent term in Eq. (3) of (), and the other from the momentum-dependent term in Eq. (3) of (). The latter parameters determine the isoscalar and isovector effective masses as CBHMS97 ()
where we have used, for simplicity, the average nucleon mass . Thus the values for the effective masses determine the parameters , while the total coefficients of the EoSs of SNM and PNM can remain unchanged. This is different from the traditional Skyrme force model that has a priori , making the whole term to be momentum-dependent.
The unknowns to be determined for nuclei, besides , are then the momentum dependence proportions in and and, correspondingly, the precise values of and . Our procedure is to
determine the momentum-dependent terms by fitting to the energy and charge radius of \nuclideCa, with initialized to null,
determine from the energies and radii of the nuclei \nuclideCa and \nuclidePb,
iterate, i.e., examine \nuclideCa with the new value of and again determine anew and so on.
It turns out that iteration is largely unnecessary, because the bulk properties of the spin-saturated nucleus \nuclideCa are insensitive to .
We first determine the momentum dependence part (steps 1, 3 above). A simplistic procedure we explored before GOHP17 (); GPHPO16 () is to set , and encode the momentum dependence in a single parameter , corresponding to the portion of assigned to the momentum dependence part. The value of is then determined from the properties of \nuclideCa, which gives small values of . For KIDS-ad2 we obtained , corresponding to (, ), and .
A preferable way is to retain the freedom in . We now have four parameters to be explored. We consider specific values for () which will be varied for examining the results. This leaves us with only two unknowns in each case. In practice, the parameter space is further constrained by imposing 1) that the gradient coefficient be lower than (a handy and loose enough rule of thumb HPDBD13 ()) and 2) that polarized neutron matter remains stable at high densities KW94b (). From the acceptable combinations we choose the one that gives the best results for \nuclideCa. Finally, for each parameter set [i.e., essentially, for each pair of () and best fit to \nuclideCa] we determine the best value of by fitting to the energies and radii of \nuclideCa and \nuclidePb.
In the following applications we retain for comparison the KIDS-ad2 EoS with , henceforth labeled KIDS0:
KIDS0: , , , , , , , , , , MeV.
Parameters are also derived, as already described, for the following cases:
K240: Same EoS as KIDS0 for SNM and PNM, i.e., same , but allowing for non-vanishing () and for (with ) or of (with ).
K220: Same EoS as KIDS0 except that the SNM compression modulus is MeV; thus , , , , with different values of (for ).
In the above, all values are given in units . Resulting Skyrme-type parameters are collected in Table 1.
|GSkI ADK06 ()|
|SLy4 CBHMS98 ()|
Results from HF calculations for the input nuclei \nuclide[40,48]Ca, \nuclidePb and for other (semi-)magic nuclei are shown in Fig. 3 along with the available data. Here, all results for \nuclideO, \nuclideO, \nuclideCa, \nuclideZr, \nuclideSn, and \nuclideU are predictions. We compute a mean absolute deviation of the calculated observable ( or ) with respect to data defined as
where the sum runs over nuclei considered here for which data exist. Results are shown in Table 1.
We observe that in the scale of the graphs the results for the bulk properties are practically indistinguishable and, apart from \nuclideO, in excellent agreement with available data and on a par with other models as the values of and indicate. From these results, we draw two conclusions: (1) A good-quality Skyrme model can easily be reverse-engineered from a good-quality EoS without extensive fitting, while (2) bulk and static quantities are practically independent from the effective mass. The second conclusion is a most unusual, though intuitively unsurprising, result, which would not have been revealed had we ascribed the term fully to the kinetic energy from the outset. Let us note that, in that case, for KIDS-ad2 we would have obtained and .
The insensitivity is also examined in Table 1 for the exotic nucleus Ca. Some dependence is observed in the much finer cases of the neutron skin thickness, in particular, of \nuclidePb, \nuclideU, and \nuclideZr, likely attributable to structural details that should be examined further in subsequent studies. We also find a tendency that the MeV parameterizations perform better than those with MeV. Systematic studies will be reported elasewhere.
The effective mass values can of course affect dynamical properties such as removal and capture energies (single-particle spectrum) and nuclear collective motion. In Fig. 4 we compare obtained single particle energies with data and other model calculations, UNEDF2 KMNO13 (), GSkI ADK06 (), and SLy4 CBHMS98 (). In the first two models (UNEDF2 and GSkI) single particle energy levels of \nuclidePb are used in the fitting, so they are expected to give better results. For a more precise comparison we consider a mean absolute deviation,
with the sum executed on the whole states shown in Fig. 4. This value is shown under the name of the model in Fig. 4. The accuracy of KIDS models with high effective mass is similar to those of GSkI and UNEDF2 models. Strictly speaking, only the highest occupied state of a many-body system can be considered as an observable Bechstedt (); DV (). The comparison is nonetheless interesting, in confirming that higher values of may be needed to reproduce the single-particle spectrum of \nuclidePb.
We presented and validated a method for extracting a generalized Skyrme-type EDF for nuclei from a given, immutable EoS. The scheme utilizes a natural and versatile EoS for SNM and PNM which is “agnostic” with respect to effective masses. We have shown that 1) a good-quality Skyrme model can easily be reverse-engineered from a good-quality EoS without extensive fitting, and that 2) bulk and static quantities are practically independent of effective masses. To our knowledge these are unique results obtained by bridging EDF for finite and homogeneous systems. Future applications abound: Our methodology will allow us to study the effect of the EoS parameters and of the effective mass on nuclear observables while retaining the freedom to vary major EoS parameters and the effective mass independently. An exploration of symmetry-energy parameters is underway Ahn18 (). Finally, it remains feasible to constrain the momentum dependence based on microscopic calculations of the effective mass or polarized matter and the spin-orbit coupling from, e.g., relativistic approaches.
Acknowledgements.We thank B. K. Agrawal for providing us with the correct values of GSkI and GSkII parameters. This work was supported by the National Research Foundation of Korea under Grant Nos. NRF-2015R1D1A1A01059603 and NRF-2017R1D1A1B03029020. The work of P.P. was supported by the Rare Isotope Science Project of the Institute for Basic Science funded by Ministry of Science, ICT and Future Planning and the National Research Foundation (NRF) of Korea (2013M7A1A1075764).
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