Relativistic mean field interaction with density dependent meson-nucleon vertices based on microscopical calculations
Although ab-initio calculations of relativistic Brueckner theory lead to large scalar isovector fields in nuclear matter, at present, successful versions of covariant density functional theory neglect the interactions in this channel. A new high precision density functional DD-ME is presented which includes four mesons , , , and with density dependent meson-nucleon couplings. It is based to a large extent on microscopic ab-initio calculations in nuclear matter. Only four of its parameters are determined by adjusting to binding energies and charge radii of finite nuclei. The other parameters, in particular the density dependence of the meson-nucleon vertices, are adjusted to non-relativistic and relativistic Brueckner calculations of symmetric and asymmetric nuclear matter. The isovector effective mass derived from relativistic Brueckner theory is used to determine the coupling strength of the -meson and its density dependence.
Structure properties of nuclei described in the framework of effective mean-field interactions are remarkably successful over almost the entire periodic table (1); (2); (3); (4); (5); (6); (7); (8); (9); (10); (11); (12). Relativistic and non-relativistic versions of this approach enable an effective description of the nuclear many-body problem as an energy density functional. These energy functionals are usually adjusted to a variety of finite nuclei and infinite nuclear matter properties. Although all these effective interactions are based on the mean-field approach, some differences will generally appear between them due to the specific ansatz of the density dependence adopted for each interaction. For instance, predictions in the isovector channel of existing functionals differ widely from one another and, as a consequence, the density dependence of the symmetry energy is far from being fully determined. This has an impact on finite nuclei properties as, for example, the neutron skin thickness. Mean field models which accurately describe the charge radius in Pb, predict neutron radii between 6.6 and 5.8 fm. For these reasons, one of the main goals in Nuclear Physics is to build a universal density functional theory based on microscopical calculations (13); (14). This functional should be able to explain as many as possible measured data within the same parameter set and to provide reliable predictions for properties of nuclei far from stability not yet or never accessible to experiments in the laboratory. It should be derived in a fully microscopic way from the interactions between bare nucleons. At present, however, attempts to derive such a density functional provide only qualitative results for two reasons: first the three-body term of the bare interaction is not known well enough and second the methods to derive such a functional are not precise enough to achieve the required accuracy. Note that a 1 ‰ error in the binding energy per particle of symmetric nuclear matter leads to an error of several MeV in the binding energy of heavy nuclei, an error which is an order of magnitude larger than required by astrophysical applications. Therefore, at present the most successful functionals are derived either in a fully phenomenological way from a very large set of experimental data, as for instance more than 2000 nuclear binding energies (8) or, more recently, by an adjustment to a combination of microscopic results and to a set of characteristic experimental data (15); (16); (17); (18).
With the same goal, in the more recent past a different approach was tried. Baldo, Schuck and Viñas employed the same route as in condensed matter physics and constructed a functional (BCP) (12) where the bulk part is based on the fully microscopic calculations of Ref. (19). In this more fundamental calculations, Baldo and collaborators investigate the nuclear and infinite neutron matter on the basis of the Bethe-Brueckner approach including three-body correlations of the Bethe-Faddeev type (20). Their results for the Equation of State (EoS) of symmetric and neutron matter are believed to be among the most accurate in the literature. Then, the BCP functional took as a benchmark the calculations of Ref. (19) by means of a polynomial fit. In this way an accurate and analytic EoS as a function of neutron and proton densities were constructed covering the whole range from symmetric nuclear matter to pure neutron matter in density ranges from zero to about two times saturation density. Subsequently a finite range surface term dependent on three parameters together with the strength of the spin-orbit term (four parameters in total) were added to the bulk part of the functional. The pairing correlations needed to describe open-shell nuclei were accounted by a density dependent -force with an effective mass equal to the nucleon mass that simulates pairing calculations in symmetric nuclear matter computed with the Gogny interaction (21) in the channel. Adjusting these free parameters to some selected nuclear experimental data yielded excellent results for nuclear masses and reasonable charge radii of the whole nuclear chart. In addition it has also been shown that the deformation properties of BCP functionals are similar to the ones found using the Gogny D1S force (22); (23) in spite of the fact that both models are clearly different. A recent review on the BCP functionals can be found in Ref. (24)
In general, symmetries of nature help to reduce the number of parameters and to simplify the description. One of the underlying symmetries of QCD is Lorentz invariance and therefore covariant versions of density functionals are of particular interest in nuclear physics. This symmetry allows one to describe the spin-orbit coupling, which has an essential influence on the underlying shell structure in finite nuclei, in a consistent way. Moreover it also puts stringent restrictions on the number of parameters in the corresponding functionals without reducing the quality of the agreement with experimental data. Self-consistent mean-field calculations starting from relativistic Lagrangians have been very successful in describing nuclear properties (25); (26); (27); (28); (29). They arise from a microscopic treatment of the nuclear many-body problem in terms of nucleons and mesons carrying the effective interaction between nucleons. Moreover, since the theory is relativistically invariant and the field and nucleon equations of motion are solved self-consistently, they preserve causality and provide a self-consistent description of the spin-orbit term of the nuclear effective force and of the bulk and surface parts of the interaction. In addition these functionals include nuclear magnetism (30), i.e. a consistent description of currents and time-odd fields, important for odd-mass nuclei (31), excitations with unsaturated spins, magnetic moments (32) and nuclear rotations (33). No new parameters are required for the time-odd parts of the mean fields. In non-relativistic functionals the corresponding time-odd parts are usually difficult to adjust to experimental data and even if there are additional constraints derived from Galilean invariance and gauge symmetry (34) these constraints are usually not taken into account in the successful functionals commonly used in the literature. The earlier versions of covariant density functional theory were based on the Walecka model (35); (36); (37); (38) with phenomenological non-linear meson-interactions proposed by Boguta and Bodmer (39) introducing in this way a phenomenological density dependence (4); (9); (40). Later the non-linear models have been replaced by an explicit density dependence of the meson-nucleon vertices. This density dependence has first been determined in a phenomenological way (6); (7); (10) These models have shown considerable improvements with respect to previous relativistic mean-field models in the description of asymmetric nuclear matter, neutron matter and nuclei far from the stability valley. On the other hand one has tried to derive this density dependence in a microscopic way from Brueckner calculations in nuclear matter at various densities (41); (42); (43); (44). An example is Density Dependent Relativistic Hadron Field theory (42) where the specific density dependence of the meson-nucleon vertices is mapped from Dirac-Brueckner calculations where the in-medium interaction is obtained from nucleon-nucleon potentials consistent with scattering experiments. Therefore, if this ansatz is adopted, the effective theory is derived fully from first-principle calculations. Of course, the accuracy of the results obtained in this way is by no means satisfactory for modern nuclear structure calculations and a fit of additional free parameters is still needed. This fact allows to constrain the different possibilities and keeps the compatibility, at least theoretically, with more fundamental calculations of infinite nuclear matter.
As mentioned, there exist ab initio calculations of the nuclear EoS over a wide range of nucleon densities, i.e. far from densities currently reachable at the laboratory. In this sense, apart from the experimental data needed in the fitting procedure for determining an effective interaction, further steps on building a universal density functional may need to implement such ab initio information as it was done in the BCP model. Therefore, this EoS calculated from first principles can be understood as a temporary benchmark at supra- and sub-saturation densities of the energy per particle at different asymmetries. Furthermore, from a theoretical point of view, consistency is desirable between predictions of both theories. Regrettably, to make them compatible is not only a problem of including the EoS derived from realistic nucleon-nucleon potentials within the fitting procedure. It is also a problem of taking into account the proper density dependence in the different terms of the functional. For that, first-principle calculations are also thought to be the best candidate to help in building a universal energy density functional, at least the bulk part of it since many-nucleon calculations are not feasible yet. Hence, for an effective and self-consistent treatment of the nuclear many body problem, we propose here a novel and improved relativistic mean-field interaction with an explicit density dependence of the meson-nucleon vertices in all the four spin-isospin channels compatible with fully microscopical calculations.
The essential breakthrough of density functional methods in the description of quantum mechanical many-body problems was Kohn-Sham theory (45); (46), where the exact density functional of Hohenberg and Kohn (47) was mapped in an exact way on an effective potential in a single particle Schroedinger equation, which forms the starting point of all modern applications of density functional theory. In covariant density functional theory this effective potential corresponds to the self-energy in the Dirac equation, which can be decomposed into four channels characterized by the relativistic quantum numbers of spin and isospin, the scalar isoscalar channel (), the vector isoscalar channel () the scalar isovector channel () and the vector isovector channel (). In the Walecka model these channels are connected with the exchange of mesons carrying the corresponding quantum numbers.
However, in nearly all the present successful phenomenological applications of covariant density functional theory to nuclear structure based on the relativistic Hartree model only the three mesons , , and are taken into account. The scalar isovector -meson () causing a splitting of the effective mass between protons and neutrons is neglected, because it has turned out that usual data such as binding energies and radii of finite nuclei do not allow to distinguish scalar and vector fields in the isovector channels. Allowing independent parameters for the - and the -mesons leads to redundancies in the fit. By the same reasons also modern relativistic Hartree-Fock (48); (49); (50) and Hartree-Fock-Bogoliubov (51) calculations neglect the -meson in the Lagrangian and, as a consequence, in the direct term. Of course the Fock term of these calculations contains also contributions to the scalar isovector channel. Microscopic investigations by Huber et al. (52); (53) and phenomenological studies (54); (55); (56); (57); (58); (59); (60); (61); (62); (63); (64) in the the literature stressed that mean-field models which neglect the -meson are likely to miss important ingredients in describing properly very asymmetric nuclear matter, in particular at high densities. The proton fraction of -stable matter in neutron stars can increase and the splitting of the effective mass can affect transport properties in neutron stars and heavy ion reactions. However, as long as the parameters of this meson are not fixed, such investigations are somewhat academic. Therefore we derive in this manuscript the -nucleon vertex and its density dependence from modern microscopic calculations based on the bare nucleon-nucleon force of the Tübingen group (65).
The relativistic mean field model DD-ME obtained in this way is an extension of the DD-ME model developed by the Munich group (7); (10) based on the density dependent relativistic Hartree theory. The DD-ME model has the following degrees of freedom: the proton, the neutron and three mesons carrying the nuclear interaction, namely -, - and -mesons. In addition to these degrees of freedom, we include here a new one, the -meson by the reasons pointed out before. Since this meson provides a treatment of the isospin more close to the microscopic investigations we can hope that it improves the reliability of the models for predictions in nuclei far from stability with large isospin - planned to be studied experimentally at the new Rare Ion Beam Facilities (66). Apart from the inclusion of the -meson in DD-ME the DD-ME model differs from the earlier DD-ME models in that the parameters of DD-ME were all adjusted to experimental data based on finite nuclei properties, whereas those of DD-ME are largely based on microscopic ab-initio calculations in nuclear matter. Only four of the parameters of DD-ME are fitted to finite nuclei.
The paper is organized as follows: after establishing the DD-ME model in Sect. II, we discuss in Sect. III the strategies to determine the parameters of the Lagrangian and compare in Sect. IV the results of this novel effective interaction with the experiment and with the non-relativistic BCP model (12) and the completely phenomenological DD-ME2 model (10), in particular binding energies, charge radii and neutron skins in spherical nuclei.
Ii Density Dependent Hadron Field Theory
ii.1 Lagrangian and equations of motion
Density dependent relativistic hadron field theory which forms the basis of the DD-ME interaction has been formulated and extensively discussed in Refs. (67); (68); (42). Here we present only the essential features of the mean-field equations of motion derived from such a theory. The relativistic Lagrangian includes neutrons and protons represented by the Dirac spinors of the nucleon, the four mesons (, , and ) carrying the effective nuclear strong interaction represented by the fields , , , and , and the photon field accounting for the electromagnetic interaction. The index indicates the time- and space-like components of the vector fields and the arrow indicates the vector nature of a field in isospin space. As mentioned, the -meson should be included if one wants to follow the theoretical indications of Dirac-Brueckner calculations in asymmetric nuclear matter and so we do. The Lagrangian has the following parts
where is the nucleonic free Lagrangian
is the Lagrangian of free mesons
and is the Lagrangian describing the interactions. Its algebraic expression is
where is the nucleon mass (commonly taken as MeV), the field strength tensors for the vector fields are
and correspondingly and . The electric charge is for protons and zero for neutrons. The meson-nucleon vertices are denoted by for , , , and . Since covariance is required and the quantity is in the rest frame identical to the baryon density , the nucleon-meson vertices generally depend on this quantity. Because of the relatively small velocities the difference between and is negligible in all practical applications. The subindex or is used to indicate whether we are considering neutrons or protons respectively.
The equations of motion are derived from the classical variational principle and we obtain for the nucleon spinors the Dirac equation
where is the effective Dirac nucleon mass and and are the vector and scalar self-energies defined as follows,
here, indicates the usual definition of the vector self-energy and the rearrangement term of the vector self-energy
Here is the direct term of the Coulomb potential. As in most RMF models we neglect in these investigations the Coulomb exchange term which plays an important role in -RPA calculations (70). The static mean field approximation used throughout this investigation preserves the third component of the isospin. As a consequence the other two components of the densities and fields carrying isospin vanish. In Eq. 9 and the following equations represents the time-like component of the -meson field, whereas and represent the isovector part of the baryon density and of the scalar density. The rearrangement term is a contribution to the vector self-energy due to the density dependence of the meson-nucleon vertices. The equations of motion for the mesons are,
where the different densities and currents are the ground-state expectation values defined as,
ii.2 Asymmetric infinite nuclear matter
Energy density and pressure
In infinite nuclear matter we neglect the electromagnetic field. Because of translational invariance, the Dirac equations can be solved analytically in momentum space and we obtain the usual plane-wave Dirac spinors (69). Filling up to the Fermi momenta for or , we find the densities
and the meson fields
where and where the Fermi energy of neutrons and protons is given by . Now, we calculate the energy density () and pressure () from the energy-momentum tensor,
where runs over all possible fields,
Only the pressure has a rearrangement contribution. We have checked that the pressure derived from the energy-momentum tensor coincides with the thermodynamical definition: and that the energy-momentum tensor is conserved .
The symmetry energy:
Assuming charge symmetry for the strong interaction (the and interactions are identical but different, in general, from the interaction), the total energy per particle in asymmetric nuclear matter can be written as follows,
where is the baryon density and measures the neutron excess. The term proportional to is the so called symmetry energy of infinite matter and terms proportional to (and higher) can be neglected to very good approximation. The symmetry energy is defined as,
Models including the -meson provide a richer description of the isovector sector of the nuclear strong interaction. For that reason it is important to understand its effects on asymmetric nuclear matter and for that we give the analytic expressions for the symmetry energy of the model discussed in the last section (56); (57):
The quantity will be needed below. In these equations we used the fact that for we have , , , and we have defined and . In symmetric nuclear matter, the effective mass and the scalar density read
respectively. Close to the saturation density is a very good approximation and we find in this case an analytical approximation for
and, therefore, the contribution to the symmetry energy coming from the nuclear strong interaction (potential part) as described by this kind of models can be written in the simple form,
The symmetry energy is often expanded around the saturation density
where is the symmetry energy at saturation, and and are proportional, respectively, to the slope and the curvature of the symmetry energy at saturation.
Using the analytical expressions (33) we find
where the functions , and depend on and :
The strength of the - and -nucleon vertices is quite well determined by experimental data as compared with the strength of the isovector meson-nucleon vertices. On the other side, with only the -nucleon vertex, one is able to reproduce properties of finite nuclei (71) and to account for the symmetry energy around saturation in rather good agreement with available empirical indications. However, to reproduce nucleon-nucleon scattering measurements in the vacuum, one needs to incorporate a scalar-isovector meson into the parameterization of the two-body nuclear interaction (72). Microscopic derivations of the nuclear fields using relativistic Brueckner theory (52); (53); (42); (73); (74); (75); (65) or non-relativistic Brueckner theory (43); (44) show clearly that the scalar field in the nuclear interior has an isovector part. These reasons motivate one to incorporate the -meson also in models of covariant density functional theory and to study its influence on properties such as the symmetry energy, the effective mass splitting between protons and neutrons in asymmetric matter, the isospin dependence of the spin-orbit potential and the spin-orbit splittings far from stability.
ii.3 Density dependence of the meson-nucleon vertices
Here we describe the density dependence of the meson-nucleon vertices used for the new interaction DD-ME. We start from modern fully microscopic calculations in symmetric nuclear matter and pure neutron matter at various densities and try to determine the density dependence of the vertices by fitting to those data. Of course, it is well known that successful density functionals can, at present, not be determined completely from ab-initio calculations. Therefore, we introduce in the fit not only results of microscopic calculations but also a set of data on binding energies and radii in specific finite nuclei.
In a first step we have to choose a form of the density dependence of the various vertices, which is flexible enough to reproduce the microscopic calculations. In Refs. (41); (76) the meson-nucleon vertices of density dependent RMF theory have been related to the scalar and vector self-energies obtained from Dirac-Brueckner (DB) calculations in infinite nuclear matter. The density dependence deduced from DB calculations is
As in Refs. (7); (10); (16) we use the value fm. In fact, this choice is very close to the saturation density obtained in the following fit. As we see from the ansatz (47) the actual value of is irrelevant for the calculations. It can be completely absorbed in the values of the parameters , , , and . It is only used to make them dimensionless. By definition, the parameters are constrained by the condition . In the earlier applications (6); (7); (10) this ansatz was only used for the - and -meson. The density dependence of the isovector coupling was described by an exponential and the -meson was neglected. Here we use the same ansatz (47) also for the isovector mesons and . This turned out to be necessary in order to obtain a density dependence of the -coupling similar to that derived from microscopic ab-initio calculations in Refs. (76); (77); (42). We impose as in Ref. (6) the constraints , , and . We work with meson masses MeV, MeV and MeV. The nucleon mass is MeV. All in all, the model has 14 adjustable parameters. Namely, the 4 coupling constants in the 4 relativistic channels (Lorentz-scalar, Lorentz-vector, isoscalar and isovector), 9 parameters describing the density dependence in the functions , and the -mass allowing for a finite range and a proper description of the nuclear surface.
ii.4 Calculation of finite nuclei
The self-consistent results for masses include a microscopic estimate for the center-of-mass correction:
where is the total momentum of a nucleus with nucleons. The expression
is used for the charge radius. The description of open shell nuclei requires pairing correlations. We introduce this through the BCS approach with a seniority zero force in the soft pairing window described in Ref. (78). For the fit of the parameters of the Lagrangian described in the next section the constant gap approximation (79) has been used and the gap parameters have been derived from the experimental binding energies by a 3-point formula.
Iii The parameters of the functional DD-ME
In this section we describe the determination of the parameters of DD-ME. Earlier fits of relativistic Lagrangians have shown that the usual set of experimental ground state properties in finite nuclei, such as binding energies and radii do not allow to determine more than 7 or 8 parameters (4). Two of them ( and ) determine the saturation energy and the saturation density of symmetric nuclear matter (37), one of them () is fixed by the radii in finite nuclei and another one of them () determines the symmetry energy at saturation. The additional parameters (as for instance and in the nonlinear meson coupling models NL1 (3) or NL3 (4) or the three parameters in the ansatz (47) for density dependence in the isoscalar channel of DD-ME1 (7) and DD-ME2 (10)) are determined by the isoscalar surface properties and are necessary to describe deformations and the nuclear incompressibility properly. Finally one parameter ( in DD-ME1 or DD-ME2) is needed to describe the density dependence of the symmetry energy by a fit to the experimental data on the neutron skin thickness.
In order to calibrate the 14 free parameters of the DD-ME functional we therefore added pseudo-data in the form of results of modern microscopic non-relativistic and relativistic Brueckner calculations. To this end, we selected the EoS of symmetric nuclear matter and of neutron matter (see Fig. 1) derived by Baldo et al. (19) in a state-of-the-art non-relativistic Brueckner calculation including relativistic corrections and three-body forces. We also used as a benchmark the isovector part of the effective Dirac mass (see Fig. 2) derived by the Tübingen group (65) in relativistic Dirac-Brueckner theory. The use of non-relativistic results for the EoS and of relativistic results for the isovector effective mass may seem somewhat arbitrary. However we have to keep in mind that the non-relativistic calculations of the Catania group are more sophisticated than presently available Dirac-Brueckner calculations, because they include not only relativistic effects but also three-body forces. On the other side it is very complicated to deduce Dirac masses from a non-relativistic calculation which does not distinguish between Lorentz scalars and vectors. This is in principle possible (43); (44), but it is difficult and connected with additional uncertainties. With this caveat in mind, we decided to use a reliable relativistic Brueckner calculation (65) providing directly the effective Dirac masses and and their difference, a quantity directly connected with the scalar isovector part of the self energy. The isovector part of the effective Dirac mass depends only on the -meson. It vanishes for all the conventional Lagrangians without -meson. The density dependence of this quantity is therefore the optimal tool to get information about the density dependence of the -meson vertex .
Keeping this in mind, we determine 10 of the 14 parameters in the Lagrangian of DD-ME by these pseudo-data obtained from ab-initio calculations of nuclear matter. These parameters define the density dependence for the various meson-nucleon vertices (9 parameters) and the strength of the -meson. Only a reduced set of 4 parameters (, , , and ) are fitted to the masses and charge radii of finite nuclei.
iii.1 Strategy of the parameter fit
Since the mean field equations of motion have to be solved self-consistently, we need a good starting parameter set before fixing the -meson coupling to the above mentioned calculations fully. The density-dependent meson coupling model DD-ME2 (10) provides us with an excellent description of nuclei all over the periodic table. Though DD-ME2 neglects the -meson, it is based in the isoscalar channel on the same ansatz (47). Therefore, we used DD-ME2 as a starting point of our investigations. We proceeded in three steps:
In the first step we performed an overall fit with all the 14 parameters. For the data we have chosen on one side the three microscopic curves for the EoS in Figs. 1 and 2 and on the other side the same set of data of finite nuclei which has been used in Ref. (10) for the determination of the parameter set DD-ME2 (see Table II of this reference), i.e. 12 binding energies of spherical nuclei distributed all over the periodic table and 9 charge radii. Due to the fact that the density dependence in the isovector channel is determined by the equation of state of neutrons it was not necessary to include neutron skin thicknesses () data. This fit provides us with a relatively stable starting point for a subsequent fine tuning of the model. Moreover, since the -meson is little influenced by the overall fit to finite nuclei both and (4 parameters) are relatively well determined already in this step and we need only a fine tuning of the remaining parameters in the next two steps.
In the second step we keep the four meson masses and the four parameters describing the density dependent vertex of the -meson fixed and determine the 9 parameters describing the Typel-Wolter ansatz for the density dependent vertices of the remaining three mesons (, and ) by a very accurate fit to the nuclear matter data shown in Figs. 1 and 2. Involving only nuclear matter data, this is a relatively fast calculation and as a result we obtain the three density dependent vertices for , , . In this way we describe with high precision the EoS for symmetric nuclear matter and pure neutron matter as well as the isovector part of the effective Dirac mass .
In the last step we keep the -meson parameters as determined in step 1 and the density-dependent functions for are frozen at the values found in step 2. We refine the remaining 4 parameters , , and to the binding energies of 161 spherical nuclei and the charge radii of 86 nuclei shown in Table 5 taking into account in this fit also the pseudo-data of the nuclear matter properties used in step 1 and 2 with certain weights. It turns out that the values of , , and obtained in this fit differ only slightly from the values determined in step 1 and that the nuclear matter results (EoS in symmetric nuclear matter and pure neutron matter as well as the isovector Dirac mass) differ only marginally from the results obtained in step 2. As a consequence the procedure involving step 2 and 3 does not have to be repeated.
The final parameter set DD-ME obtained in this way is given in Table 1. It is compared with the parameter set DD-ME2 in Table 2. We observe a large difference in the value of the the -nucleon vertex . This can be understood by the fact that we have in DD-ME2 only one meson () in the isovector channel with a very different density dependence. We also observe a very different density dependence for the isoscalar mesons and . All in all this has only a minor effect in the low density region but it has a large effect at high densities as shown in Fig. 3.
Our fit is performed through a test of the form
where is the number of data points and the weight associated to each data point. is the experimental value for finite nuclei and the pseudo data obtained by ab-initio calculations in nuclear matter. The observables in finite nuclei used for the fit are the binding energies of nuclei and the charge radii of nuclei given in Table 5. All of the isotopes are spherical even-even nuclei and the data are taken from the literature (80). In the standard definition of a test, the weights should be inversely proportional to the experimental uncertainties. However, in the case of energies these are usually so small, that they cannot be used as relevant quantities. We therefore used the weights given in Table 3, i.e. MeV for the masses and fm for the radii. For the fit to the results of ab-initio calculations in nuclear matter we use mesh points in a certain density range (see Table 3) and we assume a relative accuracy of 3 %. The minimization of is carried out by means of a variable metric method algorithm included in the MINUIT package of Ref. (81).
In the first step of the fit discussed in the last section we minimize the quantity
At this stage all the 14 parameters of the model are varied and the data of finite nuclei are restricted to the masses and charge radii of the 12 nuclei used in the fit of the parameter set DD-ME2 in Ref. (10). As we have described in the last section, the parameters for the -meson obtained from this fit are no longer changed. We given them in the third line of Table 1.
In the second step we minimize
for nuclear matter data with respect to the 6 constants characterizing the density dependence of the , and mesons and the 3 couplings , , . (Here, the -meson and are hold at the values found in the first step.) The obtained values for the constants that define the density dependence of the meson-nucleon vertices are given in Table 1.
In the third step we minimize for the nuclear matter data and the binding energies and charge radii given in Table 5:
Now we fit only a restricted set of 4 parameters, i.e., the 3 couplings , , , and . The resulting values are given in the first two columns of Table 1.
We have to emphasize that only four free parameters , , and have been used in the final fit to the experimental data in finite nuclei. The other 10 parameters are derived from ab-initio calculations. This is in contrast to the typical relativistic and non-relativistic fits of mean-field interactions, where commonly around free parameters are adjusted to data in finite nuclei. It is also worth to remember that adding the -meson has improved our theoretical picture of the nucleus and of the EoS of asymmetric nuclear matter.
So far we have used in the fit only nuclei with spherical shapes. The pairing correlations are treated in the first step in the constant gap approximation with gap parameters derived from the odd-even mass differences. For a full description of nuclei all over the periodic table which includes also regions where the experimental binding energies are not known, we introduce a more general description of pairing by means of a monopole force with a constant matrix element fitted to reproduce the experimental binding energies of the nuclei in Table 5. We obtain for the set DD-ME the values MeV for neutrons and MeV for protons. In order to have a fair comparison for the results in finite nuclei we treated in the following the pairing properties of the set DD-ME2 also by a monopole force. In a similar fit we found for DD-ME2 the strength parameters MeV and MeV. In all these calculations the soft pairing window described in Ref. (78) has been used.
iv.1 Nuclear and Neutron Matter Equations of State
The nuclear matter properties at saturation computed with the DD-ME functional are given in Table 4. These properties do not fully coincide with the ones of the fully microscopic calculation in (19). The reason for that is that in the microscopic calculation, the EoS is very flat around saturation density and some deviation between the microscopic results and the DD-ME fit appear. These differences remain within the uncertainty of the state of the art of present numerical microscopic calculations. They are too small to be seen on the scale of Fig. 1. They are, however, important for a fine tuning of the results.
In order to investigate the quality of the predictions of the density functional DD-ME in the high density domain, we show in Fig. 3 the pressure (27) computed with this functional as a function of the density. It is compared with the pressure derived from the microscopic calculation of Ref. (19) as well as with the results derived from the non-linear meson coupling models NL3 (4) and FSUGold (9) and from DD-ME2 (10). We see that both, microscopic and DD-ME calculations, are within the shaded area which corresponds to the ”experimental region” estimated from simulations of heavy-ion collisions (83). The standard non-linear - model NL3 is outside of this region while the FSUGold model – which has an additional non-linear - coupling that softens the symmetry energy (see below) – is inside the shaded area in rather good agreement with DD-ME and the microscopic results. The results of the parameter set DD-ME2 are slightly outside of the shaded area.
An important quantity in nuclear physics and astrophysics, directly related with the EoS of asymmetric nuclear matter, is the symmetry energy (30). The value of the symmetry energy derived from successful mean field models lies roughly in a window of MeV at saturation. However, the density dependence of the symmetry energy is much more uncertain. This fact entails important consequences for a number of isospin-dependent observables. As a paradigmatic example, one may recall that different accurate mean-field models which reproduce well the binding energy and charge radius of the nucleus Pb predict largely different values for the neutron skin thickness of this isotope, ranging from fm. This fact points out that the isovector properties of the different models are, actually, not well constrained by the binding energies and charge radii of stable finite nuclei used to fit the effective interactions.
In nuclear mean field models, a strong linear correlation exists (84); (85) between the size of the neutron skin thickness of a heavy neutron-rich nucleus such as Pb and the parameter defined in Eq. (39), i.e., the slope of the symmetry energy at saturation. Recent constraints on the L parameter have been obtained using a variety of observables such as, for instance, isospin diffusion (86); (87); (88) and isoscaling (89); (90); (91); (92); (93) in heavy ion reactions, some collective excitations in nuclei (71); (94); (95); (96) and the neutron skin thickness in finite nuclei (97); (98) measured in antiprotonic atoms (99); (100). The analysis of all these results suggests that the parameter is roughly within the window 45 - 75 MeV (97). The new experimental efforts to measure the neutron radius of Pb may turn out to be helpful to deduce in the future narrower constraints on the slope L of the symmetry energy through the correlation of with the neutron skin thickness (101); (85).
The value predicted by our novel DD-ME functional is MeV, close to the result of the microscopic calculation of MeV in Ref. (103). The density dependence of the symmetry energy exhibited by DD-ME, is displayed in Fig. 4. We see that DD-ME predicts a rather soft density dependence of the symmetry energy which lies inside the shaded region derived from the empirical law MeV imposing the range discussed above: MeV MeV. It turns out that the density dependence of DD-ME and DD-ME2 is practically the same. This fact is not trivial, first because DD-ME2 has not been adjusted to nuclear matter data, but only to the experimental skin thickness of several finite nuclei (10) and second, the full isospin dependence is determined by the -meson, whereas in DD-ME it is distributed over the and the -meson.
The reason for this good agreement can be understood from the upper panel of Fig. 5 where the different contributions to the symmetry energy are displayed, the kinetic part as well as those provided by the and the -meson. We can see that the contributions of these mesons have opposite sign and thus a noticeable cancellation appears between them over the entire range of densities under consideration. Thus, it is conceivable (see Eq. (40)) that if the -meson is not considered in the functional (as it is the case of DD-ME2) its contribution to the symmetry energy can be accounted for by the -meson (with a reduced strength of the coupling constant, see Ref. (10) and Table 2).
The lower panel shows similar decompositions of the symmetry energy for other density functionals, such as NL3 (4) and DD-ME2 (10). The parameter set NL3 (black) has no density dependence in the isovector channel. Therefore the contribution of the -meson is very stiff and proportional to the density. The parameter set DD-ME2 includes only one isovector meson, the -meson and its contribution to the symmetry energy is very close the the sum of both the - and the -meson for the set DD-ME which compensate each other to a large extend. Small differences in these curves at densities above saturation density can be traced back to the different ansatz for the density dependence of the -meson in these two parameter sets, the Typel-Wolter ansatz (47) for DD-ME and an exponential density dependence for DD-ME2 (see Eq. (7) in Ref. (10)).
iv.2 Ground-state properties of finite nuclei
As described in section III.1 the experimental masses of and the charge rms radii of even-even spherical nuclei (see Table 5) have been taken into account in the fitting procedure of the DD-ME functional.
We display in Figs. 6 and 7 the difference between theoretical results computed with the functionals DD-ME, DD-ME2 and BCP and experimental data. For DD-ME we obtain a rms deviation of MeV for the binding energies and of fm for the charge radii. These results are close to the rms deviations MeV and fm obtained with the DD-ME2 functional for the same set of data when pairing correlations are introduced by the monopole force discussed at the end of Sect. III.2. It has to be emphasized, however, that using the density functional DD-ME2 in connection with the pairing part of the finite range Gogny force D1S instead of the monopole force and taking into account spherical as well as deformed nuclei one has found rms deviations of 900 keV and 0.017 fm for the binding energies and charge radii of typical sets of 200 (10) or 300 (105) even-even nuclei.
The charge radii (defined in Eq. (49)) of Pb isotopes and their isotope shifts have been a matter of detailed discussion within the framework of mean field theories (106); (107); (108); (109). In Fig. 8 we show the isotope shifts in a chain of Pb isotopes as a function of the neutron number . The nucleus Pb has been taken as the reference point: . With a gradual addition of neutrons, the empirical charge radii of isotopes heavier than Pb do not show the trend of the lighter isotopes and at the doubly magic nucleus Pb one observes a pronounced kink (110). Conventional non-relativistic Skyrme- and Gogny forces fail to reproduce this kink (106), whereas all the relativistic models are successful in describing this kink properly (107). In Refs. (108); (109) this difference between the non-relativistic Skyrme functional and the relativistic models has been traced back to the isospin dependence of the spin-orbit force. In conventional relativistic models it is determined by the -meson vertex and it is relatively weak. In Fig. 8 we see that the parameter set DD-ME reproduces the kink in the isotope shifts rather well as all the other relativistic models do. The non-relativistic set BCP of Ref. (12) which has the same spin-orbit force as conventional Skyrme and Gogny functionals fails in this context.
Finally we show in Fig. 9 values for the neutron skin thickness of a large set of nuclei as a function of the relative neutron excess and compare the results obtained with the parameter set DD-ME with those of the set DD-ME2 and with experimental values (99). Both theoretical calculations are in rather good agreement and within the range of the experimental error bars.
iv.3 Impact of the -meson on the spin-orbit splitting
In this work, we have included the -meson in our theoretical treatment of the nucleus motivated by microscopic calculations (42); (43); (65) and by the importance of a scalar-isovector meson of the nucleon-nucleon potentials for describing the nucleon-nucleon scattering data in the vacuum (72). In our investigation of the properties of nuclear matter we have seen in Fig. 5 that the influence of the -meson on the symmetry energy can be largely compensated by renormalizing the -meson coupling constant in the DD-ME2 model. The same seems to be true also for the masses (Fig. 6), radii (Fig. 7) and skin thicknesses (Fig. 9) in finite nuclei. Obviously this also applies for all the other successful covariant density functionals without the -meson degree of freedom.
In order to get a better understanding of these results we follow Ref. (111) and eliminate the small components of the spinor in the Dirac equation (5). For the large components we are left with a Schroedinger like equation
It contains the potentials
The potential MeV corresponds to the conventional potential in the corresponding non-relativistic Schrödinger equation. In theories containing the and the -mesons it can be decomposed into an isoscalar and an isovector part
where , , , and are the corresponding meson fields. In theories without -meson the -meson vertex has to be renormalized and we find for the isovector part a pure -field