ЯДЕРНАЯ ФИЗИКА, 2007, том 70, № 8, с. 1388-1393
ЯДРА. Теория
CRITICAL ASSESSMENT OF MEAN FIELD MODELS BASED ON EFFECTIVE INTERACTIONS
© 2007 G. Colo *
Dipartimento di Fisica dell' University degli Studi di Milano and INFN, Milano, Italy
Received October 31, 2006
Mean field schemes, from simple Hartree—Fock plus Random Phase Approximation calculations of the ground and excited states to more sophisticated approaches which include pairing as well, have been popular since quite a long time. In these models, the input is an effective interaction. We still lack a precise link between this interaction and a more fundamental theory; however, there have been various new recent attempts to correlate empirical pieces of evidence about nuclear (and neutron) matter, or experimental results, with the properties of the effective interactions. In this contribution we claim that, while we have indeed made some progress in our understanding of certain features of the interactions, we still miss a clue about its proper density dependence and about its isovector properties.
PACS:21.60.Jz, 21.30.Fe, 24.30.Cz
1. INTRODUCTION
Despite our progress in understanding the origin and the features of the bare nucleon—nucleon (NN) interaction, and the success of the so-called "ab-initio" calculations based on that interaction for light nuclei, we still need to resort to effective interactions in order to understand the medium-heavy nuclei, and in particular their excitation spectra. Among the effective interactions, the Skyrme forces have been introduced first [1], and have been employed and improved for about three decades. Among the latest developments, we mention that some interactions have been fitted so to reproduce the outcome of realistic calculations of pure neutron matter [2], whereas others have been forced to reproduce with good accuracy the systematics of nuclear masses [3]. Both Hartree—Fock (HF) calculations of the ground state, and Random Phase Approximation (RPA) calculations of the vibrational states have been available for long time. If the forces are supplemented by an effective pairing interaction, HF—BCS or HFB calculations can be performed for open-shell nuclei (cf., e.g., [4]). Self-consistent quasiparticle RPA (QRPA) calculations have had a kind of blooming in recent years.
At the same time, also calculations based on other kinds of effective forces have been substantially improved. QRPA calculations (on top of HFB) with the finite-range Gogny force have become recently available [5]. A large amount of theoretical work has been carried out in the domain of relativistic
E-mail: colo@mi.infn.it
(or covariant) mean field models (RMF) based on effective Lagrangians (we confer the reader to [6] for a review).
These facts have modified considerably our understanding of the mean field models. It is generally believed that all these models can be viewed as (approximate) realizations of a Density Functional Theory (DFT) for atomic nuclei [7]. The DFT is based on the Hohenberg—Kohn theorem [8] which states that, for every Fermi system subject to a given potential, it is possible to build a functional E[q] of the density q such that the ground state of the system is obtained by minimizing the value of the functional (with the constraint on the number of fermions). One can generalize the theorem by considering that the systems may possess other degrees of freedom: for instance, the electrons have 1/2 spin and the polarization density has to be taken into account in addition to the totaldensity. In the nuclear case, other generalizations are in order (see [7] for details). Below, we discuss the isospin densities.
The important fact to be stressed, is that in the nuclear case we miss a precise link between the many-body theory and the parameters of the energy functional. This link is possible, to a good extent, in the electronic case where, for instance, the correlations entering the functional can be calculated exactly in the high density limit. In the nuclear case, the parameters characterizing E[q] have to be fitted.
In this unfavourable situation, we can nonetheless gain an increasing understanding of the variety of nuclear systems by improving the energy functionals, assessing the accuracy of the calculations based
Fig. 1. Symmetry energy S(g) (cf. Eq. (2)) as a function of the density g for some of the Skyrme forces built in [9]. In particular, these are forces of the same kind of SLy4, that is, they have the exponent of the density-dependent part, a, equal to 1/6 and have K^ = 240 MeV. See the text for a discussion, in particular on why the curves overlap in the small rectangle which is displayed.
on them, and providing new links with the known phenomenology. This paper is written along these lines. In particular, we will focus on the problem of the so-called "symmetry" part of E[p].
The outline of the paper is the following. In Section 2 we specify better what parts of the energy functional E[g] are highly debated nowadays. In the following, we present some calculations of the vibrational states which may shed light on the open problems. In Section 3, the basic formalism for these calculations is sketched, whereas in Sections 4 and 5 we discuss, respectively, how the data on the isoscalar giant monopole resonance (ISGMR) constrain the curvature of E [g] in symmetric matter and how the data on the isovector giant dipole resonance (IVGDR) are related to the so-called "symmetry energy". Finally, we draw our conclusions in Section 6.
2. THE SYMMETRY ENERGY
In the nucleus both neutrons and protons are present, with the corresponding densities gn and gp. Usually, the isoscalar (or total) density g and the isovector density g- are defined as
Q = Qn + Qp, Q- = Qn - Qp.
(1)
A Q Q-] = A [q] + ^(Q)
Q— Qo
(2)
The above formula is based on the fact that for a given density the energy is minimum if there is no proton—neutron asymmetry (namely, g- =0), and
around that minimum a quadratic approximation is appropriate. The quantity S(g) is called symmetry energy and characterizes the isovector part of the energy functional.
In symmetric nuclear matter, only the isoscalar part of the functional appears. Empirically, we know that this quantity is minimum at the saturation density go = 0.16 fm-3 and assumes the value A [go] = = —16 MeV. These two inputs are indeed used when all the effective interactions, or covariant La-grangians, are fitted. The curvature of the energy around this minimum is associated to the so-called "nuclear matter incompressibility", that is,
*» = 9Q§ d E/A )
dQ2
(3)
Q0
The energy per particle can be written as the sum of a term which depends only on the total density, plus another term which includes g-, that is,
We claim that our understanding of this quantity has significantly improved in recent years.
On the other hand, if neutron—proton asymmetry enters the discussion, our knowledge of the equation of state is much limited. In particular, the symmetry energy and its density dependence are poorly constrained. This is the main motivation for further studying the nuclear systems with neutron excess: the goal is to reach a unified picture of stable nuclei, unstable nuclei and neutron stars.
In order to make visible what we mean by uncertainty on the density dependence of the symmetry energy, we display S(g) in the case of different Skyrme functionals in Fig. 1. The curves correspond to a class of functionals which have been fitted in [9]. All of them fit quite well, not only the saturation point of symmetric nuclear matter but the binding energies and charge radii of a number of finite nuclei as well.
2
1390
COLO
These nuclei are 40>48Ca, 56Ni, 132Sn, 208Pb (i.e., systems with neutron excess are considered). From Fig. 1 we notice the following features:
1. At saturation density, the value of the symmetry energy (that we denote by J while other authors may employ different notations like aT or a4) is not constrained and lies between 26 and 40 MeV.
2. At a lower value of the density (about 0.10 fm"3) all functionals converge. This happens not by chance, but as a consequence of the fact that for nuclei with neutron—proton asymmetry mainly surface properties are fitted, so that a density which is intermediate between zero and the bulk is involved.
3. As a consequence, there is a strong correlation between the value of the symmetry energy at saturation (J) and its slope.
Similar considerations can be found in previous papers [10].
In the Skyrme forces, the density dependence arises from the well known term which carries the parameters (t3, x3) and has a functional form of the type ~ga. There are no strong arguments to fix the value of a, except for the fact that it should probably lie below 2/3 (cf. [11]; moreover, it is known that Skyrme functionals associated with larger values of a have too large compressibilities).
The Gogny interaction borrows the density-dependent part from Skyrme. As far as the RMF Lagrangians are concerned, again the density dependence comes purely from a (reasonable) ansatz. In this sense, one cannot say that the density-dependent part of any of these functionals is on firm grounds. We should also mention, for the sake of completeness, that a different ansatz (not based on the underlying Hamiltonian picture) characterizes the energy functional built by Fayans and collaborators [12].
In the present contribution, we discuss a few connections between the problem of the density dependence in the energy functionals and the experimental data on the giant resonances. In particular, concerning the ISGMR, we will show in Section 4 that the recent accurate data are very instrumental in order to extract the proper value of K^ but the remaining uncertainty is in fact related to our poor knowledge of the density dependence of S(g) (because the measurements are performed in nuclei with neutron excess). In Section 5 we show correlations between some quantities ass
Для дальнейшего прочтения статьи необходимо приобрести полный текст. Статьи высылаются в формате PDF на указанную при оплате почту. Время доставки составляет менее 10 минут. Стоимость одной статьи — 150 рублей.