Nuclear Physics A 701 (2002) 503c–508c

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Relativistic description of medium-heavy nucleifar from stability

P. Ring∗, G.A. Lalazissis, D. Vretenar1

a Physik Department der Technischen Universität München, D-85748 Garching, Germany

Abstract

Exotic nuclei far form the line of beta-stability have gained considerable interest in recent yearsboth on the experimental as on the theoretical side. New phenomena are predicted such as anincreasing surface diffuseness, a quenching of the spin–orbit term, and changes in the nuclear shellstructure. Approaching the drip-lines one expects on the proton-rich side proton radioactivity, andfor N = Z nuclei proton–neutron pairing and on the neutron-rich side the formation of neutronskins and neutron halos. A reliable description of the spin–orbit term and its isospin dependence istherefore necessary. We review the investigations in the framework of Relativistic Mean Field (RMF)and Relativistic Hartree–Bogoliubov (RHB) theory in the medium-heavy nuclei far from the valleyof stability. In particular, we show as an application the quenching of the spin–orbit splitting withincreasing distance from the valley of beta-stability. 2002 Elsevier Science B.V. All rights reserved.

Keywords: Exotic nuclei; RHB theory; Spin–orbit splitting

1. Introduction

New accelerators with radioactive beams allow the experimental study of nuclei far fromstability. Theoretical investigations predict considerable changes in the single particle levelsplittings, which are the basis of many effects in nuclear structure.

In recent years Relativistic Mean-Field (RMF) models have been successfully applied incalculations of nuclear matter and properties of finite nuclei throughout the periodic table,for a review, see Refs. [1,2]. Taking into account basic effects of relativistic dynamics in the

* Corresponding author. Invited speaker at the V Int. Conf. on Radioactive Beams, April 3–8, 2000, Divonne,France.

E-mail address: ring@physik.tu-muenchen.de (P. Ring).1 On leave from the University of Zagreb, Croatia.

0375-9474/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0375-9474(01)01635-9

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nucleus seriously, they allow a fully self-consistent description of many nuclear propertieswith high accuracy. As compared to other nonrelativistic mean-field approximations suchas density dependent Hartree–Fock calculations, these models have the advantage toprovide a consistent description of the spin–orbit term and its isospin dependence.

For open-shell nuclei, pairing correlations play an important role. Usually they aretaken into account in the frozen gap approximation, where only BCS-type occupationfactors are included in the calculation of the corresponding densities. This method brakesdown, close to the drip-lines. One has to use in this cases relativistic Hartree–Bogoliubovtheory based on a self-consistently determined pairing field. In principle such a theoryhas been developed in Ref. [3], however at present there is no fully microscopic set ofmeson parameters which gives a reasonable description of pairing properties in nuclei.Since pairing is a genuine nonrelativistic phenomenon, we therefore have used withgreat success the phenomenological Gogny force in the pairing channel. This is a finiterange force which does not require a pairing cut-off and its parameters are given in theliterature. Nonrelativistic calculations have shown, that it provides an excellent descriptionof pairing properties in finiteβ-stable nuclei. Form earlier investigations one knows thatthis is also the case where Gogny pairing is applied in connection with relativistic mean-field theory [4]. One therefore has good reasons to hope that this description of pairingcorrelations yields also reliable results in the region of exotic nuclei far from the valley ofbeta-stability.

2. The relativistic Hartree–Bogoliubov model

In comparison with conventional nonrelativistic approaches, relativistic models ex-plicitly include mesonic degrees of freedom and describe the nucleons as Dirac parti-cles. Nucleons interact in a relativistic covariant manner through the exchange of virtualmesons: the isoscalar–scalarσ -meson, the isoscalar–vectorω-meson and the isovector–vector ρ-meson. The model is based on the one-boson-exchange description of thenucleon–nucleon interaction [5]. It has turned out that linear forms of such theories arenot able to describe in a proper way the surface properties of nuclei. One needs a den-sity dependence of the coupling constants, which is in most of the applications taken intoaccount by nonlinear coupling terms between the mesons [6].

Besides the Dirac Hamiltonianh produced by the mesons, one has a self-consistentlydetermined pairing field∆ in the relativistic Hartree–Bogoliubov equations,(

h − λ ∆

−∆∗ −h + λ

)(Uk

Vk

)= Ek

(Uk

Vk

), (1)

which have to be solved by iteration. Details are given in Refs. [4,7]. Its eigenvectors(U,V ) determine the densitiesρab and κab and therefore the fieldsS,V,∆ in a self-consistent way.

In the following, we discuss applications of this theory in the region far from the valleyof beta-stability.

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3. Application to the surface diffuseness and the reduction of the spin–orbit

The spin–orbit interaction plays a central role in the physics of nuclear structure. It isrooted in the basis of the nuclear shell model, where its inclusion is essential in order to re-produce the experimentally established magic numbers. In nonrelativistic models based onthe mean-field approximation, the spin–orbit potential is included in a phenomenologicalway. Of course, such an ansatz introduces an additional parameter, the strength of the spin–orbit interaction. The value of this parameter is usually adjusted to the experimental spin–orbit splittings in spherical nuclei, for example16O. On the other hand, in the relativisticframework the nucleons are described as Dirac spinors. This means that in the relativisticdescription of the nuclear many-body problem, the spin–orbit interaction arises naturallyfrom the Dirac–Lorentz structure of the effective Lagrangian. No additional strength para-meter is necessary, and relativistic models reproduce the empirical spin–orbit splittings.

Nonrelativistic models and the relativistic mean-field theory predict very similar resultsfor many properties of beta-stable nuclei. However, cases have been found where thenonrelativistic description of nuclear structure fails. An example is the anomalous kink inthe isotope shifts of Pb nuclei [8]. This phenomenon could not be explained neither by theSkyrme model, nor by the Gogny approach. Nevertheless, it is reproduced very naturally inrelativistic mean-field calculations. A more careful analysis [9] has shown that the originof this discrepancy is the isospin dependence of the spin–orbit term.

In the following, we present results for the chain of Sn and Ni isotopes. We find that inthe framework of relativistic mean-field theory, the magnitude of the spin–orbit potentialis considerably reduced in light drip-line nuclei [10]. With the increase of the neutronnumber, the effective one-body spin–orbit interaction becomes weaker. This result in areduction of the energy splittings between spin–orbit partners. The reduction of the spin–orbit potential is especially pronounced in the surface region, and does not depend on aparticular parameter set used for the effective Lagrangian. These results are at variancewith those calculated with the nonrelativistic Skyrme model. It has been shown that thedifferences have their origin in the isospin dependence of the spin–orbit terms in the twomodels. If the spin–orbit term of the Skyrme model is modified in such a way that itdoes not depend so strongly on the isospin, the reduction of the spin–orbit potential iscomparable to that observed in relativistic mean-field calculations.

In Fig. 1 we display the one- and two-neutron separation energies

Sn(Z,N) = Bn(Z,N) − Bn(Z,N − 1) and

S2n(Z,N) = Bn(Z,N) − Bn(Z,N − 2)

for Ni (24 � N � 50) isotopes, respectively. The values that correspond to the self-consistent RHB ground states are compared with experimental data and extrapolated valuesfrom Ref. [11]. The theoretical values reproduce in detail the experimental separationenergies. The model describes not only the empirical values within one major neutronshell, but it also reproduces the transitions between major shells (for details, see Ref. [12]).

In Fig. 2 we show the self-consistent ground-state neutron densities for the Sn andNi nuclei. The density profiles display shell effects in the bulk and a gradual increaseof neutron radii. In the insert of Fig. 2 we include the corresponding values for the

506c P. Ring et al. / Nuclear Physics A 701 (2002) 503c–508c

Fig. 1. One-neutron and two-neutron separation energies in Ni isotopes, calculated with the parameter set NL3.

Fig. 2. Self-consistent RHB single-neutron density distributions for Sn (50� N � 82) and Ni (28� N � 50)nuclei, calculated with the NL3 effective interaction.

P. Ring et al. / Nuclear Physics A 701 (2002) 503c–508c 507c

Fig. 3. Energy splittings between spin–orbit partners for neutron levels in Ni and Sn isotopes, as functions ofneutron number.

surface thickness and diffuseness parameter. The surface thicknesst is defined to be thechange in radius required to reduceρ(r)/ρ0 from 0.9 to 0.1 (ρ0 is the maximal value ofthe neutron density). The diffuseness parameterα is determined by fitting the neutrondensity profiles to the Fermi distribution. By adding more units of isospin the valueof the neutron surface thickness increases and the surface becomes more diffuse. andtherefore the spin–orbit coupling, which is determined by the slope of the potentials isdecreasing [12]. The effect is reflected in the calculated spin–orbit splittings of the neutronlevels∆Els = En,l,j=l−1/2 − En,l,j=l+1/2.

In Fig. 3 we display the energy splittings of spin–orbit neutron partners for Ni and Sn,respectively. The calculated spacings are shown as function of the neutron number. Weonly include the spin–orbit doublets for which one of the partners is an intruder orbitalin a major shell. These doublets display the largest energy splittings. We notice in Fig. 3that the spacing between spin–orbit partners decreases with neutron number. The effect isstronger in Ni than in Sn.

Acknowledgements

This work has been supported in part by the Bundesministerium für Bildung undForschung under the project 06TM979, by the Deutsche Forschungsgemeinschaft and bythe Gesellschaft für Schwerionenforschung (GSI) Darmstadt.

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[10] G.A. Lalazissis, D. Vretenar, W. Pöschl, P. Ring, Phys. Lett. B 418 (1998) 7.[11] G. Audi, A.H. Wapstra, Nucl. Phys. A 595 (1995) 409.[12] G.A. Lalazissis, D. Vretenar, P. Ring, Phys. Rev. C 57 (1998) 2294.