Mean–Field Models
and Superheavy Elements
Abstract
We discuss the performance of two widely used nuclear meanfield models, the relativistic mean–field theory (RMF) and the nonrelativistic Skyrme–Hartree–Fock approach (SHF), with particular emphasis on the description of superheavy elements (SHE). We provide a short introduction to the SHF and RMF, the relations between these two approaches and the relations to other nuclear structure models, briefly review the basic properties with respect to normal nuclear observables, and finally present and discuss recent results on the binding properties of SHE computed with a broad selection of SHF and RMF parametrisations.
Institut für Theoretische Physik II,
Universität Erlangen–Nürnberg,
Staudtstraße 7,
D–91058 Erlangen, Germany
Joint Institute for HeavyIon Research,
Oak Ridge National Laboratory,
Oak Ridge, TN 37831–6374, U.S.A.
Gesellschaft für Schwerionenforschung,
Planckstraße 1,
D–64921 Darmstadt, Germany
Institut für Theoretische Physik,
Universität Frankfurt am Main,
Robert–Mayer–Straße 8–10,
D–60325 Frankfurt, Germany
1 Introduction
Nuclear structure models are available at various levels of description. There is the macroscopic view in terms of the liquiddrop model (LDM) LDM . The macroscopicmicroscopic (macmic) method combines the rich phenomenological experience summarised in the LDM with a finetuning through shell effects estimated in a properly chosen singleparticle potential micmac ; micmac2 . And there is the broad family of selfconsistent meanfield approaches (Hartree or Hartree–Fock) employing effective energy functionals on which we will concentrate here. All these models presently enjoy a revival due to a world of new experimental information emerging from the production and measurement of exotic nuclei and new elements. In fact, it is more than three decades ago that speculations on the possible existence of shellstabilized superheavy elements (SHE) Mosel ; SuperNils have motivated the construction of dedicated heavyion accelerators. The production of SHE turned out to be the most tedious task in the field of exotic nuclei. It took about two decades to reach the first island of shellstabilised deformed SHE in the region of dubna ; GSI ; GSI2 . Recent experiments give first evidence for nuclei even closer to the expected island of spherical SHE. The synthesis of the neutronrich isotopes , Z114 , and Z116 were reported from Dubna and at Berkeley three decay chains attributed to the even heavier were observed Z118 . While earlier superheavy nuclei could be unambiguously identified by their decay chains leading to already known nuclei, the decay chains of the newfound superheavy nuclei cannot be linked to any known nuclides. The new discoveries still have to be viewed carefully, see the critical discussion in Arm00a . While for the heaviest systems only their mere existence and a few decay properties are established, the first spectroscopic data become available for nuclei at the lower end of the superheavy region, e.g. lowlying states in Rf isotopes from the analysis of decay finestructure Hessberger and rotational bands of nuclei around No which were found to be stable against fission at least up to Reiter ; Leino . Interpretation of data and planning of future experiments call for a significant refinement in the modeling of SHE. As their mere existence emerges from a delicate balance between the Coulomb instability of the liquid drop against fission and stabilisation through shell effects, SHE provide a demanding testing ground for nuclear structure models, probing all their details. The aim of this review is to examine the performance of current nuclear meanfield models under the particular perspective of SHE. In this context we concentrate on the two most widely used brands, the relativistic meanfield model (RMF) and the nonrelativistic SkyrmeHartreeFock approach (SHF). We ought to mention that there are also other models like the nonrelativistic Gogny force gogny ; gogny2 employing finiterange terms in the interaction, the energy functionals of Fayans et al. fayans which are similar to SHF but use variants of density dependence and pairing interaction, or the pointcoupling variant of the RMF madland which can be viewed as the relativistic analogue of the Skyrme interaction. As none of these was widely used for the calculation of SHE so far we omit them from our discussion.
2 Framework
2.1 Meanfield models in the hierarchy of approaches
Selfconsistent meanfield models are intermediate between the fully microscopic manybody theories as, e.g., Brückner–Hartree–Fock (BHF) BHF and semiclassical models as the macmic approach micmac ; micmac2 . The microscopic approaches have made considerable progress over the past decades BHF ; HenPan , yet the actual precision in describing nuclear properties is still limited. Moreover, application to finite nuclei is extremely expensive. Thus fully microscopic methods are presently not used for largescale nuclear structure calculations. They provide, however, useful guidelines for the construction of effective meanfield theories Tmatexp .
On the other side are the macroscopic approaches which are inspired by the idea that the nucleus is a drop of nuclear liquid, giving rise to the liquiddrop model (LDM) and the more refined droplet model droplet . With a mix of intuition and systematic expansion one can write down the corresponding energy functional even including finiterange effects of the nuclear interaction FRDM ; FY . There remain a good handful of free parameters, as e.g. the coefficients for volume energy , symmetry energy , incompressibility , or surface energy . These have to be adjusted to a multitude of nuclear bulk properties such that modern droplet parametrisations deliver an excellent description of average trends LDM . Actual nuclei, however, deviate from the average due to quantum shell effects, so that shell corrections are added, which are related to the level density near the Fermi surface and can be computed from a well tuned nuclear singleparticle potential. Macroscopic energy plus shell corrections constitute the macmic approach which is enormously successful in reproducing the systematics of known nuclear binding energies micmac ; micmac2 ; FRDM ; FY . One has to admit, though, that the macmic method relies strongly on phenomenological input. This induces uncertainties when extrapolating to exotic nuclei. Particularly uncertain is the extrapolation of the singleparticle potential because this is not determined selfconsistently but added as an independent piece of information.
Selfconsistent meanfield models do one big step towards a microscopic description of nuclei. They produce the appropriate singleparticle potential corresponding to the actual density distribution for a given nucleus. Still, they cannot be handled as an ab initio treatment because the genuine nuclear interaction induces huge shortrange correlations. Selfconsistent meanfield models deal with effective energy functionals. The concept has much in common with the successful densityfunctional theory for electronic systems GD90 ; Petkov . Taking up the notion from there, we can speak of nuclear KohnSham models as synonym for meanfield models. The difference is, however, that electronic correlations are well under control and that reliable electronic energydensity functionals can be derived ab initio. Nuclear manybody theories, as discussed above, have not yet reached sufficient descriptive power to serve as immediate input for effective meanfield models, but serve as motivation and source for the basic features of the meanfield approach. This sets the framework, the actual energy functionals are then constructed by systematic expansion considering symmetries Dob96a , their parameters are adjusted phenomenologically. (For a recent review on nuclear correlations and their relation to effective meanfield theories see Rei94a .)
The connection between selfconsistent meanfield models and macmic approaches is much better developed. There are several attempts from either side. The ETFSI approach starts from SHF and derives an effective macmic model by virtue of a semiclassical expansion ETFSI . From the macroscopic side there is an attempt to induce more selfconsistency by virtue of a ThomasFermi approach TFmyers . The investigation of these links is useful to gain more insight into the crucial constituents of either models.
2.2 Skyrme–Hartree–Fock model
The concept of an effective interaction for meanfield calculations can be justified within manybody theories. For example, the matrix in BHF is the actual effective force for the underlying mean field. It was a formal matrix expansion Tmatexp which gave theoretical support to the first working selfconsistent model Skfirst using an effective interaction introduced much earlier by T. H. R. Skyrme skyrme ; skyrmeLS . The original idea was that a convenienttouse effective interaction can be obtained from a momentumspace expansion of any finiterange interaction which leads to a zerorange force plus momentumdependent terms. A density dependence has to be added to incorporate manybody correlations in an effective way (note that the matrix depends strongly on density) and, last not least, a (zerorange) spinorbit force skyrmeLS is added to account for the strong spinorbit splitting in nuclei.
This concept has an intimate relation to energydensity functionals. The energy expectation value of such an effective interaction is precisely a functional of the local density. Thus the well developed densityfunctional theory GD90 adds support for SHF. We present here the SHF functional complemented by an obvious graphical illustration:
The first two columns parametrise a density functional marked . Note the distinction between total (isoscalar) density and isovector density , and similarly for the kinetic density and the spinorbit current . The leading term is the twobody interaction . It is attractive in the isoscalar channel and repulsive for the isovector part, just as the genuine twobody force is. The necessary densitydependent interaction is parametrised in the next term, for traditional reason in the form of an extra . These two terms together set up a localdensity functional which is represented graphically as where a heavy dot with an appended circle stands for the density. That density functional allows already to fix all relevant nuclear bulk properties. Finite systems require a finetuning of surface properties which is achieved by the gradient correction term. They are represented graphically by the rightleft arrow atop the symbol. Moreover, the strong dressing of nucleons in matter calls for an effective mass effmass which can be achieved by the kinetic correction term where . This term requires derivatives within the density summation which is indicated by an updown arrow in one of the two densities. Last but not least, a strong spinorbit splitting is crucial for a correct description of singleparticle spectra and shell closures lsfirst . This is guaranteed by the spinorbit term where the spinorbit current is indicated graphically by a circle with arrow around the densitydot. Note that each term comes twice, once in an isoscalar form and another time in an analogous isovector form.
The nice feature of the Skyrme functional is that each term can be related to a corresponding bulk property (or LDM feature). This is indicated in the last two lines of the above sketch. The two isoscalar densitydependent parts together are related to bulk binding , equilibrium density , and incompressibility . The isovector part complements this by the symmetryenergy coefficient and its derivative . The gradient corrections relate naturally to the isoscalar and isovector surfaceenergy coefficients. The kinetic terms adjust the isoscalar effective mass and the isovector effective mass . The latter modifies the ThomasReicheKuhn sum rule RingSchuck by an enhancement factor and one often parametrises in terms of this enhancement factor, see e.g. SLyx . No bulk property can be associated directly with the spinorbit term. This term is related to the shell structure, i.e. to the singleparticle spectrum, of finite nuclei. What looks here like a quickly drawn and superficial analogy to the LDM, has indeed deep theoretical foundations. One can, in fact, derive the micmac method from SHF by virtue of semiclassical expansions SHF2micmac .
There is a subtle difference between the force concept and the energy functional concept which concerns the spinorbit force. Derivation of the energy functional from a Skyrme force yields (usually small) extra terms and which emerge from the exchange part of the kinetic terms or . Some Skyrme parametrisations include these terms, some do not. We will specify that later when presenting the parametrisations. The actual HartreeFock (or KohnSham) equations are derived variationally from the given energy functional, see e.g. SkIx ; RPAnucl .
2.3 Relativistic meanfield model
The history of the RMF has similarities to SHF. After an early first conjecture duerr , it was only in the seventies that this model was lifted to a competitive meanfield model RMF1 ; RMF2 . The starting point is, at first glance, different from SHF. RMF is conceived as a relativistic theory of interacting nucleonic and mesonic fields. The mesonic fields are approximated to mean fields (realnumber fields rather than field operators), a feature which is reflected in the name RMF. Moreover, the antiparticle contributions in the Dirac fields for the nucleons are suppressed (“no–sea” approximation). Again, the meanfield approximation is not valid in connection with the true physical meson fields. The meson fields of the RMF are effective fields at the same level as the forces in SHF are effective forces. The RMF is the relativistic cousin of SHF, and the same strategy applies: relativistic BHF is still not precise enough to allow an ab initio derivation of the RMF; the model is postulated from a mix of intuition and theoretical guidelines with the parameters to be fixed phenomenologically.
The RMF is usually formulated in terms of an effective Lagrangian, as any relativistic theory. For stationary problems it can be mapped to an energy functional Schmid . We discuss it here in terms of an effective energy functional and we confine ourselves to a graphical presentation because the RMF is well documented in several reviews Rei89 ; Ser92a ; Rin96a . The RMF functional can be sketched as follows:
The ansatz looks at first glance conceptually even simpler than that of SHF. One merely writes down the basic nucleonmeson couplings RMF1 . The mesons can be characterised by their internal quantum numbers. Scalar and vector fields are taken into account (the most famous pion field does not contribute in Hartree approximation because there is no finite pseudoscalar density in the ground state). For the isovector part, one employs only the vector field associated with the meson. The isovectorscalar field, the meson, should appear at the same level (and plays a role, indeed, in the nucleonnucleon force). It can be omitted for purely phenomenological reasons because it does not improve the performance of the model when included. But a simple series of mesonnucleon couplings does not suffice to deliver a highprecision model. We know from microscopic theory that the effective interaction needs density dependence to effectively incorporate manybody correlations. Such had been introduced into the RMF via nonlinear terms (cubic and quartic) in the scalar meson field RMF2 , as indicated by the second line in the above sketch. This leads to a model with the same descriptive power as SHF Rei89 ; Ser92a ; Rin96a . This choice to introduce densitydependence was originally motivated by the aim to maintain renormalisibility of the theory. This is not a very stringent condition in connection with an effective meanfield theory which incorporates manybody effects, but the ansatz delivers an empirically wellworking scheme and thus there was little pressure for modifications, for exceptions and variants of the modeling see PL40 ; TM1 .
At first glance, the RMF functional looks quite different from SHF, and indeed, the pieces have been put together in a different fashion. The concept of the RMF seems to emerge naturally from a field theoretical perspective whereas the manybody aspects are not so transparent. These had been more obvious in SHF, particularly the relation to nuclear bulk properties, but it is possible to draw straight connections from RMF to SHF. These can be established by considering the nonrelativistic and zero–range limit of RMF Rei89 ; thies . It can be sketched as follows:
The left upper two diagrams summarise the RMF from the previous sketch. The right upper box indicates the two independent steps of expansion: a expansion of the scalar density and a gradient expansion of the meson propagator. The scalar density delivers as leading term the normal density (zero component of vector density) and as corrections the kineticenergy density as well as the divergence of the spinorbit current . The finite range of the mesons is expanded as leading zerorange coupling and gradient correction. Inserting these approximations yields the functional as represented by the diagrams in the second line of the sketch. It looks almost identical to the Skyrme functional. The interpretation in terms of bulk properties then proceeds as in case of SHF. It is repeated here for the sake of completeness. There is one aspect, however, which is very hard to map: it is the form of the density dependence. The mechanism in the RMF goes through nonlinear meson coupling and is much different from the SHF with its straightforward expansion in powers of density . A thorough comparison of the density dependences is still a task for future research. As in case of the SHF, we skip a detailed derivation of the coupled field equations and refer the reader to Rei89 ; Ser92a ; Rin96a .
2.4 Further ingredients
The above two subsections have outlined the main body of SHF and RMF. There are several further details which are handled similarly in both approaches. The direct part of the Coulomb interaction is given by the standard expression
A further crucial ingredient are pairing correlations. There are several recipes in the literature differing in the variational principle used, the correction for the particlenumber uncertainty of the BCS state and the effective pairing interaction. Nowadays, for the latter a zerorange twobody pairing force with separately adjustable strengths for protons and neutrons is most widely used. For most calculations reported here the matrix elements of this force are used in the BCS equations (as approximation to Hartree–Fock–Bogoliubov). For a recent discussion of this and competing recipes see gappaper and references therein.
The mean field localises the nucleus in space. This violates translation invariance. Centerofmass projection restores that symmetry RingSchuck . It turns out that a secondorder estimate for the centerofmass correction is fully sufficient. The correction is performed by subtracting from the calculated binding energy where is the nucleon mass and the mass number, see cmpaper for details. The term is usually subtracted a posteriori to circumvent twobody terms in the meanfield equation. For some parameterizations this is even simplified further. One approach is to use the harmonic oscillator estimate , another to use only the diagonal part of , i.e. . The latter recipe leads to a simple renormalisation of the nucleon mass and is usually included in the variational equations. Different groups are using different recipes for the centerofmass correction and thus one has to keep track which recipe is employed with a given parametrisation.
In fact, centerofmass correction is already one step, although the most trivial, beyond the meanfield approach. There are many other correlation effects possibly to be considered, particlenumber projection, angularmomentum projection in case of deformed nuclei, vibrational corrections. These aspects are being investigated intensively at present. Here we stay at the strict meanfield level (plus c.m. correction).
2.5 Parametrisations
As already mentioned above, nuclear manybody theory is not yet precise enough to allow an ab initio derivation of the effective energy functionals for SHF or RMF from nucleonnucleon interactions. Theory, with a spark of intuition, sets the frame and defines the form of the functional. The remaining free parameters have to be adjusted phenomenologically. Different groups have different biases in selecting the observables to which a force should be fitted. One usually restricts the fits to a few spherical nuclei with at least one magic nucleon number (an exception is a recent largescale fit to all known nuclear masses pearstond ). All fits take care of binding energy and r.m.s. charge radii . From then on, different tracks are pursued. SHF fits invoke extra information on spinorbit splittings. RMF generally does not need that because the spinorbit interaction is a relativistic effect that emerges naturally from relativistic models. Some groups add information on nuclear matter, some even on neutron matter. Some groups give a weight to isovector trends. Others make a point to include more information from the electromagnetic formfactor, in terms of a diffraction radius and surface thickness Fri86a . A detailed discussion of a fitting strategy can be found, e.g., in SkIx ; Rei89 ; Fri86a .
In view of these different prejudices entering the determination of a force, it is no surprise that there exists a world of different parametrisations for SHF as well as RMF. We confine the discussion to a few well adjusted and typical sets. For SHF we consider the parametrisations SkM* , SkP SkP , SkT6 Tx , Z Fri86a , SkI3, SkI4 SkIx , and SLy6 SLyx . The forces SkM, SkT6, SkP, and Z can be called the second generation forces which emerged in the mid eighties and which delivered for the first time a well equilibrated highprecision description of nuclear ground states. The force was the first to deliver acceptable incompressibility and fission properties. It also provides a fairly good description of surface thickness although this type of data was not fitted explicitly. The force SkT6 is a fit with constraint on . It did take into account the nuclear surface energy and thus also provides a satisfying surface thickness ( electromagnetic formfactor). The force SkP uses effective mass and is designed to allow a selfconsistent treatment of pairing. We will skip this pairing feature and use an appropriately adjusted delta pairing force. The force Z stems from a leastsquares fit including diffraction radius and surface thickness but without any reference to pseudodata from nuclear matter. The forces SLy6, SkI3, and SkI4 have been developed in the nineties. They take care of new data (e.g. from exotic nuclei) and new aspects. The force SLy6 stems from a recent attempt to cover properties of pure neutron matter together with normal nuclear ground state properties, sacrificing the quality of surface thickness somewhat to achieve this. All Skyrme forces up to here use the spinorbit coupling in the particular combination which is dictated by deriving the spinorbit energy from a twobody zerorange spinorbit force skyrmeLS . The forces SkI3/4 employ a spinorbit force with isovector freedom to simulate the relativistic spinorbit structure. SkI3 contains a fixed isovector part analogous to the RMF, whereas SkI4 is adjusted allow free variation of the isovector spinorbit force. The modified spinorbit force was introduced because no conventional SHF force was able to reproduce the isotope shifts of the m.s. radii in heavy Pb isotopes, see SkIx and references cited therein. The isovectormodified spinorbit force in SkI3 and SkI4 solves this problem. It then has, of course, a strong effect on the spectral distribution in heavy nuclei and thus for the predictions of SHE.
For the RMF we consider the parametrisations NL–Z NLZ , NL3 NL3 and TM1 TM1 . The force NL–Z comes from fits with the choice of observables quite similar to those of SkI3 and SkI4, with in particular the charge formfactor taken care of. NL3 is fitted without looking at the formfactor but more emphasis on the isovector trends. TM1 is an extended version of the RMF including a quartic nonlinear selfcoupling of the isoscalarvector field.
Each parametrisation is complemented by Coulomb, pairing and centerofmass correction as outlined in section 2.4. The pairing strengths need to be adjusted separately to comply with the level density of the force. Table 1 provides the actually used pairing strengths. The table indicates also the type of centerofmass correction used.
SkM  SkT6  SkP  Z  SLy6  SkI3  SkI4  NLZ  NL3  TM1  

c.m. 
3 Results and discussion
3.1 Basic properties
Before coming to a discussion of SHE we review briefly the basic properties of the various parametrisations, their performance with respect to normal nuclei and their nuclear matter properties (which are equivalent to the coefficients of the LDM expansion). They are summarised in Fig. 1. Note that only a subset of these data are used in actual fits. The left panels deal with finite nuclei. They show the r.m.s. errors of the basic groundstate properties (relative errors in ). Note that diffraction radius and surface thickness are key quantities determining the electromagnetic formfactor Fri86a . For the nuclei discussed here , and are linked by the Helm model in such a way that only two values are independent FriVoe . The lowest panel adds a more specific piece of information: the isotopic shift in heavy Pb isotopes. The energy (uppermost panel) is very well reproduced. All chosen forces have an error of only or below. More differences are seen concerning the reproduction of radii and surface thicknesses. The correlation is obvious: quantities which had been included in the adjustment (full dots) are usually well reproduced. Those which had not been fitted tend to show larger errors. Exceptions from the rule to some extent are SkT6 and SkM which yield acceptable and without having fitted them. Both forces, however, include a fitted surface energy which is related to reasonable surface thickness Fri86a . A less positive exception is the comparatively large error in for NL3, which includes this observable in the fit, but the other RMF forces have similar problems with . It seems that this is a principal problem of the RMF in its present form. It could be related to the somewhat curious form of shaping the density dependence in that approach. After all, one can conclude that the error in radii, or , is certainly below , often half of that. Surface thicknesses can be reproduced within if used in the fit; moreover, gives a handle on the surface tension (and subsequently on good fission barriers SkM* ; cmpaper ). The lowest right panel in Fig. 1 shows the isotopic shift in heavy Pb isotopes. It is obvious that all conventional Skyrme forces (i.e. those with ) fall short of the experimental value of . All RMF forces hit that value very well as a prediction. It was worked out that this is due to the particular form of the spinorbit force in the RMF SkIx ; Ringls . Extending the SHF to allow for yields an equally good reproduction of these isotopic shifts, see SkI3 and SkI4 in Fig. 1. But the values need to be included as fit data because the spinorbit force is added “by hand” in SHF whereas it is an intrinsic feature of the nucleonic Dirac equation in RMF.
The right panels of Fig. 1 show nuclear matter properties. There is general agreement about the volume energy, although the RMF forces seem to prefer slightly smaller values. The equilibrium density is almost the same for all SHF forces while RMF again prefers slightly smaller values. This systematic difference in extrapolation to nuclear matter is most probably related to the very different way in which the densitydependence is modeled in SHF and RMF. A thorough study of those effects is still lacking.
The effective mass shows a clear trend to values lower than one. It is, however, a rather vaguely fixed property. For example, SkT6 has fixed and is still able to provide good overall quality (see right panels). It is said that fits which concentrate on binding energies automatically prefer pearstond . On the other hand, fits which include the formfactor ( and ) prefer lower . And the RMF always prefers particularly low values. It is yet an open point what the best value for should be for nuclear meanfield models. Exotic nuclei, and particularly SHE, may help towards an answer.
Concerning the incompressibility , the SHF forces almost all gather nicely around the generally accepted value of incomp , SkP being an exception with a rather low value of . The RMF forces make quite different predictions. NLZ produces too low , which results from the fit, while NL3 comes up with a rather large value, which is to some extent a bias entering the adjustment. The actual number is probably at the upper edge of presently accepted values. A similarly large value is produced by TM1.
The largest variations are seen for the asymmetry energy . The LDM predicts values around . Indeed, most SHF forces reproduce that nicely, with the exception of SkI3 which comes out too high and Z which yields a somewhat low value, but the RMF forces generally yield a very large value for . One then wonders what the properties of the isovector dipole giant resonance might be. It turns out that its position depends not only on but also on the isovector effective mass, or sum rule enhancement factor , respectively. Most Skyrme forces have rather low , yielding correct resonance frequencies for . The RMF forces have much larger and here the value is appropriate. For a detailed discussion of these somewhat surprising interconnections see varenna . It remains that there is a substantial difference between SHF and RMF in that respect. The reasons are as yet unclear; it is probably again caused by the different form of densitydependence.
3.2 Binding Energies of Superheavy Nuclei
We now proceed to the discussion of SHE. The first feature to look at is, of course, the binding energy. Fig. 2 shows the relative error on binding energies for a selection of already known SHE. One sees at first glance, that the errors stretch out towards underbinding. The RMF forces remain very well within the desired error bands. The two SHF forces with extended spinorbit splitting also stay just within the bounds, and all conventional SHF forces fall below the margin. This is most probably not caused by the underlying bulk properties but related to shell effects. Note that Fig. 2 presents the same data in two different fashions to disentangle different trends in the error stemming from the isoscalar ( const.) and isovector ( const) channel of the interaction dubna . We look first at the left panel where the trends with are drawn. It is gratifying to see that all SHF forces basically follow a horizontal line which implies that the isoscalar bulk properties are described correctly. The RMF lines, however, have visible slopes, showing that the trends with are not perfectly reproduced. Such a feature had already been hinted at in Fig. 1 where the volume parameters and from the RMF differed from those of SHF and from the typical LDM values. This again most probably indicates a deficiency of the densitydependence in RMF.
The right panel of Fig. 2 displays the isovector trends. Clearly almost no force hits these trends correctly. SkP shows the most horizontal lines and thus seems to incorporate some correct isovector features. It is, on the other hand, a strange surprise that SLy6 deviates so much from the experimental isovector trends. This force was intended to perform particularly well in the isovector channel. The feature has yet to be fully understood. Keeping in mind that the actual trends are a mix of isovector bulk properties and shell effects it is most probable that the shell effects cause these deviations. SkI3 has as bad trends as SLy6 while SkI4 performs a bit better. This again is an accident because these trends had not been included in the fit. The RMF forces also fail with respect to isovector trends, The force NL3 performing a bit better than NLZ, possibly because isovector trends of binding energies had been included in the fit. It is even more surprising that NL3 does not perform better. There are still open problems with a proper parametrisation of the isovector channel in the RMF. Remembering that there is only one isovector field taken into account, one would like to also incorporate the scalarisovector field (the meson) to achieve a better isovector performance in the RMF, but this channel probably needs nonlinear couplings because a simple linear ansatz did not lead to improvements RutzDiss .
All these results on this apparently innocuous observable binding energy hint that new information from SHE sheds new light on meanfield models. A thorough study of the reasons for underbinding and unresolved trends has yet to come and will certainly help to deduce new constraints on the parametrisations.
3.3 Shell Effects
Shell effects are constitutive for the existence of SHE and they play a crucial role in determining the actual stability against fission. It is thus worthwhile to have a closer look at shell effects. A prominent feature is the occurrence of shell closures or magic numbers, respectively, in the singleparticle spectrum. One way to characterise them is to examine the twonucleon separation energies, e.g. . They display a sudden drop at shell closures because it is easier to remove nucleons from the next open shell (the former valence shell). The size of the step is a measure for the “magicity” of the shell closure. It is given by the twonucleon shell gaps, e.g. for the neutrons
(1)  
and similarly for the protons. This quantity is a way to access the gap between last occupied and first unoccupied singleparticle states, see e.g. Ben99a . Peaks in or indicate a shell closure. Fig. 3 shows proton and neutron shell gaps for a large range of SHE and for a subselection of forces. It is done for simplicity with spherical calculations. This suffices when searching for spherical shell closures. Deformation might change the picture in details and adds deformed shell closures, e.g. or , see Bue98a . The left panels show . The dark horizontal stripes thus indicate the closed proton shells. The right panels show , the dark vertical stripes there stay for closed neutron shells. The different forces show quite different patterns. This holds particularly for the proton shell closures. The RMF force NL–Z and the most RMFlike SHF force SkI3 predict a magic whereas SkI4 prefers and SkP shows no pronounced proton shell closures at all. For the neutrons, all SHF forces predict a shell while RMF prefers . That is not mutually exclusive. Several forces, SHF and RMF, have both closures.
The shell gaps are very useful when searching shell closures, but they are not directly related to the “shell effect” that stabilizes SHE against Coulomb fission. This quantity is provided by the shell correction energy
(2) 
High level density around the Fermi surface yields positive which corresponds to reduced binding. Smaller–than–average level density, in turn, corresponds to negative , i.e. extra binding from shell effects LDM ; micmac ; micmac2 . Fig. 4 shows an example of the individual shell correction of protons and neutrons in comparison with the twonucleon shell gaps . While in mac–mic models the shell correction is an constitutive part of the calculation of the binding energy, the values presented here are a posteriori analysed from the actual singleparticle spectra of fully selfconsistent calculations as a measure of the shell effect shelcor . Maximum (negative) values of the shell corrections coincide with the peaks in , but there is also a significant difference. While the twonucleon shell gaps show isolated peaks, the shell corrections appear as rather broad valleys of shell stabilised nuclei. The valley is broader than that around magic shells for normal nuclei. The stabilizing effect of the shell correction is given by the sum of the shell corrections for proton and neutrons. The mechanism is sketched in the right panel of Fig. 5. The dashed line indicates the smooth deformation energy curve corresponding to the LDM background. It is repulsive for SHE which means that they all would be fissionunstable in a LDM world. The full line has the shell corrections added. They oscillate with deformation and this generates minima which are stabilised against fission. The amplitude of the oscillations corresponds to the height of the fission barrier. Thus the depth of the shellcorrection valley is a rough measure for fission stability. The total shell correction energy is given in the left panel of Fig. 5. There emerges a broad region of shell stabilised SHE. The positive aspect of these findings is that one has good chances to hit longliving SHE in a variety of entrance channels. The negative aspect is that the quest for doublymagic SHE is misleading. Magicity is not very pronounced out there, a feature which was already seen in macmic models FY . The crucial features for SHE are large shell corrections, and these exist; even better, they appear for a broad range of and which makes the search for SHE in some sense comfortable.
3.4 Single–particle structure
Both previous sections have pointed out signatures of shell effects. In this section we sketch the actual singleparticle spectra of SHE. The left panel of Fig. 6 shows the proton levels near the Fermi surface for and varying . The overall trend is obvious: proton levels become more deeply bound with increasing , but note the change of the gap at along the isotopic chain which is coupled to the magic neutron number; a slight shift of the singleparticle energies destroys the around . This illustrates the same effect seen in the in Fig. 3. It is a typical selfconsistency effect not seen in earlier macmic calculations caused by the strong coupling of bulk properties and singleparticle structure. We will come back to that in Section 3.5. The right lower panel of Fig. 6 tries to visualize the competition between the and shell.
The lower right panel of 6 displays the proton spectra for . Most interactions predict the same levelordering in the superheavy region, the different bias on 114 and 120 among the forces found in Fig 3 is related to slight changes in the relative distances of the levels. The reason for this behaviour is clearly apparent: the shell corresponds to large spinorbit splitting of the proton levels, while the shell requires small splitting. Because the selfconsistency makes the level scheme depend strongly on and (cf. the left panel of Fig. 6) this graph is by itself not fully conclusive, but a more careful examination shows the conclusion to be valid Ben99a .
It is thus the spinorbit force which decides on the preferred shell closure. To estimate the reliability of the spinorbit splitting, we look at its performance in normal nuclei, see the right upper panel in Fig. 6. It shows the relative error in spinorbit splittings in selected proton levels of doublymagic nuclei (only “safe” splittings have been chosen according to the study of Rut98a ). There is a systematic difference between SHF and RMF. The RMF forces give a very satisfactory description of the data in all cases which emerges without any special fit to spectral data. It is a natural outcome of the Dirac equation combined with two strong fields (scalar versus vector) which counteract in the potential but cooperate in the spinorbit force duerr . The nonrelativistic interactions fall into two groups which can be distinguished by their performance for spinorbit splittings: those where the spinorbit interaction is adjusted to several nuclei throughout the chart of nuclei (FY, SkP, SkT6) and those fitted solely to O. The latter reproduce the proton splitting in O but overestimate the splittings in heavier nuclei. This is an unpleasant common feature of all nonrelativistic models. Owing to the fit strategy the errors are centered around zero for SkP and SkT6 which gives a better overall performance but does not cure the problem. The other forces show even more serious discrepancies, particularly SkI4. This makes the (among selfconsistent models) unique prediction of a spherical shell (that is directly related to the spinorbit splitting) of this interaction very questionable. This is not necessarily a defect of the SHF as such. Better performing parametrizations are feasible but have yet to be fully worked out. The mismatch in the spinorbit splittings should be a warning that extrapolations to detailed features of SHE have to be taken with care because these depend sensitively on shell effects. Note in that context that the much celebrated micmac approach gives also questionable predictions for shell closures as it displays also rather large errors, see the column FY in Fig. 6.
3.5 Density Profiles
The unusual spinorbit splitting for seen in the lower right panel of Fig. 6 is related to an unusual density profile of this nucleus, see the left panel of Fig. 7. The pattern can be understood as a cooperative effect from Coulomb repulsion and shell fluctuations (where the shell effect actually takes the lead). The dip at the center may be understood at first glance from Coulomb repulsion. But note that the depth of the dip differs substantially amongst the forces. Meanfield interactions with effective mass (SkP) show only a shallow minimum whereas those with low display a deep central depression, halfway to a bubble nucleus bubble . This is perfectly consistent with a shell fluctuation giving rise to the oscillation of the density in the interior shellflu and the low effective masses in the RMF make particularly large fluctuations. Shell structure thus dominates the Coulomb effects on the densities which is confirmed looking at the FY predictions where this effects appears even without the selfconsistent feedback between densities and potentials.
The right panels of Fig. 7 show the spinorbit potentials for . The dominant part of is located at the surface, where we see that SkI3 has a larger amplitude. That agrees with the larger spinorbit splittings found in the upper right panel of Fig. 6. At second glance, we see that the maximum of is shifted to smaller for the RMF force NL–Z. This is a tiny, but systematic, effect in all spinorbit potentials which we have looked at, and is probably one ingredient for the superior performance of the RMF with respect to spinorbit splittings. As is approximately in selfconsistent models, it directly reflects the shell oscillation of the density distribution. The central depression of the density leads to the large positive peak in around . This may lead to a disappearance of the splitting for states with low (which are sensitive to the interior) that is crucial for the appearance of the spherical and shells in some of the models (sometimes even inversion of sign, see the proton states in the lower right panel of Fig. 6 as predicted by NL–Z, NL3 and TM1). Again this is a selfconsistent effect which cannot be described with (current) micmac models. The spinorbit potential from the FY model is proportional to the gradient of the parameterized average potential and that gradient disappears inside the nucleus. Note that in this case the peak of is at larger radii, much narrower and of larger amplitude than in all selfconsistent models which causes the differences in the spinorbit splittings visible in Fig. 6.
3.6 Potential Energy Surfaces for Fission
Fig. 8 shows the deformation energy curves, usually called potentialenergy surfaces (PES), for fission of . For large deformations both models give very similar predictions, obvious differences (related to spherical shell structure) appear for smaller deformations only. Small deformations () unambiguously prefer (reflection) symmetric shapes. Distinct symmetric and asymmetric fission paths develop for larger , which is a general feature of very heavy nuclei PESfiss . Both also show a very low second minimum, in fact rather a saddle point, since there is no second barrier. A very interesting feature is the octupole softness around , the PES in octupole direction is almost perfectly flat between and . The asymmetric path continues then with negligible barrier. This is very different from the pronounced doublehumped structure of actinide nuclei. For some lighter nuclides the asymmetric path even lowers the first barrier as well as triaxial shapes do PESfiss ; Cwi96a . The inner barrier is still very high, showing that SHE could very well be relatively stable against spontaneous fission. All of these are generic features of fission paths in shellstabilized SHE PESfiss . There are, of course, some differences between the two forces shown in Fig. 8. is a small closure for NL–Z and accordingly we have a well developed spherical minimum, but SkI4 already produces a deformed minimum. Note that this does little harm to the stability. The first barrier is even higher than for NL–Z. As seen already in Fig. 4, shell stabilisation works very well even somewhat remote from spherical shell closures and even for deformed shapes Ben00a .
The PES of SkI4 is instructive in another respect. We see at least three almost degenerate minima. That is a typical example of shape coexistence. And it demonstrates once more that exotic nuclei are very likely to display shape coexistence shapeiso ; 186Pb . The example of with NL–Z showing a clear cut spherical minimum is rather the exception than the rule, see also Ber96a .
3.7 Recent –Decay Chains
The preferred decay mode of shell stabilised SHE is decay. A key quantity there is the value for the reaction which is defined as
(3) 
Recent experiments Z114 ; Z116 ; Z118 have reached the lower bounds of the island of spherical SHE which is expected somewhere around and depending on the model, as discussed above. It is interesting to compare the new data on with predictions from meanfield models. As –decay chains have constant , the isoscalar channel mainly determines the slope of the and the isovector channel the offset. Shell effects bend the curves locally, leading to kinks and peaks. Recent investigations of throughout the region of SHE with SLy4 in Cwi99a and NL–Z2 in Ben00a show a good overall description of the data by these two forces, although none of the interactions reproduces all details of the data. Most of the recently synthesized SHE are odd nuclei where the unpaired nucleon complicates the theoretical description, see Cwi99a ; Ben00a ; Cwi94a . Fig. 9 compares data with calculations in the selfconsistent SHF and RMF models (using the forces SLy4 and NL–Z2 respectively) and the macmic FRDM+FY and YPE+WS models. In view of the uncertainties, SLy4 and NL–Z2 give a very good description of the data for the decay chain of and reproduce the shell effect, which cannot be seen in the FRDM+FY predictions. While all models give similar predictions for this wellestablished chain, the spread among the models is much larger for the new chains. All models with the exception of macroscopic YPE+WS model show spherical or deformed shells which cannot be seen in the data. The right panel of Fig. 9 compares predictions with the recent data for the eveneven 116 decay chain (which still have to be viewed as preliminary). It is most interesting that the data agree with calculated values from interactions SkI4, SLy6 and NL3, although these three forces make different predictions for the spherical magic numbers, i.e. SkI4 (, ), SLy6 (, ), and NL3 (, ). All other interactions show wrong overall trends of the or pronounced deformed shells in disagreement with the data or even both. This demonstrates that predictions for spherical shell closures and binding energy systematics are fairly independent.
The experimental values follow a smooth trend while most meanfield results have a pronounced kink at . This is a shell effect related to predicted deformed shells which is not reflected in the data. The resolution of that puzzle lies in correlation effects. The PES of these SHE are rather soft (see also the previous subsection). The ground state then explores large fluctuations in quadrupole deformation and the pure meanfield minimum is insufficient in such a situation. One has at least to evaluate the correlation effects for motion, for an example from stable nuclei see e.g. girod . Applying such a scheme to the chain of SHE indeed yields a smooth trend of the Rei00a , but this aspect of correlations goes beyond the scope of this paper.
4 Conclusions
We have reviewed selfconsistent models for nuclear structure, thereby concentrating on the two most widely used brands: the relativistic meanfield model (RMF) and the nonrelativistic Skyrme–Hartree–Fock model (SHF). A brief discussion of the formal properties has shown the relation between these two models and to nuclear bulk properties. Each term in the SHF energy can be directly connected with a bulk property (as e.g. volume energy, asymmetry energy, etc). An exception is, of course, the spinorbit term which disappears in bulk matter. The RMF can be connected with SHF by virtue of a nonrelativistic expansion in orders of and in orders of meson range. The basic structures are fully comparable whereas the details of density dependence differ. An advantage of the RMF is that the spinorbit force is automatically included without the need of separate tuning. The SHF, on the other hand, is superior in its flexibility to accommodate isovector forces. Both models contain a good handful of free parameters which need to be adjusted phenomenologically. Different demands and bias in the adjustment has led to a world of different forces, in SHF as well as in RMF. We have selected an overseeable set of typical forces to display the possible variations in the results. The basic properties for normal nuclei are all very well reproduced by all chosen forces. An interesting detail is the isotope shift of radii in Pb. The remarked kink at the magic Pb is immediately reproduced by the RMF but can only be described within SHF after an appropriate extension of the (isovector) spinorbit force.
The discussion of results has concentrated on superheavy elements (SHE). The average error of binding energies in known SHE covers a larger span than the error in stable nuclei. This is an expected result because extrapolations always tend to scatter the errors. The positive aspect is that all errors remain in bearable bounds (safely below 1 ) and that there remain even several forces which maintain the quality found in stable nuclei. A different feature is given by looking at the relative errors of binding energies nucleus by nucleus, thus displaying the trends in these errors. The quality in reproducing the trends is found to be independent from the average quality of the binding energies. Large differences between the forces are seen for the trends in mass number and neutron excess where the SHF forces generally perform better with respect to these trends. A different way to look at trends is provided by the twonucleon separation energies and values. The for recently discovered chains of SHE are nicely reproduced within MeV for the more recent and well adjusted parametrisations in SHF and RMF. Somewhat more variance is found for the separation energies (not shown in this paper). The trend of the trends is given with the twonucleon shell gaps, i.e. the difference of adjacent separation energies. They depend predominantly on shell structure (unlike binding and separation energies which are also influenced by the bulk properties of a force). Large gaps serve as an indicator for magic shell closures. The predictions on the twonucleon shell gaps vary substantially, to the extend that different forces predict shell closures at different proton numbers. The differences look less dramatic if one realises that the overall size of the twonucleon shell gaps is small in any case. The concept of magic shell closures seems to fade away in SHE. One has to remind that magic shells had been looked for as a simple guideline where to find SHE which are sufficiently shellstabilised against Coulomb pressure towards spontaneous fission. The ultimate, but hard to evaluate, criterion is the height of the fission barrier. Simpler to compute is the shellcorrection energy which can serve as a rough estimate: large negative shellcorrection energy is a necessary condition for stability against fission. We find for all forces a broad valley of large shell correction energies. This a welcome feature as it leaves some freedom in the choice of the reaction channels. Thus we see good chances to hit many more SHE in near future experiments, in spite of the fact that pronounced doubly magic systems will not be found.
The investigations have demonstrated the high descriptive power of nowadays meanfield models. They have also revealed some weak points where further finetuning is needed, taking advantage of the many new data from exotic nuclei in general and SHE in particular. Last not least, one explores the limits of meanfield models when going towards the limits of stability. Shape coexistence and subsequent need for correlation effects shows up notoriously for the less well bound nuclei.
Acknowledgements
We would like to thank our collaborators T. Bürvenich, S. Ćwiok, D. J. Dean, J. Dobaczewski, P. Fleischer, W. Greiner, A. Kruppa, W. Nazarewicz, Ch. Reiß, K. Rutz, T. Schilling, M. R. Strayer, and T. Vertse for their contributions and helpful hints. We also acknowledge many inspiring discussions with our experimental colleagues J. Friedrich, S. Hofmann G. Münzenberg, V. Ninov, and Yu. Ts. Oganessian. This work was supported by Bundesministerium für Bildung und Forschung (BMBF), Project No. 06 ER 808 and by Gesellschaft für Schwerionenforschung (GSI). The Joint Institute for Heavy Ion Research has as member institutions the University of Tennessee, Vanderbilt University, and the Oak Ridge National Laboratory; it is supported by the members and by the Department of Energy through Contract No. DE–FG05–87ER40361 with the University of Tennessee.
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