Superheavy nuclei: relativistic mean field outlook Anatoli Afanasjev University of Notre Dame Motivation 2. Reliability of RMF parametrizations 3. Is that possible to find some common ingredient in the predictions of magic gaps in superheavy spherical nuclei?  Central depression in the density and its impact on shell structure. 4. Pairing in superheavy nuclei. 5. Conclusions Disclaimer: few results with Gogny and Skyrme forces used in the present presentation are from J.-F. Berger et, NP A685 (2001) 1c and M. Bender et, PRC 60 (1999) 034304 In collaboration with Stefan Frauendorf

Magic gaps in spherical superheavy nuclei Nucleonic densities Magic gaps in spherical superheavy nuclei Self-consistency Self-consistent theories give the largest variations in the predictions of magic gaps at Z=120, 126 and 172, 184 Single- particle spectra Potentials: nucleonic, spin-orbit 1. A pair of the RMF sets which indicate Z=114 and N=184 gaps are eliminated as candidates by performing the analysis of quasiparticle spectra in deformed nuclei in actinide region, A.V.Afanasjev et, PRC 67 (2003) 024303 protons neutrons Self- consistency 2. Skyrme SkI4 also predicts Z=114 (K. Rutz et al, PRC 56 (1997) 238) but it reproduces spin-orbit splitting very badly (M. Bender et, PRC 60 (1999) 034304) Circles are broken in the macroscopic- microscopic method which consistently predicts Z=114, N=184

Reliability of parametrizations Skyrme > 80 RMF > 20 “All animals [theories, parametrizations] are equal, but some animals [theories, parametrizations] are more equal than others” George Orwell, ‘Animal farm’’ Goal: To find the RMF parametrizations best suited for the description of superheavy nuclei Method: perform detailed analysis of the spectroscopic data in the A~250 deformed mass region in the framework of cranked relativistic Hartree-Bogoliubov (CRHB) theory see A.V.Afanasjev et al, PRC 67 (2003) 024309 for results

CRHB+LN results sometimes called as “2-nucleon gap” An analysis of experimental data only in terms of these quantities can be misleading Example of NLSH: indicates Z=100 gap but between wrong s-p states

is calculated fully self- For each configuration (blocked solution) the binding energy Ei is calculated fully self- consistently (including also time-odd mean fields)

RMF analysis of single-particle energies in spherical Z=120, N=172 nucleus corrected by the empirical shifts obtained in the detailed study of quasiparticle spectra in odd-mass nuclei of the deformed A~250 mass region (PRC 67 (2003) 024309) Self-consistent solution Pseudospin doublets 2g7/2-3d5/2 172 172 1i11/2-2g9/2 1h9/2-2f7/2

Additional constraint by the study single-particle Accuracy of the description of single-particle energies Best sets (NL1,NL3,NL-Z2) ---- for most subshells better than 0.5 MeV for a few subshells the discrepancies reach 0.7-1 MeV Worst sets (NLSH, NLRA1) --- much higher errors in the energies of single-particle states (the only sets which predict Z=114 as shell gap)  should be excluded Important: no-single particle information is used in the fit of the RMF forces Additional constraint by the study single-particle states in nuclei with N~162 and/or Z~108 Single-particle states are observed

Macroscopic + microscopic approach predicts Z=114 and N=184, but basic assumptions (see below) are VIOLATED. pairing liquid drop quantum (shell) correction The radial density dependence used in liquid drop model FLAT DENSITY distribution in the central part of nucleus Woods-Saxon potential R0=r0A1/3 r0~1.2 fm a ~ 0.5 fm V0~50 MeV

Lesson from quantum mechanics: spherical harmonic oscillator A B A: the radial wave function B: effective radial potential, i.e. with the centrifugal term added.

Densities of superheavy nuclei: spherical RMF calculations with the NL3 force x

distribution in spherical superheavy nuclei. The clustering of single-particle states into the groups of high- and low-j subshells is at the origin of the central depression in the nuclear density distribution in spherical superheavy nuclei.

A.V.Afanasjev and S.Frauendorf, PRC 71, 024308 (2005) ‘g-s’ – ground state configuration ‘exc-s’ – excited state configuration Occupied state Unoccupied state p = + p = - 3p1/2 126 particle-hole excitation leading to flatter neutron and proton density distribution 3d5/2 Self-consistency effects related to density redistribution define the shell structure: Z=114 shell gap can be excluded Spherical RMF calculations with NL3 forces

Impact of particle-hole excitations on the densities and potentials (nucleonic, spin-orbit) densities densities General conclusion (tested on large # of particle-hole excitations in different nuclei): potentials Large density depression in the central part of nucleus: shell gaps at Z=120, N=172 2. Flat density distribution in the central part of nucleus: Z=126 appears, N=184 becomes larger and Z=120 (N=172) shrink

Which role effective mass plays??? Skyrme SkP [m*/m=1] double shell closure at Z=126, N=184 (SkM*, ???? mass m*/m~0.8-1.0 Large effective Skyrme SkI3 [m*/m=0.57] gaps at Z=120, N=184 no double shell closure, SLy6 Which role effective mass plays??? Gogny D1S Z=120, N=172(?) Z=126, N=184 Low effective mass m*/m ~ 0.65 RMF double shell closure at Z=120,N=172

Why macroscopic+microscopic models are working so well in known superheavy nuclei??? They are deformed Deformation leads to more equal distribution of the states emerging from high- and low-j subshells (and, thus, removes the clustering of high-j subshells seen in spherical superheavy nuclei). This leads to almost flat density distribution in the central part of nucleus. Single-particle features are fitted to experimental data CRHB+LN results

The importance of particle number projection (PNP) in spherical superheavy nuclei Most of other calculations were performed with no PNP Pairing collapse at “magic” gaps Approximate PNP by Lipkin-Nogami (LN): no pairing collapse Calc. with no PNP overestimate d2p (“2-proton shell gap”)

Stability against fission RMF - low barriers, Skyrme HF – high barriers Experiment: fission barrier heights in heavy actinides (A~240) RMF: somewhat underestimates Skyrme HF – overestimates by few MeV From M.Bender et al, PRC (2003)

Conclusions The central depression in density distribution of spherical superheavy nuclei plays an important role in the definition of their shell structure: Large density depression in the central part of nucleus favors shell gaps at Z=120, N=172 2. Flat density distribution in the central part of nucleus favors Z=126 and N=184 Intermediate situations appear dependent on the quality of force with respect of single-particle degrees of freedom (position of j-subshells, spin-orbit splitting, pseudospin splitting) The check of quality of force with respect of deformed single-particle states is MUST BE 2. 3. Particle number projection is important for proper description of pairing properties of spherical superheavy nuclei and calculation of indicators of low-level density (d2n – “2-nucleon gaps”)

A.V.Afanasjev et al, PRC 67 (2003) 024309 Discrepancy between theory and experiment   empirical shifts for the positions of spherical subshells

The Sleep of Reason Produces Monsters. (Caprichos, no. 43, Goya)

How magic are “magic” shell gaps? Calculated spherical Z=120 gap versus experimental deformed Z=100 gap Similar relation for neutron spherical N=172 and deformed N=152 gap It might be that the effect of spherical shell gaps in superheavy nuclei is only 30-40% more pronounced than the effect of deformed gaps in the A~250 region