1. 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)
In collaboration with Stefan Frauendorf

2. 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)
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) )
Circles are broken in the macroscopic-
microscopic method which consistently
predicts Z=114, N=184

3. 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) for results

4. 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

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

6. 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) )
Self-consistent
solution
Pseudospin doublets
2g7/2-3d5/2
172
172
1i11/2-2g9/2
1h9/2-2f7/2

7. 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 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

8. Macroscopic + microscopic approach predicts Z= 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

9. 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.

10. Densities of superheavy nuclei: spherical RMF calculations with the NL3 forcex

11. 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.

12. 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

13. 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

14. Which role effective mass plays???
Skyrme SkP [m*/m=1]
double shell closure
at Z=126, N=184
(SkM*, ????
mass m*/m~
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

15. 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

16. 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”)

17. 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)

18. 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”)

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

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

21. 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