Superheavy nuclei: relativistic mean Superheavy nuclei: relativistic mean field outlookfield outlook

Anatoli AfanasjevAnatoli Afanasjev

University of Notre University of Notre DameDame

Disclaimer:Disclaimer: few results with Gogny and Skyrme forces used in the present few results with Gogny and Skyrme forces used in the present presentation are from J.-F. Berger et, NP A685 (2001) 1c and presentation are from J.-F. Berger et, NP A685 (2001) 1c and M. Bender et, PRC 60 (1999) 034304M. Bender et, PRC 60 (1999) 034304

1.1. MotivationMotivation2. Reliability of RMF parametrizations2. Reliability of RMF parametrizations3. Is that possible to find some common ingredient in the predictions3. Is that possible to find some common ingredient in the predictions of magic gaps in superheavy spherical nuclei? of magic gaps in superheavy spherical nuclei? Central depression Central depression in the density and its impact on shell structure.in the density and its impact on shell structure.4. Pairing in superheavy nuclei.4. Pairing in superheavy nuclei.5. Conclusions5. Conclusions

In collaboration with Stefan FrauendorfIn collaboration with Stefan Frauendorf

Self-consistencySelf-consistency

Magic gaps in spherical superheavyMagic gaps in spherical superheavy nucleinuclei

Potentials:Potentials: nucleonic, nucleonic, spin-orbitspin-orbit

Single-Single-particle particle spectraspectra

NucleonicNucleonicdensitiesdensities

Self-consistent theories give theSelf-consistent theories give thelargest variations in the predictions largest variations in the predictions of magic gapsof magic gaps at Z=120, 126 and 172, 184 at Z=120, 126 and 172, 184

1. 1. A pair of the RMF sets which indicate A pair of the RMF sets which indicate Z=114 and N=184 gaps are eliminated Z=114 and N=184 gaps are eliminated as candidates by performing the analysis as candidates by performing the analysis of quasiparticle spectra in deformed nuclei of quasiparticle spectra in deformed nuclei in actinide region,in actinide region,A.V.Afanasjev et, PRC 67 (2003) 024303A.V.Afanasjev et, PRC 67 (2003) 024303

2. Skyrme SkI4 also predicts Z=1142. Skyrme SkI4 also predicts Z=114(K. Rutz et al, PRC 56 (1997) 238)(K. Rutz et al, PRC 56 (1997) 238)but it reproduces spin-orbit splitting but it reproduces spin-orbit splitting very badly (M. Bender et, PRC 60very badly (M. Bender et, PRC 60 (1999) 034304)(1999) 034304)

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

Self-Self-consistencyconsistency neutronsneutronsprotonsprotons

““All animals [All animals [theories, theories, parametrizationsparametrizations] are equal, ] are equal, but some animals [but some animals [theoriestheories, , parametrizationsparametrizations] are ] are more equal than others” more equal than others” George Orwell, ‘Animal farm’’George Orwell, ‘Animal farm’’

Goal:Goal: To find the RMF parametrizations best suited for theTo find the RMF parametrizations best suited for the

description of superheavy nucleidescription of superheavy nuclei

Method:Method: perform detailed analysis of the spectroscopic data perform detailed analysis of the spectroscopic data in the A~250 deformed mass region in the frameworkin the A~250 deformed mass region in the framework

of cranked relativistic Hartree-Bogoliubov (CRHB) theoryof cranked relativistic Hartree-Bogoliubov (CRHB) theorysee A.V.Afanasjev et al, PRC 67 (2003) 024309 for results

Reliability of parametrizationsReliability of parametrizations

# of parametrizations:# of parametrizations:Skyrme > 80Skyrme > 80

RMF > 20RMF > 20

),2(),(),(2 NZBNZBNZS p

)2,(),(2),2(

),2(),(),( 222

NZBNZBNZB

NZSNZSNZ ppp

An analysis of An analysis of experimental data onlyexperimental data only in terms of these in terms of these quantities can be quantities can be misleadingmisleading

Example of NLSH:Example of NLSH:indicates Z=100 gap butindicates Z=100 gap butbetween wrong s-p statesbetween wrong s-p states

sometimes called assometimes called as““2-nucleon gap”2-nucleon gap”

CRHB+LN resultsCRHB+LN results

nE,...,E,E 21

statelowestElow

For each configuration For each configuration (blocked solution) the (blocked solution) the binding energy Ebinding energy Ei i

is calculated fully self- is calculated fully self- consistently (including consistently (including alsoalso time-odd mean fields)time-odd mean fields)

lowiexc EEE

stateparticleEE.Vif

stateholeEE.Vif

excqp

excqp

50

502

2

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)

Pseudospin doublets

1h1h9/29/2-2f-2f7/27/2

2g2g7/27/2-3d-3d5/25/2

1i1i11/211/2-2g-2g9/29/2

Self-consistentSelf-consistent solutionsolution

RMF analysis of single-particle energies in spherical Z=120, N=172 nucleusRMF analysis of single-particle energies in spherical Z=120, N=172 nucleus

172172172172

Accuracy of the description of single-particle energiesAccuracy of the description of single-particle energies

Best sets (NL1,NL3,NL-Z2) ---- for most subshells better than 0.5 Best sets (NL1,NL3,NL-Z2) ---- for most subshells better than 0.5 MeVMeV

for a few subshells the discrepancies reach 0.7-1 MeV for a few subshells the discrepancies reach 0.7-1 MeV Worst sets (NLSH, NLRA1) --- much higher errors in the energiesWorst sets (NLSH, NLRA1) --- much higher errors in the energiesof single-particle states (the only sets which predict Z=114 as of single-particle states (the only sets which predict Z=114 as

shell gap) shell gap) should be excluded should be excludedImportant: no-single particle information is usedImportant: no-single particle information is used

in the fit of the RMF forcesin the fit of the RMF forces

Additional constraint by Additional constraint by the study single-particle the study single-particle

states in nuclei with states in nuclei with N~162 and/or Z~108N~162 and/or Z~108

Single-particleSingle-particlestates are states are observedobserved

Woods-Saxon potential

R0=r0A1/3 r0~1.2 fm

V0~50 MeV a ~ 0.5 fm

The radial density dependence used in liquid drop model

Macroscopic + microscopic approach predicts Z=114 and N=184, but basic assumptions (see below) are

VIOLATED.EEEE pairshellLDtot

liquid drop

quantum (shell) correction

pairing

FLAT DENSITY distribution in the central part of nucleus

)Mr/()( 22 21

Lesson from quantum mechanics:Lesson from quantum mechanics: spherical harmonic oscillatorspherical harmonic oscillator

AA

BB

A: the radial wave functionA: the radial wave function

B: effective radial potential, i.e. with the centrifugal termB: effective radial potential, i.e. with the centrifugal term

)(R

added.added.

Densities of superheavy nuclei: spherical RMF calculations with the Densities of superheavy nuclei: spherical RMF calculations with the NL3 forceNL3 force

xx

The clustering of single-particle states into the groups of high- and low-j 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 densitysubshells is at the origin of the central depression in the nuclear density

distribution in spherical superheavy nuclei.distribution in spherical superheavy nuclei.

‘g-s’ – ground state configuration

‘exc-s’ – excited state configuration

Occupied state

Unoccupied state

= += -

particle-hole excitationleading to flatter neutron and proton density distribution

Spherical RMF calculations with NL3 forces

3p1/2 126

Self-consistency effects related to density redistribution define the shell structure:Z=114 shell gap can be excluded

3d5/2

A.V.Afanasjev and S.Frauendorf, PRC 71, 024308 (2005)A.V.Afanasjev and S.Frauendorf, PRC 71, 024308 (2005)

densitidensitieses

Impact of particle-holeImpact of particle-hole excitations on the densitiesexcitations on the densitiesand potentials (nucleonic,and potentials (nucleonic, spin-orbit)spin-orbit)

General conclusionGeneral conclusion (tested (tested on large # of particle-holeon large # of particle-hole excitations in different nuclei):excitations in different nuclei):

1.1. Large density depressionLarge density depression in the central part of nucleus:in the central part of nucleus: shell gaps at Z=120, shell gaps at Z=120, N=172N=172

2. Flat density distribution2. Flat density distribution in the central part of nucleus:in the central part of nucleus: Z=126 appears, Z=126 appears, N=184 becomes larger N=184 becomes larger and Z=120and Z=120 (N=172) shrink(N=172) shrink

densitidensitieses

potentialspotentials

RMFRMFdouble shell closure double shell closure at Z=120,N=172at Z=120,N=172

Skyrme SkP [m*/m=1]Skyrme SkP [m*/m=1] double shell closuredouble shell closure at Z=126, N=184at Z=126, N=184

(SkM*, (SkM*, ????????

Skyrme SkI3 [m*/m=0.57]Skyrme SkI3 [m*/m=0.57] gaps at Z=120, N=184gaps at Z=120, N=184 no double shell closure,no double shell closure, SLy6SLy6

Gogny D1SGogny D1S Z=120, Z=120, N=172(?)N=172(?) Z=126, N=184Z=126, N=184

Low

eff

ect

ive m

ass

Low

eff

ect

ive m

ass

m*/

m ~

0.6

5m

*/m

~ 0

.65

Larg

e e

ffect

ive

Larg

e e

ffect

ive

mass

m*/

m~

0.8

-1.0

m

ass

m*/

m~

0.8

-1.0

Which role effective mass plays???Which role effective mass plays???

Why Why macroscopic+microscomacroscopic+microscopic pic

models are working so models are working so well well

in known superheavy in known superheavy nuclei??? nuclei???

They are deformedThey are deformed Deformation leads to more Deformation leads to more

equal equal distribution of the states distribution of the states

emerging from high- and emerging from high- and low-j subshells (and, thus, low-j subshells (and, thus, removes the clustering of removes the clustering of high-j subshells seen in high-j subshells seen in spherical superheavy spherical superheavy nuclei).nuclei).

This leads to almost This leads to almost flat flat density distributiondensity distribution in the in the central part of nucleus.central part of nucleus.

Single-particle features are Single-particle features are fitted fitted

to experimental datato experimental data

CRHB+LN resultsCRHB+LN results

The importance of particle number projection (PNP) in spherical The importance of particle number projection (PNP) in spherical superheavy nuclei superheavy nuclei

Most of other Most of other calculations were calculations were

performed with no PNPperformed with no PNP

Pairing collapsePairing collapseat “magic” gapsat “magic” gaps

Approximate PNP byApproximate PNP byLipkin-Nogami (LN):Lipkin-Nogami (LN):

no pairing collapseno pairing collapse

Calc. with no PNPCalc. with no PNP

overestimate overestimate 2p2p

(“2-proton shell gap”)(“2-proton shell gap”)

)(TrEpairing 2

1

Stability against fissionStability against fission

RMF - low barriers,RMF - low barriers,Skyrme HF – high barriersSkyrme HF – high barriers

Experiment: fission Experiment: fission barrier heightsbarrier heights

in heavy actinidesin heavy actinides(A~240)(A~240)

RMF: RMF: somewhat underestimatessomewhat underestimates

Skyrme HFSkyrme HF– – overestimatesoverestimates

by few MeVby few MeV

From M.Bender et al,From M.Bender et al,PRC (2003)PRC (2003)

1.1. The central depression in density distribution of spherical superheavy The central depression in density distribution of spherical superheavy nuclei plays an important role in the definition of their shell structure:nuclei plays an important role in the definition of their shell structure:

1.1. Large density depressionLarge density depression in the central part of nucleus favorsin the central part of nucleus favors shell gaps at Z=120, N=172shell gaps at Z=120, N=172

2. Flat density distribution2. Flat density distributionin the central part of nucleus in the central part of nucleus favorsfavors Z=126 and N=184Z=126 and N=184

Intermediate situations appear dependent on the quality of forceIntermediate situations appear dependent on the quality of forcewith respect of single-particle degrees of freedomwith respect of single-particle degrees of freedom

(position of j-subshells, spin-orbit splitting, pseudospin splitting) (position of j-subshells, spin-orbit splitting, pseudospin splitting)

ConclusionsConclusions

The check of quality of force with respect of deformed The check of quality of force with respect of deformed single-particle states is MUST BEsingle-particle states is MUST BE

2.2.

3. Particle number projection is important for proper description of3. Particle number projection is important for proper description ofpairing properties of spherical superheavy nuclei and calculation pairing properties of spherical superheavy nuclei and calculation

of indicators of low-level density (of indicators of low-level density (2n2n – “2-nucleon gaps”) – “2-nucleon gaps”)

A.V.Afanasjev et al, PRC 67 (2003) 024309

Discrepancy between theory and experiment Discrepancy between theory and experiment empirical shifts for the positions of spherical subshellsempirical shifts for the positions of spherical subshells

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

How magic are “magic” shell gaps?How magic are “magic” shell gaps?

Calculated spherical Z=120 gap versus experimental deformed Z=100 gapCalculated spherical Z=120 gap versus experimental deformed Z=100 gap

Similar relation for neutron spherical N=172 and deformed Similar relation for neutron spherical N=172 and deformed N=152 gapN=152 gap

It might be that the effect of spherical shell gaps in superheavy nuclei is only 30-40% 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 regionmore pronounced than the effect of deformed gaps in the A~250 region