Projected-shell-model study for the structure oftransfermium nuclei

Yang SunShanghai Jiao Tong University

Beijing, June 9, 2009

The island of stability

What are the next magic numbers, i.e. most stable nuclei? Predicted neutron magic number: 184 Predicted proton magic number: 114, 120, 126

Approaching the superheavy island

Single particle states in SHE Important for locating the island Little experimental information available

Indirect ways to find information on single particle states Study quasiparticle K-isomers in very heavy nuclei Study rotation alignment of yrast states in very heavy nuclei

Deformation effects, collective motions in very heavy nuclei

gamma-vibration Octupole effect

Single-particle states

R. Chasmanet al., Rev. Mod. Phys. 49, 833 (1977)R. Chasmanet al., Rev. Mod. Phys. 49, 833 (1977)

neutronsprotons

Nuclear structure models

Shell-model diagonalization method Most fundamental, quantum mechanical Growing computer power helps extending applications A single configuration contains no physics Huge basis dimension required, severe limit in applications

Mean-field method Applicable to any size of systems Fruitful physics around minima of energy surfaces No configuration mixing States with broken symmetry, cannot be used to calculate

electromagnetic transitions and decay rates

Bridge between shell-model and mean-field method

Projected shell model Use more physical states (e.g. solutions of a deformed mean-

field) and angular momentum projection technique to build shell model basis

Perform configuration mixing (a shell-model concept) • K. Hara, Y. Sun, Int. J. Mod. Phys. E 4 (1995) 637

The method works in between conventional shell model and mean field method, hopefully take the advantages of both

The projected shell model

Shell model based on deformed basis

Take a set of deformed (quasi)particle states (e.g. solutions of HFB, HF + BCS, or Nilsson + BCS)

Select configurations (deformed qp vacuum + multi-qp states near the Fermi level)

Project them onto good angular momentum (if necessary, also particle number, parity) to form a basis in lab frame

Diagonalize a two-body Hamiltonian in projected basis

Model space constructed by angular-momentum projected states

Wavefunction:

with a.-m.-projector:

Eigenvalue equation:

with matrix elements:

Hamiltonian is diagonalized in the projected basis

IM K

IM Pf

DDdI

P IM K

IM K

28

12

0''

''

fENH II

'''''' I

KK

II

KK

I PNPHH

IM KP

Hamiltonian and single particle space

The Hamiltonian Interaction strengths

is related to deformation by

GM is fitted by reproducing moments of inertia

GQ is assumed to be proportional to GM with a ratio ~ 0.15

Single particle space Three major shells for neutrons or protons

For very heavy nuclei, N = 5, 6, 7 for neutrons

N = 4, 5, 6 for protons

PPGPPGQQHH QM

20

ppnn QQ 00

3/2

Building blocks: a.-m.-projected multi-quasi-particle states

Even-even nuclei:

Odd-odd nuclei:

Odd-neutron nuclei:

Odd-proton nuclei:

,0ˆ,0ˆ,0ˆ,0ˆ I

MKIMK

IMK

IMK PPPP

,0ˆ,0ˆ,0ˆ,0ˆ I

MKIMK

IMK

IMK PPPP

,0ˆ,0ˆ,0ˆ I

MKIMK

IMK PPP

,0ˆ,0ˆ,0ˆ I

MKIMK

IMK PPP

Recent developments in PSM

Separately project n and p systems for scissors mode Sun, Wu, Bhatt, Guidry, Nucl. Phys. A 703 (2002) 130

Enrich PSM qp-vacuum by adding collective d-pairs Sun and Wu, Phys. Rev. C 68 (2003) 024315

PSM Calculation for Gamow-Teller transition Gao, Sun, Chen, Phys. Rev, C 74 (2006) 054303

Multi-qp triaxial PSM for -deformed high-spin states Gao, Chen, Sun, Phys. Lett. B 634 (2006) 195 Sheikh, Bhat, Sun, Vakil, Palit, Phys. Rev. C 77 (2008) 034313

Breaking Y3 symmetry + parity projection for octupole band Chen, Sun, Gao, Phys. Rev, C 77 (2008) 061305

Real 4-qp states in PSM basis (from same or diff. N shell) Chen, Zhou, Sun, to be published

Very heavy nuclei: general features

Potential energy calculation shows deep prolate minimum A very good rotor, quadrupole + pairing interaction dominant Low-spin rotational feature of even-even nuclei can be well

described (relativistic mean field, Skyrme HF, …)

Yrast line in very heavy nuclei

No useful information can be extracted from low-spin g-band (rigid rotor behavior)

First band-crossing occurs at high-spins (I = 22 – 26)

Transitions are sensitive to the structure of the crossing bands

g-factor varies very much due to the dominant proton or neutron contribution

Band crossings of 2-qp high-j states

Strong competition between 2-qp i13/2 and 2qp j15/2 band crossings (e.g. in N=154 isotones)

Al-Khudair, Long, Sun, Phys. Rev. C 79 (2009) 034320

MoI, B(E2), g-factor in Cf isotopes

-crossingdominant

-crossingdominant

-crossingdominant

-crossingdominant

MoI, B(E2), g-factor in Fm isotopes

-crossingdominant

-crossingdominant

MoI, B(E2), g-factor in No isotopes

-crossingdominant

-crossingdominant

-crossingdominant

K-isomers in superheavy nuclei

K-isomer contains important information on single quasi-particles

e.g. for the proton 2f7/2–2f5/2 spin–orbit partners, strength of the spin–orbit interaction determines the size of the Z=114 gap

Information on the position of 1/2[521] is useful Herzberg et al., Nature 442 (2006) 896

K-isomer in superheavy nuclei may lead to increased survival probabilities of these nuclei

Xu et al., Phys. Rev. Lett. 92 (2004) 252501

Enhancement of stability in SHE by isomers

Occurrence of multi-quasiparticle isomeric states decreases the probability for both fission and decay, resulting in enhanced stability in superheavy nuclei.

K-isomers in 254No

The lowest k = 8- isomeric band in 254No is expected at 1–1.5 MeV

Ghiorso et al., Phys. Rev. C7 (1973) 2032

Butler et al., Phys. Rev. Lett. 89 (2002) 202501

Recent experiments confirmed two isomers: T1/2 = 266 ± 2 ms and 184 ± 3 μs

Herzberg et al., Nature 442 (2006) 896

What do we need for K-isomer description?

K-mixing – It is preferable to construct basis states with good angular momentum I and

parity , classified by K to mix these K-states by residual interactions at given I and to use resulting wavefunctions to calculate electromagnetic

transitions in shell-model framework

A projected intrinsic state can be labeled by K With axial symmetry: carries K

defines a rotational band associated with the intrinsic K-state

Diagonalization = mixing of various K-states

IMKP̂

III NHE /

K-isomers in 254No - interpretation

Projected shell model calculation

A high-K band with K = 8- starts at ~1.3 MeV

A neutron 2-qp state:

(7/2+ [613] + 9/2- [734])

A high-K band with K = 16+ at 2.7 MeV

A 4-qp state coupled by two neutrons and two protons:

(7/2+ [613] + 9/2- [734]) +

(7/2- [514] + 9/2+ [624])

Herzberg et al., Nature 442 (2006) 896

Prediction: K-isomers in No chain

Positions of the isomeric states depend on the single particle states

Nilsson states used:

T. Bengtsson, I. Ragnarsson, Nucl. Phys. A 436 (1985) 14

Predicted K-isomers in 276SgPredicted K-isomers in 276Sg

A superheavy rotor can vibrate

Take triaxiality as a parameter in the deformed basis and do 3-dim. angular-momentum-projection

Microscopic version of the -deformed rotor of

Davydov and Filippov, Nucl. Phys. 8 (1958) 237

’~0.1 (~22o)

Data: Hall et al., Phys. Rev. C39 (1989) 1866

-vibration in very heavy nuclei

Prediction: -vibrations (bandhead below 1MeV) Low 2+ band cannot be explained by qp excitations

Sun, Long, Al-Khudair, Sheikh, Phys. Rev. C 77 (2008) 044307

Multi-phonon -bands

Multi-phonon -vibrational bands were predicted for rear earth nuclei

Classified by K = 0, 2, 4, … Y. Sun et al, Phys. Rev. C61

(2000) 064323

Multi-phonon -bands also predicted to exist in the heaviest nuclei

Show strong anharmonicity

Bands in odd-proton 249Bk

Nilsson parameters of T. Bengtsson-Ragnarsson

Slightly modifiedNilsson parameters

Ahmad et al., Phys. Rev. C71 (2005) 054305

Bands in odd-proton 249Bk

Nature of low-lying excited states

N = 150 Neutron states

N = 152 Proton states

Octupole correlation: Y30 vs Y32

Strong octupole effect known in the actinide region (mainly Y30 type: parity doublet band)

As mass number increases, starting from Cm-Cf-Fm-No, 2- band is lower

Y32 correlation may be important

Triaxial-octupole shape in superheavy nuclei

Proton Nilsson Parameters of T. Bengtsson and Ragnarsson

i13/2 (l = 6, j = 13/2), f7/2 (l = 3, j = 7/2) degenerate at the spherical limit

{[633]7/2; [521]3/2}, {[624]9/2; [512]5/2} satisfy l=j=3,K=2

Gap at Z=98, 106

Yrast and 2- bands in N=150 nuclei

Chen, Sun, Gao, Phys. Rev. C 77 (2008) 061305

Summary

Study of structure of very heavy nuclei can help to get information about single-particle states.

The standard Nilsson s.p. energies (and W.S.) are probably a good starting point, subject to some modifications.

Testing quantities (experimental accessible) Yrast states just after first band crossing Quasiparticle K-isomers Excited band structure of odd-mass nuclei

Low-lying collective states (experimental accessible) -band Triaxial octupole band