1. Projected-shell-model study for the structure of transfermium nuclei Yang Sun Shanghai Jiao Tong University Beijing, June 9, 2009

2. 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

3. 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

4. Single-particle states neutrons protons

5. 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

6. 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

7. 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

8. 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

9. Hamiltonian and single particle space The Hamiltonian Interaction strengths  is related to deformation  by G M is fitted by reproducing moments of inertia G Q is assumed to be proportional to G M 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

10. Building blocks: a.-m.-projected multi-quasi-particle states Even-even nuclei: Odd-odd nuclei: Odd-neutron nuclei: Odd-proton nuclei:

11. 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 Y 3  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

12. 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, …)

13. 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

14. Band crossings of 2-qp high-j states Strong competition between 2-qp  i 13/2 and 2qp j 15/2 band crossings (e.g. in N=154 isotones) Al-Khudair, Long, Sun, Phys. Rev. C 79 (2009) 034320

15. MoI, B(E2), g-factor in Cf isotopes  -crossing dominant  -crossing dominant  -crossing dominant  -crossing dominant

16. MoI, B(E2), g-factor in Fm isotopes  -crossing dominant  -crossing dominant

17. MoI, B(E2), g-factor in No isotopes  -crossing dominant -crossing dominant -crossing dominant

18. K-isomers in superheavy nuclei K-isomer contains important information on single quasi-particles e.g. for the proton 2f 7/2 –2f 5/2 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

19. 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.

20. K-isomers in 254 No The lowest k  = 8 - isomeric band in 254 No 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: T 1/2 = 266 ± 2 ms and 184 ± 3 μs Herzberg et al., Nature 442 (2006) 896

21. 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

22. K-isomers in 254 No - interpretation

23. 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

24. 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

25. Predicted K-isomers in 276 Sg

26. 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 (  ~22 o ) Data: Hall et al., Phys. Rev. C39 (1989) 1866

27.  -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

28. 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

29. Bands in odd-proton 249 Bk Nilsson parameters of T. Bengtsson-Ragnarsson Slightly modified Nilsson parameters Ahmad et al., Phys. Rev. C71 (2005) 054305

30. Bands in odd-proton 249 Bk

31. Nature of low-lying excited states N = 150 Neutron states N = 152 Proton states

32. Octupole correlation: Y 30 vs Y 32 Strong octupole effect known in the actinide region (mainly Y 30 type: parity doublet band) As mass number increases, starting from Cm-Cf-Fm-No, 2 - band is lower Y 32 correlation may be important

33. Triaxial-octupole shape in superheavy nuclei Proton Nilsson Parameters of T. Bengtsson and Ragnarsson i 13/2 (l = 6, j = 13/2), f 7/2 (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

34. Yrast and 2 - bands in N=150 nuclei Chen, Sun, Gao, Phys. Rev. C 77 (2008) 061305

35. 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