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Projected-shell-model study for the structure of transfermium nuclei

Projected-shell-model study for the structure of transfermium nuclei. Yang Sun Shanghai 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

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Projected-shell-model study for the structure of transfermium nuclei

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  1. Projected-shell-model study for the structure oftransfermium 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 protons neutrons

  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 • c is related to deformation e 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

  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 g-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 Y3m 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 pi13/2 and 2qp nj15/2 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 p-crossing dominant p-crossing dominant p-crossing dominant p-crossing dominant

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

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

  18. 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 p1/2[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 adecay, resulting in enhanced stability in superheavy nuclei.

  20. K-isomers in 254No • The lowest kp = 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

  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 p, classified by K • to mix these K-states by residual interactions at given I and p • 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 254No - interpretation

  23. Projected shell model calculation • A high-K band with Kp = 8- starts at ~1.3 MeV • A neutron 2-qp state: (7/2+ [613] + 9/2- [734]) • A high-K band with Kp = 16+ at 2.7 MeV • A 4-qp state coupled by two neutrons and two protons: n (7/2+ [613] + 9/2- [734]) + p (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 276Sg Predicted K-isomers in 276Sg

  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 g-deformed rotor of Davydov and Filippov, Nucl. Phys. 8 (1958) 237 • e~0.25, e’~0.1 (g~22o) Data: Hall et al., Phys. Rev. C39 (1989) 1866

  27. g-vibration in very heavy nuclei • Prediction: g-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 g-bands • Multi-phonon g-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 g-bands also predicted to exist in the heaviest nuclei • Show strong anharmonicity

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

  30. Bands in odd-proton 249Bk

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

  32. 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

  33. 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 Dl=Dj=3,DK=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) • g-band • Triaxial octupole band

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