超重原子核的结构 孙 扬 上海交通大学 合作者：清华大学 龙桂鲁， F. Al-Khudair 中国原子能研究院 陈永寿，高早春 济南，山东大学, 2008 年 9 月 20 日

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

Explore the island Single particle states for SHE Important for locating the island Little experimental information available Indirect ways to find information on single particle states Study of rotation alignment of yrast states in very heavy nuclei Study of quasiparticle K-isomers in very heavy nuclei Deformation effects, collective motions in SHE gamma-vibration (Triaxial) octupole effect

Single-particle states neutrons protons

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 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

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

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 ~ 0.13 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

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  i 13/2 and 2qp j 15/2 band crossings (e.g. in N=154 isotones)

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

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

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

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) Recent experiments confirmed two isomers: T 1/2 = 266 ± 2 ms and 184 ± 3 μs Herzberg et al., Nature 442 (2006) 896 Tandel, et al., Phys. Rev. Lett. 97 (2006)

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])

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

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

 -vibration in very heavy nuclei Prediction:  -vibrations (bandhead below 1MeV) Low 2 + band cannot be explained by qp excitations

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

Bands in odd-proton 249 Bk

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

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

Yrast and 2 - bands in N=150 nuclei

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