Projected-shell-model study for the structure of transfermium nuclei
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Projected-shell-model study for the structure oftransfermium nuclei
Yang SunShanghai Jiao Tong University
Beijing, June 9, 2009
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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
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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
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Single-particle states
R. Chasmanet al., Rev. Mod. Phys. 49, 833 (1977)R. Chasmanet al., Rev. Mod. Phys. 49, 833 (1977)
neutronsprotons
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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
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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
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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
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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
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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
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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
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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
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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, …)
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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
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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
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MoI, B(E2), g-factor in Cf isotopes
-crossingdominant
-crossingdominant
-crossingdominant
-crossingdominant
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MoI, B(E2), g-factor in Fm isotopes
-crossingdominant
-crossingdominant
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MoI, B(E2), g-factor in No isotopes
-crossingdominant
-crossingdominant
-crossingdominant
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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
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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.
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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
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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 /
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K-isomers in 254No - interpretation
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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
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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
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Predicted K-isomers in 276SgPredicted K-isomers in 276Sg
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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
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-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
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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
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Bands in odd-proton 249Bk
Nilsson parameters of T. Bengtsson-Ragnarsson
Slightly modifiedNilsson parameters
Ahmad et al., Phys. Rev. C71 (2005) 054305
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Bands in odd-proton 249Bk
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Nature of low-lying excited states
N = 150 Neutron states
N = 152 Proton states
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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
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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
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Yrast and 2- bands in N=150 nuclei
Chen, Sun, Gao, Phys. Rev. C 77 (2008) 061305
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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