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
Predicted proton magic number: 114, 120, 126
Approaching the superheavy island

Single particle states in SHE



Indirect ways to find information on single particle states



Important for locating the island
Little experimental information available
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
protons
neutrons
R. Chasman et al., Rev. Mod. Phys. 49, 833 (1977)
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 meanfield) 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



 
with a.-m.-projector:
I

2I  1
I
PMK 
d DMK  D 
2 
8
Eigenvalue equation:
 H
I
'
'
I
with matrix elements: H 

I
f PMK 
Wavefunction:
I
M
'

 ENI ' f '  0
 I
  HPKK '  '
I
N '
I
  PKK '  '
Hamiltonian is diagonalized in the projected basis

I
PMK 

Hamiltonian and single particle
space
 The Hamiltonian H  H 0  

Interaction strengths





c

Q
Q  GM P  P  GQ  P P

2

c is related to deformation e by c  

2 / 3 e  
n Q0
  p Q0
GM is fitted by reproducing moments of inertia
GQ is assumed to be proportional to GM with a ratio ~ 0.15
Single particle space

n
Three major shells for neutrons or protons
For very heavy nuclei, N = 5, 6, 7 for neutrons
N = 4, 5, 6 for protons
p
Building blocks: a.-m.-projected
multi-quasi-particle states

Even-even nuclei:
Pˆ
I
MK

Odd-odd nuclei:
Pˆ
I
MK


I
I
I
 0 , PˆMK
 0 , PˆMK
 0 , PˆMK
 0 ,
Odd-neutron nuclei:
Pˆ
I


MK 


I
I
I
0 , PˆMK
 0 , PˆMK
 0 , PˆMK
 0 ,
Odd-proton nuclei:
Pˆ
I
MK

I
I
0 , PˆMK
 0 , PˆMK
 0 ,

I
I
 0 , PˆMK
 0 , PˆMK
 0 ,
Recent developments in PSM

Separately project n and p systems for scissors mode

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Enrich PSM qp-vacuum by adding collective d-pairs
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
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


Gao, Sun, Chen, Phys. Rev, C 74 (2006) 054303
Multi-qp triaxial PSM for g-deformed high-spin states


Sun and Wu, Phys. Rev. C 68 (2003) 024315
PSM Calculation for Gamow-Teller transition
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
Sun, Wu, Bhatt, Guidry, Nucl. Phys. A 703 (2002) 130
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
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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


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No useful information can be
extracted from low-spin gband (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
-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 superheavy nuclei

K-isomer contains important information on single
quasi-particles



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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

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I
 can be labeled by K
A projected intrinsic state P̂MK



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
With axial symmetry:  carries K
I
I
EI  H
/ N
defines a rotational band associated with
the intrinsic K-state 
Diagonalization = mixing of various K-states
K-isomers in 254No - interpretation
Projected shell model calculation

A high-K band with K = 8starts 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 276Sg
A superheavy rotor can vibrate

Take triaxiality as a parameter
in the deformed basis and do
3-dim. angular-momentumprojection


Microscopic version of the gdeformed 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
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
Multi-phonon g-bands

Multi-phonon g-vibrational
bands were predicted for
rear earth nuclei


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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
Bands in odd-proton 249Bk
Nilsson parameters of
T. Bengtsson-Ragnarsson
Ahmad et al., Phys. Rev.
C71 (2005) 054305
Slightly modified
Nilsson parameters
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 Dl=Dj=3,DK=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)


g-band
Triaxial octupole band