Diapositive 1

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Triaxial deformations some results from
mean-field based calculations
J.-P. Delaroche, M. Girod, H. Goutte, S.
Hilaire, S. Péru, N. Pillet CEA, DAM,
DIF Collaboration with J. Libert IPN
Orsay G. Bertsch Univ. Seattle, USA A.
Gillibert, A Gorgen, L. Jungvall, W. Korten, A.
Obertelli, Ch. Theisen CEA, DSM, IRFU
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Outline

• The 5-dimensional collective Hamiltonian
  approach (5DCH)
• Figure of merit of the 5DCH systematics of 21
  excitations
• - Examples of manifestations of the triaxial
  degree of freedom
• - Shape coexistence in Kr and Se isotopes
• -  Intermediate  K0 and K2 structures in
  250Cm
• - Superdeformed collective levels in actinides
• - First barriers in actinides
• - Conclusion

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Mean-field-based approaches

• Main advantages
• ? to study all the nuclei including exotic ones
  produced at SPIRAL,
• and SPIRAL2.
• Predictions possible
• ? to give a geometrical image in terms of nuclear
  deformation,
• upon which spectroscopy strongly depends.

Ground-state deformation from axial HFB
calculations Gogny D1S force
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Different approaches
Short range
Long range
Large amplitude
Small amplitude
pairing
Short range correlations
Name
Collective correlations
Coupling HF g.s. and 1p/1h states
Coupling HF g.s and S0 T1 2p/2h
Repulsive part of the short range of the nuclear
force
Origin
Shape coexistence
Independent quasi-particles
Independent particles
W.f. of the g.s.
G.s w.f more general than w.f. of independent
particles or qp
HFB
GCM 5DCH
Hartree-Fock theory effective force
Method
RPA/QRPA
mp-mh configuration mixing
Particle -vibration coupling
Soft
Rigid
Open shell nuclei
all
Which nuclei ?
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5-dimensional collective Hamiltonian from GCMGOA
Microscopic collective Hamiltonian (full
quadrupole)
Rotation Three moments of inertia
Thouless-Valatin prescription
Vibration Three mass parameters Inglis-Belyaev
prescription
Potential energy
ZPE correction
Eigenstates are solutions of
Matrix elements use of the local approximation
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Figure of merit of the 5-dimensional collective
Hamiltonian approach
Systematics of the first 2 excitation
energy with the Gogny interaction, G. Bertsch, et
al. PRL 99, 032502 (2007)
Elimination of 39 nuclei among the 557 of the
compilation of Raman Light nuclei with Z or
N lt 8 (16) doubly magic nuclei (23)
Z 80 82 , N 104
Strongly deformed actinides
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Shape coexistence in N 104 Hg and Pb isotopes
See also J.P. Delaroche et al., PRC 50 2332 (1994)
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Systematics of B(E2,0 -gt 2) with the Gogny
interaction
Strongly deformed actinides
It is clear that 5DCH theory performs much
better for strongly deformed nuclei than for
others.
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Systematics of quadrupole moments with the Gogny
interaction
160Dy, 170,174,176Yb and 180W the sign of their
moments is not exp. known -gt they are here all
predicted to be prolate with a negative
quadrupole moment
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Statistics for the performance of the 5DCH theory
using the Gogny D1S interaction
Classification of Sabbey et al. PRC 75 044305
(2007) deformed ? lt?gt is larger than the
r.m.s. fluctuations in lt?gt, Semimagic ? either
Z or N 8, 20, 28, 50, 82 or 126 Other ? the
remaining nuclei
R ? log(th/exp) ? ? lt ?R2gt1/2
Excitation energy
Transition matrix elements
Deformed nuclei results are within 5 of the
exp. values Semimagic are the poorest Global
performance Averages within 13 and
dispersions lt 40
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Shape coexistence in krypton isotopes

• Experimental evidence for shape coexistence
• ? Observation of a low-lying 02 state in
  even-even nuclei
• ? Irregularities in the g.s band at low spins
• (due to the mixing of the 2 rotational bands)
• ? Decrease of the B(E2) from the high spin states
  (6) to the g.s.

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Shape coexistence in krypton isotopes experiment
Shape coexistence in light krypton isotopes was
studied in low energy Coulomb excitation
experiments using radioactive 74Kr and 76Kr beams
from the SPIRAL facility at GANIL ( g.s band up
to 8 via multistep Coulomb excitation and
several non-yrast states)
In both isotopes, the spectroscopic quadrupole
moments for the g.s. and the bands based on
excited 02 states are found to have opposite
signs.
E. Clément, et al., PRC 75 (2007) 054313.
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Shape coexistence in krypton isotopes results
(mostly) prolate oblate K2 ?
vibration
E. Clément et al., Phys. Rev. C 75, 054313
(2007).
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Shape coexistence in krypton isotopes comments
G.s. band well defined
A grouping of the non-yrast states above the 02
state into band structures is not straightforward
22,31, 42, 51 and 62 ?-vibrational
band (see K 2 components)
In this scenario the 23 states in both isotopes
can be interpretated as rotational states with
oblate character built on the excited 02
states (see excitation energy and transition
strengths , sign of the diagonal matrix
elements). But the strong mixing of K0 and K2
components for 22 and 23 states blurs a clear
classification of these states. (Problem in the
decay of the 22 state)
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Theoretical observables
How do we proceed to analyze the theoretical
results from the five dimensional collective
Hamiltonian ? We look at the

• Energies
• Mean axial and triaxial deformations lt?gt, lt?gt,
  and fluctuations lt??gt, lt??gt
• Transition strength
• K components of the wave function

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G.s band
Gamma vibrational band
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Shape coexistence in neutron-deficient krypton
isotopes conclusion

• 76Kr and 74Kr are both prolate in the g.s.
• Excited 22 states are interpreted as
  ?-vibrational states
• Excited 23 states as oblate rotational states
• A clear classification of the states is difficult
  due to the mixing of
• prolate and oblate configurations,
• K0 and K2 configurations.

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Shape coexistence in light selenium isotopes
Theory
Theory
Theory
Exp.
Exp.
Exp.
B(E2) decreases from the 6 to the g.s.
(increase of oblate and prolate configuration
mixing for the low-spin states)
J. Ljungvall et al., Phys. Rev. Lett. 100,
102502 (2008).
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Shape transition in light selenium isotopes
?70-72Se Shape transition in the yrast band
(red and blue) 68Se G.s. band oblate
(black) excited prolate shapes (green)
J. Ljungvall et al., Phys. Rev. Lett. 100,
102502 (2008).
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Shape coexistence conclusions and remarks

• ?The explicit treatment of triaxiality is
  essential to describe shape coexistence.
• ? New regions of shape coexistence ? N 28, A
  100
• ? New developpements
• Configuration mixing of angular-momentum and
  particle-number projected triaxial HFB states
  using the Skyrme energy density functional.
• M. Bender, and P.H. Heenen, PRC 78 024309 (2008)

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 Intermediate  structure in 250Cm
These states would manifest as broad structures
in fission probability measurements
H.C. Britt et al., PRL 40 (1978) 1010.
J.-P. Delaroche et al., NPA 771 (2006) 103.
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SD collective levels in actinides
? Very good agreement with exp. data for shape
isomers
? Overprediction of quadrupole one-phonon
energies especially ? phonons
? Changing the mass parameter cures the K0 1
phonon but the ? phonon energies remain
unaltered.
?challenging issue ?extension of the collective
model space to include other collective
coordinates Ex hexadecapole predicted to be
important for the stability of trans-actinide
nuclei. Z. Patyk, A. Sobiczewski, NPA 533
(1991) 132
J.-P. Delaroche et al., NPA 771 (2006) 103.
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First barriers in actinides
Least energy paths from constrained HFB
calculations

• First barrier triaxial
• Second barrier asymmetric N lt 152
• SD minima washed out for N gt 156

Axial and symmetric Triaxial Asymmetric
No gap found at the SD minimum
J.P. Delaroche, M. Girod, H. Goutte, J. Libert,
NPA 771 (2006) 103.
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Least action path in (axial, triaxial) quad.
plane Inner barrier
Search for the trajectory L, which minimizes the
action S

• Least action path goes
• through axial saddles in many actinides
• Full quadrupole treatment is esential

Potential energy in the well
Mass parameter
Least Action Path Lowest Energy
Path
J.P. Delaroche, M. Girod, H. Goutte, J. Libert,
NPA 771 (2006) 103.
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Conclusion

• ? 5DCH an approach which can be applied in more
  than 90 of the nuclei
• ? Work in progress a global study of a few
  excited levels
• (g.s. rotational band, ?-band, ?-band )
• ? Improvement
• beyond the Inglis Belyaev approximation for
  the mass parameters
• using the Quasi-Particle Random Phase
  Approximation (see S. Pérus talk)