Beyond mean-field versus shell model

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Beyond mean-field versusshell model

• The beyond mean-field method
• 16O a nice meeting point
• Existing codes
• Developments, open questions, possible answers

P. Bonche, H. Flocard, P.-H. Heenen M. Bender
(Bordeaux) T. Duguet (MSU)
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• Starting point
• The mean-field method
• Skyrme HFB
• Pairing with a density-dependent zero range
  interaction
• Solution of the HF equations on a 3D cartesian
  mesh
• (unrestricted nuclear shape)
• Triaxiality always included, same accuracy for
  any shape,
• Octupole,.
• Lipkin Nogami method to simulate a Variation
• after N,Z projection

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• Beyond mean-field
• Set of mean-field wave functions depending on a
  collective
• 
  variables
• Projection on N, Z, J

• New wave functions by mixing on q

with fJ,k (q) determined by minimizing the energy
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16O mean-field and projection (axial Q)
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Single particle levels
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Configuration mixing
Energies Wave functions
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How to analyze the wave functions in shell model
terms?
Mean-field basis changes for each
deformation Construction of a basis formed by
- the spherical mf state - np-nh
deformed HF states projected on J0 After
convergence, the np-nh are below the Fermi
levels and cause the deformation of the
mean-field state.
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Decomposition in ph components
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Self-consistent HF states
Configuration mixing states
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Conclusion

• PLUS
• -Very rich basis with many ph components
• (more precisely qp excitations with respect)
• with respect to a spherical basis
• GMC is not limited to small amplitudes as the
  QRPA
• MINUS
• Kind of ph determined by the constraint
• Only time reversed pairs are excited

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• Projection on angular momentum
• From intrinsic to laboratory frame of
  reference
• No approximation based on the collective model
• for transition probabilities.

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CODES

• Existing
• In (active) preparation
• What should be done

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Mean-field Codes
tetrahedral shapes stability?
qp excitations in transfermium
T0 pairing
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From static to dynamics
tailored HW program
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From static to dynamics
HW program!
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From static to dynamics
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The Skyrme functionals (TRI)
10 parameters
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The densities (TRI)
qn or p
The sp wave functions are obtained from the HF
equation
Static to dynamic densities replaced by mixed
densities
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24Mg, g30, Q0 80 fm2, cranking state Jz2
reconstruction of all matrix elements from the
projected ones (very
strict test of codes!)
without
summation
projection on J from J0 to 8
norm 0.279534
0.279186 total energy
-194.463 -194.560 r2a
1387.79
1387.80 pairing neutron -1.99581
-2.01915 pairing proton
-1.98091 -1.99062
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Esperance the new code
Projection of triaxial wave functions
(ok) Projection of cranking wave functions
(nearly ok?)
e-e nuclei
Nearly ready for odd nuclei, any kind of qp
excitations,
Same range of applications as the shell
model Tool to test models (from the Bohr
Hamiltonian to
tunneling in the (Q, g) plane