Pr

1
Role of deformation on giant resonances within
the QRPA and the Gogny force
S.PÉRU
2
Formalism
HFBQRPA
b b quasi-particle (qp) creation and
annihilation operators. In spherical symmetry,
QRPA states q are obtained for each block J?
. In axial symmetry , they are obtained for each
block K? .
KpJp
QRPA states are solutions of
In our approach, The effective interaction D1S
is used both in the HFB mean field and in the
QRPA matrix.

3
Restoration of rotational symmetry for deformed
states
We want to calculate
for all QRPA states (K J)
For example Jp 2
In intrinsic frame
Using rotational approximation and relations for
3j symbols
Using time reversal symmetry, three independent
calculations (Kp 0, 1, 2) are needed.
4
Role of deformation on giant resonances within
the QRPA and the Gogny force, SP and H. Goutte,
PRC 77, 044313 (2008)
Potential Energy Surfaces
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Pairing Energies
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IV Dipole
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IsoVector Giant Dipole resonance
MeV Th. Syst.
22Mg 23.09 23.4
24Mg 23.18 22.9
26Mg 23.13 22.5
28Mg 23.03 22.1
26Si 22.91 22.5
28Si 22.80 22.1
30Si 23.17 21.7
Systematic law EIVGDR 31.2 A-1/3 3 20.6 A-1/6
MeV
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Monopole
D.H. Youngblood, Y.-W. Lui, and H.L. Clark,
Phys.Rev.C 60 (1999) 014304
D.H. Youngblood, Y.-W. Lui, and H.L.Clark, Phys.
Rev. C, 65,(2002) 034302
0 10 20 30 40 MeV
0 10 20 30 40
MeV
9
Monopole
10
Giant Monopole resonance
MeV Th. Syst. Exp.
22Mg 21.11 28.5
24Mg 21.46 27.7 21.0 0.6
26Mg 21.93 27.0
28Mg 21.19 26.3
26Si 21.47 27.0
28Si 21.56 26.3 21.25 0.38
30Si 21.66 25.7
Systematic law EGMR 80 A-1/3 MeV
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Quadrupole
D.H. Youngblood, Y.-W. Lui, and H.L.Clark, Phys.
Rev. C, 65, (2002) 034302
D.H. Youngblood, Y.-W. Lui, and H.L. Clark,
Phys.Rev.C 60 (1999)014304
0 10 20 30 40 MeV
12
Quadrupole
13
Giant Quadrupole resonance
MeV Th. 63 A-1/3 Exp.
22Mg 20.62 22.5
24Mg 20.54 21.8 16.9 0.6
26Mg 20.97 21.3
28Mg 21.41 20.7
26Si 20,63 21.3
28Si 20.41 20.7 18.54 0.25
30Si 21.44 20.3
Systematic law EGQR 63 A-1/3 MeV
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Octupole
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Theoretical Octupole resonances
MeV LEOR Theor. LEOR Syst. HEOR Theor.
22Mg 12.37 10.70 34.48
24Mg 13.49 10.40 34.83
26Mg 12.13 10.12 35.72
28Mg 11.18 9.87 35.23
26Si 11.88 10.12 34.97
28Si 11.54 9.87 34.97
30Si 11.67 9.65 35.63
Systematic law ELEOR 30 A-1/3 MeV
Systematic law For heavy nuclei EHEOR 110 A-1/3
MeV
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Theoretical Predictions in Neons
J.Gibelin, Ph D Thesis Orsay 2005 ORSAY-RIKEN
collaboration
Systematic, E1 Strength predicted by QRRPA
calculations
1.2
26Ne (Elt10 MeV)
E 8 MeV
Spygmy /STRK 5

S e2fm2/MeV
28Ne
0
26Ne
0
24Ne
0
22Ne
0
20Ne
0
Cao L.-G. and Ma Z.-Y. Phys. Rev. C 71, 034305
(2005)
E MeV
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J.Gibelin, Ph D Thesis Orsay 2005 ORSAY-RIKEN
collaboration
L1 L2 Sum
B(E1) ? 0.49 0.16 e2 fm2 STRK 4.9 1.6 _at_
9 MeV J. Gibelin et al, PRL 101, 212503 (2008)
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responses for 26 Ne
IV Dipole
M1/M023.13 MeV
Quadrupole
Monopole
M1/M020.73 MeV
M1/M019.75 MeV
63 A-1/3 21.26 MeV
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Transition densities for pygmy
26Ne
Core is not inert
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Main components in pygmy wave function Proton
2qp components 15.97 MeV 2s1/2 1p1/2
17.60 MeV 1d5/2 1p3/2 17.48 MeV 1d5/2
1f7/2 Neutron 2qp components 10.52 MeV
2p3/2 2s1/2 13.68 MeV 1f7/2
1d5/2 12.43 MeV 2p1/2 2s1/2 10.82 MeV
2p3/2 1d5/2 18.50 MeV 1d3/2
1p1/2
Neutron
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responses for 26 Ne
IV Dipole
M1/M023.13 MeV
Quadrupole
Monopole
M1/M020.73 MeV
M1/M019.75 MeV
63 A-1/3 21.26 MeV
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HFBGCM with Gogny force D1S
26Ne is found to be deformed ltbgt0.3, ltggt30
A. Obertelli, SP, J.-P. Delaroche, A. Gillibert,
M. Girod et H. Goutte, Phys. Rev. C 71, 024304
(2005)
J. Gibelin et al, PRC 75, 057306 (2007)
QRPA
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CONCLUSION
A fully consistent microscopic axially-symmetric-d
eformed (ASD) QRPA approach using the D1S Gogny
force as been applied to light even-even Mg and
Si isotopes. In deformed nuclei IVGDR is found
split into 2 major components K0 and K1. In
prolate nuclei K0 components are found in the
low energy part of the spectra, and K1 in
oblate ones. Isoscalar monopole resonances also
display a splitting in deformed nuclei. Isoscalar
quadrupole and octupole resonances are found to
be well fragmented in particular in well-deformed
nuclei. Results obtained in 24Mg and in 28Si
show that the Gogny interaction qualitatively
reproduces known resonances without resorting to
any readjustment of parameters. Good agreement
between experimental and QRPA results
And coherence with GCMGOA calculations using
the same interaction (26Ne and Ni isotopes)
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Test of QRPA wave functions in proton inelastic
scattering
Perspectives
36S
HFBGCMGOA E (21) 2.34 MeV
B(E2) 375
e2fm4 HFBQRPA E (21)
3.29 MeV
B(E2) 139.7 e2fm4 Exp
E (21) 3.29 MeV
B(E2) 100 e2fm4
E. Bauge and S. Péru