1. Low energy nuclear collective modes and excitations
Low energy excitations in Neon isotopes, N=16 isotones and 68Ni within QRPA and Gogny force
Sophie Péru
Marco Martini

2. Beyond “mean field” … with RPA or QRPA
RPA approaches describe
all multipolarties and all parities,
collective states and individual ones,
low energy and high energy states
with the same accuracy.
A l’approximation de faible amplitude , i.e. pour des noyaux « harmoniques » E
δ2E/δq2>0
δ2E/δq2
δE/δq=0
qμν 2

3. Formalism
HFB+QRPA
{b+ b} quasi-particle (qp) creation and annihilation operators.
In axial symmetry , QRPA states {q+} are obtained for each block K
(Kp≤Jp)
They are solution of
In our approach,
The effective interaction D1S is used both in the HFB mean field and in the QRPA matrix.

4. High energy collective states: giant resonances
Giant resonances are related to nuclear matter properties
Monopole
Dipole
Quadripole
Octupole
IS GMR
spurious state
GQR
IV GMR
IV GDR

5. Spurious states « treatment »

6. Approach limited to Spherical nuclei with no pairing
RPA in spherical symmetry
Giant resonances in exotic nuclei:
100Sn, 132Sn, 78Ni; S. Péru, J.F. Berger, and P.F. Bortignon, Eur. Phys. Jour. A 26, (2005)
Monopole
Dipole
Approach limited to Spherical nuclei with no pairing
 Such study have shown the role of the consistence between mean field and RPA matrix.

7. Potential Energy Surfaces
QRPA in axial symmetry :
Potential Energy Surfaces

8. Formalism Restoration of rotational symmetry for deformed states
For example: Jπ = 2+
In intrinsic frame
Using time reversal symmetry,
three independent calculations (Kπ = 0+, 1+, 2+) are needed.

9. IV Dipole

10. Quadrupole D.H. Youngblood, Y.-W. Lui, and H.L. Clark,
Phys.Rev.C 60 (1999)014304
Quadrupole
D.H. Youngblood, Y.-W. Lui,
and H.L.Clark,
Phys. Rev. C, 65, (2002)

11. dipole response for Neon isotopes
Increasing neutron number
PDR and shift to low energies
Increasing of fragmentation
20Ne
22Ne
18Ne
PDR
proton
24Ne
26Ne
28Ne
PDR
PDR
PDR

12. B(E1) distributions

14. Evolution with A
Néon isotopes
N=16 Isotones Ne Ne Ne Ne Mg Si

15. ρ δρ 26Ne Pygmy w.f.(10.7 MeV) Proton: 2qp contrib. [e fm^-3]
15.97 MeV 2s1/2 1p1/2 X2=0.04
17.60 MeV 1d5/2 1p3/2 X2=0.02
17.48 MeV 1f7/2 1d5/2 X2=0.01
Neutron: 2qp contrib.
10.52 MeV 2p3/2 2s1/2 X2=0.67
13.68 MeV 1f7/2 1d5/2 X2=0.10
12.43 MeV 2p1/2 2s1/2 X2=0.09
10.82 MeV 2p3/2 1d3/2 X2=0.03
18.50 MeV 1d3/2 1p1/2 X2=0.01
[e fm^-3]
[fm^-3]

16. ρ δρ 28Mg Pygmy w.f. (11.6 MeV) [e fm^-3] Proton: 2qp contrib. [fm^-3]
MeV 2s1/2 1p1/2 X2=0.06
MeV 1f7/2 1d5/2 X2=0.02
Neutron: 2qp contrib.
MeV 2p3/2 2s1/2 X2=0.40
MeV 2p3/2 1d3/2 X2=0.33
MeV 1f7/2 1d5/2 X2=0.06
MeV 2p1/2 2s1/2 X2=0.06
MeV 2p1/2 1d5/2 X2=0.03
[e fm^-3] δρ
[fm^-3]

17. ρ δρ 30Si Pygmy w.f. (12.2 MeV) [e fm^-3] Proton: 2qp contrib
MeV 2s1/2 1p1/2 X2=0.19
MeV 1f7/2 1d5/2 X2=0.07
MeV 2p3/2 1d5/2 X2=0.01
Neutron: 2qp contrib.
MeV 2p3/2 2s1/2 X2=0.44
MeV 1f7/2 1d5/2 X2=0.11
MeV 2p3/2 1d3/2 X2=0.07
MeV 2p1/2 2s1/2 X2=0.07
[e fm^-3] δρ
[fm^-3]

18. ρ δρ 18Ne Pygmy w.f.(14.2 MeV) Proton: 2qp contrib.
MeV 1f7/2 1d5/2 X2=0.26
MeV 2p3/2 1d5/2 X2=0.23
MeV 2s1/2 1p1/2 X2=0.12
MeV 2p3/2 2s1/2 X2=0.09
16.95 MeV 1d5/2 1p3/2 X2=0.09
MeV 2p1/2 2s1/2 X2=0.04
Neutron: 2qp contrib.
MeV 1d3/2 1p3/2 X2=0.04
MeV 2s1/2 1p1/2 X2=0.04 ρ δρ
[e fm^-3]
[fm^-3]

20. Correlations between PDR and symmetry energy
Carbone et al. Phys. Rev. C (2010)
Effective Lagrangians
Skyrme Forces
EWSR [%]
L [MeV]
D1S
1,2
22,5
D1N
1,5
33,6
D1M
1,3
25,3
Similar study with Gogny:
(absent in literature)
M. Martini

21. Dipole Resonances in 68Ni
Gogny
Effective Lagrangian
Comparison among models and forces
Skyrme
M. Martini

22. Dipole response for Zr isotopes :
B(E1)
B(M1)
86Zr
88Zr
90Zr
M1 γ strength for 92Zr,
H. Utsunomiya et al,
PRL 100, (2008)
92Zr
94Zr
96Zr
S.Goriely,
H. Goutte,
S. Hilaire,
M. Martini,
S. Péru, …

23. = 5 Dimension Collective Hamiltonian
Beyond mean field … with GCM
(GCM+GOA 2 vibr. + 3 rot.)
= 5 Dimension Collective Hamiltonian
5DCH

24. HFB+QRPA / HFB+5DCH with the same interaction:
A. Obertelli, et al, Phys. Rev. C 71, (2005)
N=16 isotones
S. Péru,AIPS Conference proceedings No. 1165,NSD09, (Melville, New York, 2009), p165

25. QRPA/5DCH
Sn isotopes

26. Ni isotopes
S. Péru,AIPS Conference proceedings No. 1165,NSD09, (Melville, New York, 2009), p165

27. Spectroscopy in neutron rich Ni isotopes within QRPA
L. Gaudefroy, S. Péru …

28. Spectroscopy in neutron rich Ni isotopes within QRPA

29. 0+ states in 68Ni within QRPA
M. Martini & S. Péru

30. 0+ states in 68Ni within QRPA Transition densities
Protons
Neutrons
Transition densities
Protons
Neutrons
Protons
Neutrons
M. Martini, S. Péru …

31. … beyond the nuclear structure :
Test of QRPA and 5DCH (GCM) wave functions in proton inelastic scattering…
36S
HFB+5DCH E (2+1) = 2.34 MeV
B(E2) = 375 e2fm4
HFB+QRPA E (2+1) = 3.29 MeV
B(E2) = e2fm4
Exp E (2+1) = MeV
B(E2) = 100 e2fm4
E. Bauge and S. Péru

32. differential cross sections for 14.1 MeV neutron on 238 U (a,c).
Comparison between experimental data (circles) and one-step contributions (full curves) to the double-
differential cross sections for 14.1 MeV neutron on 238 U (a,c).
M. Dupuis ,E. Bauge,L. Bonneau,J.-P. Delaroche ,T. Kawano ,S. Karataglidis and S. Péru,
Proceedings of the Second International Workshop on Nuclear Compound Reactions and Related Topics, (2010).