Some results obtained with Gogny force
included in HFB, QRPA as well as in
configuration mixing GCM like approach
Sophie Péru
M. Dupuis, S. Hilaire, F. Lechaftois (CEA, DAM),
M. Martini (Ghent University, Belgium),
S. Goriely (Université Libre de Bruxelles, Belgium),
I. Deloncle (CSNSM, Orsay)
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
Short Reminder
Gogny
Static mean field (HFB)
for Ground State Properties :
• Masses
• Deformation
• (Single particle levels)
DAM, DIF, S. Péru
Amedee database :
http://www-phynu.cea.fr/HFB-Gogny_eng.htm
S. Hilaire & M. Girod, EPJ A33 (2007) 237
Beyond static mean field approximation (5DCH or QRPA)
for description of Excited State Properties
• Low-energy collective levels
• Giant Resonances
FUSTIPEN, 16th-20th March 2015
Beyond mean field … with 5DCH
(2 vibr. + 3 rot.) = 5 Dimension Collective Hamiltonian
Potential energy surfaces
Wave function
of the collective 01+
h2
ˆ
H coll =
2
DAM, DIF, S. Péru
Triaxial
axial
3
∑
k =1
2
Iˆk
h2
−
Jk
2
∑
m, n = 0 and 2
D −1 / 2
∂
−1 ∂
D 1 / 2 (B mn )
+ V (a 0 , a 2 ) − ∆ V (a 0 , a 2 )
∂a m
∂a n
FUSTIPEN, 16th-20th March 2015
Some exploitation of 5DCH with Gogny forces
Systematics studies
D1S: J. P. Delaroche et al. PRC 81, 014303 (2010)
D1M: S. Hilaire et al. PRC 86, 064317 (2012)
S. Goriely, S. Hilaire, M. Girod, S. Péru , PRL 102, 242501 (2009)
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
Shell closure studies with 5DCH
E (2+1) Theo. = 1.32 MeV
E
(2+1) Exp.
Exp.
N=20, 32Mg
S. Péru, M. Girod, JF. Berger ,
Eur. Phys. J. A 9,35-47, (2000)
DAM, DIF, S. Péru
= 885 keV
N=28, 44S
E(2+1)Theo. = 1.46 MeV,
E(2+1) Exp. = 1.297 (0.018)
0.018)MeV,
B(E2,0+gs→2+1) = 420 e2fm4
B(E2,0+gs→2+1) = 314 (88)
88) e2fm4
FUSTIPEN, 16th-20th March 2015
Further analysis : moments of inertia
32Mg
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
Further analysis : moments of inertia
32Mg
Sulfur isotopes
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
Shell closure studies with 5DCH
N=16
5DCH
J. Gibelin et al,
PRC 75, 057306 (2007)
A. Obertelli, S. Péru, J.-P. Delaroche, A. Gillibert, M. Girod et H. Goutte, Phys. Rev. C 71, 024304 (2005)
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
N=40
Shell closure studies with 5DCH
+ 5DCH
Expt
(a)
+ 5DCH
31
D1S, 5DCH
1.5
+ 5DCH
42
4
0.5
2.5x10 3
B(E2) (e 2 fm4)
22
5
E (MeV)
E (MeV)
2.5
(b)
1.5
+ Expt
22
+ Expt
31
+ Expt
42
+ Expt
5+1 5DCH 51
+ 5DCH
62
3
+ Expt
62
2
0.5
22
24
26
28
30
32
34
36
38
40
1
42
Z
Expt
E (MeV)
2
22 24 26 28 30 32 34 36 38 40 42
Z
D1S, 5DCH
1.5
1
0.5
First excited 0+
22 24 26 28 30 32 34 36 38 40 42
Z
rotor
0
2.5
2
γ soft
-0.5
Qs (eb)
E(41+)/E(21+)
3
vibrator
Q2+(Expt)
1
-1
Expt
1.5
D1S, 5DCH
Q2+(5DCH)
1
Q4+(5DCH)
1
Q6+(5DCH)
1
22 24 26 28 30 32 34 36 38 40 42
22 24 26 28 30 32 34 36 38 40 42
Z
DAM, DIF, S. Péru
Z
L. Gaudefroy et al, Phys.Rev. C 80, 064313, (2009)
FUSTIPEN, 16th-20th March 2015
Beyond static mean field … with 5DCH or QRPA
5 Dimension Collective Hamiltonian
describes ground state and excited states
within configuration mixing :
quadrupole vibration
and rotational degrees of freedom.
HFB+D1S
(Q)RPA approaches describe
all multipolarities and all parities,
collective states and individual ones,
low energy and high energy states
with the same accuracy.
But small amplitude approximation
i.e. « harmonic » nuclei
E
δE/δq=0
SM
Exp.
δ2E/δq2>0
5DCH
S.Péru and M. Martini, EPJA (2014) 50: 88. D. Sohler et al, PRC 66, 054302 (2002)
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
HFB+QRPA versus HFB+5DCH with the same interaction
N=16 isotones
Sn isotopes (Z=50)
5DCH : A. Obertelli, et al, Phys. Rev. C 71, 024304 (2005)
S.Péru and M. Martini, EPJA (2014) 50: 88.
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
HFB+QRPA versus HFB+5DCH with the same interaction
Ni isotopes (Z=28)
Two shell (N= 28, 50) and one sub-shell (N=40) closures
! For deformed nuclei
the first 2+ state is rotational
78Ni
is predicted doubly magic
S.Péru and M. Martini,
EPJA (2014) 50: 88.
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
Beyond mean field… with QRPA
RPA approaches describe
all multipolarties and all parities,
collective states and individual ones,
low energy and high energy states
with the same accuracy.
Within the small amplitude approximation,
approximation, i.e. « harmonic » nuclei
Spherical RPA with Gogny force
E
J. Dechargé and L.Sips, Nucl. Phys. A 407,1 (1983)
J.P. Blaizot, J.F. Berger, J. Dechargé, M. Girod, Nucl. Phys. A 591, 435 (1995)
S. Péru, JF. Berger, PF. Bortignon, Eur. Phys. J. A 26, 25-32, (2005)
δE/δq=0
δ2E/δq2>0
Axially symetric deformed QRPA with Gogny force
S. Péru, H. Goutte, Phys. Rev. C 77, 044313, (2008)
M. Martini, S. Péru and M. Dupuis, Phys. Rev. C 83, 034309 (2011)
S. Péru et al, Phys. Rev. C 83, 014314 (2011)
RPA approaches are well adapted for describing giant resonances
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
QRPA Formalism
2 p1/2
2 p3/2
p-p
RPA
Particle-hole excitations
+
ν
ah + a p
Qν = ∑ X νph a p+ a h + Y ph
ph
QRPA
2 quasi-particle excitations
+
ν + +
Qν = ∑ X ij η i η j + Y ijν η jηi
ij
ηi+ = ∑ uiα aα+ − viα aα
α
Hartree-Fock Bogoliubov: ε, u, v
QRPA: ω, X, Y
Ground state properties
Excited states properties
p-h
1 f7/2
1 d3/2
2 s1/2 Fermi
1 d5/2
1 p1/2
1 p3/2
Neutron’s HF
h-h
levels
26Ne
1 s1/2
Same interaction (Gogny) in HFB and QRPA
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
RPA in spherical symmetry
Giant resonances in exotic nuclei:
100Sn, 132Sn, 78Ni;
Monopole
S. Péru, J.F. Berger, and P.F. Bortignon, Eur. Phys. Jour. A 26,
26 25-32 (2005)
Dipole
Quadrupole
Such study have shown
the role of the consistence
between mean field and RPA matrix.
matrix.
Approach limited to Spherical nuclei with no pairing
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
Axially-symmetric deformed QRPA
J
Restoration of rotational symmetry for deformed states
JM (K )
to calculate:
=
~ ˆ
0 Qλµ JM (K )
K
2J +1
J −K
J
d
Ω
D
(
Ω
)
R
(
Ω
)
θ
+
(
−
)
DMJ − K (Ω ) R (Ω ) θ K
MK
K
∫
4π
for all QRPA states (K ≤ J)
Qˆ λµ ∝ ∑ r λ (Yλµ )
λ 2
r 2Yλµ = ∑ Dµυ
r Yλυ
υ
In intrinsic frame
We use rotational approximation and relations for 3j symbols
For example: Jπ = 2+
~ ˆ
0Q
JM (K ) =
1
3
3
0 Qˆ 20 θ K δ K , 0 +
0 Qˆ 2−1 θ K δ K , ±1 +
0 Qˆ 22 θ K δ K , ±2
20
5
5
5
Using time reversal symmetry, three independent calculations (Kπ = 0+, 1+, 2+) are needed.
Main features:
• Possibility to treat axially-symmetric deformed nuclei
• Pairing correlations consistently included
• Use of an unique nuclear force:
force finite range Gogny force
- same interaction for all the nuclei
- same interaction for ground state and excited states (self-consistency)
essential features to treat consistently isotopic chains from drip line to drip line
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
QRPA in axial symmetry
IV Dipole
dipole
Potential
Energy
Surfaces
0.4
0.4
26
Si
0.4
28
Si
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
30
Si
-
K =0
K =1
Kπ=0-
K=0
0.4
0.4
22
Mg
K=1
J
Kπ=10.4
24
Mg
0.4
26
Mg
0.3
0.3
0.3
0.3
0.2
0.2
0.2
0.2
0.1
0.1
0.1
0.1
28
Mg
K
0
10
20
30
40 MeV
0
10
20
30
40 MeV
0
10
20
30
40 MeV
0
10
20
30
40 MeV
S. Péru and H. Goutte, Phys. Rev. C 77, 044313 (2008).
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
Dipole response for Neon isotopes
Increasing neutron number
•Low energy dipole resonances and shift to low energies
•Increasing of fragmentation
Ne : B(E1) = 0.49 ± 0.16 e fm
26
18Ne
24Ne
2
2
%STRK = 4.9 ± 1.6 @ 9 MeV
J. Gibelin et al, PRL 101, 212503 (2008)
20Ne
22Ne
26Ne
28Ne
B(E1)=0.173 e2fm2
@ 10.69 MeV
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
Dipole response for Neon isotopes and N=16 isotones
GDR
PDR
M. Martini, S. Péru and M. Dupuis, Phys. Rev. C 83, 034309 (2011)
Increasing |N-Z| number :
•Low energy dipole resonances shift to low energies
•Increasing of fragmentation and collectivity
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
Multipolar responses for 238U
Heavy deformed nucleus
0+
massively parallel computation
1J
K
2+
3-
S. Péru et al, PRC 83, 014314 (2011)
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
Beyond the nuclear structure
(n, x n) cross section on
238
U
Problem of underestimation of
n emission cross section at high energy
QRPA provides
enough collective contribution
M. Dupuis, S.Péru, E. Bauge and T. Kawano,
13th International Conference on Nuclear Reaction Mechanisms, Varenna 2012
CERN-Proceedings-2012-002, p 95
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
Nuclear Masses
Comparison with experimental data (2149 nuclei: Audi, Wapstra & Thibault 2003)
Gogny D1M
r.m.s = 0.798 MeV
S. Goriely, S. Hilaire, M. Girod, S. Péru , PRL 102, 242501 (2009)
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
Dipole excitations in QRPA and photoabsorption results
D1S versus D1M
Spherical nuclei
First peak
Second peak
Deformed nuclei
We calculate E1 strength for all the nuclei for which photoabsorption data exist
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
Dipole excitations in QRPA and photoabsorption results
Size of the basis
Spherical nuclei
First peak
Second peak
Deformed nuclei
We calculate E1 strength for all the nuclei for which photoabsorption data exist
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
Dipole excitations in QRPA and photoabsorption results
With folding and shift ….
60Ni
28Si
90Zr
94Zr
70Ge
94Mo
74Ge
98Mo
76Ge
116Sn
80Se
120Sn
σ [mb]
Exp.
QRPA
124Te
128Te
138Ba
142Ce
144Nd
148Nd
144Sm
150Sm
158Gd
174Yb
206Pb
232Th
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
Dipole excitations in QRPA and photoabsorption results
To take into account complex configurations as well as coupling with phonons, the deformed QRPA
strength SE1(w) is folded with a Lorentzian function.
The width of the Lorentzian is adjusted on experimental data (RIPL-3 IAEA) while the energy shift is
related to the number of 4qp keeping in this way a connection with our microscopic calculation.
Calculations performed also for isotopic chains of astrophysical interest (e.g. Sn).
S .Goriely,
M. Martini,
DAM, DIF,
S. Péru S. Hilaire, F. Lechaftois and S. Péru
FUSTIPEN, 16th-20th March 2015
γ- ray strength functions predictions for exotic nuclei
e.g. Sn isotopes
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
Photoneutron cross sections for Mo isotopes
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
Photoabsorption cross sections for Mo isotopes
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
Dipole electric and magnetic excitations for Zr isotopes
E1
M1
E1
M1
H. Utsunomiya et al. (2008)
H. Utsunomiya et al, PRL 100, 162502 (2008)
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
Nuclear Excitations
Photo-absorption
Electron scattering
e-
(N,Z)
γ
(p,p) or (n,n)
p
(N,Z)
γ
e-
n
or
π0
(N,Z)
p
n
p n
Charge exchange:
β decay
e-
Neutrino scattering
e-
ν
n
(N-1,Z+1)
W+
(N,Z)
(p,n) or (3He,t)
(N-1,Z+1)
W+
(N,Z)
(N-1,Z+1)
π+
(N,Z)
p
n
ν
p
Spin-isospin nuclear excitations (in particular GT)
• Crucial role in nuclear physics, astrophysics and particle physics
• Experimentally studied via charge exchange reactions , e.g. (p,n) and β decay
• Theoretical models to study the nuclei experimentally inaccessible
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
pnQRPA
General expression
QRPA
p
n
pnQRPA
p
DAM, DIF, S. Péru
n
FUSTIPEN, 16th-20th March 2015
Main charge exchange excitations
p
Isobaric Analog Resonance (IAR)
n
isospin flip τ
S=0
T=1
Jπ=0+
p
n
p
n
Gamow Teller (GT)
isospin flip τ
spin flip σ
DAM, DIF, S. Péru
S=1
T=1
Jπ=1+
FUSTIPEN, 16th-20th March 2015
Checks
When Coulomb repulsion is out of HFB calculations,
the energy of the IAR dives to zero.
Sum Rules :
Ikeda
Fermi
DAM, DIF, S. Péru
F: 13.9999993456112
114Sn
GT/3: 13.9999993456064
F: 32.0000000000001
132Sn
GT/3: 32.0000000000001
FUSTIPEN, 16th-20th March 2015
Gogny pnQRPA Strength Distributions
M. Martini, S. Péru and S. Goriely, Phys. Rev. C 89, 044306 (2014)
Good agreement with experimental data
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
An example of deformed nucleus : 76Ge
GT Jπ=1+ distributions obtained by adding twice the Kπ=1+ result to the Kπ=0+ one
pnQRPA-D1M
HFB
D1M
2
β2(min. HFB) = 0.15
β2 (0+1:5DCH) = 0.26
prolate
γ(min.HFB) = 0°
γ(0+1:5DCH) = 26°
Experiment
Thies et al., Phys. Rev. C 86, 014304 (2012)
•
•
•
The deformation tends to increase the fragmentation
Displacements of the peaks
Deformation influences the low energy strength hence β decay half-lives are expected to be affected
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
β- decay half-life T1/2
Comparison with experimental data for 145 eveneven-even nuclei
•
•
G.Audi et al.
Chinese Phys. C 36,
36
1157 (2012)
Deviation with respect to data rarely exceeds one order of magnitude
Larger deviations for nuclei close to the valley of β-stability, as found in most models
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
Comparison with other models
Our model
Other models
FRDM: Moller et al., ADNDT, 66,131 (1997)
GT2:Tachibana
et al.
GT2:
Prog. Theor. Phys., 84, 641 (1990)
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
β- decay half-lives of deformed isotopic chains
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
β- decay half-lives of the N=82, 126, 184 isotones
Relevance for the r-process nucleosynthesis
Shell Model : MartinezMartinez-Pinedo et al., PRL 83, 4502 (1999)
DAM, DIF, S. Péru
DF3+cQRPA : Borzov et al., PRC 62, 035501 (2000)
FUSTIPEN, 16th-20th March 2015
Even and odd systems, deformed and spherical nuclei
Preliminary results
Extension to odd systems
in collaboration with
Isabelle Deloncle (CSNSM) Orsay
28
Recent experimental results
Z.Y. Xu et al, PRL 113, 032505 (2014)
β-decay Half lives of 76,77Co, 79,80Ni and 81Cu :
Experimental indication of a Doubly Magic 78Ni
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015
To summarize
Great successes using the finite range Gogny force:
5DCH : good reproduction of collective low energy spectra and shell effects
QRPA : good description of pygmy and giant resonances in spherical or deformed nuclei
QRPA and 5DCH complete each other.
Some results were shown:
* Self-consistent QRPA approach has been applied to the deformed nuclei up to 238U.
* IsoScalar-IsoVector mixing for low-lying dipole states in Ne isotopes and N=16 isotones.
* QRPA results successfully used in reaction models : (n, x n) , photoabsorption …
Extension of QRPA to charge exchange :
* For magic spherical nuclei, IAR and GT results in good agreement with data.
* The role of the intrinsic deformation has been shown for prolate 76Ge.
* Predictions of the β decay half-life time are compatible with experimental data.
* Satisfactory agreement with experimental half-lives which justifies the additional study on the
exotic neutron-rich N = 82, 126 and 184 isotonic chains (r-process).
Promising preliminary results for odd nuclei.
DAM, DIF, S. Péru
FUSTIPEN, 16th-20th March 2015