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Atelier de l’Espace de Structure Nucléaire Théorique, Saclay February 4 – 6, 2008. Nuclear structure far from stability. Marcella Grasso. General interest: Correlations in finite fermion many-body systems. Adopted approaches: Microscopic mean field approaches and extensions. - PowerPoint PPT Presentation
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Atelier de l’Espace de Structure Nucléaire Théorique, Saclay
February 4 – 6, 2008
Marcella Grasso
Nuclear structure far from stability
General interest: Correlations in finite fermion
many-body systems
Adopted approaches: Microscopic mean field
approaches and extensions
DIFFERENT TOPICS:
Nuclear structure. Exotic nuclei (properties of exotic nuclei, pairing, continuum coupling, shell structure evolution along isotopic chains,…)
Mean field, HF, HFB + QRPA
Collaborations: Elias Khan, Jerome Margueron, Nguyen Van Giai, IPN-Orsay
Nicu Sandulescu, Bucarest
Nuclear astrophysics. Neutron star crusts (pairing, excitation modes, specific heat,…)
Mean field, HFB + QRPA
Collaborations: Elias Khan, Jerome Margueron, Nguyen Van Giai, IPN-Orsay
Extensions of RPA (avoiding the quasi-boson approximation)
Collaborations: Francesco Catara, Danilo Gambacurta, Michelangelo Sambataro, Catania
Interdisciplinary activity: ultra-cold trapped Fermi gases
Mean field, finite temperature HFB and QRPA
Collaborations: Elias Khan, Michael Urban, IPN-Orsay
Second meeting:
May 21 2007
Next meeting: to be fixed (2008)
Noyaux riches en neutrons – Approche self-consistante champ moyen + appariement Hartree – Fock – Bogoliubov (HFB)
Etats du continuum: comportement asymptotique (états de diffusion) et largeur des résonances
Isotopes of Ni
Drip line
A
S2n (MeV)
Neutron drip line position?
S2n(N,Z)=E(N,Z)-E(N-2,Z)
Microscopic mean field approach. Pairing is included in a self-consistent way (Bogoliubov quasiparticles): Hartree-Fock-Bogoliubov (HFB)
Two-neutron separation energy
Last observed isotope
Boundary conditions of scattering states for the wave functions of continuum states
Exp. values
Grasso et al, PRC 64, 064321 (2001)
Pairing and continuum coupling in neutron-rich nuclei. What to
look at?
Direct reaction studies: pair transfer? (LoI GASPARD for Spiral2)
Reduction of spin-orbit splitting for neutron p states in 47Ar
Gaudefroy, et al. PRL 97, 092501 (2006)
Transfer reaction 46Ar(d,p)47Ar: energies and spectroscopic factors of neutron states p3/2, p1/2 and f5/2 in 47Ar. Comparison with 49Ca: reduction the spin – orbit splitting for the f and p neutron states
Energy difference between the states 2s1/2 and 1d3/2
Grasso, Ma, Khan, Margueron, Van Giai, PRC 76, 044319 (2007)
Effect due to the tensor contribution with SLy5
Grasso, Ma, Khan, Margueron, Van Giai, PRC 76, 044319 (2007)
HF proton density in 46Ar with SkI5
Khan, Grasso, Margueron, Van Giai, NPA 800, 37 (2008)
INVERSION
Perspectives
• Particle-phonon coupling
• Extensions of RPA (to include correlations that are not present in a standard mean field approach). Applications to nuclei
B(E2;0+ g.s. -> 21
+) (e2fm4)
Riley, et al. PRC 72, 024311 (2005)
Raman, et al., At. Data Nucl. Data Tables 36, 1 (2001)
218 31 e2 fm4
SkI5 SLy4
Inv. No inv.
B (E2) (e2 fm4) 256 24
Khan, Grasso, Margueron, Van Giai,
NPA 800, 37 (2008)
Inversion of s and d proton states
Theoretical analysis. Relativistic mean field (RMF). 48Ca et 46Ar
48Ca Z=20
1d3/2 2s1/2
2s1/2 1d3/2
1d5/2 1d5/2
46Ar Z=18
1d3/2 2s1/2
2s1/2 1d3/2
1d5/2 1d5/2
Todd-Rutel, et al., PRC 69, 021301 (R) (2004)
Kinetic, central and spin – orbit contributions to the energy difference between the states 2s1/2 and 1d3/2
Grasso, Ma, Khan, Margueron, Van Giai, PRC 76, 044319 (2007)
Extension of RPA: starting from the Hamiltonian a boson image
is introduced via a mapping procedure (Marumori type)
Approximation:
Degree of expansion of the boson Hamiltonian (quadratic -> standard RPA)
If higher-order terms are introduced the RPA equations are non linear (the matrices A and B depend on the amplitudes X and Y)
Test on a 3-level Lipkin modelGrasso et al.
Diag. of HB in B
RPA
Extension
'00 222 qqq
qSO
WWV
q, q’ -> proton or neutron
Spin – orbit potential
Non relativistic case and standard Skyrme forces
Relativistic case
The potential is proportional to 'qq
Hartree-Fock equations with the equivalent potential.
rrrVrr
llr
dr
d
mljeq
,
1
2 22
22
rU
m
rmm
rm
rmm
rm
rmm
rmrU
m
rmrV
ljso
ljeq
)(
)(1
22
)(
22,
*
*''
*
2*
2'
*
2
2
2*
0
*
Equivalent potential:
Central term
Veqcentr
m
rmVVT so
centreq
)(*1
rU
m
rmm
rm
rmm
rm
rmm
rmrU
m
rmrV
ljso
ljeq
)(
)(1
22
)(
22,
*
*''
*
2*
2'
*
2
2
2*
0
*
dso
d
d
centreq
ds
centreq
s
dd
ss
ds
Vmrm
Vmrm
Vmrm
Tmrm
Tmrm
/)(*
1
/)(*
1
/)(*
1
/)(*
1
/)(*
1
Kinetic contribution
Central contribution
Spin-orbit contribution
Important contributions of the HF potential
qxxt 000 2122
1
2213
133 21222
24
1npqxxt
Central term
Density-dependent term
It favors the inversion
Against the inversion
…and the tensor contribution?
• Shell model : T. Otsuka, et al., PRL 95, 232502 (2005)
• Relativistic mean field: RHFB : W. Long, et al., PLB 640, 150 (2006)
• Non relativistic mean field:• Skyrme : G. Colò, et al., PLB 646, 227 (2007)• Gogny : T. Otsuka, et al., PRL 97, 162501
(2006)
Variation of the energy density (dependence on J)
pnpn JJJJH 22
2
1
)(4
31112
4
1)( 2
3rvlljjj
rrJ iiiii
iiq
''0 22 qq
qqqSO JJ
dr
d
dr
dWU
TC TC
221121 8
1
8
1xtxtttC 22118
1xtxtC
J -> spin density
The spin – orbit potential is modified: