Atelier de l’Espace de Structure Nucléaire Théorique, Saclay February 4 – 6, 2008

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: