M. Girod, F.Chappert, CEA Bruyères-le-Châtel

Neutron Matter and Binding Energies with a

New Gogny Force

Purpose of our study

, 0=0.16 fm-3

D1S Gogny force does not reproduce the EOS for neutron matter

Fit NM with Skyrme forces: PhD E. Chabanat Sly4

I) The fit to Neutron Matter EOS New Gogny force: D1N

II) Properties of D1N in nuclei

Contents

I) The fit to Neutron Matter EOS

New Gogny force: D1N

The Gogny force

Coulomb

.rr.W.i

2

rrrrPx1t

PPMPHPBW.exp

rrV

21122112ls

212100

2,1jjjjj

rr

21

2

j

2

21

�

14 parameters (Wi, Bi, Hi, Mi, i) for i=1,2; t0, x0, Wls

14 parameters 14 parameters 14 equations

• B.E. and radii: 16O et 90Zr G.S. properties• 2 pairing matrix elements pairing properties

• 48Ca: sym=2sN - 2s

P N-P asymmetry

• ………….

14 parameters determined

parameter « sym » Neutron Matter   ?

Test of the interaction: 0 , E0, K, Esurf, meff, Esym….

Link between the parameter symand the Neutron Matter EOS?

sym=2sN - 2s

P in 48Ca

Results in neutron matter: D1S, D1N

Results in nuclear matter with D1N?

Nuclear matter properties D1S-D1N

D1S D1N

0 (fm-3) 0.163 0.159

E0/A (MeV) -15.93 -15.96

K (MeV) 210 229

Esurf (MeV) 20.0 19.3

meff 0.70 0.75

Esym (MeV) 32.0 32.9

Wls (MeV) 130 115

Results in nuclei with D1N?

II) Properties of D1N in nuclei

1) Pairing properties

2) Binding energies

II) Properties of D1N in nuclei

1) Pairing properties

Pairing properties: D1, D1S, D1N

A1A1A B

2

BBB

, A odd

correlations

Pairing gap (Satula et al.)

Pairing energy in Sn isotopes

2

1E pair

HFB

Moment of Inertia in 244Pu

II) Properties of D1N in nuclei

2) Binding energies

Binding Energies: Sn isotopesE=EHFB-Eexp

D1S

D1N

Neutron Matter fit Drift of Binding Energies?

Binding Energies: more precise study

Binding Energies: Sm isotopesE=EHFB-Eexp

D1S

D1N

Binding EnergiesB=BHFB-Bexp

Conclusion Aim: build a new Gogny force which fits Neutron Matter EOS

D1N

Properties in nuclei:

Same pairing properties as D1S if not better(moments of inertia)

The drift of B.E. with N has disappeared

I) PAIRING

II) BINDING ENERGIES (B.E.)

Other calculations are being done: beyond mean-field

D1N should be soon validated: D1S D1N

Acknowledgements

Nuclear Structure Theory group:

J.F. Berger, M.Girod

B.Ducomet, H.Goutte, S.Peru, N.Pillet, V.Rotival

Results in neutron matter: D1S, D1N

Neutron Matter EOS with Gogny forces:

Pairing properties

Scattering lengths S=0, T=1:

18.50 fm Experimental value13.51 fm D1 12.12 fm D1S 10.51 fm D1N

Experiment: pairing force ~ bare force (Paris, AV18, ….)

Semi-empirical (Weiszäcker) mass formula

Empirical values:

av=-15.68, as=18.56, ac=0.717, aI=28.1 [MeV]

energysymmetry

2

I

energyCoulomb

3/12c

energysurface

3/2s

energyvolume

v A

)ZN(a

4

1AZaAaAa)Z,N(E

Calculation of the coefficients (av,as,aI)with the built interaction?

Pairing propertiesFull HFB calculation

Odd A: blocking approximation is used

Deviation with experiment:

Blocking approximation

B.E. of odd nuclei under-estimated

when quasi-particle-vibration coupling present

Kuo et al: few hundred keV correction

Pairing propertiesFull HFB calculation

Odd A: blocking approximation is used

Deviation with experiment:

Blocking approximation

B.E. of odd nuclei under-estimated

when quasi-particle-vibration coupling present

Kuo et al: few hundred keV correction

Inertia momenta in 232Th1

Inertia momenta in 232Th

1 12 2

Neutron Matter EOS:the variational method

non interacting WF

Trial wave-function:

A

jiijrfF F

kfkf

k

f(rij)f(rij)

f(rij) is varied until Evar is minimum

FF

HFFHEvar

Variational procedure:

f(rij) f(rij)

FF

HFFHE

rfF,F

var

A

jiij

Pairing properties: D1N

A1A1A B

2

BBB

, A odd

~200 keV

Binding Energies: D1S

B=BHFB-Bexp

Binding EnergiesB=BHFB-Bexp

Binding Energies: D1S

B=BHFB-Bexp

Binding EnergiesB=BHFB-Bexp

Binding EnergiesB=BHFB-Bexp

Results in neutron matter: D1S, D1N

Neutron Matter EOS with Gogny forces:

Pairing properties: D1

A1A1A B

2

BBB

, A odd

~300 keV

Pairing gap (Satula et al.)

Pairing properties

GS energy

HFB

correlations

Exp.?

Beyond HFB

Odd nucleus

Ecorr

Kuo et al. D1 D1S D1N

corr[keV] Few ~100 ~300 ~200 ~200

Corr.

HFB

GS

1st excited states

Eoddeven odd

Eeven

ST sub-spaces