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Variational Multiparticle-Multihole Configuration Mixing Method with the D1S Gogny force. Nathalie Pillet (CEA Bruyères-le-Châtel, France). Collaborators: J.-F. Berger, E. Caurier and H. Goutte. [email protected]. INPC2007, Tokyo, 06/06/2007. Independent particles. Shell model - PowerPoint PPT Presentation
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Variational Multiparticle-Multihole
Configuration Mixing Method
with the D1S Gogny force
INPC2007, Tokyo, 06/06/2007
Nathalie Pillet(CEA Bruyères-le-Châtel, France)
Collaborators: J.-F. Berger, E. Caurier and H. Goutte
Nucleus = A interacting nucleons
N-N interaction
(QCD not yet usable)
Many-body problem
Bare forces
INPC2007, Tokyo, 06/06/2007
Numerical solution of exact equations A ≤
12-14
Approximations
In medium forces(Phenomenol
Shell model
Mean field and beyond
Independent particles
Variational mpmh configuration mixing
o Unified description of correlations beyond the HF approximation {mainly Pairing + RPA + particle vibration}
o Conservation of particle numbers and respect of the Pauli principle
o Treatement on the same footing of even-even, odd and odd-odd nuclei
o Description of both ground and excited statesINPC2007, Tokyo, 06/06/2007
Beyond mean field approach to the many-body problem
Theoretical motivations
...AAAh2p2
h2p2
h1p1
h1p1h0p0
h0p0
o Mixing coefficients
o Single particle orbitals
Variational parameters
Trial wave function: Superposition of Slater Determinants
INPC2007, Tokyo, 06/06/2007
m
1)kl(h0p0lk )aa(
N
1iia
Formalism
=> Simultaneous solution of both sets of equations (full self-consistency) => renormalization of HF field
INPC2007, Tokyo, 06/06/2007
Variational principle
o Functional ][H
o Determination of variational parameters
One-body density matrix of the
correlated state
ˆ
Mixing coefficients
0A
A][H][HA
Secular equation
Optimized single particle states
0i
,G]],[h[Generalized HF equations
+
Phenomenological effective D1S* Gogny force
CentralDensity-dependent
Spin-orbit
Coulomb
o The two ranges simulate a “molecular potential”
o Density dependence necessary for saturation in nuclear matter
o Spin-orbit necessary for magic numbers
14 parameters adjusted on nuclear matter properties and some stable nuclei
*J.-F. Berger, M. Girod and D. Gogny, Comput. Phys. Commun. 63 (1991) 365.
INPC2007, Tokyo, 06/06/2007
Study of “usual” Pairing correlations
INPC2007, Tokyo, 06/06/2007
o No proton-neutron residual interaction o Correlated wave function
o Spin-Isospin components of the D1S Gogny force
S=0 T=1 S=1 T=1 S=0 T=0 S=1 T=0
Central x x x x
Density x
Spin-Orbit x
Coulomb x x
Residual interactio
n
“Usual” pairing S=0
T=1
n npairs
npairs
n npairs
npairspair CC
...AAApairs2
pairs2
pair1
pair1pair0
pair0pair
A pair : two nucleons in time-reversed
states
Usual Pairing in 116Sn, 106Sn and 100Sn ground states
INPC2007, Tokyo, 06/06/2007
o 116Sn, 106Sn and 100Sn: spherical nuclei
o Correlated wave function: up to 2 pair excitations (3 pair excitation negligibles)
o Correlation energy:
HF][HHF][HE pairpaircorr
Example: 116Sn (non-selfconsistent mpmh calculations)
Proton valence space: 286 levels
Number of neutron individual levels
1 pair
2 pairs
BCS
-Ecorr
(MeV
)
=> Majority of correlations comes from single particle levels closest to the Fermi level=> Majority of correlations comes from configurations associated to 1 pair excitations=> Convergence of correlation energy (finite ranges of the central term)
=> More correlations than in the BCS approach
INPC2007, Tokyo, 06/06/2007
(MeV)
116Sn 5.44
106Sn 4.62
100Sn 3.67
totalcorrE
4.67
3.91
2.97
0ScorrE
3.06
2.48
1.62
neutroncorrE
3.10
1.45
0.00
BCScorrE
Without residual Coulomb interaction
4.68
3.92
2.98
totalcorrE
o Residual Coulomb: non-negligible effecto mpmh induced correlations: S=0 => dominant pairing correlations
S=1 => negligible contributiono BCS method is a better approximation in strong pairing regime (116Sn)
o Conservation of particle numbers: very important in weak and medium pairing
regimes (100Sn and 106Sn)
Usual Pairing in 116Sn, 106Sn and 100Sn ground states
INPC2007, Tokyo, 06/06/2007
Correlated wave function components (%)
116Sn 106Sn 100Sn
HF 65.38 67.44 90.85
1n pair 26.04 25.29 5.02
1p pair 4.50 3.62 3.70
2n pairs 2.68 2.53 0.16
1n+1p pairs 1.23 0.90 0.18
2p pairs 0.17 0.10 0.09
116Sn occupation probabilities
2pairs BCS
Neutron single particle states
Neu
tron
occu
pati
on
p
rob
ab
ilit
ies
2pairs BCS
Proton single particle statesP
roto
n o
ccu
pati
on
p
rob
ab
ilit
ies
Self-consistency effect - 116Sn Preliminar results ([h[ρ],ρ]=0)
o Correlation energy
(MeV) 1 pair
No self-consistency 4.47
Approximate self-consistency
5.07
(%) HF 1 pair
No self-consistency 87.29 12.71
Approximate self-consistency
82.60 17.40
o Correlated wave function components
Energy gain
INPC2007, Tokyo, 06/06/2007
Summary and Perpectives
INPC2007, Tokyo, 06/06/2007
o Self-consistent mpmh approach (new in nuclear physics)
-unifies the description of important correlations beyond mean field in nuclei (Pairing, RPA, Particle vibration)
-now tractable for medium-heavy nuclei with present computers (pairing hamiltonian) -still have to solve exactly the generalized HF equations
o First applications to nuclear superfluidity quite encouraging
o Future applications: collective vibrations, exotic light nuclei
o Re-definition of effective N-N interaction needed in T=0 channel
(based on the PhD thesis work of F.Chappert -> Gogny force with a finite range density-dependent term)
