Br ¨uckner-Hartree-Fock with Realistic Nucleon-Nucleon Potentials

Brückner-Hartree-Fock with
Realistic Nucleon-Nucleon Potentials
Patrick Hedfeld
Institut für Kernphysik, TU Darmstadt

Overview
■ Realistic Potentials & Effective Interactions
■ Unitary Correlation Operator Method (UCOM)
■ Hartree-Fock + Perturbation Theory
■ Brückner-Hartree-Fock (BHF)
■ Summary & Outlook

Realistic Potentials & Effective Interactions
Realistic Potentials
■ QCD motivated: based on meson exchange picture or chiral effective field
theory
■ fitted to properties of the deuteron and phase shifts
■ need a short-range repulsion and tensor force
Examples: AV18, CD Bonn, Chiral N 3 LO . . .
Effective Interactions
■ simple model spaces cannot describe short-range correlations
■ adapt realistic potentials to
model spaces while conserving phase shifts of the realistic potential

Unitary Correlation Operator Method
Correlation Operator
introduce short-range correlations by
means of a unitary transformation with respect
to the relative coordinates of all pairs
C = exp[−i G] = exp − i
gij Correlated States
ψ = C ψ
i

φ
.
φ r Cr
.
φ r CΩCr
.
Correlated States: The Deuteron
0.15
0.1
0.05
0
0.15
0.1
0.05
0
0.15
0.1
0.05
0
L = 0
1 2 3 4 5
r [fm]
1 2 3 4 5
L = 0 r [fm]
L = 2
1 2 3 4 5
r [fm]
central
correlations
tensor
correlations
[fm]
.
0.2
0.15
0.1
0.05
0
0.08
0.06
0.04
0.02
s(r)
1 2 3 4
r [fm]
ϑ(r)
. 0
1
only short-range tensor
correlations treated by CΩ
2 3
r [fm]
4

Binding Energies & Radii
E/A [MeV]
.
Rch [fm]
.
0
-2
-4
-6
-8
5
4
3
2
1
4 He 16O
24O 40
34Si
Ca48Ca48
Ni56Ni68
Ni78Ni88
Sr90Zr100
Sn114Sn
132Sn146Gd 208Pb experiment ● HF

.
.
Binding Energies & Radii
E/A [MeV]
Rch [fm]
0
-2
-4
-6
-8
5
4
3
2
1
4 He 16O
24O 40
34Si
Ca48Ca48
Ni56Ni68
Ni78Ni88
Sr90Zr100
Sn114Sn
132Sn146Gd 208Pb experiment ● HF HF+PT2 RPA
long-range
correlations are
perturbative
(PT, BHF, RPA,...)

Bethe-Goldstone-Equation
Equation
G ω = W + W 1 Q
G
2 ω − H0
Implementation
Inversion
Iteration
G ω = W + W 1
G ω = [ − 1
2
Q
2 ω − H0
Explanation
W = T − Tcm + VUCOM + VC
ω : starting energy
Q : Pauli operator
H0 : single particle Hamiltonian
Q
W]
ω − H0
−1 W
W + W 1
4
Q
ω − H0
W Q
W + . . .
ω − H0

Correlations I
Vab[MeV]
.
10
5
0
-5
ǫFermi = 1ω
-10
-10 -5 0
Gab[MeV]
5 10
ǫmax=6ω
J = 1
T = 0

Correlations II
V ab[MeV]
ǫmax=6ω
.
V ab[MeV]
.
10
5
0
-5
ǫ Fermi = 6ω
-10
-10 -5 0
Gab[MeV] 5 10
10
5
0
-5
ǫ Fermi = 3ω
-10
-10 -5 0
Gab[MeV] 5 10
V ab[MeV]
.
V ab[MeV]
.
10
5
0
-5
ǫ Fermi = 5ω
-10
-10 -5 0
Gab[MeV] 5 10
10
5
0
-5
ǫ Fermi = 2ω
-10
-10 -5 0
Gab[MeV] 5 10
V ab[MeV]
.
V ab[MeV]
.
10
5
0
-5
ǫ Fermi = 4ω
-10
-10 -5 0
Gab[MeV] 5 10
10
5
0
-5
ǫ Fermi = 1ω
-10
-10 -5 0
Gab[MeV] 5 10

Fully Self-Consistent BHF
until EB
converged
single particle
energies
and states
Hartree-Fock
Bethe-
Goldstone-
Equation
(ω)
VUCOM
as first input
matrix elements
of the effective
interaction

Results
E/A[MeV]
.
-4
-5
-6
-7
-8
-9
EExp
40 Ca
-10
2 4 6 8 10
ǫmax ω
EExp
90 Zr
4 6 8 10
ǫmax ω
EExp
208 Pb
4 6 8 10
ǫmax ω

ǫmax=10ω
Binding Energy & Radii
preliminary
results
.
.
E/A[MeV]
Rch[fm]
-4
-6
-8
5
4
3
2
4He16O 24O34Si 40Ca48Ca 56Ni 88Sr90Zr 100Sn114Sn 132Sn146Gd 208Pb 68Ni
experiment ● HF BHF HF+PT2

Summary & Outlook
■ UCOM describes short-range correlations and ’tames’ the
interaction; can be used in HF (BHF, RPA . . . )
■ fully self-consistent BHF based on the realistic two body interaction
VUCOM
■ results for the energy with BHF are in good agreement with
experiment and second-order perturbation theory but radii
are not much improved
■ Outlook: other ’strategies’ for the starting energy, Padè approximants
in the iterative scheme

Epilogue...
My Collaborators
■ R. Roth, P. Papakonstantinou, H.Hergert, A. Zapp
Institut für Kernphysik, TU Darmstadt
References
■ R. Roth, P. Papakonstantinou, N.Paar, and H. Hergert, Phys. Rev. C73, 044312 (2006)
■ R. Roth, H.Hergert, P. Papakonstantinou, T.Neff and H. Feldmeier, Phys. Rev. C72, 034002 (2005)
■ C. Barbieri, N.Paar, R. Roth and P. Papakonstantinou, arXiv: nucl-th/0608011 (2006)
■ http://crunch.ikp.physik.tu-darmstadt.de/tnp/