Description of nuclear structures in light nuclei with Brueckner-AMD

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Description of nuclear structures in light nuclei
with Brueckner-AMD
Tomoaki Togashi
Yuhei Yamamoto Kiyoshi Kato
Division of Physics, Hokkaido University
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The 19th International IUPAP Conference on
Few-Body Problems in Physics
_at_Bonn, 31.Aug. - 5.Sep. 2009
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Introduction (1)
One of the recent remarkable developments in
theoretical nuclear physics ab initio
calculations based on the realistic nuclear force

3,4-body systems can be solved strictly by
Faddeev, Gauss Expansion Method, and so on
R.B.Wiringa et al., PRC62 (2000) 014001

• For heavier nuclei,
• Greens function Monte Carlo (GFMC)
• no-core shell model (NCSM)
• coupled-cluster (CC) method
• unitary-model operator approach
  (UMOA)
• Fermionic molecular dynamics
• (FMD) unitary correlation
• operator method (UCOM)
• tensor optimized shell model (TOSM)
• and so on

We can discuss cluster structures based on the
realistic nuclear force.
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Introduction (2)
Recently, the studies with the Antisymmetrized
Molecular Dynamics (AMD) have been developed.
Y.Kanada-Enyo and H.Horiuchi, PTP Supple 142
(2001), 205
Advantages of AMD - We can treat alpha-nuclei
and non-alpha nuclei - The wave function is
written in a Slater determinant form - Both
states of shell model and cluster model can be
described - no assumption for configurations
AMD has succeeded in understanding and predicting
many properties of both stable and unstable
nuclei.
However,
the AMD calculations have been performed with
phenomenological interactions.
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Our strategy
We develop the new AMD framework based on
realistic nuclear forces.
We apply the Brueckner theory to AMD and
calculate G -matrix in AMD starting from
realistic nuclear forces.
Brueckner-AMD the Brueckner theory AMD
T.Togashi and K.Kato Prog. Theor. Phys. 117
(2007) 189.
T.Togashi, T.Murakami and K.Kato Prog. Theor.
Phys. 121 (2009) 299.
In this framework, we introduce no
phenomenological corrections.
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AMD wave function
A-body wave function
Nucleon-nucleon (NN) correlations cannot be
described sufficiently.
(Slater determinant)
single-particle wave functions
We calculate G-matrix in AMD.
Cooling method
A.Ono, H.Horiuchi, T.Maruyama, A.Ohnishi, PTP 87
(1992) 1185
Energy Variation of A-body problem

The initial configuration is chosen randomly.
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Brueckner-AMD
The single-particle orbits in AMD
Bethe-Goldstone equation
H.Bando, Y.Yamamoto, S.Nagata, PTP 44 (1970) 646
AMD Hartree-Fock method
A.Dote, Y.Kanada-Enyo, H.Horiuchi, PRC56 (1997)
1844
single-particle w.f. of AMD
starting energy
self-consistent
Q -operator
the eigenstate of Hartree-Fock Hamiltonian
occupied states
single particle energy(s.p.e.)
Cooling method
Gmatrices are changed because of dependence on
s.p.e.
The configuration is changed.
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Correlation functions in Brueckner-AMD
correlation function
Bethe-Goldstone Equation
Model (AMD) pair wave function
Solution of the Bethe-Goldstone equation
G -matrix element
Correlation function
Correlation functions are determined for every
pair.
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Decomposition of G -matrix
G-matrix can be decomposed into central, LS,
tensor by correlation functions.
S.Nagata, H.Bando, Y.Akaishi, PTPS 65 (1979) 10
correlation function
? rank of operators
G-matrix with operators can be defined as
.
In the practical calculations, we use this
potential with operators.
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Application to light nuclei
Interactions
We use Argonne v8 (AV8) as a realistic nuclear
interaction.
AV8 P.R.Wringa and S.C.Pieper, PRL89 (2002),
182501.
( cutting off electromagnetic interaction)
Projection
variation after projection (VAP)
Parity
Spin (J ) projection after variation (PAV)
Parity eigenstate obtained with VAP
(K -mixing)
? We calculate the binding energy of the ground
state and the energy levels of the excited
states.
We present the Brueckner-AMD (B-AMD) results of
some light nuclei (4He, 8Be, 7Li, 9Be, and 12C).
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Result of 4He 8Be level scheme
Argonne v8'
Result of 4He
4He
?/2
0.1
4
0.08
0.06
Y
0.04
0.02
fm-3
-4
4
B.E. -44.0 (MeV)
B.E. -56.5 (MeV)
Z
binding energy(0)?
-24.6(MeV)?
EXP
Few-body cal
-28.3(MeV)?
aa cluster
-25.9(MeV)?
H. Kamada et al. , PRC64 (2001) 044001
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7Li level scheme
Argonne v8'
B.E. -29.6 (MeV)
B.E. -39.2 (MeV)
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9Be (parity -) level scheme
Argonne v8'
B.E. -41.9 (MeV)
B.E. -58.2 (MeV)
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9Be level scheme
Argonne v8'
-41.9 (MeV)
-58.2 (MeV)
aan picture
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12C level scheme
Argonne v8'
3a cluster
-70.4 (MeV)
-92.2 (MeV)
3a-like
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Higher 0 state of 12C
The shell-model approaches have difficulty in
describing the higher 0 state of 12C.
Huge shell-model model space is needed.
P.Navratil et.al., PRL84 (2000) 5728
On the other hand, cluster models have been
succeeded in describing the 02 state of 12C.
Ex) E.Uegaki et.al., PTP 62 (1979) 1621
Cluster models indicate that a number of spatial
configurations are needed to describe the 02
state of 12C.
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Description of higher 0 state of 12C
In order to describe the higher 0 state, it is
necessary to superpose the intrinsic
configurations different from 0 ground state.
We perform the energy variation with the
orthogonality to the lowest state.
Y.Kanada-Enyo, PTP117 (2007) 655
We perform the energy variation where the
inertial axes are fitted in order to prevent us
from obtaining the solution of rotated ground
state.
Intrinsic state for the excited state
parity-projected states
In order to create various spatial
configurations, we perform the above variation
under r.m.s.-radius and deformation parameter ß
constraint.
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12C level scheme
All states contribute coherently.
Argonne v8'
B.E. -73.7 (MeV)
B.E. -92.2 (MeV)
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Summary Proceeding works
Summary

• We developed the framework of AMD with realistic
  interactions based on the Brueckner theory.

• In the Brueckner-AMD, we can construct the
  G-matrix and correlation functions in AMD
  self-consistently. That means this method has no
  phenomenological corrections.

• We applied Brueckner-AMD to some light nuclei
  and succeeded to describe not only ground states
  but also excited states with cluster structures
  which shell model approaches cannot describe.

Proceeding works

• Roles of tensor force for 8Be (discussed by
  Yuhei Yomamoto)

• The way how to introduce genuine 3-body forces

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Appendix
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G -matrix (central)
1E
3E
1O
3O
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G -matrix (spin-orbit, tensor)
3O LS
3O tensor
3E LS
3E tensor
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Superposition of Slater determinants in
Brueckner-AMD
The G -matrix between the different
configurations in bra and ket states is necessary.
However, the G matrix can be determined for only
a single configuration.
The G -matrix is calculated with the correlation
functions.
Correlation functions derived from bra and ket
states
T.Togashi, T.Murakami and K.Kato Prog. Theor.
Phys. 121 (2009) 299.
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Features of G -matrix in Brueckner-AMD
?. The G-matrix can be solved in AMD framework
because the single-particle orbits in AMD can be
defined and applied to the Brueckner theory.
starting energy
?. The G-matrix in Brueckner-AMD is changed with
the structures through the starting-energy
dependence.
The G-matrix in Brueckner-AMD is different from
the case of nuclear matter and considers the
property of finite nuclei.
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Approximated ß parameter
the moment of the inertia matrix
Approximated ß parameter
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11B (parity-) level scheme
Argonne v8'
B.E. -54.5 (MeV)
B.E. -76.3 (MeV)
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Whats the origin of spin-orbit splitting ?
No-core shell model calculations without genuine
3-body forces also indicate the inversion between
3/2- and 1/2- in 11B.
C.Forssen et.al., PRC71 (2005) 044312
3-body forces may be essential to reproduce the
energy levels of some nuclei.