?????

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???????????????? ?? (????????????)
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Excitation energy
decay threshold to clusters
Cluster-shell competition
Shell structure single-particle motion of
protons and neutrons
Nuclear structure
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3 alpha model for 12C
T. Yoshida, N. Itagaki, and T. Otsuka, Phys. Rev.
C 79 034308 (2009).
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a-cluster model
We assume each 4He as (0s)4 configuration with
respect to some localized position Non-central
interactions do not contribute
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In the low-lying state..
Spin-orbit interaction is the driving force
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Cluster model covers the model space of the
shell model?
y
x
y
x
This is (s)4(px)4(py)4, but not (s1/2)4(p3/2)8
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How we can include jj-coupling shell model-like
correlations in the cluster model?

• If we treat all the Gaussian centers parameters
  of the nucleons independently and superpose
  different Slater determinants, this is possible
  as in anti-symmetrized molecular dynamics (AMD)
  and Fermionic molecular dynamics (FMD)

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• How we can describe cluster-shell competition in
  a simple way?
• Is there a simple transformation
• from Brinks cluster model wave function
• to the lowest configuration of jj-coupling
  shell model
• within a single Slater determinant?

We introduce a general and simple model to
describe this transition
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How we can include the spin-orbit
contribution In the cluster model??
spatial part of the single particle wave function
exp-?( r Ri )2
In the cluster model, 4 nucleons share the same
Ri value in each alpha cluster
The spin-orbit interaction (r x p) s
lt r gt Gaussian center parameter Ri
lt p gt imaginary part of Ri
For the nucleons in the quasi cluster Ri ?
Rii ? (e_spin x Ri)
Simplified method to include the spin-orbit
contribution (SMSO) N. Itagaki, H. Masui, M.
Ito, and S. Aoyama, Phys. Rev. C 71 064307
(2005).
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• 12C case

3alpha model ? 0
2alphaquasi cluster ? finite
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-Y axis
Z axis
X axis
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Single particle wave function of nucleons in
quasi cluster (spin-up)
Quasi cluster is along x Spin direction is along
z Momentum is along y
the cross term can be Taylor expanded as
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the single particle wave function in the quasi
cluster becomes
for the spin-up nucleon (complex conjugate for
spin-down)
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12C
Various configurations of 3as with ?0
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12C
? ? 0
Various configurations of 3as with ?0
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We must measure the breaking of alpha cluster in
each state
What is the operator related to the a breaking?
 
one-body spin-orbit operator for the proton part
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12C
0.03 0.30 0.28 0.64 one body ls
? ? 0
Various configurations of 3as with ?0
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For C isotopes

• The R and ? values for the 12C core are randomly
  generated
• The positions of Gaussian center parameters for
  the valence neutrons are randomly generated, and
  we superpose many different configurations for
  the valence neutrons

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16C
One-body LS 0.44 0.51 1.45 1.39
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Breaking all the clusters
Introducing one quasi cluster
Rotating both the spin and spatial parts of the
quasi cluster by 120 degree (rotation does not
change the j value)
Rotating both the spin and spatial parts of the
quasi cluster by 240 degree (rotation does not
change the j value)
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Single particle energy of spin-up protons
p3/2
Energy MeV
s1/2
?
R 0.01 fm
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Energy sufaces
0 energy
LS force
Minimum point R 0.9 fm, ? 0.2 - 89.6 MeV
Tadahiro Suhara, Naoyuki Itagaki, Jozsef Cseh,
and Marek Ploszajczak arXiv nucl-th 1302.5833
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A single state has different features
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Comparison with ß-? constraint AMD
SMSO AMD
energy - 89.6 MeV - 90.1 MeV
of freedom 2 (R, ?) 6A(32A)
xyz complex
overlap
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Tensor contribution in C isotopesusing SMSO
(MeV) 11C 12C 13C 14C 16C
With tensor -64.8 -89.3 -93.8 -106.2
-109.2 Without T -65.6 -91.4 -95.5 -106.3
-111.6 ? 0.9 2.1 1.7
0.1 2.4
We can take into account first order-like tensor
contribution, if we break alphas to take into
account the spin-orbit effects
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How we can incorporate the 2p2h nature of the
tensor contributionin the cluster modelin a
simple way?
Simplified method to include the tensor
contribution in alpha-cluster model N.
Itagaki, H. Masui, M. Ito, S. Aoyama, and K.
Ikeda Phys. Rev. C 73 034310 1-7 (2006).
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We would like to break the a cluster in a
simplified way
However, the tensor interaction does not work
even if we break ato 3 nucleons 1 nucleon
system
p?
Tensor interaction works between proton spin-up
and neutron spin-up, but this is cancelled by the
spin-sown neutron located at the same position
n?
n?
Necessary to break aas 211 system (2p2p)
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-33.83 MeV compared to -27.57 MeV of the (0s)4
model
Model 2
p?
Spin (z) direction
p?
n?
n?
Simplified method to include Tensor contribution
(SMT)
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12C
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Summary

• Cluster-shell competition and role of spin-orbit
  interaction in light nuclei can be studied in a
  simplified way
• We can transform Brinks wave function to
  jj-coupling shell model (jj subclosure states) by
  introducing two parameters
• The model is rather general and we can apply to
  heavier nuclei
• Description of particle-hole states is going on
• First-order-like effect of the tensor interaction
  can be taken into account if we break the alpha
  clusters to taken into account the spin-orbit
  contribution
• To take into account the 2p2h nature of the
  tensor interaction, we need another model, which
  is under development

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THSR wave function
For the study of heavier systems, we must
simplify the THSR wave function
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Virtual THSR wave functionN.Itagaki., M.
Kimura, M. Ito, C. Kurokawa, and W. von Oertzen,
PRC 75 037303 (2007)
Gaussian center parameters are randomly generated
by the weight function of
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3a
Solid, dotted, dashed, dash-dotted ? s 2,3,4,5
fm
r.m.s. radius of 12C (fm)
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s 4 fm
s 3.5 fm
s 3 fm