AMD HartreeFock

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A. Dote (KEK), Y. Kanada-Enyo(KEK), H. Horiuchi
(Kyoto univ.), Y. Akaishi (KEK), K. Ikeda (RIKEN)

• Introduction
• Requests for AMDs wave function to treat the
  tensor force
• Further requests found in the study of 4He
• Use of wave packets with different
  width-parameters
• Importance of angular momentum projection
• Single particle levels in 4He
• Summary and future plan

RCNP??????????? 04.03.22 at RCNP
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Introduction
Why tensor force?
1, large contribution in nuclear forces
ex) deuteron bound by the tensor force
In microscopic models, tensor force is
incorporated into central force.
AMD, Hartree-Fock without tensor
force Relativistic Mean-Field approach no
pmeson
How in finite nuclei? Affection to nuclear
structure?
2, creation and annihilation of clustering
structure
Tensor force works well on nuclear surface
more work in more developed clustering structure?
3, Development of effective interaction for AMD

• Unified effective interaction for light to heavy
  nuclei.

• reduction of calculation costs

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Introduction
Y.Akaishi, H.Bando, S.Nagata, PTP 52 (1972) 339
Previous study of the tensor force says

• In the nuclear matter, the 3E effective
  interaction
• containing the tensor force is weakened.

Saturation property
????
The tensor force might be weakened in heavy
nuclei rather than in light nuclei, and inside
of nuclei rather than near nuclear surface.
Tensor force might favor such a structure that
the ratio of the surface is large, namely
well-developed clustering structure.
The incorporation of tensor force into central
force is sensitive to the starting energy.

• In the perturbation theory,

The effective central force changes,
corresponding to the nuclear structure.
If the nuclear structure changes, the starting
energy changes also. Therefore, the effective
central force changes.
Shell
Tensor force
Our scenario
Following the tensor force, each of nuclei
chooses shell- or clustering-structure, which
are qualitatively different from each other.
Cluster
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Requests for AMDs wave function to treat the
tensor force

• Parity-violated mean field

tensor force
Change the parity of a single particle wave
function

• Superposition of wave packets with spin

tensor force
Strong correlation between spin and space

• Changeability of isospin wave func.

tensor force
Change the charge of a single particle wave
function
Ex.) Furutani potential
i
ii
iii
i
iii
ii
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Further requests found in the study of 4He
Interaction Tensor force is not incorporated
into central force.
Akaishi potential
Based on Tamagaki potential, The repulsive core
of the central force is treated with G-matrix
method. Tensor force bare interaction.

• Wave packets with different width-parameters

28 MeV

• Gain the binding energy without shrinkage

25 MeV

• ? for S-wave ? ? for P-wave

17 MeV

• Angular momentum projection

10 MeV
Because of very narrow wave packets, only a
little mixture of J?0 components makes the
kinetic energy increase.
0
1
2
3

• J projection after J constraint variation

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Wave packets with different width-parameters
Tensor force
Radius
Kinetic energy
One nucleon wave function superposition of
Gaussian wave packets with different ?s




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Why different ?s ?
SP max
Tensor max
vs
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Test for different ?s

• 4He
• 3 wave packets
• Vc1.0/ Vt3.0
• Frictional cooling
• spin/isospin free

(0.2, 0.2, 0.2)
(0.5, 0.5, 0.5)
(0.2, 0.5, 0.9)
(0.9, 0.9, 0.9)
By using the wave packets with different ?s,
the binding energy can be gained without
shrinkage.
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Result of 4He

• 4 wave packets
• Vc1.0/ Vt2.0
• Frictional cooling
• spin/isospin free

0. 4 wave packets with common ?s - ?0.6 -
1. 4 wave packets with different ?s - ?0.31.5
geometric ratio -
2. Angular momentum projection onto J0
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Importance of angular momentum projection
details of energy gain
?in case of ? 0.31.5
Contributions from various J components to the
kinetic energy
Because of very narrow wave packets, J?0
components have large kinetic energy. Only 1
mixture of a J?0 component increases the kinetic
energy by 1 MeV.
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J projection after J constraint variation

• 4He
• 4 wave packets
• ? 0.3 1.5
• Vc1.0/ Vt2.0
• Frictional cooling
• spin/isospin free
• J projection (VBP)

No constraint B.E. -25.3 MeV Rrms 1.32 fm
J2 constraint B.E. -28.6 MeV Rrms 1.35 fm
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Single particle levels in 4He
What happens in the 4He obtained by the AMD
calculation?
See single particle levels!
?
Extract the single particle levels from the
intrinsic state with AMDHF method.
Single Slater determinant
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How to extract single particle levels (AMDHF)
one nucleon wave function in AMD wave function

• Prepare orthonormal base.

Diagonalize the overlarp matrix
2. Mimicking Hartree-Fock, construct a single
particle Hamiltonian.
3. Diagonalizing h, get single particle levels.
Single particle energy
Single particle state
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Single particle levels

• 2 groups
• Lower neutron-like, P(-)15
• Upper proton-like, P(-)19

P-state

• High L?

If S(80)P(20), L20.4. But L20.83.
7 D-state?

• High J?

If 0s1/20p1/2, J20.75. But J21.16.
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Summary

• To treat the tensor force in AMD framework,
  following points are needed
• 1, superposition of wave packets with
  spin,
• 2, changiability of charge wave
  function.
• Other points are found to be important, by the
  study of 4He
• 3, wave packets with different ?s
• gain the B.E. without
  shrinkage.
• ? for S-wave is rather
  different from that for P-wave.
• 4, J projection
• Because of very narrow wave
  packets,
• J?0 components have large
  kinetic energy.
• 4, J projection after J constraint
  variation
• leads to better solution.
• Result of 4He
• 
  Akaishi potential, Vt2.0.

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Summary and Future plan

• We have investigated the single particle levels
  in 4He.
• Characteristics of our S.P. levels
• 1, two groups (22)
• 2, 1520 negative parity component
  (P-state) is mixed.
• 3, including some components except 0s1/2
  0p1/2
• higher L, higher J state.

• More detailed analysis of single particle levels.

How is each component? proton-parity ,
proton-parity -, neutron-parity ,
neutron-parity -

• Vt2 ?

Treat the short range part by tensor correlator
method (Neff Feldmeier)
or
Cut the high momentum component by G-matrix
method (Akaishi-san)