# Presentation: 009431

## T. Nakatsukasa (Univ. of Tsukuba)Collaborators:S. Shinohara, K. Yabana (Univ. of Tsukuba)H. Ohta (Sumitomo Chemical)ORNL2007/3/5-8 

- T. Nakatsukasa (Univ. of Tsukuba)*Collaborators*:S. Shinohara, K. Yabana (Univ. of Tsukuba)H. Ohta (Sumitomo Chemical)ORNL2007/3/5-8

- Complete spectroscopy with the Skyrme functional beyond mean field

## Exploring the Skyrme DFT and more

- Correlation beyond the mean field is often important for description of ground-state and excited states.
- RPA, Projection, and GCM are typical methods, but with a limited applicability.

- Toward the _*theoretical complete spectroscopy*_ of low-lying states with *a single Hamiltonian* and with a *very large model space*:

- “Stochastic” approach to configuration mixing with the Skyrme Hamiltonian

## Generation and selection of Slater det’s in the 3D Cartesian Coordinate space

Projection on good Jπ (3D rotation)

Solution of generalized eigenvalue eq.

- Projection on good *J**π** *(3D rotation)
- Solution of generalized eigenvalue eq.

- _Stochastic configuration mixing with parity and angular momentum projection_

## Skyrme Hartree-Fock in a mesh space

- 3D space is discretized in lattice
- Single-particle orbital:

- Imaginary-time Method

- Variation for a Slater determinant

- 20Ne

## Generation of many S-det’s

- 16O (BKN)

- Gaussian wave packets (n & p) whose positions are determined by random numbers.

- Initial state

- Imaginary-time evolution

*n*

- 3D real space

## Selection of Slater det’s

- Every 50 iterations, a Slater det is stored in a *basket*.
- Check the linear dependence, then select some of them in a bag and throw away the rest.

- If overlap

- with all possible exchange of axes (24), we put them in a bag.

## Further Selection ...

- _Further Selection ..._

- Eigenvalues of the norm matrix

- smaller than 10-3

*Garbage box*

***Small norm only***

- BKN

## Many paths on a potential surface

- 16O

- Randomly-generated Slater determinants

- soft

- HF state

## Many configurations included

- in the calculation in 16O

- Local minima  and soft modes automatically appear.

## 16O (SLy4)

- Jπ=0+

- Jπ=1-

- Jπ=2+

- Jπ=3-

- Number of basis states

- Number of basis states

- Number of basis states

- Number of basis states

## 16O (SLy4)

## 16O

- Calculation (SLy4)

- Experiment

- 16O

- π=+

- π=−

- Jπ

- Jπ

- Jπ

- Jπ

## Energy spectrum in 16O

- SGII

## Energy convergence in 12C

- Jπ=0+

- Jπ=3-

- Jπ=2+

- Jπ=1-

- SGII

## Energy spectrum in 12C

- SGII

## Energy spectrum in 12C

- SGII

## Variation after parity projection

- Spin-orbit interaction is weakened by 20% to fit the g.s. band

- HF

- VAPP(+)

- VAPP(-)

*Ohta, Yabana, Nakatsukasa, PRC70 (2004)  014301*

- SGII

- 12C

## Energy spectrum in 20Ne

- SGII

- (0p1/2)-1(sd)5

## Energy spectrum in 20Ne

| B(E2) (e2fm4) |  |  |
| --- | --- | --- |
|  | cal | exp |
| 2+→0+ | 52.09 | 57±4 |
| 4+→2+ | 64.52 | 71±7 |
| 6+→4+ | 50.04 | 66±8 |
| 3-→1- | 114.19 | 164±26 |
| 5-→3- | 120.20 |  |

## Energy spectrum in 20Ne

| B(E2) (e2fm4) |  |  |
| --- | --- | --- |
|  | cal | exp |
| 3-→2- | 106.21 | 113±29 |
| 4-→3- | 77.8 | 77±16 |
| 5-→4- | 84.5 | <808 |
| 6-→5- | 29.9 | 32±13 |

## Variation after parity projection

- Negative-parity states are well reproduced.
- The band termination of the g.s. band is not reproduced because the band is constructed from a single intrinsic state.

- HF

- VAPP(+)

- VAPP(-)

*Ohta, Yabana, Nakatsukasa, PRC70 (2004)  014301*

- VAPP(-)

- SGII

- 20Ne

## Summary

- Complete low-lying spectroscopy with the Skyrme Hamiltonian
- Capable of describing a variety of excited states in a unified way, such as vibrational excitations, cluster excitations, single-particle excitations.
-       12C: ground-state band, 2nd 0+, negative-parity bands (3-, 1- , 2-)
-     16O: 2nd Kπ=0+ & 0- bands, Kπ=3- ,1- ,  2- bands
-     20Ne: Kπ=0+ & 0- bands, Kπ=2- band
- Non-empirical calculation without any restriction on shape of wave functions
_*Problems*_
- Still overestimate the excitation energy of 2nd 0+ state in 16O by about 3 MeV, and underestimate its B(E2)
-             Center of mass?    Weak-coupling phenomena?
- Overestimate the moment of inertia of the ground-state band in 20Ne
-      Pairing?

- Shinohara et al, PRC **74**, 054315 (2006)

## 12C

- Jπ

- Jπ

- Jπ

- Jπ

- Calculation (SIII)

- Experiment

- π=+

- π=−

## Initial Slater determinant

- Gaussian single-particle wave functions are distributed.
- Positions of the Gaussian-centers are determined by random numbers to remove any prejudice.

- Gaussian wave packet

- iteration

- iteration

## 40 check points

## 12C

- 12C

**cal**

**exp**

## 20Ne

- 20Ne

**cal**

**exp**