1. Triaxial shapes in the sd and fp shell nuclei with realistic shell model Hamiltonians
Zao-Chun Gao(高早春)
China Institute of Atomic Energy
Mihai Horoi
Central Michigan University
Y. S. Chen (陈永寿)

2. Contents Introduction. Variation After 3D Angular Momentum Projection.
Examples of 26Mg and 28Si with the USD interaction.
Systematic calculations for the sd and pf shell nuclei with the USD and GXPF1A interactions.
PCI calculations for 52Fe and 56Ni.
Summary.

3. Shell model Introduction Good:
Very successful in the description of various observables.
Good shell model Hamiltonians. USD, GXPF1A, etc.
Bad:
Huge dimension.
(Treatments: Mont Carlo shell Model, VAMPIR, etc)
No intrinsic structure (deformation)
24Mg and 26Mg are triaxially deformed!
[D. Kurath, Phys. Rev. C 5, 768 (1972)]

4. Mean Field (HF or HFB) Good: Very clear intrinsic structure.
Applied to the whole nuclear region.
Bad:
No good angular momentum.
Missing correlations beyond mean field.

5. 3D Angular Momentum Projection (3DAMP)
The key tool to transform the mean-field wave function from the intrinsic to the laboratory frame of reference.
A intrinsic state with triaxial deformation
projected states differed by

6. Theories relate to the 3DAMP for 24Mg
Skyrme energy density functional
Gogny force
RMF
M. Bender etc Phys. Rev. C78, (2008)
J. M. Yao, etc Phys. Rev. C81, (2010)
T. Rodríguez etc Phys. Rev. C81, (2010)

7. What we are going to do :
Obtaining shapes from well established shell model Hamiltonians. USD and GXPF1A
Two ways:
Hartree-Fock mean field
Variation after Projection (VAP)

8. Variation After Projection : VAMPIR
The only standard method where variation after angular-momentum projection is exactly considered (together with restoration of N,Z,and parity).
Very complicated.
Too much time consuming due to the five-fold integration. (3 for AMP and 2 for N,Z projection)
No explicit discussions about the intrinsic triaxial shapes.

9. Present VAP is much simpler !
Using HF type Slater determinant
(HFB type in VAMPIR)
Real HF transformation
(Complex HFB in VAMPIR)
Time reversal symmetry is imposed.
Gamma degree of freedom is allowed.
The adopted interactions: USD and GXPF1A

10. Basic idea of the present VAP
HF type Slater determinant
Wik are real here, determined by minimization of EPJ(I)
Where fK satisfy

11. Algorithm of the present VAP (L-BFGS quasi-Newton method)
Thouless Theorem:

12. Problem:The VAP SD may have an arbitrary orientation in the space.
Treatment: The 3 principle axes of the triaxial shape have to be in accord with the laboratory axes.
Time reversal symmetry and Q21=0

13. Quantities of Q and g for the deformed VAP Slater determinant

14. There may have several possible solutions in the VAP calculations
There may have several possible solutions in the VAP calculations. Needs to try many times to find out all minima.
No EPJ(I=0) Q g
(MeV)
Ground state (I=0) of 26Mg
with USD interaction

15. All possible VAP and HF solutions for 26Mg

16. Another example of 28Si

17. In HF: Most nuclei are axial. In VAP:No nulceus is axial !
With USD interaction
Hartree-Fock
VAP
Q and g values in the sd shell nuclei with 10N,Z18

18. The same situation for the fp shell nuclei!
With GXPF1A interaction
VAP
Hartree-Fock
Q and g values in the fp shell nuclei with 22Z32, 22N38

19. Configuration interaction including Particle -Hole states (Triaxial Projected CI)
Selection of the particle hole excited SDs
(Gao, Horoi, PRC 79,014311(2009)
VAP SD
Particle –Hole excitations on top of the VAP SD
A Preliminary method!

20. With GXPF1A interaction
PCI for 52Fe
With GXPF1A interaction
GXPF1A interaction

21. With GXPF1A interaction
PCI for 56Ni
With GXPF1A interaction

22. Comparison with MCSM and VAMPIR
With FPD6 interaction

23. Future work
VAP calculationd with unpaired SD for odd-spin states and odd mass nuclei (keeping the time reversal symmetry).
Non-axial octupole deformation will be included.
VAP for the excited states.
Try to improve the PCI algorithm.

24. Summary
Variation After Projection (VAP) calculations have been performed using Hartree-Fock type Slater determinant in the shell model space.
Using USD and GXPF1A interaction, VAP calculations show that all the sd and fp shell nuclei are triaxial, while most nuclei are exactly axial in the HF calculations.
Preliminary triaxial PCI calculations has been carried out and compared with the MSCM and VAMPIR.

25. Thanks for your attention!