Triaxial shapes of the sd and fp shell nuclei with realistic shell model Hamiltonians Zao-Chun Gao (高早春) China Institute of Atomic Energy Contents Introduction. Variation After 3D Angular Momentum Projection. Examples of 26Mg and 28Si with the USD interaction. Triaxial shapes of the sd and pf shell nuclei with the USD and GXPF1A interactions. Summary. Introduction Nuclear Deformation 1. 2. 3. A very important concept in the understand of the nuclear structure. Determined by various methods: Macro-micro PES calculations (LDM+SC). Mean field. ( RMF, HF, HFB). Beyond mean field. (Variation After Angular Momentum Projection, Project Energy Surfaces ). Nuclear Hamiltonian and Shell Model As a many-body quantum system, the structure of the nucleus is determined by the nuclear Hamiltonian H. The nuclear deformation also can be obtained from H. In the shell model calculations, good shell model Hamiltonians are crucial in the successful description of various nuclear properties. sd shell: USD, USDA,USDB, etc. fp shell: FPD6, KB3,GXPF1A, etc. Shell Model calculations:Excitation energies Shell Model calculations: Binding energies Even-Z Odd-Z GXPF1 interaction GXPF1 interaction (Honma etal PRC69 034335(2004) Shell Model calculations: Electro-magnetic moments GXPF1 interaction GXPF1 interaction Honma etal PRC69 034335(2004) Shell Model calculations: Spetroscopic Factors M.B. Tsang et al PRL 102 062501(2009) USDA and USDB GXPF1A Where is the nuclear deformation in the shell model calculations? In the Shell Model calculations, the basis is spherical. There is NO concept of the nuclear deformation in the Shell Model. Even in the case of describing rotational bands. Example: 48Cr [Caurier etal PRL 75 2466(1995)] Nuclear deformation is not an observable. Hartree-Fock calculations with SM Hamiltonians USD interaction GXPF1A interaction Mean Field Good: Bad: Very clear intrinsic structure. Applied to the whole nuclear region. No good angular momentum. Missing correlations beyond mean field. Potential Energy Curves from CHFB and Projected CHFB 32Mg Gogny Interaction (D1S) From: et al. Motivation of this work: From SM H,we should also use the method beyond mean field to obtain the nuclear shapes , and compare them with those shapes obtained from the HF method. Beyond mean field :Variation After 3Dimensional Angular Momentum Projection. 3-Dimensional 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 2I 1 I I ˆ ˆ PMK d D MK R 2 8 projected states differed by K I , I 1,...I Theories relate to the 3DAMP for 24Mg Skyrme energy density functional RMF M. Bender etc Phys. Rev. C78, 024309 (2008) J. M. Yao, etc Phys. Rev. C81, 044311 (2010) Gogny force T. Rodríguez etc Phys. Rev. C81, 064323 (2010) Variation After Projection : VAMPIR The only standard method where variation after angularmomentum 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. 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 SM interactions: USD and GXPF1A Basic idea of the present VAP HF type Slater determinant Wik are real here, determined by minimizing EPJ(I) Where fK satisfy Algorithm of the present VAP (L-BFGS quasi-Newton method) Thouless Theorem: 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 Quantities of Q and g for the deformed VAP Slater determinant There may have several possible solutions in the VAP calculations. Needs to try many times to find out all minima. Ground state (I=0) of 26Mg with USD interaction No EPJ(I=0) (MeV) 1 -103.954 2 -103.954 3 -103.954 4 -103.954 5 -103.954 6 -103.954 7 -103.287 8 -103.954 9 -103.954 10 -103.287 Q 16.372 16.372 16.372 16.372 16.372 16.372 15.424 16.372 16.372 15.424 g 32.017 -87.982 87.983 -152.018 -32.017 32.017 -52.229 -32.018 32.017 -52.229 All possible VAP (I=0) and HF solutions for 26Mg 26 90 120 g (deg) -96 60 20 30 150 Q 10 0 180 0 10 330 210 Energy (MeV) Mg -98 300 270 VAP -97.694 -98.396 -100 -102 -103.287 -104 -103.954 20 240 HF -105.536 -106 SM Another example of 28 Si 90 120 g (deg) -122 60 -124 30 Q 10 0 180 0 10 330 210 20 240 300 270 Energy (MeV) 20 150 28Si -126 VAP HF -126.031 -126.300 -128 -129.612 -130 -132 -132.228 -134.089 -134 -134.133 -135.938 -136 SM In HF: Most nuclei are axial. In VAP:No nulceus is axial ! Hartree-Fock With USD interaction VAP Q and g values in the sd shell nuclei with 10 N,Z 18 The same situation for the fp shell nuclei! Hartree-Fock With GXPF1A interaction VAP Q and g values in the fp shell nuclei with 22Z 32, 22 N 38 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. Thanks for your attention! Including Particle-hole excitations on top of the VAP SD: Projected Configuration Interaction (PCI) HF VAP 52 Fe g (deg) 45 30 15 0 0 5 10 15 20 25 30 Q2 Energy (MeV) 60 GXPF1A interaction -146 GXPF1A -148 I=6 -150 I=4 I=2 -152 I=0 HF VAP PCI SM 56Ni 60 HF VAP 56 Ni With GXPF1A interaction g (deg) -196 45 30 15 0 0 5 10 15 20 25 30 Q2 Energy (MeV) PCI for -198 -200 I=6 -202 I=4 I=2 -204 -206 I=0 HF VAP PCI SM Comparison with MCSM and VAMPIR With FPD6 interaction 60 -202.0 -202.2 Energy (MeV) 45 HF VAP VAP g (deg) 30 -202.4 15 -202.6 -202.8 0 5 0 10 15 20 25 30 Q2 -203.0 1 SD 2 SDs 3 SDs -203.2 PCI VAMPIR (2002) MCSM (2010) SM