1. Yu Zhang(张宇), Feng Pan(潘峰)
Nucleon shape phase transition at the nucleon level within SD-pair shell model
Yanan Luo(罗延安), Lei Li(李磊)
School of Physics, Nankai University, Tianjin
Yu Zhang(张宇), Feng Pan(潘峰)
Department of Physics, Liaoning Normal University, Dalian
J. P. Draayer
Department of Physics and Astronomy, LSU, BTR
Lanzhou, Sept.6, 2015

2. Outline I. Nuclear Shapes and Shape Phase Transition
II. Status of the Research on the nuclear shape
Phase Transition
III. SD-pair shell model(SDPSM)
IV. Nuclear shapes in SD-pair shell model
V. Nucear shape phase transition in SD-
pair shell model
VI. Summary

3. I. Nuclear Shape-Phases and Shape-Phase Transition
Modes of nuclear collective motion and the symmetries
 Shape of Nucleus
 Sphere
 Deforemation
quadrupole
octupole
hexadecupole
 Modes of Nuclear Collective Motion
vibration
axial rotation ( prolate, oblate)
-soft rotation
triaxial rotation ••• ••• ••• •••
From Prof. Yuxin Liu

4.  Correspondence between collective motion and symmetry
Vibration U(5)
Axial Rotation SU(3) ( SU*(3))
γ-soft rotation O(6)
 Shape Phase Transition and the States at the Critical
Points of the Phase Transitions
Vibration – γ-soft Rotation E(5)
Vibration – Prolate Axial Rotation X(5)
Prolate – Oblate Axial Rotations Y(5)

5. Iachello, PRL 91, (’03)

6. The shape-phase transition in the U(5)-SU(3) transitional region
X(5)
SU(3)
Harmonic Vibrator
Soft Liquid-drop
Rigid Rotor

7. II. Status of the research on nuclear shape phase transition
System:
Even-even nuclei, Odd-even nuclei, Odd-odd nuclei
Theoretical method:
IBM, IBFM, IBFFM
Geometric Collective models
Fermionic approaches
FDSM, shell model, HFB
••••••••••••

8. The most useful observables

9. Two neutron seperation energies
E0 transition
Isotopes shifts
Two neutron transfer cross sections
……………….

10. III. SD-pair shell model(SDPSM)
Because of the success of the IBM, the full space
was truncated into SD-pair subspace, r=0, 2
The Hamiltonian can be diagonolized directly in the Fermion space!

11. All one and two-body matrix elements
can be expressed in terms of overlap
For eg.

12. The overlap between two N-pair basis

13. The new pair structure coefficient

15. The hierarchy of various nuclear model
SDPSM
NPSM

16. IV. Nuclear shape phases in SDPSM

17. Rotational spectrum

19. V. Nuclear shape phase transition in SDPSM

20. Vibration to rotational transitional pattern

22. Liu YX, et. al., Phys. Lett. B688,298(2010)
Boson Mapping (Dyson)
It is found that the nuclear shape phase transitional pattern can be
Produced by changing the interactional strength!
Problem: is it true without Bosonization approximation?

23. The Hamiltonian used in the SD-pair shell model
Toy case: N=9 oscillator shell

24. The effect of monopole pairing interaction
Quadrupolel-quadrupole interactional strength kappa

26. The effect of quadrupole-quadrupole interaction

29. Summary
The nuclear shape phases can be produced very well in the SDPSM
The nulcear shape phase transitional pattern as in the IBM can be produced in the SDPSM
The properties of the critical symmetry can be produced in the SDPSM

30. Thanks !

31. Spectrum of E(5) Spectrum of X(5) Spectrum of Y(5) Symmetry Symmetry Symmetry
( Iachello, PRL 85, 3580 (2000) ) ( Iachello, PRL 87, (2001)) ( Iachello, PRL 91, (2003))
134Ba, 108Pd, 130Xe, ··· Sm, 154Gd, 156Dy, 150Nd, ··· Er, 168Er, ···
( Casten, PRL 85, 3584 (2000); ( Casten, PRL 87, (2001); (PRC 68 , (2003); …
Ginocchio, PRL 90, (2003) ; Capirio, PRC 66, (2002);
Liu, PRC 65, (2002), … ) Tonev, PRC 69, (2004); …)

32. Characteristic of Evolution of Energy Spectrum for the
Transition from U(5) to SU(3) Through X(5)
SU(3) (Rotation) X(5) U(5) (Vibration)
( Pan, Draayer, et. al., PLB 576, 297 (2003) )

33. Summary for the shape-phase transition in the U(5)-SU(3) transitional region
Harmonic Vibrator
Soft Liquid-drop
Rigid Rotor

34. Extension: Unified description of the shape phase structure and
phase transition of odd-A nuclei
F. Iachello, Phys. Rev. Lett. 95, (2005);
Result of U(6,4) model
135Ba, E(5/4) Symmetry, M.S. Fetea, et al., Phys. Rev. C 73, (2005)

39. Nuclear shape-phase and phase transition in IBM-2
M. A. Caprio and F. Iachello，Phys. Rev. Lett. 93, (2004)