1. Origin of prolate dominance of nuclear deformation - an analysis with Woods-Saxon potential -
*S. Takahara1,　N. Tajima2,　Y. R. Shimizu3
1Kyorin Univ., 2Fukui Univ., 3Kyushu Univ.
RIKEN Symposium 2006
Methods of many-body systems: mean field theories and beyond
March 21, 2006

2. Contents Introduction Nilsson Strutinsky method
Woods-Saxon Strutinsky method
Summary

3. Basic question: prolate dominance
Most of nuclei are deformed into prolate shapes
Why nuclei prefer prolate shapes?
Deformation: shell structure of single particle spectrum
Relation between Hamiltonians and prolate shapes?
Only consider mean single-particle potential

4. Principal features of single particle potential
Spin-orbit coupling
Radial profile　(H.O. ⇔ square-well) Frisk: classical periodic orbits (1990)
square-well wo LS: prolate dominance
Pairing
What happens if these features are changed?
Strutinsky method with Nilsson and Woods-Saxon potential

6. pseudospin symmetry at μN=0.5
e.g. p3/2, d3/2 are degenerate exactly

7. Nilsson
LL / LS
Results with Nilsson

8. Nilsson
pairing
0.7 1
1.2

9. Nilsson potential : results
PES(ε2, ε4) over the nuclear chart (1834 even-even nuclei)
Proportion of prolate nuclei =
# of prolate / (# of prolate + # of oblate)
Standard potential: 86% are prolate
Potential profile:
Rp(fll) along fls=0: increasing trend, support Frisk theory
Spin-orbit:
Does not favor prolate or oblate: oscillating
Strong interference with l2 term
Relation with pseudospin symmetry
Pairing
Enhances both prolate and oblate dominances

11. Calculation with W. S. Strutinsky method
Easier than HF but costs 10 times longer than Nilsson
Construct a cluster of ten PCs
Job control with scripts
PES(β2, β4) over the nuclear chart (8<Z<126, 8<N<184, 1834 even-even nuclei)
As a function of spin-orbit, pairing, (diffuseness) strength, calculate the proportion of prolate nuclei

12. Woods-Saxon: spin-orbit / pairing

13. optimized
Ramon-Wyss 1
universal
Ramon-Wyss 2

14. Woods-Saxon : results Spin-orbit / pairing
6 types of parameter set (global properties on the nuclear chart are different. E.g. driplines)
Essentially the same results
Spin-orbit / pairing
Similar results with Nilsson
Oscillates with spin-orbit strength
Pairing enhances prolate/oblate dominances
f_SO=±1,0: prolate dominance
f_SO=±1/2: Nilsson, oblate dominance
W.S., more than 50% prolate

15. Woods-Saxon : in progress
Effect of diffuseness
Some difficulties
Existence of continuum
Restriction on the combination of diffuseness and depth
Arita, rα-potential
Diffuseness and spin-orbit
Pseudospin symmetry

16. spin-orbit/diffuseness
Preliminary　result

17. Summary approach to explain the origin of prolate dominace
Nilsson Strutinsky: published PRC64,037301(2001)
Interference between spin-orbit and radial profile
Woods-Saxon Strutinsky:
Same as Nilsson qualitatively
Effect of diffuseness: in progress