1. ROLE OF THE NON-AXIAL OCTUPOLE DEFORMATION IN THE POTENTIAL ENERGY OF HEAVY AND SUPERHEAVY NUCLEI XVI NUCLEAR PHYSICS WORKSHOP Kazimierz Dolny 23. – 27.09.2009 PIOTR JACHIMOWICZ, MICHAŁ KOWAL, PIOTR ROZMEJ, JANUSZ SKALSKI, ADAM SOBICZEWSKI

2. ● Introduction ● Introduction ● Method of calculations ● Method of calculations ● More about motivation ● More about motivation ● Results and discussion ● Results and discussion ○ Ground – state energy ○ Ground – state energy ○ Potential – energy surfaces ○ Potential – energy surfaces ● Conclusions ● Conclusions Plan of the presentation :

3. Motivation : a 32 (Y 3 2 + Y 3 -2 ) a 32 (Y 3 2 + Y 3 -2 ) ● The importance of nonaxial octupole (tetrahedral) deformation in atomic nuclei was (tetrahedral) deformation in atomic nuclei was suggested some time ago (J. Dudek et al.). suggested some time ago (J. Dudek et al.). ● One searches for local (global) minima with large (or sizable) a 32 on total energy surfaces. large (or sizable) a 32 on total energy surfaces.

4. Macroscopic-microscopic approach: Macroscopic-microscopic approach: E = E tot ( β λ µ ) – E MACRO ( β λ µ = 0) E = E tot ( β λ µ ) – E MACRO ( β λ µ = 0) E MACRO ( β λ µ )+ E MICRO ( β λ µ ) ○ E MACRO ( β λ µ ) = Yukawa + exp Method of the calculation : ○ E MICRO (β λ µ ) = Woods – Saxon + pairing BCS ○ E MICRO (β λ µ ) = Woods – Saxon + pairing BCS

5. ○ We studied energy vs. a 32 (a one-dimensional calculation) ○ E def = E(0) – E(a 32 ) Our try from one year ago : (a 32 ) ○ values of deformation a 32

6. ○ We studied energy vs. a 32 (a one-dimensional calculation) ○ E def = E(0) – E(a 32 ) (a 32 ) ○ values of deformation a 32 Our try from one year ago :

7. ○ E def = E MICRO (0) – E MICRO (a 32 ) E def = E def + E def MICROMACRO MICRO ○ E def = E MACRO (0) – E MACRO (a 32 ) MACRO (a 32 )

8. Now we include many deformations trying to answer the following: ● Does a tetrahedral (a 32 ) effect survive competition with other deformations competition with other deformations in heavy and superheavy nuclei ? in heavy and superheavy nuclei ? ● How large is the effect of a 32 on the potential – energy surfaces on top potential – energy surfaces on top of the axial deformations β 2 … β 8 ? of the axial deformations β 2 … β 8 ?

9. E def is much larger than E def ○ One can suspect that the deformations {β λ }, will strongly decrase or even eliminate the effect of a 32. Results : ○ E def = E(0) – E GS (β λ ) ○ E def = E(0) – E min (a 32 ) (β2…β8)(β2…β8)(β2…β8)(β2…β8) (a 32 ) (β2…β8)(β2…β8)(β2…β8)(β2…β8)

10. ○ E def = E(0) – E min (a 32 ) (a 32 ) ○ E def = E(0) – E GS (β λ ) (β2…β8)(β2…β8)(β2…β8)(β2…β8) (a 32 ) Results : (β2…β8)(β2…β8)(β2…β8)(β2…β8) E def is much larger than E def ○ One can suspect that the deformations {β λ }, will strongly decrase or even eliminate the effect of a 32.

11. ○ The GS energies obtained from the minimization in the 8-dimensional deformation space {a 32,β 2, …, β 8 } ○ E def = E(0) – E GS (a 32, β λ ) (a 32, β 2 …β 8 ) ○ values of deformation a 32 Results :

12. ○ E def = E(0) – E(a 32, β λ ) (a 32, β 2 …β 8 ) ○ values of deformation a 32 Results : ○ The GS energies obtained from the minimization in the 8-dimensional deformation space {a 32,β 2, …, β 8 }

13. Results : ○ The map from the 6-dimensional minimization over {β 3, …, β 8 } at each point

14. ○ The map from the 6-dimensional minimization over {β 3, …, β 8 } at each point Results :

15. ○ E def = E(0) – E(a 32 ) Preliminary results in the region Z ≥110, N ≥146 : (a 32 ) ○ values of deformation a 32 ○ first the one-dimensional calculation:

16. ○ E def = E(0) – E(a 32 ) ○ values of deformation a 32 (a 32 ) Preliminary results in the region Z ≥110, N ≥146 : ○ first the one-dimensional calculation:

17. ○ E def = E MICRO (0) – E MICRO (a 32 ) ○ E def = E MACRO (0) – E MACRO (a 32 ) MICROMACRO Energy decomposition into micro and macro parts: E def = E def + E def (a 32 ) MICROMACRO

18. E def is larger than E def (β2…β8)(β2…β8)(β2…β8)(β2…β8) ○ E def = E(0) – E GS (β λ ) (β2…β8)(β2…β8)(β2…β8)(β2…β8) ○ E def = E(0) – E min (a 32 ) (a 32 ) Comparison between effects of (β 2 … β 8 ) and a 32 alone (a 32 )

19. E def is larger than E def (β2…β8)(β2…β8)(β2…β8)(β2…β8) (a 32 ) ○ E def = E(0) – E min (a 32 ) (a 32 ) ○ E def = E(0) – E GS (β λ ) (β2…β8)(β2…β8)(β2…β8)(β2…β8) Comparison between effects of (β 2 … β 8 ) and a 32 alone

20. ○ The map from the 6-dimensional minimization over {β 3, …, β 8 } at each point

21. Conclusions : ● Deformation a 32 significantly lowers energy of some heavy nuclei with respect to the energy at some heavy nuclei with respect to the energy at the spherical shape. the spherical shape. ● Since these nuclei are strongly deformed with E def ≈ 8 MeV, the a 32 efect manifests itself mostly as E def ≈ 8 MeV, the a 32 efect manifests itself mostly as a local minimum in the energy surface, appearing a local minimum in the energy surface, appearing high above the global minimum. high above the global minimum. ● Around 228 Fm we can find global minima with a 32, but those minima are very shallow. but those minima are very shallow. ● In the superheavy region we didn't find any global a 32 minimum, a 32 minimum, BUT: one has to note that a 32 was the only nonaxial BUT: one has to note that a 32 was the only nonaxial deformation included. deformation included.