ROLE OF THE ROLE OF THE NON-AXIAL OCTUPOLE DEFORMATIONNON-AXIAL OCTUPOLE DEFORMATION

IIN THE POTENTIAL ENERGY N THE POTENTIAL ENERGY OF HEAVY AND SUPERHEAVY NUCLEIOF HEAVY AND SUPERHEAVY NUCLEI

XVI NUCLEAR PHYSICS WORKSHOPXVI NUCLEAR PHYSICS WORKSHOP

Kazimierz Dolny 23. – 27.09.2009Kazimierz Dolny 23. – 27.09.2009

PIOTR JACHIMOWICZ, MICHAŁ KOWAL, PIOTR ROZMEJ,PIOTR JACHIMOWICZ, MICHAŁ KOWAL, PIOTR ROZMEJ,

JANUSZ SKALSKI, ADAM SOBICZEWSKIJANUSZ SKALSKI, ADAM SOBICZEWSKI

●● Introduction Introduction ●● Method of calculationsMethod of calculations ●● More about motivationMore about motivation ● ● Results and discussionResults and discussion

○ ○ GroundGround –– state energystate energy ○ ○ PotentialPotential –– energy surfacesenergy surfaces

●● ConclusionsConclusions

Plan of the presentation :Plan of the presentation :

Motivation :Motivation :

aa3232 (Y(Y33 2 2 + Y+ Y33 -2-2))

●● The importance of The importance of nonaxial octupole nonaxial octupole (tetrahedral) (tetrahedral) deformation in atomic nuclei was 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 One searches for local (global) minima with large (or sizable) large (or sizable) aa3232 on total energy surfaces. on total energy surfaces.

Macroscopic-microscopic approach:Macroscopic-microscopic approach:

EE = E= Etottot((ββλλ µµ) ) – – EEMACRO MACRO ((ββλλ µµ == 0)0)

EEMACROMACRO((ββλλ µµ)+ E)+ EMICROMICRO((ββλλ µµ))

○○ EEMACROMACRO((ββλλ µµ) ) == Yukawa + exp Yukawa + exp

Method of the calculation :Method of the calculation :

○ ○ EEMICROMICRO((ββλλ µµ) ) == Woods – Saxon + pairing BCS Woods – Saxon + pairing BCS

○○ We studiedWe studied energy vs. energy vs. aa3232

(a one-dimensional(a one-dimensional calculation) calculation)

○○ EEdefdef = E(0) – E(a = E(0) – E(a3232))

Our try from one year ago :Our try from one year ago :

(a(a3232))

○ ○ values of deformation avalues of deformation a3232

○○ We studiedWe studied energy vs. energy vs. aa3232

(a one-dimensional(a one-dimensional calculation) calculation)

○ ○ EEdefdef = E(0) – E(a = E(0) – E(a3232))(a(a3232))

○○ values of deformation values of deformation aa3232

Our try from one year ago :Our try from one year ago :

○○EEdefdef == E E MICROMICRO(0) – E (0) – E MICROMICRO (a(a3232))

EEdefdef == E Edefdef + + EEdefdef

MICROMICRO MACROMACRO

MICROMICRO

○○EEdefdef == E E MACROMACRO (0) – E (0) – E MACROMACRO (a(a3232))MACROMACRO

(a(a3232))

Now we include many deformationsNow we include many deformationstrying to answer the following:trying to answer the following:

● ● Does a Does a tetrahedraltetrahedral ( (aa3232) ) effect surviveeffect survive competition with other deformations competition with other deformations in in heavyheavy and and superheavysuperheavy nuclei ? nuclei ?

● ● How large is the effect of How large is the effect of aa3232 on the on the potential – energy surfaces on top potential – energy surfaces on top of the axial deformations of the axial deformations ββ2 2 … … ββ88??

EEdef def is is much largermuch larger than E than Edefdef

○○ One can suspect that the deformations One can suspect that the deformations {{ββλλ}}, ,

will strongly will strongly decrasedecrase or or even eliminate even eliminate the effect ofthe effect of a a32 32 ..

Results :Results :○○ EEdef def = E(0) – E = E(0) – EGSGS((ββλλ))

○ ○ EEdef def = E(0) – E = E(0) – Eminmin(a(a3232))

((ββ22……ββ88))

(a(a3232))

((ββ22……ββ88)) (a(a3232))

○○ EEdef def = E(0) – E = E(0) – Eminmin(a(a3232))(a(a3232))

○ ○ EEdef def = E(0) – E = E(0) – EGSGS((ββλλ))

((ββ22……ββ88))

(a(a3232))

Results :Results :

((ββ22……ββ88))EEdef def is is much largermuch larger than E than Edefdef

○○ One can suspect that the deformations One can suspect that the deformations {{ββλλ}}, ,

will strongly will strongly decrasedecrase or or even eliminate even eliminate the effect ofthe effect of a a32 32 ..

○○ The GS energies obtained from the minimization in theThe GS energies obtained from the minimization in the8-dimensional8-dimensional deformation space deformation space {a{a3232,,ββ22, … , , … , ββ88}}

○○ EEdef def = E(0) – E = E(0) – EGSGS(a(a32 32 ,, ββλλ))(a(a32,32,ββ22……ββ88))

○ ○ values of deformation avalues of deformation a3232

Results :Results :

○ ○ EEdef def = E(0) – E(a = E(0) – E(a32 32 , , ββλλ))(a(a32,32,ββ22……ββ88))

○○ values of deformation values of deformation aa3232

Results :Results :

○○ The GS energies obtained from the minimization in theThe GS energies obtained from the minimization in the8-dimensional8-dimensional deformation space deformation space {a{a3232,,ββ22, … , , … , ββ88}}

Results :Results :

○○ The map from the The map from the 6-dimensional6-dimensional minimization over minimization over {{ββ33, … , , … , ββ88} } at each pointat each point

○○ The map from the The map from the 6-dimensional6-dimensional minimization over minimization over {{ββ33, … , , … , ββ88} } at each pointat each point

Results :Results :

○○ EEdefdef = E(0) – E(a = E(0) – E(a3232))

Preliminary results in the region Z Preliminary results in the region Z ≥≥110, N 110, N ≥≥146 :146 :

(a(a3232))

○ ○ values of deformation avalues of deformation a3232

○○ first the first the one-dimensionalone-dimensional calculation: calculation:

○ ○ EEdefdef = E(0) – E(a = E(0) – E(a3232))

○○ values of deformation values of deformation aa3232

(a(a3232))

Preliminary results in the region Z Preliminary results in the region Z ≥≥110, N 110, N ≥≥146 :146 :

○○ first the first the one-dimensionalone-dimensional calculation: calculation:

○○EEdefdef == E E MICROMICRO(0) – E (0) – E MICROMICRO (a(a3232)) ○○EEdefdef == E E MACROMACRO (0) – E (0) – E MACROMACRO (a(a3232))MICROMICRO MACROMACRO

Energy decomposition into micro and macro parts:Energy decomposition into micro and macro parts:

EEdefdef == E Edefdef + + EEdefdef

(a(a3232)) MICROMICRO MACROMACRO

EEdef def is is largerlarger than than EEdefdef

((ββ22……ββ88))

○○ EEdef def = E(0) – E = E(0) – EGSGS((ββλλ))((ββ22……ββ88))

○ ○ EEdef def = E(0) – E = E(0) – Eminmin(a(a3232))(a(a3232))

Comparison between effects of (Comparison between effects of (ββ22… … ββ88) and ) and aa3232 alonealone

(a(a3232))

EEdef def is is largerlarger than than EEdefdef

((ββ22……ββ88)) (a(a3232))

○○ EEdef def = E(0) – E = E(0) – Eminmin(a(a3232))(a(a3232))

○ ○ EEdef def = E(0) – E = E(0) – EGSGS((ββλλ))((ββ22……ββ88))

Comparison between effects of (Comparison between effects of (ββ22… … ββ88) and ) and aa3232 alonealone

○○ The map from the The map from the 6-dimensional6-dimensional minimization over minimization over {{ββ33, … , , … , ββ88} } at each pointat each point

Conclusions :Conclusions :

● ● Deformation Deformation aa3232 significantly lowers energy of 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 Since these nuclei are strongly deformed with EEdefdef ≈≈ 8 MeV8 MeV, the , the aa3232 efect manifests itself mostly as 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 Around 228228FmFm we can find global minima with we can find global minima with aa3232, , but those minima but those minima are very shalloware very shallow..

● ● IIn the sn the superheavyuperheavy regionregion we we didn't finddidn't find any any global global aa3322 minimum, minimum, BUTBUT: one has to note that : one has to note that aa32 32 was the was the only nonaxial only nonaxial deformation included. deformation included.