Tetrahedral Symmetry in Nuclei:Theory and Experimental Criteria

Jerzy DUDEK

Department of Subatomic Research, CNRS/IN2P3

andUniversity of Strasbourg I

14th October 2008

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

The Most Fundamental Issue In Particle Physicsare fundamental objects: quarks, leptons, gauge-bosons

* * *

The Most Fundamental Issue in Nuclear Physics

is nuclear existence: nuclear masses and thus nuclear stability

* * *

In the following we address the new concepts of stability:theoretical ideas and examples of an experimental evidence

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

The Most Fundamental Issue In Particle Physicsare fundamental objects: quarks, leptons, gauge-bosons

* * *

The Most Fundamental Issue in Nuclear Physics

is nuclear existence: nuclear masses and thus nuclear stability

* * *

In the following we address the new concepts of stability:theoretical ideas and examples of an experimental evidence

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

The Most Fundamental Issue In Particle Physicsare fundamental objects: quarks, leptons, gauge-bosons

* * *

The Most Fundamental Issue in Nuclear Physics

is nuclear existence: nuclear masses and thus nuclear stability

* * *

In the following we address the new concepts of stability:theoretical ideas and examples of an experimental evidence

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

The Most Fundamental Issue In Particle Physicsare fundamental objects: quarks, leptons, gauge-bosons

* * *

The Most Fundamental Issue in Nuclear Physics

is nuclear existence: nuclear masses and thus nuclear stability

* * *

In the following we address the new concepts of stability:theoretical ideas and examples of an experimental evidence

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

The Most Fundamental Issue In Particle Physicsare fundamental objects: quarks, leptons, gauge-bosons

* * *

The Most Fundamental Issue in Nuclear Physics

is nuclear existence: nuclear masses and thus nuclear stability

* * *

In the following we address the new concepts of stability:theoretical ideas and examples of an experimental evidence

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Pathways for Future Facilities and Astrophysics ...

Neutron Number

Pro

ton

Nu

mb

er

with new [tetrahedral] SH magic numbers !

Superheavy Nuclei

New Islands of Stability

Nuclei Known Today

[special astrophysical consequences]

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

COLLABORATORS:

Dominique CURIENJacek DOBACZEWSKIAndrzej GOZDZFlorent HAASDaryl HARTLEYAdam MAJKasia MAZUREKHerve MOLIQUELee RIEDINGERNicolas SCHUNCK

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Group Theory and Geometry for Historians [I]

Recall: Platonic Figures = Polyhedra whosefaces are identical regular convex polygons

Allowed polygons are equilateral triangles, orsquares or regular pentagons

Plato 428-347

Tetrahedron, Cube, Octahedron, Icosahedron, Dodecahedron

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Group Theory and Geometry for Historians [I]

Recall: Platonic Figures = Polyhedra whosefaces are identical regular convex polygons

Allowed polygons are equilateral triangles, orsquares or regular pentagons

Plato 428-347

Tetrahedron, Cube, Octahedron, Icosahedron, Dodecahedron

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Group Theory and Geometry for Historians [I]

Recall: Platonic Figures = Polyhedra whosefaces are identical regular convex polygons

Allowed polygons are equilateral triangles, orsquares or regular pentagons

Plato 428-347

Tetrahedron, Cube, Octahedron, Icosahedron, Dodecahedron

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Group Theory and Geometry for Historians [I]

Recall: Platonic Figures = Polyhedra whosefaces are identical regular convex polygons

Allowed polygons are equilateral triangles, orsquares or regular pentagons

Plato 428-347

Tetrahedron, Cube, Octahedron, Icosahedron, Dodecahedron

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Group Theory and Geometry for Historians [II]

The symbol of ’beauty in symmetry’ are five Platonic Figures

There exist only five regular convex (=platonic) polyhedra:tetrahedron, cube, octahedron, icosahedron & dodecahedron

As it seems, neolithic people from Scotland have developed thefive Platonic solids about 1000-3000 years before Plato (stonemodels in Ashmolean Museum, Oxford) ... in religious context

In what follows we stick to the aspect of beauty and nuclearreality - similarity to any other context will be purely accidental

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Group Theory and Geometry for Historians [II]

The symbol of ’beauty in symmetry’ are five Platonic Figures

There exist only five regular convex (=platonic) polyhedra:tetrahedron, cube, octahedron, icosahedron & dodecahedron

As it seems, neolithic people from Scotland have developed thefive Platonic solids about 1000-3000 years before Plato (stonemodels in Ashmolean Museum, Oxford) ... in religious context

In what follows we stick to the aspect of beauty and nuclearreality - similarity to any other context will be purely accidental

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Group Theory and Geometry for Historians [II]

The symbol of ’beauty in symmetry’ are five Platonic Figures

There exist only five regular convex (=platonic) polyhedra:tetrahedron, cube, octahedron, icosahedron & dodecahedron

As it seems, neolithic people from Scotland have developed thefive Platonic solids about 1000-3000 years before Plato (stonemodels in Ashmolean Museum, Oxford) ... in religious context

In what follows we stick to the aspect of beauty and nuclearreality - similarity to any other context will be purely accidental

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Non-Trivial Discrete Symmetries in Nuclei: Td

Let us recall one of the magic forms introduced long time by Plato.The implied symmetry leads to the tetrahedral group denoted Td

A tetrahedron has four equal walls.Its shape is invariant with respect to24 symmetry elements. Tetrahedronis not invariant with respect to theinversion. Of course nuclei cannot berepresented by a sharp-edge pyramid

... but rather in a form of a regular spherical harmonic expansion:

R(ϑ, ϕ) = R0 c({α})[1 +λmax∑

λ

λ∑µ=−λ

αλ,µ Yλ,µ(ϑ, ϕ)]

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Non-Trivial Discrete Symmetries in Nuclei: Td

Let us recall one of the magic forms introduced long time by Plato.The implied symmetry leads to the tetrahedral group denoted Td

A tetrahedron has four equal walls.Its shape is invariant with respect to24 symmetry elements. Tetrahedronis not invariant with respect to theinversion. Of course nuclei cannot berepresented by a sharp-edge pyramid

... but rather in a form of a regular spherical harmonic expansion:

R(ϑ, ϕ) = R0 c({α})[1 +λmax∑

λ

λ∑µ=−λ

αλ,µ Yλ,µ(ϑ, ϕ)]

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Non-Trivial Discrete Symmetries in Nuclei: Td

Let us recall one of the magic forms introduced long time by Plato.The implied symmetry leads to the tetrahedral group denoted Td

A tetrahedron has four equal walls.Its shape is invariant with respect to24 symmetry elements. Tetrahedronis not invariant with respect to theinversion. Of course nuclei cannot berepresented by a sharp-edge pyramid

... but rather in a form of a regular spherical harmonic expansion:

R(ϑ, ϕ) = R0 {1+α3+2(Y3+2 + Y3−2)| {z }one parameter 3rd order

+ α72

ˆ(Y7+2 + Y7−2) −

q1113

(Y7+6 + Y7−6)˜| {z }

one parameter 7th order

}

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Introducing Nuclear Octahedral Symmetry

Let us recall one of the magic forms introduced long time by Plato.The implied symmetry leads to the octahedral group denoted Oh

An octahedron has 8 equal walls. Itsshape is invariant with respect to 48symmetry elements that include in-version. However, the nuclear surfacecannot be represented in the form ofa diamond → → → → → → → →

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... but rather in a form of a regular spherical harmonic expansion:

R(ϑ, ϕ) = R0 c({α})[1 +λmax∑

λ

λ∑µ=−λ

αλ,µ Yλ,µ(ϑ, ϕ)]

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Introducing Nuclear Octahedral Symmetry

Let us recall one of the magic forms introduced long time by Plato.The implied symmetry leads to the octahedral group denoted Oh

An octahedron has 8 equal walls. Itsshape is invariant with respect to 48symmetry elements that include in-version. However, the nuclear surfacecannot be represented in the form ofa diamond → → → → → → → →

... but rather in a form of a regular spherical harmonic expansion:

R(ϑ, ϕ) = R0

˘11+α40

ˆY40 +

q514

(Y4+4 + Y4−4)˜

| {z }one parameter 4th order

+ α60

ˆY60 −

q72(Y6+4 + Y6−4)

˜| {z }

one parameter 6th order

¯

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Group Theory and Geometry for Pedestrians [I]

Consider a nuclear surface with tetrahedral deformation ...

This is what we call in jargon: ‘nuclear pyramid’ = tetrahedron

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Group Theory and Geometry for Pedestrians [II]

... and another surface with octahedral deformation...

This is what we call in jargon: ‘nuclear diamond’ = octahedron

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Group Theory and Geometry for Pedestrians [III]

... or even better, compare them directly ...

You most likely1 clearly see the difference between the two shapes:the pyramid [left] and the diamond [right]

1As one student said: ”Pyramids are in Egypt” and ”Diamonds Are a Girl’s Best Friend”;

... but she studied bio-physics...

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Group Theory and Geometry for Pedestrians [IV]

Surprise-Surprise ! Difficult to believe !Superposing the tetrahedral- and octahedral-symmetry surfaces givesus a new surface but again of tetrahedral symmetry: new richness

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Group Theory and Geometry for Pedestrians [IV]

Although by far not intuitive - but it is strict mathematical truth !

It is a manifestation of the theorem saying that:

”Tetrahedral group is a sub-group of the octahedral one”

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

A few words before starting:

Why Should We Be Interested in All That?

First class of reasons:

• Within nuclear mean-field theory the tetrahedral and octahedralsymmetries imply the highest degeneracies of the nucleonic levels →big single-particle gaps and high nuclear stability

• Most economic search for nuclear stability as compared to the oldspherical harmonic expansions [only very few degrees of freedom]

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

A few words before starting:

Why Should We Be Interested in All That?

Second class of reasons:

• Tetrahedral and octahedral symmetries are only a forefront ofgroup-theory based families of new nuclear symmetries

• The approach uses two most powerful tools: group theory and thenuclear mean-field theory → the best what we have at the beginningof the XXIst century

• All other so far known results, criteria of nuclear stability etc. arejust a particular case of the new formulation

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Outline of the New Theory of Nuclear Stability

Part I

Outline of the New Theory of Nuclear Stability

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Outline of the New Theory of Nuclear StabilitySingle-Particle Structure and Global Stability of NucleiExamples of Most Attractive Point-Group Symmetries

Summarising: Global Stability vs. Gaps in SP Spectra

Consider a typical outcome of theMean-Field calculation: the shellstructures and the total energies

Presence of sufficiently strong gapscorrelates with local minima of thetotal nuclear energy

The ‘Deformation Parameter’ axisrepresents several deformations ofthe mean field e.g. {Qλµ}, {αλµ},the usual approach in constrainedmean-field calculations

GAP

GAP

Deformation Parameter

Nuc

leon

Ene

rgie

sDeformation Deformation

Nucleus ’1’ Nucleus ’2’

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Outline of the New Theory of Nuclear StabilitySingle-Particle Structure and Global Stability of NucleiExamples of Most Attractive Point-Group Symmetries

Summarising: Global Stability vs. Gaps in SP Spectra

Consider a typical outcome of theMean-Field calculation: the shellstructures and the total energies

Presence of sufficiently strong gapscorrelates with local minima of thetotal nuclear energy

The ‘Deformation Parameter’ axisrepresents several deformations ofthe mean field e.g. {Qλµ}, {αλµ},the usual approach in constrainedmean-field calculations

GAP

GAP

Deformation Parameter

Nuc

leon

Ene

rgie

sDeformation Deformation

Nucleus ’1’ Nucleus ’2’

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Outline of the New Theory of Nuclear StabilitySingle-Particle Structure and Global Stability of NucleiExamples of Most Attractive Point-Group Symmetries

Summarising: Global Stability vs. Gaps in SP Spectra

Consider a typical outcome of theMean-Field calculation: the shellstructures and the total energies

Presence of sufficiently strong gapscorrelates with local minima of thetotal nuclear energy

The ‘Deformation Parameter’ axisrepresents several deformations ofthe mean field e.g. {Qλµ}, {αλµ},the usual approach in constrainedmean-field calculations

GAP

GAP

Deformation Parameter

Nuc

leon

Ene

rgie

sDeformation Deformation

Nucleus ’1’ Nucleus ’2’

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Outline of the New Theory of Nuclear StabilitySingle-Particle Structure and Global Stability of NucleiExamples of Most Attractive Point-Group Symmetries

Symmetries, Representations and Degeneracies

Given Hamiltonian H and a group: G = {O1,O2, . . . Of }Assume that G is a symmetry group of H i.e.

[H,Ok ] = 0 with k = 1, 2, . . . f

Let irreducible representations of G be {R1,R2, . . . Rr}Let their dimensions be {d1, d2, . . . dr}, respectively

Then the eigenvalues {εν} of the problem

Hψν = ενψν

appear in multiplets d1-fold, d2-fold ... degenerate

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Outline of the New Theory of Nuclear StabilitySingle-Particle Structure and Global Stability of NucleiExamples of Most Attractive Point-Group Symmetries

Symmetries, Representations and Degeneracies

Given Hamiltonian H and a group: G = {O1,O2, . . . Of }Assume that G is a symmetry group of H i.e.

[H,Ok ] = 0 with k = 1, 2, . . . f

Let irreducible representations of G be {R1,R2, . . . Rr}Let their dimensions be {d1, d2, . . . dr}, respectively

Then the eigenvalues {εν} of the problem

Hψν = ενψν

appear in multiplets d1-fold, d2-fold ... degenerate

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Outline of the New Theory of Nuclear StabilitySingle-Particle Structure and Global Stability of NucleiExamples of Most Attractive Point-Group Symmetries

Symmetries, Representations and Degeneracies

Given Hamiltonian H and a group: G = {O1,O2, . . . Of }Assume that G is a symmetry group of H i.e.

[H,Ok ] = 0 with k = 1, 2, . . . f

Let irreducible representations of G be {R1,R2, . . . Rr}Let their dimensions be {d1, d2, . . . dr}, respectively

Then the eigenvalues {εν} of the problem

Hψν = ενψν

appear in multiplets d1-fold, d2-fold ... degenerate

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Outline of the New Theory of Nuclear StabilitySingle-Particle Structure and Global Stability of NucleiExamples of Most Attractive Point-Group Symmetries

Symmetries, Representations and Degeneracies

Given Hamiltonian H and a group: G = {O1,O2, . . . Of }Assume that G is a symmetry group of H i.e.

[H,Ok ] = 0 with k = 1, 2, . . . f

Let irreducible representations of G be {R1,R2, . . . Rr}Let their dimensions be {d1, d2, . . . dr}, respectively

Then the eigenvalues {εν} of the problem

Hψν = ενψν

appear in multiplets d1-fold, d2-fold ... degenerate

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Outline of the New Theory of Nuclear StabilitySingle-Particle Structure and Global Stability of NucleiExamples of Most Attractive Point-Group Symmetries

Symmetries, Representations and Degeneracies

Given Hamiltonian H and a group: G = {O1,O2, . . . Of }Assume that G is a symmetry group of H i.e.

[H,Ok ] = 0 with k = 1, 2, . . . f

Let irreducible representations of G be {R1,R2, . . . Rr}Let their dimensions be {d1, d2, . . . dr}, respectively

Then the eigenvalues {εν} of the problem

Hψν = ενψν

appear in multiplets d1-fold, d2-fold ... degenerate

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Outline of the New Theory of Nuclear StabilitySingle-Particle Structure and Global Stability of NucleiExamples of Most Attractive Point-Group Symmetries

Symmetries and Gaps in Nuclear Context: Schematic

Schematic illustration: Levels of 6 irreps and average spacings/gaps

ap

G

Irrep.1 Irrep.2 Irrep.3 Irrep.4 Irrep.5 Irrep.6 All Irreps.

Roughly: The average level spacings within an irrep increase by afactor of 6. The total spectrum may present big unprecedented gaps.

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Outline of the New Theory of Nuclear StabilitySingle-Particle Structure and Global Stability of NucleiExamples of Most Attractive Point-Group Symmetries

Symmetries and Gaps in Nuclear Context: Schematic

Schematic illustration: Levels of 6 irreps and average spacings/gaps

ap

G

Irrep.1 Irrep.2 Irrep.3 Irrep.4 Irrep.5 Irrep.6 All Irreps.

Roughly: The average level spacings within an irrep increase by afactor of 6. The total spectrum may present big unprecedented gaps.

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Outline of the New Theory of Nuclear StabilitySingle-Particle Structure and Global Stability of NucleiExamples of Most Attractive Point-Group Symmetries

Symmetries and Gaps in Nuclear Context: Summary

To increase the chances of having big gaps in the spectra we eitherlook for point groups with high dimension irreps or with many irreps

This implies that we need to verify in the group-theory literature:Which groups have many irreps? ... and/or high dimension irreps?

In other words:

We suggest replacing the multipole expansion in favour ofa selection of point groups when studying nuclear stability

It turns out that the above ideas are well confirmed by calculations

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Outline of the New Theory of Nuclear StabilitySingle-Particle Structure and Global Stability of NucleiExamples of Most Attractive Point-Group Symmetries

Symmetries and Gaps in Nuclear Context: Summary

To increase the chances of having big gaps in the spectra we eitherlook for point groups with high dimension irreps or with many irreps

This implies that we need to verify in the group-theory literature:Which groups have many irreps? ... and/or high dimension irreps?

In other words:

We suggest replacing the multipole expansion in favour ofa selection of point groups when studying nuclear stability

It turns out that the above ideas are well confirmed by calculations

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Outline of the New Theory of Nuclear StabilitySingle-Particle Structure and Global Stability of NucleiExamples of Most Attractive Point-Group Symmetries

Example: Octahedral Symmetry - Proton Spectra

Double group ODh has four 2-dimensional and two 4-dimensional

irreducible representations → six distinct families of levels

-.35 -.25 -.15 -.05 .05 .15 .25 .35

-12

-10

-8

-6

-4

-2

0

2

Octahedral Deformation

Pro

ton

En

erg

ies

[MeV

]

α 40(m

in)=

-.350,

α 40(m

ax)=

.350

α 44(m

in)=

-.209,

α 44(m

ax)=

.209

Str

asb

ou

rg, A

ugu

st 2

002

Dir

ac-

Wood

s-S

axon

5252

5658

64

70

7282

8888

94

{09}[4,3,1] 3/2{13}[4,0,4] 9/2{09}[4,2,0] 1/2{11}[4,1,3] 5/2{12}[4,0,2] 5/2{15}[4,3,1] 1/2{17}[5,2,3] 7/2{08}[4,3,1] 1/2{08}[4,2,2] 3/2{13}[4,0,4] 7/2{23}[5,1,4] 9/2{10}[5,3,2] 5/2{11}[5,1,2] 5/2{07}[5,2,3] 5/2{15}[5,1,4] 7/2{11}[4,0,0] 1/2{18}[4,0,2] 3/2{19}[4,4,0] 1/2{13}[5,0,5] 11/2{10}[5,4,1] 3/2{10}[5,0,3] 7/2{08}[5,0,3] 7/2{10}[5,0,5] 11/2

{10}[4,3,1] 1/2{24}[3,1,2] 3/2

{09}[4,3,1] 3/2{16}[4,1,3] 7/2{25}[4,2,2] 5/2{08}[5,0,3] 7/2{13}[3,0,1] 3/2{08}[3,3,0] 1/2{09}[4,2,2] 3/2{10}[4,3,1] 1/2

{10}[5,0,5] 11/2{21}[4,2,0] 1/2{11}[5,4,1] 3/2{07}[5,1,0] 1/2{10}[5,0,5] 9/2{09}[5,1,2] 5/2{24}[4,2,2] 3/2{17}[4,1,3] 7/2{21}[4,1,3] 5/2{08}[5,0,5] 9/2{18}[5,1,4] 9/2{16}[5,3,2] 5/2{09}[4,0,2] 5/2

160Yb 90 70

Figure: Full lines correspond to 4-dimensional irreducible representations -

they are marked with double Nilsson labels. Observe huge gap at Z=70.

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Outline of the New Theory of Nuclear StabilitySingle-Particle Structure and Global Stability of NucleiExamples of Most Attractive Point-Group Symmetries

Example: Octahedral Symmetry - Neutron Spectra

Double group ODh has four 2-dimensional and two 4-dimensional

irreducible representations → six distinct families of levels

-.35 -.25 -.15 -.05 .05 .15 .25 .35

-12

-10

-8

-6

-4

-2

Octahedral Deformation

Neu

tron

En

erg

ies

[MeV

]

α 40(m

in)=

-.350,

α 40(m

ax)=

.350

α 44(m

in)=

-.209,

α 44(m

ax)=

.209

Str

asb

ou

rg, A

ugu

st 2

002

Dir

ac-

Wood

s-S

axon

82

86

8888

94 94

100

110

114 116

118126

{21}[4,0,2] 3/2{22}[4,4,0] 1/2{12}[4,0,0] 1/2{10}[5,2,3] 5/2{21}[5,1,4] 7/2{13}[5,0,5] 11/2{12}[5,0,5] 11/2{13}[5,2,1] 1/2{07}[5,0,5] 11/2{10}[5,3,0] 1/2{09}[5,0,3] 7/2{16}[5,4,1] 1/2{11}[5,0,1] 3/2{16}[6,2,4] 9/2{07}[5,4,1] 3/2{20}[6,1,5] 11/2{19}[6,3,3] 7/2{10}[5,4,1] 3/2{21}[5,0,5] 9/2{18}[6,1,5] 11/2{08}[5,0,5] 9/2{09}[6,4,2] 5/2{17}[6,1,3] 7/2

{08}[5,4,1] 1/2{08}[5,0,5] 11/2{23}[4,2,2] 3/2{13}[4,1,1] 1/2{21}[4,1,3] 5/2

{08}[5,0,5] 9/2

{16}[5,1,4] 9/2{14}[5,3,2] 5/2{21}[5,2,3] 7/2{08}[5,3,2] 5/2{09}[6,0,4] 9/2{08}[4,0,2] 5/2{07}[8,8,0] 1/2{11}[5,3,0] 1/2{11}[5,2,1] 3/2{12}[5,3,2] 3/2{07}[5,1,4] 7/2

{17}[6,0,6] 13/2{11}[6,5,1] 3/2{10}[6,1,3] 7/2{06}[8,0,2] 5/2{07}[6,4,0] 1/2

{08}[6,0,6] 11/2

160Yb 90 70

Figure: Full lines correspond to 4-dimensional irreducible representations -

they are marked with double Nilsson labels. Observe huge gap at N=114.

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral Symmetry

Part II

Focus on Nuclear Tetrahedral Symmetry

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

First Goal: Obtain Tetrahedral Magic Numbers ...

1 After inspecting many single-particle diagrams in function oftetrahedral deformation we read-out all the magic numbers

2 The tetrahedral symmetric nuclei are predicted to beparticularly stable around magic closures:

{Zt ,Nt} = {32, 40, 56, 64, 70, 90, 136}

3 ... and more precisely around the following nuclei:

6432Ge32,

7232Ge40,

8832Ge56,

8040Zr40,

11040Zr70,

11256Ba56,

12656Ba70,

14656Ba90,

13464Gd70,

15464Gd90,

16070Yb90,

22690Th136

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

First Goal: Obtain Tetrahedral Magic Numbers ...

1 After inspecting many single-particle diagrams in function oftetrahedral deformation we read-out all the magic numbers

2 The tetrahedral symmetric nuclei are predicted to beparticularly stable around magic closures:

{Zt ,Nt} = {32, 40, 56, 64, 70, 90, 136}

3 ... and more precisely around the following nuclei:

6432Ge32,

7232Ge40,

8832Ge56,

8040Zr40,

11040Zr70,

11256Ba56,

12656Ba70,

14656Ba90,

13464Gd70,

15464Gd90,

16070Yb90,

22690Th136

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

First Goal: Obtain Tetrahedral Magic Numbers ...

1 After inspecting many single-particle diagrams in function oftetrahedral deformation we read-out all the magic numbers

2 The tetrahedral symmetric nuclei are predicted to beparticularly stable around magic closures:

{Zt ,Nt} = {32, 40, 56, 64, 70, 90, 136}

3 ... and more precisely around the following nuclei:

6432Ge32,

7232Ge40,

8832Ge56,

8040Zr40,

11040Zr70,

11256Ba56,

12656Ba70,

14656Ba90,

13464Gd70,

15464Gd90,

16070Yb90,

22690Th136

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Tetrahedral Stability; Tetrahedral Magic Numbers

90

80

70

60

50

40

30

100

30 40 50 60 70 80 90 100 120 140

Z=40

Z=56

Z=64

Z=70

N=40

N=112

N=56

N=70

160 180

110

120

Z=90

Z=118N=178

Z=112

Tetrahedral Symmetry Induced Magic Numbers

Neutron Number

Pro

ton

Nu

mb

er

N=136

N=90

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Tetrahedral Stability; Tetrahedral Magic Numbers

90

80

70

60

50

40

30

100

30 40 50 60 70 80 90 100 120 140

Z=40

Z=56

Z=64

Z=70

N=40

N=112

N=56

N=70

160 180

110

120

Z=90

Z=118N=178

Z=112

Tetrahedral Symmetry Induced Magic Numbers

Neutron Number

Pro

ton

Nu

mb

er

N=136

N=90

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Experimental Hints: Where Are Tetrahedra?

-.1 .0 .1 .2 .3 .4 .5

-6

-4

-2

0

2

4

6

8

10

Deformation a[2,0]

Ene

rgy

[MeV

]

Total Energy (WS Universal)

IPA

RA

M=

4 Z

0=10

0 N

0=13

6 Z

pl=

96

Npl

=13

6 IS

OSP

I=0

Oct

ober

200

2; N

POL

YN

= 6

HO

MFA

C=

3.20

MeV

130

132134136

138140142144146150148

136134132130138140142144146148150

Z=96

-.1 .0 .1 .2 .3

-6

-4

-2

0

2

4

6

8

10

Tetrahedral Deformation

Ene

rgy

[MeV

]

Total Energy (WS Universal)

IPA

RA

M=

4 Z

0=10

0 N

0=13

6 Z

pl=

96

Npl

=13

6 IS

OSP

I=0

Oct

ober

200

2; N

POL

YN

= 6

HO

MFA

C=

3.20

MeV

130132134

136138140142144146148150

Z=96

• An original competition between quadrupole and tetrahedral shapes

• Observe that tetrahedral minima may lie very high

• Finding optimum experimental conditions is a matter of a compromise

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Experimental Hints: Where Are Tetrahedra?

Cold Rea

ctions, C

oulex, ..

.

2

4

0 20

Yrast

Line

Tetrahedral

and

octhedral

bands

Exc

itat

ion

Ene

rgy

10Spin

The phenomenon discussed is a low-spin and relatively high-energy effect

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Axial Nuclei Rotate?

Spin-Orientation Probability

Z=

40,N

=40

,Jx=

24.5

,Jy=

24.5

,Jz=

18.8

;No-

high

er-d

efs.

I=2, n=1; Erot=1.00000000000 (A1)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability in the yrast states of the axially symmetric

nuclei - observe flattening of the distribution with increasing spin value

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Axial Nuclei Rotate?

Spin-Orientation Probability

Z=

40,N

=40

,Jx=

24.5

,Jy=

24.5

,Jz=

18.8

;No-

high

er-d

efs.

I=2, n=1; Erot=1.00000000000 (A1)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability in the yrast states of the axially symmetric

nuclei - observe flattening of the distribution with increasing spin value

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Axial Nuclei Rotate?

Spin-Orientation Probability

Z=

40,N

=40

,Jx=

24.5

,Jy=

24.5

,Jz=

18.8

;No-

high

er-d

efs.

I=6, n=1; Erot=1.00000000000 (A1)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability in the yrast states of the axially symmetric

nuclei - observe flattening of the distribution with increasing spin value

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Axial Nuclei Rotate?

Spin-Orientation Probability

Z=

40,N

=40

,Jx=

24.5

,Jy=

24.5

,Jz=

18.8

;No-

high

er-d

efs.

I=8, n=1; Erot=1.00000000000 (A1)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability in the yrast states of the axially symmetric

nuclei - observe flattening of the distribution with increasing spin value

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Axial Nuclei Rotate?

Spin-Orientation Probability

Z=

40,N

=40

,Jx=

24.5

,Jy=

24.5

,Jz=

18.8

;No-

high

er-d

efs.

I=10, n=1; Erot=1.00000000000 (A1)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability in the yrast states of the axially symmetric

nuclei - observe flattening of the distribution with increasing spin value

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Axial Nuclei Rotate?

Spin-Orientation Probability

Z=

40,N

=40

,Jx=

24.5

,Jy=

24.5

,Jz=

18.8

;No-

high

er-d

efs.

I=12, n=1; Erot=1.00000000000 (A1)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability in the yrast states of the axially symmetric

nuclei - observe flattening of the distribution with increasing spin value

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Axial Nuclei Rotate?

Spin-Orientation Probability

Z=

40,N

=40

,Jx=

24.5

,Jy=

24.5

,Jz=

18.8

;No-

high

er-d

efs.

I=14, n=1; Erot=1.00000000000 (A1)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability in the yrast states of the axially symmetric

nuclei - observe flattening of the distribution with increasing spin value

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Axial Nuclei Rotate?

Spin-Orientation Probability

Z=

40,N

=40

,Jx=

24.5

,Jy=

24.5

,Jz=

18.8

;No-

high

er-d

efs.

I=14, n=1; Erot=1.00000000000 (A1)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at the constant spin I of an axially symmetric

nucleus - observe the evolution with nuclear excitation at fixed value I=14

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Axial Nuclei Rotate?

Spin-Orientation Probability

Z=

40,N

=40

,Jx=

24.5

,Jy=

24.5

,Jz=

18.8

;No-

high

er-d

efs.

I=14, n=5; Erot=1.00571226367 (A1)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at the constant spin I of an axially symmetric

nucleus - observe the evolution with nuclear excitation at fixed value I=14

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Axial Nuclei Rotate?

Spin-Orientation Probability

Z=

40,N

=40

,Jx=

24.5

,Jy=

24.5

,Jz=

18.8

;No-

high

er-d

efs.

I=14, n=9; Erot=1.02284905470 (A1)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at the constant spin I of an axially symmetric

nucleus - observe the evolution with nuclear excitation at fixed value I=14

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Axial Nuclei Rotate?

Spin-Orientation Probability

Z=

40,N

=40

,Jx=

24.5

,Jy=

24.5

,Jz=

18.8

;No-

high

er-d

efs.

I=14, n=13; Erot=1.05141037307 (A1)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at the constant spin I of an axially symmetric

nucleus - observe the evolution with nuclear excitation at fixed value I=14

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Axial Nuclei Rotate?

Spin-Orientation Probability

Z=

40,N

=40

,Jx=

24.5

,Jy=

24.5

,Jz=

18.8

;No-

high

er-d

efs.

I=14, n=17; Erot=1.09139621878 (A1)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at the constant spin I of an axially symmetric

nucleus - observe the evolution with nuclear excitation at fixed value I=14

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Axial Nuclei Rotate?

Spin-Orientation Probability

Z=

40,N

=40

,Jx=

24.5

,Jy=

24.5

,Jz=

18.8

;No-

high

er-d

efs.

I=14, n=21; Erot=1.14280659185 (A1)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at the constant spin I of an axially symmetric

nucleus - observe the evolution with nuclear excitation at fixed value I=14

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Axial Nuclei Rotate?

Spin-Orientation Probability

Z=

40,N

=40

,Jx=

24.5

,Jy=

24.5

,Jz=

18.8

;No-

high

er-d

efs.

I=14, n=25; Erot=1.20564149227 (A2)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at the constant spin I of an axially symmetric

nucleus - observe the evolution with nuclear excitation at fixed value I=14

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Axial Nuclei Rotate?

Spin-Orientation Probability

Z=

40,N

=40

,Jx=

24.5

,Jy=

24.5

,Jz=

18.8

;No-

high

er-d

efs.

I=14, n=29; Erot=1.27990092003 (A1)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at the constant spin I of an axially symmetric

nucleus - observe the evolution with nuclear excitation at fixed value I=14

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

What did we learn from these pictures?

• Nearly trivial case of the yrast states goes along with our intuition

• Anything else is anti-intuitive: it can be seen as a surprise !

• However: these pictures carry a very neat geometrical information

• To our knowledge nobody asks this type of questions in nuclearphysics - although there are entire books discussing the analogousproblems in molecular physics !

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

What did we learn from these pictures?

• Nearly trivial case of the yrast states goes along with our intuition

• Anything else is anti-intuitive: it can be seen as a surprise !

• However: these pictures carry a very neat geometrical information

• To our knowledge nobody asks this type of questions in nuclearphysics - although there are entire books discussing the analogousproblems in molecular physics !

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

What did we learn from these pictures?

• Nearly trivial case of the yrast states goes along with our intuition

• Anything else is anti-intuitive: it can be seen as a surprise !

• However: these pictures carry a very neat geometrical information

• To our knowledge nobody asks this type of questions in nuclearphysics - although there are entire books discussing the analogousproblems in molecular physics !

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Octahedral Nuclei Rotate?

Spin-Orientation Probability

Z=

64,N

=86

,Jx=

31.9

,Jy=

31.9

,Jz=

31.9

;H0P

.150

00H

4P.0

8964

I=6, n=1; Erot=1.00000000000 (A1)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at I = 6 and the increasing excitations of an

octahedral nucleus - observe unexpected forms of probability distributions

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Octahedral Nuclei Rotate?

Spin-Orientation Probability

Z=

64,N

=86

,Jx=

31.9

,Jy=

31.9

,Jz=

31.9

;H0P

.150

00H

4P.0

8964

I=6, n=2; Erot=1.02052394299 (T1)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at I = 6 and the increasing excitations of an

octahedral nucleus - observe unexpected forms of probability distributions

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Octahedral Nuclei Rotate?

Spin-Orientation Probability

Z=

64,N

=86

,Jx=

31.9

,Jy=

31.9

,Jz=

31.9

;H0P

.150

00H

4P.0

8964

I=6, n=3; Erot=1.02054815357 (T1)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at I = 6 and the increasing excitations of an

octahedral nucleus - observe unexpected forms of probability distributions

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Octahedral Nuclei Rotate?

Spin-Orientation Probability

Z=

64,N

=86

,Jx=

31.9

,Jy=

31.9

,Jz=

31.9

;H0P

.150

00H

4P.0

8964

I=6, n=4; Erot=1.02057345492 (T1)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at I = 6 and the increasing excitations of an

octahedral nucleus - observe unexpected forms of probability distributions

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Octahedral Nuclei Rotate?

Spin-Orientation Probability

Z=

64,N

=86

,Jx=

31.9

,Jy=

31.9

,Jz=

31.9

;H0P

.150

00H

4P.0

8964

I=6, n=5; Erot=1.04367125088 (T2)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at I = 6 and the increasing excitations of an

octahedral nucleus - observe unexpected forms of probability distributions

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Octahedral Nuclei Rotate?

Spin-Orientation Probability

Z=

64,N

=86

,Jx=

31.9

,Jy=

31.9

,Jz=

31.9

;H0P

.150

00H

4P.0

8964

I=6, n=6; Erot=1.04378815004 (T2)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at I = 6 and the increasing excitations of an

octahedral nucleus - observe unexpected forms of probability distributions

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Octahedral Nuclei Rotate?

Spin-Orientation Probability

Z=

64,N

=86

,Jx=

31.9

,Jy=

31.9

,Jz=

31.9

;H0P

.150

00H

4P.0

8964

I=6, n=7; Erot=1.04390398697 (T2)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at I = 6 and the increasing excitations of an

octahedral nucleus - observe unexpected forms of probability distributions

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Octahedral Nuclei Rotate?

Spin-Orientation Probability

Z=

64,N

=86

,Jx=

31.9

,Jy=

31.9

,Jz=

31.9

;H0P

.150

00H

4P.0

8964

I=6, n=8; Erot=1.13151128191 (A2)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at I = 6 and the increasing excitations of an

octahedral nucleus - observe unexpected forms of probability distributions

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Octahedral Nuclei Rotate?

Spin-Orientation Probability

Z=

64,N

=86

,Jx=

31.9

,Jy=

31.9

,Jz=

31.9

;H0P

.150

00H

4P.0

8964

I=6, n=9; Erot=1.15595648841 (T2)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at I = 6 and the increasing excitations of an

octahedral nucleus - observe unexpected forms of probability distributions

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Octahedral Nuclei Rotate?

Spin-Orientation Probability

Z=

64,N

=86

,Jx=

31.9

,Jy=

31.9

,Jz=

31.9

;H0P

.150

00H

4P.0

8964

I=6, n=10; Erot=1.15622141413 (T2)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at I = 6 and the increasing excitations of an

octahedral nucleus - observe unexpected forms of probability distributions

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Octahedral Nuclei Rotate?

Spin-Orientation Probability

Z=

64,N

=86

,Jx=

31.9

,Jy=

31.9

,Jz=

31.9

;H0P

.150

00H

4P.0

8964

I=6, n=11; Erot=1.15648631131 (T2)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at I = 6 and the increasing excitations of an

octahedral nucleus - observe unexpected forms of probability distributions

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Octahedral Nuclei Rotate?

Spin-Orientation Probability

Z=

64,N

=86

,Jx=

31.9

,Jy=

31.9

,Jz=

31.9

;H0P

.150

00H

4P.0

8964

I=6, n=12; Erot=1.16421998383 (E)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at I = 6 and the increasing excitations of an

octahedral nucleus - observe unexpected forms of probability distributions

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Do We Know How the Octahedral Nuclei Rotate?

Spin-Orientation Probability

Z=

64,N

=86

,Jx=

31.9

,Jy=

31.9

,Jz=

31.9

;H0P

.150

00H

4P.0

8964

I=6, n=13; Erot=1.16456302304 (E)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at I = 6 and the increasing excitations of an

octahedral nucleus - observe unexpected forms of probability distributions

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

What did we learn from these pictures?

• First message is this: we know the wave functions, we can andwe do calculate the transition probabilities that give neccessary andsufficient experimental criteria for the presence of the symmetries

• These pictures help to understand and interpret geometrically thetheoretical predictions for electromagnetic transition probabilities

• ... but this is a looong topic for yet another, separate story !

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

What did we learn from these pictures?

• First message is this: we know the wave functions, we can andwe do calculate the transition probabilities that give neccessary andsufficient experimental criteria for the presence of the symmetries

• These pictures help to understand and interpret geometrically thetheoretical predictions for electromagnetic transition probabilities

• ... but this is a looong topic for yet another, separate story !

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

What did we learn from these pictures?

• First message is this: we know the wave functions, we can andwe do calculate the transition probabilities that give neccessary andsufficient experimental criteria for the presence of the symmetries

• These pictures help to understand and interpret geometrically thetheoretical predictions for electromagnetic transition probabilities

• ... but this is a looong topic for yet another, separate story !

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Survey of Doubly-Magic Tetrahedral Nuclei

0

0

2

2

2

2

2

2

2 4

4

4

44

44

466

6

6

6

6

6

8

8

8

8

10

10

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

E(fyu)+Shell[e]+Correlation[PNP]

UN

IVE

RS_

CO

MPA

CT

(D

=3,

23)

Gp=

0.96

0 G

n=0.

980

∆Np=

35 ∆

Nn=

45

Zr56 Zr 96 40

0.000.501.001.502.002.503.003.504.004.505.005.506.006.507.007.508.008.509.009.5010.0010.5011.0011.50

E [MeV]

Deformation α20

Def

orm

atio

n t 1

Emin=-3.77, Eo=-2.55

This stable nucleus is predicted tetrahedral-unstable in the ground-state -

and yet will combine the signs of sphericity and tetrahedrality

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Survey of Doubly-Magic Tetrahedral Nuclei

2

2

2

22

22

2

2

2

4

4

4

4

4 4

4

6 6

6

6

8 8 8

888

1010

10

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.30-0.25-0.20-0.15-0.10-0.050.000.050.100.150.200.250.30

E(fyu)+Shell[e]+Correlation[PNP]

UN

IVE

RS_

CO

MPA

CT

(D

=4,

23)

Gp=

0.96

0 G

n=0.

980

∆Np=

35 ∆

Nn=

45

Th136Th226 90

0.000.501.001.502.002.503.003.504.004.505.005.506.006.507.007.508.008.509.009.5010.0010.5011.0011.50

E [MeV]

Deformation α20

Def

orm

atio

n α 3

2

Emin=-3.55, Eo= 0.95

This nucleus is predicted to manifest tetrahedral minima in competition

with the quadrupole ground-state minimum ...

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Survey of Doubly-Magic Tetrahedral Nuclei

2

2

22

2

2

4

4

4

4

4

6

6

6

6

6

6

6

8

8

8

8

8

8

8

8

10

1010

10

1010

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

E(fyu)+Shell[e]+Correlation[PNP]

UN

IVE

RS_

CO

MPA

CT

(D

=4,

23)

Gp=

0.96

0 G

n=0.

980

∆Np=

35 ∆

Nn=

45

Th136Th226 90

0.000.501.001.502.002.503.003.504.004.505.005.506.006.507.007.508.008.509.009.5010.0010.5011.0011.50

E [MeV]

Deformation α20

Def

orm

atio

n α 3

0

Emin=-6.68, Eo= 0.95

... and in competition with the pear-shape ‘old’ octupole minimum !!!

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Multipole Moments as Functionals of the Density

Nuclear surface Σ is defined in terms of multipole deformations:

Σ : R(ϑ, ϕ) = R0

[1 +

∑λ

∑µ αλµ Yλµ(ϑ, ϕ)

]Given uniform density ρΣ(~r ) defined using the surface Σ

ρΣ(~r ) =

{ρ0 : ~r ∈ Σ0 : ~r /∈ Σ

Express the multipole moments as usual by

Qλµ =∫ρΣ(~r ) r λ Yλµ d3~r

We will calculate the quadrupole moments as functions of α3µ

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Multipole Moments as Functionals of the Density

Nuclear surface Σ is defined in terms of multipole deformations:

Σ : R(ϑ, ϕ) = R0

[1 +

∑λ

∑µ αλµ Yλµ(ϑ, ϕ)

]Given uniform density ρΣ(~r ) defined using the surface Σ

ρΣ(~r ) =

{ρ0 : ~r ∈ Σ0 : ~r /∈ Σ

Express the multipole moments as usual by

Qλµ =∫ρΣ(~r ) r λ Yλµ d3~r

We will calculate the quadrupole moments as functions of α3µ

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Multipole Moments as Functionals of the Density

Nuclear surface Σ is defined in terms of multipole deformations:

Σ : R(ϑ, ϕ) = R0

[1 +

∑λ

∑µ αλµ Yλµ(ϑ, ϕ)

]Given uniform density ρΣ(~r ) defined using the surface Σ

ρΣ(~r ) =

{ρ0 : ~r ∈ Σ0 : ~r /∈ Σ

Express the multipole moments as usual by

Qλµ =∫ρΣ(~r ) r λ Yλµ d3~r

We will calculate the quadrupole moments as functions of α3µ

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Multipole Moments as Functionals of the Density

Nuclear surface Σ is defined in terms of multipole deformations:

Σ : R(ϑ, ϕ) = R0

[1 +

∑λ

∑µ αλµ Yλµ(ϑ, ϕ)

]Given uniform density ρΣ(~r ) defined using the surface Σ

ρΣ(~r ) =

{ρ0 : ~r ∈ Σ0 : ~r /∈ Σ

Express the multipole moments as usual by

Qλµ =∫ρΣ(~r ) r λ Yλµ d3~r

We will calculate the quadrupole moments as functions of α3µ

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Multipole Moments as Functionals of the Density

For small deformations we use Taylor expansion:

Qλµ(α) ≈ Qλµ

∣∣∣α=0

+ Q′λµ

∣∣∣α=0

∆α+1

2Q′′λµ

∣∣∣α=0

∆α∆α

We set λ = 2, µ = 0 and λ1 = λ2 = 3 and obtain

α30 : Q20 = 15/(2√

5π) · α 230 · ρ0 R5

0

α31 : Q20 = 15/(4√

5π) · α3+1α3−1 · ρ0 R50

α32 : Q20 = 0

α33 : Q20 = 125/(12√

5π) · α3+3α3−3 · ρ0 R50

Conclusion: Among λ = 3 defs. only α32 leads to Q2 ≡ 0 !!!

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Multipole Moments as Functionals of the Density

For small deformations we use Taylor expansion:

Qλµ(α) ≈ Qλµ

∣∣∣α=0

+ Q′λµ

∣∣∣α=0

∆α+1

2Q′′λµ

∣∣∣α=0

∆α∆α

We set λ = 2, µ = 0 and λ1 = λ2 = 3 and obtain

α30 : Q20 = 15/(2√

5π) · α 230 · ρ0 R5

0

α31 : Q20 = 15/(4√

5π) · α3+1α3−1 · ρ0 R50

α32 : Q20 = 0

α33 : Q20 = 125/(12√

5π) · α3+3α3−3 · ρ0 R50

Conclusion: Among λ = 3 defs. only α32 leads to Q2 ≡ 0 !!!

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Multipole Moments as Functionals of the Density

For small deformations we use Taylor expansion:

Qλµ(α) ≈ Qλµ

∣∣∣α=0

+ Q′λµ

∣∣∣α=0

∆α+1

2Q′′λµ

∣∣∣α=0

∆α∆α

We set λ = 2, µ = 0 and λ1 = λ2 = 3 and obtain

α30 : Q20 = 15/(2√

5π) · α 230 · ρ0 R5

0

α31 : Q20 = 15/(4√

5π) · α3+1α3−1 · ρ0 R50

α32 : Q20 = 0

α33 : Q20 = 125/(12√

5π) · α3+3α3−3 · ρ0 R50

Conclusion: Among λ = 3 defs. only α32 leads to Q2 ≡ 0 !!!

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

We Proceed to Suggest a Physical Interpretation

When looking for experimental signs of tetrahedral symmetryassume that Q2-moments of tetra-configurations nearly vanish

Strictly speaking: Exact tetrahedral and octahedral symmetriescause vanishing of quadrupole and dipole moments: the onlypermitted transitions are octupole - an extreme idea !

We know that at reduced transition probabilities B(E1) andB(E3) comparable, the E1-decay is ∼1012 more probable !!

Therefore it will be neccessary to look into problems of partialsymmetry breaking in the case of the two high-rank symmetries

We begin with the mechanism of the zero-point quadrupole mo-tion about zero-quadrupole deformation → following discussion

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

We Proceed to Suggest a Physical Interpretation

When looking for experimental signs of tetrahedral symmetryassume that Q2-moments of tetra-configurations nearly vanish

Strictly speaking: Exact tetrahedral and octahedral symmetriescause vanishing of quadrupole and dipole moments: the onlypermitted transitions are octupole - an extreme idea !

We know that at reduced transition probabilities B(E1) andB(E3) comparable, the E1-decay is ∼1012 more probable !!

Therefore it will be neccessary to look into problems of partialsymmetry breaking in the case of the two high-rank symmetries

We begin with the mechanism of the zero-point quadrupole mo-tion about zero-quadrupole deformation → following discussion

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

We Proceed to Suggest a Physical Interpretation

When looking for experimental signs of tetrahedral symmetryassume that Q2-moments of tetra-configurations nearly vanish

Strictly speaking: Exact tetrahedral and octahedral symmetriescause vanishing of quadrupole and dipole moments: the onlypermitted transitions are octupole - an extreme idea !

We know that at reduced transition probabilities B(E1) andB(E3) comparable, the E1-decay is ∼1012 more probable !!

Therefore it will be neccessary to look into problems of partialsymmetry breaking in the case of the two high-rank symmetries

We begin with the mechanism of the zero-point quadrupole mo-tion about zero-quadrupole deformation → following discussion

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

We Proceed to Suggest a Physical Interpretation

When looking for experimental signs of tetrahedral symmetryassume that Q2-moments of tetra-configurations nearly vanish

Strictly speaking: Exact tetrahedral and octahedral symmetriescause vanishing of quadrupole and dipole moments: the onlypermitted transitions are octupole - an extreme idea !

We know that at reduced transition probabilities B(E1) andB(E3) comparable, the E1-decay is ∼1012 more probable !!

Therefore it will be neccessary to look into problems of partialsymmetry breaking in the case of the two high-rank symmetries

We begin with the mechanism of the zero-point quadrupole mo-tion about zero-quadrupole deformation → following discussion

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

We Proceed to Suggest a Physical Interpretation

When looking for experimental signs of tetrahedral symmetryassume that Q2-moments of tetra-configurations nearly vanish

Strictly speaking: Exact tetrahedral and octahedral symmetriescause vanishing of quadrupole and dipole moments: the onlypermitted transitions are octupole - an extreme idea !

We know that at reduced transition probabilities B(E1) andB(E3) comparable, the E1-decay is ∼1012 more probable !!

Therefore it will be neccessary to look into problems of partialsymmetry breaking in the case of the two high-rank symmetries

We begin with the mechanism of the zero-point quadrupole mo-tion about zero-quadrupole deformation → following discussion

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Extreme-Symmetry Limit: Q2=0 and Q1=0

0 +

2 +

4 +

1 +

3 +3 −

1 −

4 −

2 −

0 − 0 −

1 +

2 −

3 +

4 −

0 +

1 −

2 +

3 −

4 +

Transitionsalways present are the octupole ones

The only transitions

Pear−shape Tetrahedral

non−zeroDipole Moment Dipole Moment

vanishes

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Quadrupole Polarisation through Zero-Point Motion

Denoting quadrupole deformations α and classical energy E we have:

E (α, α) = 12B α

2 + 12C α

2 → H = 12B

∂ 2

∂α 2 + 12C α

2,

The normalised wave functions are:

ϕn(α) =1√√

2π2 nn!· 1

A· e−α 2/2σ 2

Hn(α/σ), σdf .=√

2A,

whereA df .

= [〈ϕn=0|α 2|ϕn=0〉]1/2 = [~ 2/(4BC)]1/4

We find an effective quadrupole deformation different from zero

Most probable deformation: 〈|α|〉 ∼∫

|α| ϕ 2n (α) dα 6= 0

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Quadrupole Polarisation through Zero-Point Motion

Denoting quadrupole deformations α and classical energy E we have:

E (α, α) = 12B α

2 + 12C α

2 → H = 12B

∂ 2

∂α 2 + 12C α

2,

The normalised wave functions are:

ϕn(α) =1√√

2π2 nn!· 1

A· e−α 2/2σ 2

Hn(α/σ), σdf .=√

2A,

whereA df .

= [〈ϕn=0|α 2|ϕn=0〉]1/2 = [~ 2/(4BC)]1/4

We find an effective quadrupole deformation different from zero

Most probable deformation: 〈|α|〉 ∼∫

|α| ϕ 2n (α) dα 6= 0

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Quadrupole Polarisation through Zero-Point Motion

Denoting quadrupole deformations α and classical energy E we have:

E (α, α) = 12B α

2 + 12C α

2 → H = 12B

∂ 2

∂α 2 + 12C α

2,

The normalised wave functions are:

ϕn(α) =1√√

2π2 nn!· 1

A· e−α 2/2σ 2

Hn(α/σ), σdf .=√

2A,

whereA df .

= [〈ϕn=0|α 2|ϕn=0〉]1/2 = [~ 2/(4BC)]1/4

We find an effective quadrupole deformation different from zero

Most probable deformation: 〈|α|〉 ∼∫

|α| ϕ 2n (α) dα 6= 0

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

We Have the Smoking-Gun Signatures Almost There

caused by CoriolisValence particles

cause a certainquadrupole polarisation

Additional polarisation

spin alignments

Tot

al S

pin

Spin-alignment will cause additional quadrupole polarisation

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

We Have the Smoking-Gun Signatures Almost There ...

According to a simplified way of thinking, when all deformationstend to zero (αλµ → 0) then Q2 → 0 and Q1 → 0 and we areconfronted with an ill-defined mathematical problem

limα→0

B(E2)B(E1) =??? (undefined symbol 0

0) !!!

However, because of the residual polarisations in terms of quadrupoledeformation and of induced dipole moments at the band-heads wehave

limα→0

B(E2)B(E1) = Bres(E2)

Bres(E1) ≡ B0 6= 0

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

We Have the Smoking-Gun Signatures Almost There ...

According to a simplified way of thinking, when all deformationstend to zero (αλµ → 0) then Q2 → 0 and Q1 → 0 and we areconfronted with an ill-defined mathematical problem

limα→0

B(E2)B(E1) =??? (undefined symbol 0

0) !!!

However, because of the residual polarisations in terms of quadrupoledeformation and of induced dipole moments at the band-heads wehave

limα→0

B(E2)B(E1) = Bres(E2)

Bres(E1) ≡ B0 6= 0

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

We Have the Smoking-Gun Signatures Almost There ...

Octupole BandB(E

2)/

B(E

1)

Angular Momentum

Tetrahedral B

and

Schematic: Predictions for B(E2)/B(E1)

In other words, we expect a spin dependence: B(E2)B(E1) ∼ B0 + B1 · I

Conclusion: Tetrahedral symmetry must always be accompanied bystatic or dynamic quadrupole deformations at α20 6= 0 and α22 6= 0

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

We Have the Smoking-Gun Signatures Almost There ...

Octupole BandB(E

2)/

B(E

1)

Angular Momentum

Tetrahedral B

and

Schematic: Predictions for B(E2)/B(E1)

In other words, we expect a spin dependence: B(E2)B(E1) ∼ B0 + B1 · I

Conclusion: Tetrahedral symmetry must always be accompanied bystatic or dynamic quadrupole deformations at α20 6= 0 and α22 6= 0

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Possible El-Magnetic Signs of Tetrahedral Symmetry

Table: Experimental ratios B(E2)in/B(E1)out × 106

Spin 152Gd 156Gd 154Dy 160Er 164Er 162Yb 164Yb

19− - 50 - - - - -

17− - 16 - - - - -

15− - 6 - 60 24 - -

13− 14 7 15 18 23 - 17

11− 4 15 5 9 0 10 11

9− 4 0 - 0 - 11 10

7− 0 0 0 - - 0 0

Above: Branching ratios related to the negative parity bands areinterpreted as tetrahedral, interband transitions to g.s.band

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Focus on the Nuclear Tetrahedral SymmetryTetrahedral Magic Numbers and Nuclear BindingExpected Experimental Signs of Tetrahedral Symmetry

Possible El-Magnetic Signs of Tetrahedral Symmetry

Table: Experimental ratios B(E2)in/B(E1)out × 106

Spin 152Gd 156Gd 154Dy 160Er 164Er 162Yb 164Yb

19− - 50 - - - - -

17− - 16 - - - - -

15− - 6 - 60 24 - -

13− 14 7 15 18 23 - 17

11− 4 15 5 9 0 10 11

9− 4 0 - 0 - 11 10

7− 0 0 0 - - 0 0

Above: Branching ratios related to the negative parity bands areinterpreted as tetrahedral, interband transitions to g.s.band

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to Experiment

Part III

New Theory of Nuclear Stability: Its Realisation,Tests and Experimental Background

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

We can now formulate further experimental criteria!

The Story of the ‘Smoking Guns’

• Tetrahedral nuclei are deformed→ they produce collective rotation

• The lowest order Td−symmetry is Y3±2 → negative parity bands

• At the exact symmetry limit Q2 moments must vanish! Therefore:

• There must exist negative-parity bands without E2 transitions !!!

We suggest looking for the collective negative parity bandswithout ‘rotational ′ (E2) transitions. The question − Where?

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

We can now formulate further experimental criteria!

The Story of the ‘Smoking Guns’

• Tetrahedral nuclei are deformed→ they produce collective rotation

• The lowest order Td−symmetry is Y3±2 → negative parity bands

• At the exact symmetry limit Q2 moments must vanish! Therefore:

• There must exist negative-parity bands without E2 transitions !!!

We suggest looking for the collective negative parity bandswithout ‘rotational ′ (E2) transitions. The question − Where?

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

We can now formulate further experimental criteria!

The Story of the ‘Smoking Guns’

• Tetrahedral nuclei are deformed→ they produce collective rotation

• The lowest order Td−symmetry is Y3±2 → negative parity bands

• At the exact symmetry limit Q2 moments must vanish! Therefore:

• There must exist negative-parity bands without E2 transitions !!!

We suggest looking for the collective negative parity bandswithout ‘rotational ′ (E2) transitions. The question − Where?

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

We can now formulate further experimental criteria!

The Story of the ‘Smoking Guns’

• Tetrahedral nuclei are deformed→ they produce collective rotation

• The lowest order Td−symmetry is Y3±2 → negative parity bands

• At the exact symmetry limit Q2 moments must vanish! Therefore:

• There must exist negative-parity bands without E2 transitions !!!

We suggest looking for the collective negative parity bandswithout ‘rotational ′ (E2) transitions. The question − Where?

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Where to Look for Experimental Evidence: Summary

90

80

70

60

50

40

30

100

100 120 140

Neutron Number

Pro

ton

Nu

mb

er

Z=40

Z=56

Z=64

Z=70

~>

N=40

N~112

N=56

N=70

~>

Z 90

160

N 90 N 136~>

The Strongest Tetrahedral Islands Predicted by Theory

most of the so−called octupole bands

have never seen their E2 transitions

in experiment [detailed discussion]

In the Actinide region

40 60 80

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Where to Look for Experimental Evidence: Summary

90

80

70

50

40

30

100

100 120 140

Neutron Number

Pro

ton

Nu

mb

er

Z=40

Z=56

Z=64

Z=70

~>

N=40

N~112

N=56

N=70

~>

Z 90

160

N 90 N 136~>

The Strongest Tetrahedral Islands Predicted by Theory

40 60 80

60

In the Rare Earth Region

the Sm, Gd and Dy nuclei [Z=62,64,66]

manifest negative parity bands

without E2 transitions [see details]

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Where to Look for Experimental Evidence: Summary

90

80

70

50

40

30

100

100 120 140

Neutron Number

Pro

ton

Nu

mb

er

Z=40

Z=56

Z=64

Z=70

~>

N=40

N~112

N=56

N=70

~>

Z 90

160

N 90 N 136~>

The Strongest Tetrahedral Islands Predicted by Theory

40 60 80

60

several nuclei manifest largest everIn the Zirconium region

octupole transitions [B(E3)~(20−60)W.u.]

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Example: This Is Not Our Effect [Pear-Shape]

00

2

2

2

2

2

2

4

4

46

6

6

6

6

6

8

8

8

8

8

8

10

10

1010

10

10-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

E(fyu)+Shell[e]+Correlation[PNP]

UN

IVE

RS_

CO

MPA

CT

(D

=4,

23)

Gp=

0.96

0 G

n=0.

980

∆Np=

35 ∆

Nn=

45

Th132Th222 90

0.000.501.001.502.002.503.003.504.004.505.005.506.006.507.007.508.008.509.009.5010.0010.5011.0011.50

E [MeV]

Deformation α20

Def

orm

atio

n α 3

0

Emin=-7.58, Eo=-3.07

0 +

4 +

2 +

6 +

+8

+10

12 +

14 +

3 −

5 −

7 −

9 −

13 −

11 −

Th222

183

440467

651

750

925

1094

1255

1461

1623

1851

2016

2260

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Example: This Is Not Our Effect [Pear-Shape]

00

2

2

2

2

2

2

4

4

4

6

6

6

6

6

6

8

8

8

8

8

8

10

101010

10

10-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

E(fyu)+Shell[e]+Correlation[PNP]

UN

IVE

RS_

CO

MPA

CT

(D

=4,

23)

Gp=

0.96

0 G

n=0.

980

∆Np=

35 ∆

Nn=

45

Th132Th222 90

0.000.501.001.502.002.503.003.504.004.505.005.506.006.507.007.508.008.509.009.5010.0010.5011.0011.50

E [MeV]

Deformation α20

Def

orm

atio

n α 3

0

Emin=-7.58, Eo=-3.07

B(E2)in/B(E1)out ×106e Fm

Spin 222Th

19− +0.3

17− +0.4

15− +0.4

13− +0.3

11− +0.4

9− +0.4

7− +0.4

Experiment

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Possible El-Magnetic Signs of Tetrahedral Symmetry

Table: Experimental ratios B(E2)in/B(E1)out × 106

Spin 152Gd 156Gd 154Dy 160Er 164Er 162Yb 164Yb 222Th

19− - 50 - - - - - +0.3

17− - 16 - - - - - +0.4

15− - 6 - 60 24 - - +0.4

13− 14 7 15 18 23 - 17 +0.3

11− 4 15 5 9 0 10 11 +0.4

9− 4 0 - 0 - 11 10 +0.4

7− 0 0 0 - - 0 0 +0.4

Conclusion: Tetrahedral bands in Rare Earth nuclei behave verydifferently as compared e.g. to ’classical’ octupole 222Th band!

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Possible El-Magnetic Signs of Tetrahedral Symmetry

Table: Experimental ratios B(E2)in/B(E1)out × 106

Spin 152Gd 156Gd 154Dy 160Er 164Er 162Yb 164Yb 222Th

19− - 50 - - - - - +0.3

17− - 16 - - - - - +0.4

15− - 6 - 60 24 - - +0.4

13− 14 7 15 18 23 - 17 +0.3

11− 4 15 5 9 0 10 11 +0.4

9− 4 0 - 0 - 11 10 +0.4

7− 0 0 0 - - 0 0 +0.4

Conclusion: Tetrahedral bands in Rare Earth nuclei behave verydifferently as compared e.g. to ’classical’ octupole 222Th band!

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Partial Decay of 156Gd and Vanishing Q2-Moments

From Iπ = 9− down - no E2-transitions are observed despite tries

Process Refs Last Exp.156Eu: β-decay 43 1995

156Tb: ec-decay 38 1995

150Nd: ( 13C,A-3n-γ) 2 2001

154Sm: (A,2n-γ) 11 2001

154Gd: (t,p) 1 1989

155Gd: (n,γ) 40 2000

155Gd: (d,p) 2 1994

156Gd: (γ,γ’),(e,e’) 27 2000

156Gd: (µ,γ) 1 1971

156Gd: (n,n’γ) 3 1996

156Gd: (p,p’),(d,d’) 5 1989

157Gd: (p,d),(3He,A) 2 1984

157Gd: (d,t) 1 1993

158Gd: (p,t) 8 1982

Coulomb excitation 25 19930+2+

4+

89

1276

288

584

965

1416

1924

2476

3060

6+

8+

10+

12+

14+

16+

g. s. band

1-

5-

7-

9-

11-

13-

(15-)

1242

1408

1638

1958

2359

2829

3350

3-

n. p. band

3914 (17-)

156Gd

According to C. W. Reich, Nucl. Data Sheets 99 753 (2003) a few dozens

among those refs have been used to deduce the level scheme on the right...

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Tetrahedral/Octahedral Shapes Have No Q2-Moments

At the exact tetrahedral symmetry the quadrupole moments vanish

Equilibrium shape t1 = 0.150+2+

4+

89

1276

288

584

965

1416

1924

2476

3060

6+

8+

10+

12+

14+

16+

g. s. band

1-

5-

7-

9-

11-

13-

(15-)

1242

1408

1638

1958

2359

2829

3350

3-

n. p. band

3914 (17-)

156Gd

...but, E2-intensities are expected to grow with spin (Coriolis polarisation)

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Comparison Theory - Experiment: Alignments

Comparison between three typical hypotheses:1. Tetrahedral and Octahedral ↔ (defs. from microscopic calculations);

2. Tetrahedral and Octahedral + Zero-Point Motion (αpol.20 = 0.07);

3. Prolate ground-state deformation with α20 ≈ 0.25

Gd

Experiment

1560.0 0.1 0.2 0.3 0.4

~ω [MeV]

0

5

10

15

20

25

Nucl

ear

Spin

[~]

0.080.08

0.10 0.00

0.03

α20α32

0.07

α40

0.100.250.00

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Preliminary Results from our Jyvaskyla ExperimentPreliminary 156Gd level scheme from our experiment

g.s. bando.n.p band e.n.p band

Q.T. Doan et al., Search for Fingerprints of Tetrahedral . . . 10 / 14

1. Our first-time evidence for the non-stretched E2-transitions from therotational even-spin negative-parity band to the tetrahedral-suspect band.

2. These transitions support the interpretation of tetrahedral sequence as

a real rotational band, fed and depopulated through ∆I = 1 transitions!Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Preliminary Results from our Jyvaskyla ExperimentPreliminary 156Gd level scheme from our experiment

g.s. bando.n.p band e.n.p band

Q.T. Doan et al., Search for Fingerprints of Tetrahedral . . . 10 / 14

1. Our first-time evidence for the non-stretched E2-transitions from therotational even-spin negative-parity band to the tetrahedral-suspect band.

2. These transitions support the interpretation of tetrahedral sequence as

a real rotational band, fed and depopulated through ∆I = 1 transitions!Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

What Did We Learn So Far from Jyvaskyla Experiment

First Conclusions:

We believe that the odd-spin negative-parity band in 156Gdmust not be an octupole-vibrational band; octupole vibrationstake place around a quadrupole-deformed minimum;

It cannot be an octupole vibration about the spherical shapeeither because spherical nuclei have irregular excitation spectra

We believe it should be interpreted as tetrahedral band, there-fore K 6= 0 and the associated bands must undergo K -mixing

This band is very special: it is fed and depopulated by ∆I = 1transitions - the ∆I = 2 stretched E2-transitions are absent orevidently weak!

Does the even-spin negative-parity band have extremely lowE1’s? ... or suddenly strong E2’s? Or neither of the two - theonly mechanism being changing of the ratio?

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

What Did We Learn So Far from Jyvaskyla Experiment

First Conclusions:

We believe that the odd-spin negative-parity band in 156Gdmust not be an octupole-vibrational band; octupole vibrationstake place around a quadrupole-deformed minimum;

It cannot be an octupole vibration about the spherical shapeeither because spherical nuclei have irregular excitation spectra

We believe it should be interpreted as tetrahedral band, there-fore K 6= 0 and the associated bands must undergo K -mixing

This band is very special: it is fed and depopulated by ∆I = 1transitions - the ∆I = 2 stretched E2-transitions are absent orevidently weak!

Does the even-spin negative-parity band have extremely lowE1’s? ... or suddenly strong E2’s? Or neither of the two - theonly mechanism being changing of the ratio?

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

What Did We Learn So Far from Jyvaskyla Experiment

First Conclusions:

We believe that the odd-spin negative-parity band in 156Gdmust not be an octupole-vibrational band; octupole vibrationstake place around a quadrupole-deformed minimum;

It cannot be an octupole vibration about the spherical shapeeither because spherical nuclei have irregular excitation spectra

We believe it should be interpreted as tetrahedral band, there-fore K 6= 0 and the associated bands must undergo K -mixing

This band is very special: it is fed and depopulated by ∆I = 1transitions - the ∆I = 2 stretched E2-transitions are absent orevidently weak!

Does the even-spin negative-parity band have extremely lowE1’s? ... or suddenly strong E2’s? Or neither of the two - theonly mechanism being changing of the ratio?

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

What Did We Learn So Far from Jyvaskyla Experiment

First Conclusions:

We believe that the odd-spin negative-parity band in 156Gdmust not be an octupole-vibrational band; octupole vibrationstake place around a quadrupole-deformed minimum;

It cannot be an octupole vibration about the spherical shapeeither because spherical nuclei have irregular excitation spectra

We believe it should be interpreted as tetrahedral band, there-fore K 6= 0 and the associated bands must undergo K -mixing

This band is very special: it is fed and depopulated by ∆I = 1transitions - the ∆I = 2 stretched E2-transitions are absent orevidently weak!

Does the even-spin negative-parity band have extremely lowE1’s? ... or suddenly strong E2’s? Or neither of the two - theonly mechanism being changing of the ratio?

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

What Did We Learn So Far from Jyvaskyla Experiment

First Conclusions:

We believe that the odd-spin negative-parity band in 156Gdmust not be an octupole-vibrational band; octupole vibrationstake place around a quadrupole-deformed minimum;

It cannot be an octupole vibration about the spherical shapeeither because spherical nuclei have irregular excitation spectra

We believe it should be interpreted as tetrahedral band, there-fore K 6= 0 and the associated bands must undergo K -mixing

This band is very special: it is fed and depopulated by ∆I = 1transitions - the ∆I = 2 stretched E2-transitions are absent orevidently weak!

Does the even-spin negative-parity band have extremely lowE1’s? ... or suddenly strong E2’s? Or neither of the two - theonly mechanism being changing of the ratio?

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

What Did We Learn So Far from Jyvaskyla Experiment

First Conclusions:

We believe that the odd-spin negative-parity band in 156Gdmust not be an octupole-vibrational band; octupole vibrationstake place around a quadrupole-deformed minimum;

It cannot be an octupole vibration about the spherical shapeeither because spherical nuclei have irregular excitation spectra

We believe it should be interpreted as tetrahedral band, there-fore K 6= 0 and the associated bands must undergo K -mixing

This band is very special: it is fed and depopulated by ∆I = 1transitions - the ∆I = 2 stretched E2-transitions are absent orevidently weak!

Does the even-spin negative-parity band have extremely lowE1’s? ... or suddenly strong E2’s? Or neither of the two - theonly mechanism being changing of the ratio?

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Evidence for Vanishing E2 Transitions in Actinides

The E2 transitions not seen in 230−234U, while seen in 236U; theexperimental conditions (γ and ec) are the same or comparable

44

143

296

497

741

1024

1341

1688

45

1801

1426

1085

782

522

310

149

57

103

152

199

245

[311]

[270]

[225]

[176]

[123]

[69]

[260]

[216]

[168]

[118]

[66]

[253]

[211]

[163]

[113]

[63]

289

[346]

[298]

[334]

366

1921

1532

52

170

347

578

856

1176

157

323

541

806

1112

1454

48

1828

−

− 13

− 11

− 9

− 7

− 5

− 1

− 3

0 +2 +

4 +

6 +

+8

+10

12 +

14 +

16 +

) (11 −

) (9 −

7 −

5 −

− 3

− 1

0 +2 +

4 +

6 +

+8

+10

12 +

14 +

16 +)(13 −

)(11 −

) (9 −

) (7 −

) (5 −

− 3

− 1

0 +2 +

4 +

6 +

+8

+10

12 +

14 +

16 +)(15

−

)(13 −

)(11 −

) (9 −

) (7 −

) (5 −

3 −

1 −

0 +2 +

4 +

6 +

+8

+10

12 +

14 +

16 +

(15 ) −

15

230U

138

232U

140

234U

142

236U 144

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Evidence for Vanishing E2 Transitions in Actinides

U232U230

) (5 −

) (9 −

) (7 −

) (1 −

) (3 −

) (5 −

) (7 −

) (9 −

)(11 −

)(13 −

0 +2 +

4 +

+8

+10

6 +

2 +

4 +

+8

+10

12 +

14 +

16 +

6 +

0 +

12 +

0 +2 +

4 +

6 +

+8

+10

14 +

16 +

157

323

541

806

1112

1454

1828

1131

915

747

629 564

735

833

985

1187

14341391

−

−

3

1 691

1921

1532

1229

1540

959

734

558

435 367

52

170

347

578

856

11−

48

1176

)(15 −2023

Observe the ’tetrahedral’ band patterns (vanishing E2-transitions) in 230−232U

in both the negative and positive parities!

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Tetrahedral-Symmetry Candidates in the Actinides

Experiment: Three types of situations correspond to three colours:[red]-tetrahedral, [brown]-octupole, [green]-both.

N→ 130 132 134 136 138 140 142 144 146 148 150 152

Cm 244 246? 248

Pu 236 238 240 242? 244

U 230 232 234 236 238

Th 220 222 224 226 228 230 232 234

Ra 218 220 222 224 226 228 230

Rn 216 218 220

Nearly half of the experimental data on the Actinidae nuclei do notmanifest the E2-transitions in the negative-parity bands!

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Electromagnetic Transitions in the Actinide vs. RE

Table: Experimental ratios B(E2)/B(E1)× 106[fm2], for intra-band E2transitions vs. inter-band E1 transitions. Meaning of symbols: “-” - statehas not been observed; “?” - intensities to calculate the branching ratiosnot available; “(?)” - known information insufficient to obtain error bars.

State 220Th 222Th 224Th 226Th 228Th 152Gd 156Gd

21− no E1 0.2(?) - - - - no E119− no E1 0.3(?) - 2.0(5) - - 50(10)17− no E1 0.4(2) 0.3(1) 2.3(4) no E1 no E1 16(3)15− 0.9(2) 0.4(2) 0.4(1) 2.0(4) no E1 no E1 6(2)13− 0.2(1) 0.3(2) 0.5(1) ? 16(2) 14(?) 7(2)11− 0.5(1) 0.4(2) 0.4(1) 2.0(2) 13(1) 4(?) 15(7)9− 0.4(1) 0.4(2) ? 2.0(2) 14(1) 4(?) no E27− 0.4(1) 0.4(3) ? ? no E2 no E2 no E2

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Strong High-Rank Symmetries in Super-heavy Elements

1

11

1

1

1

2

2

2

2

22

2

2

2 2

22

2

2

3

3

3

3 3 3 3

3

3

3

3

3

3 3

3

4

4

55

-.3 -.2 -.1 .0 .1 .2 .3-.15

-.10

-.05

.00

.05

.10

.15

Tetrahedral Symmetry / Instability

UN

IVE

RS_

CO

MPA

CT

(D

=2,

23)

UN

IVE

RS_

CO

MPA

CT

(D

=2,

23)

UN

IVE

RS_

CO

MPA

CT

(D

=2,

23)

Hs152Hs260108

.25

.50

.751.001.251.501.752.002.252.502.753.003.253.503.754.004.254.504.755.005.255.505.756.00

E [MeV]

Tetrahedral Shape

Oct

ahed

ral S

hape

Emin= 6.90, Eo= 9.48

• Competition between the tetrahedral and octahedral symmetries

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Strong High-Rank Symmetries in Super-heavy Elements

12

2

2

2

2

2

2

3

3

3

3

3 3

33

3

3

3

3

3

3

3

3

3

3

4 4

4

4

45

5

6

6

-.3 -.2 -.1 .0 .1 .2 .3-.15

-.10

-.05

.00

.05

.10

.15

Tetrahedral Symmetry / Instability

UN

IVE

RS_

CO

MPA

CT

(D

=2,

23)

UN

IVE

RS_

CO

MPA

CT

(D

=2,

23)

UN

IVE

RS_

CO

MPA

CT

(D

=2,

23)

Hs154Hs262108

.25

.50

.751.001.251.501.752.002.252.502.753.003.253.503.754.004.254.504.755.005.255.505.756.00

E [MeV]

Tetrahedral Shape

Oct

ahed

ral S

hape

Emin= 6.56, Eo= 8.90

• Competition between the tetrahedral and octahedral symmetries

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Strong High-Rank Symmetries in Super-heavy Elements

11

2

2

2

2

23

3

3

3 3

3

3

4

4

4

4

4

4

4

4

4

4

4

455

5

5

566

-.3 -.2 -.1 .0 .1 .2 .3-.15

-.10

-.05

.00

.05

.10

.15

Tetrahedral Symmetry / Instability

UN

IVE

RS_

CO

MPA

CT

(D

=2,

23)

UN

IVE

RS_

CO

MPA

CT

(D

=2,

23)

UN

IVE

RS_

CO

MPA

CT

(D

=2,

23)

Hs156Hs264108

.25

.50

.751.001.251.501.752.002.252.502.753.003.253.503.754.004.254.504.755.005.255.505.756.00

E [MeV]

Tetrahedral Shape

Oct

ahed

ral S

hape

Emin= 6.08, Eo= 8.01

• Competition between the tetrahedral and octahedral symmetries

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Strong High-Rank Symmetries in Super-heavy Elements

11

1

1

22

4

4

4

4

5

5

5

5

5

5 5

5

5

6

6

6

6

6

6

-.3 -.2 -.1 .0 .1 .2 .3-.15

-.10

-.05

.00

.05

.10

.15

Tetrahedral Symmetry / Instability

UN

IVE

RS_

CO

MPA

CT

(D

=2,

23)

UN

IVE

RS_

CO

MPA

CT

(D

=2,

23)

UN

IVE

RS_

CO

MPA

CT

(D

=2,

23)

Hs158Hs266108

.25

.50

.751.001.251.501.752.002.252.502.753.003.253.503.754.004.254.504.755.005.255.505.756.00

E [MeV]

Tetrahedral Shape

Oct

ahed

ral S

hape

Emin= 5.46, Eo= 6.82

• Competition between the tetrahedral and octahedral symmetries

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Strong High-Rank Symmetries in Super-heavy Elements

11

2

6

66

6

6

6 6

66

-.3 -.2 -.1 .0 .1 .2 .3-.15

-.10

-.05

.00

.05

.10

.15

Tetrahedral Symmetry / Instability

UN

IVE

RS_

CO

MPA

CT

(D

=2,

23)

UN

IVE

RS_

CO

MPA

CT

(D

=2,

23)

UN

IVE

RS_

CO

MPA

CT

(D

=2,

23)

Hs160Hs268108

.25

.50

.751.001.251.501.752.002.252.502.753.003.253.503.754.004.254.504.755.005.255.505.756.00

E [MeV]

Tetrahedral Shape

Oct

ahed

ral S

hape

Emin= 4.66, Eo= 5.30

• Competition between the tetrahedral and octahedral symmetries

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Strong Octahedral Effects in Super-heavy Elements

(09)[1,1,5](10)[3,2,2](16)[3,0,3](14)[0,3,3](22)[0,4,2](08)[4,2,1](09)[5,0,1](09)[5,1,1](07)[2,4,1](09)[4,4,0](09)[5,5,0](08)[4,4,0](06)[4,4,1](08)[2,1,4](10)[1,5,1](18)[1,2,3](19)[2,1,3](14)[1,1,5](07)[0,0,9](13)[3,3,1](14)[2,2,2](09)[2,2,3](15)[1,2,3](10)[3,1,2](11)[1,3,2]

(08)[1,3,4](10)[4,2,1](07)[0,2,5](10)[5,1,1](07)[3,2,3](08)[5,1,1](10)[1,1,5](07)[5,0,3](14)[3,3,2](13)[1,1,5](15)[2,2,3](11)[3,1,3](11)[1,3,3](07)[1,3,3](12)[3,1,3](07)[1,4,2]

(08)[6,0,3](15)[2,4,0](11)[2,3,1](06)[0,1,5](16)[0,1,5](15)[1,5,0](07)[2,0,5](08)[0,1,5](13)[5,1,1]

296118Xx178

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

Octahedral Deformation [Rank=4]

-8

-7

-6

-5

-4

-3

Neu

tron

Lev

els

[MeV

]

162

178

184184

184

176

196

210

188

Deformed Woods-Saxon - Compact Universal Parameters

Str

asbou

rg/K

rako

wT

heo

ryG

roup,α

40

=0.

3000

,α

44

=-0

.179

3

(14)[4,1,1](08)[0,0,9](10)[0,2,4](09)[3,2,1](12)[0,0,5](06)[4,3,0](14)[4,0,1](14)[2,2,1](21)[0,3,2](05)[1,4,0](10)[0,2,4](12)[2,2,2](05)[3,2,1](07)[5,1,0](07)[1,5,0](07)[0,3,1](16)[2,0,3](21)[2,1,2](28)[1,1,3](10)[3,1,1]

(09)[4,1,1](08)[1,0,3](14)[1,1,3](16)[1,1,4]

(08)[0,0,4](16)[1,2,2]

(15)[1,1,4](11)[0,6,0](11)[3,2,2](14)[2,3,2](17)[1,2,3](14)[2,1,3](06)[1,1,4](06)[0,5,3](13)[2,0,4](21)[1,4,1]

(07)[2,2,0]

(08)[0,3,1](12)[3,0,1](09)[2,2,1](15)[1,1,3](13)[1,4,0](07)[5,0,0](16)[0,1,4](09)[1,0,4](28)[2,2,0](10)[1,4,1]

296118Xx178

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

Octahedral Deformation [Rank=4]

-4

-3

-2

-1

0

1

2

Pro

ton

Lev

els

[MeV

]

118 118

102

114

114

120

134

108

136136

126

138

100

130

Deformed Woods-Saxon - Compact Universal Parameters

Str

asbou

rg/K

rako

wT

heo

ryG

roup,

• Above: Single particle energy levels for neutrons [left] and protons [right]

• Observe neutron gap at N = 178 much stronger than the one at N = 184

• Similarly, the proton octahedral gaps at Z = 118, 120 are significantly

stronger than the ‘classical’, spherical ones at Z = 114 and/or Z = 126

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Summary

• We presented what we call the New Theory of Nuclear Stabilitybased on the nuclear mean-field concepts and group theory

• On its basis we suggest that the nuclear stability is underlined byspatial symmetries with high number of irreducible representations

• ... rather than multipole expansion, prolate/oblate shape coexis-tence etc. - although the latter are a particular case of the former

• We have illustrated the new theory of stability with two high-rankpoint-groups: octhahedral- and its tetrahedral sub-group

• We presented preliminary results of our Jyvaskyla experiment on156Gd fully confirming the expectations (E2-transitions weaker thanso far reported) and bringing new important results.

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Summary

• We presented what we call the New Theory of Nuclear Stabilitybased on the nuclear mean-field concepts and group theory

• On its basis we suggest that the nuclear stability is underlined byspatial symmetries with high number of irreducible representations

• ... rather than multipole expansion, prolate/oblate shape coexis-tence etc. - although the latter are a particular case of the former

• We have illustrated the new theory of stability with two high-rankpoint-groups: octhahedral- and its tetrahedral sub-group

• We presented preliminary results of our Jyvaskyla experiment on156Gd fully confirming the expectations (E2-transitions weaker thanso far reported) and bringing new important results.

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Conclusions

• We have demonstrated through realistic calculations that severalpoint-groups so far never considered in nuclear structure physics leadto very strong shell effects

• In particular: tetrahedral symmetry minima imply the presence ofnegative parity bands with vanishing E2 transitions

•We have found the presence of those bands in the existing literaturein full agreement with our general predictions

• It is suggested that the ’octupole effects’, considered so far in theliterature, separate into two categories: the ‘traditional’ (‘pears’)and ‘tetrahedral’ (‘pyramids’)

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Conclusions

• We have demonstrated through realistic calculations that severalpoint-groups so far never considered in nuclear structure physics leadto very strong shell effects

• In particular: tetrahedral symmetry minima imply the presence ofnegative parity bands with vanishing E2 transitions

•We have found the presence of those bands in the existing literaturein full agreement with our general predictions

• It is suggested that the ’octupole effects’, considered so far in theliterature, separate into two categories: the ‘traditional’ (‘pears’)and ‘tetrahedral’ (‘pyramids’)

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Perspectives

• We are going to study the relatively low spin states, (I ≤ 20 ~) atrelatively high excitations - in Rare Earth and Actinide Nuclei

• Of high priority are the life-time measurements of the absolutevalues of quadrupole and dipole moments of the tetrahedral bands;

All colleagues possibly interested in joining us will be most welcome

• In coming months we will perform three experiments along theselines [156Gd in Legnaro, 156Dy at Argonne and 154Gd at iThemba]

• We are extending the new symmetry ideas to the super-heavynuclei [extensive calculations for nuclei around Z∼118 and N∼178];

We expect obtaining theoretical details suited for experiment plan-ning soon and will be interested in joining the experimental efforts

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Perspectives

• We are going to study the relatively low spin states, (I ≤ 20 ~) atrelatively high excitations - in Rare Earth and Actinide Nuclei

• Of high priority are the life-time measurements of the absolutevalues of quadrupole and dipole moments of the tetrahedral bands;

All colleagues possibly interested in joining us will be most welcome

• In coming months we will perform three experiments along theselines [156Gd in Legnaro, 156Dy at Argonne and 154Gd at iThemba]

• We are extending the new symmetry ideas to the super-heavynuclei [extensive calculations for nuclei around Z∼118 and N∼178];

We expect obtaining theoretical details suited for experiment plan-ning soon and will be interested in joining the experimental efforts

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Perspectives

• We are going to study the relatively low spin states, (I ≤ 20 ~) atrelatively high excitations - in Rare Earth and Actinide Nuclei

• Of high priority are the life-time measurements of the absolutevalues of quadrupole and dipole moments of the tetrahedral bands;

All colleagues possibly interested in joining us will be most welcome

• In coming months we will perform three experiments along theselines [156Gd in Legnaro, 156Dy at Argonne and 154Gd at iThemba]

• We are extending the new symmetry ideas to the super-heavynuclei [extensive calculations for nuclei around Z∼118 and N∼178];

We expect obtaining theoretical details suited for experiment plan-ning soon and will be interested in joining the experimental efforts

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Perspectives

• We are going to study the relatively low spin states, (I ≤ 20 ~) atrelatively high excitations - in Rare Earth and Actinide Nuclei

• Of high priority are the life-time measurements of the absolutevalues of quadrupole and dipole moments of the tetrahedral bands;

All colleagues possibly interested in joining us will be most welcome

• In coming months we will perform three experiments along theselines [156Gd in Legnaro, 156Dy at Argonne and 154Gd at iThemba]

• We are extending the new symmetry ideas to the super-heavynuclei [extensive calculations for nuclei around Z∼118 and N∼178];

We expect obtaining theoretical details suited for experiment plan-ning soon and will be interested in joining the experimental efforts

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Perspectives

• We are going to study the relatively low spin states, (I ≤ 20 ~) atrelatively high excitations - in Rare Earth and Actinide Nuclei

• Of high priority are the life-time measurements of the absolutevalues of quadrupole and dipole moments of the tetrahedral bands;

All colleagues possibly interested in joining us will be most welcome

• In coming months we will perform three experiments along theselines [156Gd in Legnaro, 156Dy at Argonne and 154Gd at iThemba]

• We are extending the new symmetry ideas to the super-heavynuclei [extensive calculations for nuclei around Z∼118 and N∼178];

We expect obtaining theoretical details suited for experiment plan-ning soon and will be interested in joining the experimental efforts

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Thanks to: TetraNuc Collaboration [1]

I wish to thank for the support and participation in experiments aswell as help in theoretical development the following Colleagues:

D. Curien, J. Dudek, Ch. Beck, S. Courtin, O. Dorvaux, G. Duchene, T. Faul, B. Gall,F. Haas, F. Khalfallah, D. Lebhertz, H. Molique, J. Robin, M. Rousseau, K. Rybak,M-D. Salsac - IPHC, Strasbourg, F

A. Gozdz, A. Dobrowolski - University of Lublin, Poland

N. Dubray - CEA, Bruyeres-le-Chatel, F

N. Alahari, G. de France, M. Rejmund - GANIL, Caen, F

B. Lauss - ILL, Grenoble, F

N. Redon, Ch. Schmitt, O. Stezowski, D. Q. Tuyen - IPN, Lyon, F

F. Azaiez, B. Berthier, S. Franchoo, D. Guillemaud-Mueller, M. Lebois, F. Ibrahim,B. Mouginot, C. Petrache, I. Stefan, D. Verney - IPN, Orsay, F

A. Astier, G. Georgiev - CSNSM, Orsay, F

P.T. Greenlees, P. Jones, R. Julin, S. Juutinen, S. Ketelhut, M. Nyman, P. Rahkila,J. Sorri, M. Leino, C. Scholey, J. Saren, U. Jakobsson, J. Uusitalo - Jyvaskyla, Fi

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Thanks to: TetraNuc Collaboration [1]

I wish to thank for the support and participation in experiments aswell as help in theoretical development the following Colleagues:

D. Curien, J. Dudek, Ch. Beck, S. Courtin, O. Dorvaux, G. Duchene, T. Faul, B. Gall,F. Haas, F. Khalfallah, D. Lebhertz, H. Molique, J. Robin, M. Rousseau, K. Rybak,M-D. Salsac - IPHC, Strasbourg, F

A. Gozdz, A. Dobrowolski - University of Lublin, Poland

N. Dubray - CEA, Bruyeres-le-Chatel, F

N. Alahari, G. de France, M. Rejmund - GANIL, Caen, F

B. Lauss - ILL, Grenoble, F

N. Redon, Ch. Schmitt, O. Stezowski, D. Q. Tuyen - IPN, Lyon, F

F. Azaiez, B. Berthier, S. Franchoo, D. Guillemaud-Mueller, M. Lebois, F. Ibrahim,B. Mouginot, C. Petrache, I. Stefan, D. Verney - IPN, Orsay, F

A. Astier, G. Georgiev - CSNSM, Orsay, F

P.T. Greenlees, P. Jones, R. Julin, S. Juutinen, S. Ketelhut, M. Nyman, P. Rahkila,J. Sorri, M. Leino, C. Scholey, J. Saren, U. Jakobsson, J. Uusitalo - Jyvaskyla, Fi

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Thanks to: TetraNuc Collaboration [2]

I wish to thank for the support and participation in experiments aswell as help in theoretical development the following Colleagues:

D. Tonev - Bulgarian Academy of Sciences, Sofia, Bulgaria

J. Gerl - GSI, Darmstadt, Ge

R.P. Singh, S. Muralithar, R. Kumar, A. Jhingan, J.J. Das, R. K. Bhowmik - Inter-University Accelerator Centre, New Delhi 67, India

G. de Angelis, A. Gadea, D.R. Napoli, J.J. Valiente-Dobon, F. Della Vedova, R. Orlandi,E. Sahin - INFN, Laboratori Nazionali di Legnaro, Legnaro, Italy

D. Mengoni, F. Recchia, S. Aydin, R. Menegazzo, D. Bazzacco, E. Farnea, S. Lunardi,C. Ur - Dipartimento di Fisica and INFN, Sezione di Padova, Padova, Italy

Y. R. Shimizu - Kyushu University, Fukuoka, JP

P. Bednarczyk, B. Fornal, A. Maj, K. Mazurek - IFJ PAN, Krakow, Poland

J. Dobaczewski - University of Warsaw, Pl and JYFL, Jyvaskyla, Fi

R.A. Bark, T.E. Madiba, T.D. Singo - iThemba LABS Physics Group;D.G. Roux, J.F. Sharpey-Schafer, UWC, SA

P. Regan - University of Surrey, UK

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Thanks to: TetraNuc Collaboration [2]

I wish to thank for the support and participation in experiments aswell as help in theoretical development the following Colleagues:

D. Tonev - Bulgarian Academy of Sciences, Sofia, Bulgaria

J. Gerl - GSI, Darmstadt, Ge

R.P. Singh, S. Muralithar, R. Kumar, A. Jhingan, J.J. Das, R. K. Bhowmik - Inter-University Accelerator Centre, New Delhi 67, India

G. de Angelis, A. Gadea, D.R. Napoli, J.J. Valiente-Dobon, F. Della Vedova, R. Orlandi,E. Sahin - INFN, Laboratori Nazionali di Legnaro, Legnaro, Italy

D. Mengoni, F. Recchia, S. Aydin, R. Menegazzo, D. Bazzacco, E. Farnea, S. Lunardi,C. Ur - Dipartimento di Fisica and INFN, Sezione di Padova, Padova, Italy

Y. R. Shimizu - Kyushu University, Fukuoka, JP

P. Bednarczyk, B. Fornal, A. Maj, K. Mazurek - IFJ PAN, Krakow, Poland

J. Dobaczewski - University of Warsaw, Pl and JYFL, Jyvaskyla, Fi

R.A. Bark, T.E. Madiba, T.D. Singo - iThemba LABS Physics Group;D.G. Roux, J.F. Sharpey-Schafer, UWC, SA

P. Regan - University of Surrey, UK

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Thanks to: TetraNuc Collaboration [3]

Last but not least many thanks to an extremely dynamic contribu-tions from our colleagues from the USA

L.L. Riedinger, Department of Physics, University of Tennessee, Knoxville, TN 37996D.J. Hartley, Department of Physics, US Naval Academy, Annapolis, MD 21402N. Schunck - University of Tennessee, USAC. Beausang - Department of Physics, University of Richmond, Richmond, VA 23173M.P. Carpenter, T. Lauritsen, E.A. McCutchan - Physics Division, Argonne NationalLaboratory, Argonne, IL 60439C.J. Chiara, F.G. Kondev - Nuclear Engineering Division, Argonne National Laboratory,Argonne, IL 60439P.E. Garrett - Department of Physics, University of Guelph, Guelph, Ontario, CanadaW.D. Kulp - School of Physics, Georgia Institute of Technology, Atlanta, GA 30332M.A. Riley - Department of Physics, Florida State University, Tallahassee, FL 32306L

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Thanks to: TetraNuc Collaboration [3]

Last but not least many thanks to an extremely dynamic contribu-tions from our colleagues from the USA

L.L. Riedinger, Department of Physics, University of Tennessee, Knoxville, TN 37996D.J. Hartley, Department of Physics, US Naval Academy, Annapolis, MD 21402N. Schunck - University of Tennessee, USAC. Beausang - Department of Physics, University of Richmond, Richmond, VA 23173M.P. Carpenter, T. Lauritsen, E.A. McCutchan - Physics Division, Argonne NationalLaboratory, Argonne, IL 60439C.J. Chiara, F.G. Kondev - Nuclear Engineering Division, Argonne National Laboratory,Argonne, IL 60439P.E. Garrett - Department of Physics, University of Guelph, Guelph, Ontario, CanadaW.D. Kulp - School of Physics, Georgia Institute of Technology, Atlanta, GA 30332M.A. Riley - Department of Physics, Florida State University, Tallahassee, FL 32306L

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

The story of high-rank symmetries started some 14 years ago{X. Li and J. Dudek, Phys. Rev. C94 (1994) R1250}

and was revived some 10 years later with a series of newarticles

We had a workshop about these issues in 2006 at Trento,Italy

We had another workshop in 2007 at the ILL, Grenoble,France

There has been a recent workshop on tetrahedral symmetry,at Atlanta, GA, USA, in 2008

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

The story of high-rank symmetries started some 14 years ago{X. Li and J. Dudek, Phys. Rev. C94 (1994) R1250}

and was revived some 10 years later with a series of newarticles

We had a workshop about these issues in 2006 at Trento,Italy

We had another workshop in 2007 at the ILL, Grenoble,France

There has been a recent workshop on tetrahedral symmetry,at Atlanta, GA, USA, in 2008

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

The story of high-rank symmetries started some 14 years ago{X. Li and J. Dudek, Phys. Rev. C94 (1994) R1250}

and was revived some 10 years later with a series of newarticles

We had a workshop about these issues in 2006 at Trento,Italy

We had another workshop in 2007 at the ILL, Grenoble,France

There has been a recent workshop on tetrahedral symmetry,at Atlanta, GA, USA, in 2008

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

The story of high-rank symmetries started some 14 years ago{X. Li and J. Dudek, Phys. Rev. C94 (1994) R1250}

and was revived some 10 years later with a series of newarticles

We had a workshop about these issues in 2006 at Trento,Italy

We had another workshop in 2007 at the ILL, Grenoble,France

There has been a recent workshop on tetrahedral symmetry,at Atlanta, GA, USA, in 2008

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

There is a suggestion to hold a TetraNuc workshop inEurope soon

Italian colleagues suggest to have it in Italy, possibly in April

New mini-conference will be organized in the USA

There is a mini-conference considered in the near future atRIKEN, Japan

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

There is a suggestion to hold a TetraNuc workshop inEurope soon

Italian colleagues suggest to have it in Italy, possibly in April

New mini-conference will be organized in the USA

There is a mini-conference considered in the near future atRIKEN, Japan

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

There is a suggestion to hold a TetraNuc workshop inEurope soon

Italian colleagues suggest to have it in Italy, possibly in April

New mini-conference will be organized in the USA

There is a mini-conference considered in the near future atRIKEN, Japan

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

A Basis for Octahedral Symmetry

Only special combinations of spherical harmonics may form a basisfor surfaces with octahedral symmetry and only in even-orders:

Three Lowest Order Solutions: Rank ↔ Multipolarity λ

λ = 4 : α40 ≡ o4; α4,±4 ≡ ±q

514· o4

λ = 6 : α60 ≡ o6; α6,±4 ≡ −q

72· o6

λ = 8 : α80 ≡ o8; α8,±4 ≡q

28198· o8; α8,±8 ≡

q65198· o8

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

A Basis for Octahedral Symmetry

Only special combinations of spherical harmonics may form a basisfor surfaces with octahedral symmetry and only in even-orders:

Three Lowest Order Solutions: Rank ↔ Multipolarity λ

λ = 4 : α40 ≡ o4; α4,±4 ≡ ±q

514· o4

λ = 6 : α60 ≡ o6; α6,±4 ≡ −q

72· o6

λ = 8 : α80 ≡ o8; α8,±4 ≡q

28198· o8; α8,±8 ≡

q65198· o8

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

A Basis for Octahedral Symmetry

Only special combinations of spherical harmonics may form a basisfor surfaces with octahedral symmetry and only in even-orders:

Three Lowest Order Solutions: Rank ↔ Multipolarity λ

λ = 4 : α40 ≡ o4; α4,±4 ≡ ±q

514· o4

λ = 6 : α60 ≡ o6; α6,±4 ≡ −q

72· o6

λ = 8 : α80 ≡ o8; α8,±4 ≡q

28198· o8; α8,±8 ≡

q65198· o8

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

A Basis for Octahedral Symmetry

Only special combinations of spherical harmonics may form a basisfor surfaces with octahedral symmetry and only in even-orders:

Three Lowest Order Solutions: Rank ↔ Multipolarity λ

λ = 4 : α40 ≡ o4; α4,±4 ≡ ±q

514· o4

λ = 6 : α60 ≡ o6; α6,±4 ≡ −q

72· o6

λ = 8 : α80 ≡ o8; α8,±4 ≡q

28198· o8; α8,±8 ≡

q65198· o8

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Important to know before proceeding further:

A ‘Surprise’ Originating from Mathematics

Namely:

• Tetrahedral group is a sub-group of the octahedral one !!!

Consequences:

• Superposing octahedral and tetrahedral shapes leads to the finaltetrahedral symmetry!!!

• Does realistic theory confirm these predictions?

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Important to know before proceeding further:

A ‘Surprise’ Originating from Mathematics

Namely:

• Tetrahedral group is a sub-group of the octahedral one !!!

Consequences:

• Superposing octahedral and tetrahedral shapes leads to the finaltetrahedral symmetry!!!

• Does realistic theory confirm these predictions?

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Important to know before proceeding further:

A ‘Surprise’ Originating from Mathematics

Namely:

• Tetrahedral group is a sub-group of the octahedral one !!!

Consequences:

• Superposing octahedral and tetrahedral shapes leads to the finaltetrahedral symmetry!!!

• Does realistic theory confirm these predictions?

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Example: Results with the HFB Solutions in RE

The HFB results for tetrahedral solutions in light Rare-Earth nuclei

α4,0 ≡ o4, α4,±4 ≡ −√

5/14 o4

Z N ∆E Q32 Q40 Q44 Q40 ×√

514

(MeV) (b3/2) (b2) (b2) (b2)

64 86 −1.387 0.941817 −0.227371 +0.135878 −0.13588064 90 −3.413 1.394656 −0.428250 +0.255929 −0.25592864 92 −3.972 0.000000 −0.447215 +0.267263 −0.267262

62 86 −0.125 0.487392 −0.086941 +0.051954 −0.05195762 88 −0.524 0.812103 −0.218809 +0.130760 −0.13076362 90 −1.168 1.206017 −0.380334 +0.227293 −0.227293

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Example: Results with the HFB Solutions in RE

The HFB results for tetrahedral solutions in light Rare-Earth nuclei

α4,0 ≡ o4, α4,±4 ≡ −√

5/14 o4

Z N ∆E Q32 Q40 Q44 Q40 ×√

514

(MeV) (b3/2) (b2) (b2) (b2)

64 86 −1.387 0.941817 −0.227371 +0.135878 −0.13588064 90 −3.413 1.394656 −0.428250 +0.255929 −0.25592864 92 −3.972 0.000000 −0.447215 +0.267263 −0.267262

62 86 −0.125 0.487392 −0.086941 +0.051954 −0.05195762 88 −0.524 0.812103 −0.218809 +0.130760 −0.13076362 90 −1.168 1.206017 −0.380334 +0.227293 −0.227293

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Example: Results with the HFB Solutions in Actinide

The HFB results for tetrahedral solutions in the Actinide nuclei

α4,0 ≡ o4, α4,±4 ≡ −√

5/14 o4

α6,0 ≡ o6, α6,±4 ≡ +√

7/2 o6

Octahedral deformations of the second order that are compatible with thetetrahedral deformation in 226Th with two Skyrme parameterisations

Force Q32 Q40

√5/14 Q44 Q60

√7/2 Q64

SkM* 3.4166 0.5582 0.5583 0.1537 0.1538SLy4 3.3353 0.5471 0.5617 0.1306 0.1341

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Example: Results with the HFB Solutions in Actinide

The HFB results for tetrahedral solutions in the Actinide nuclei

α4,0 ≡ o4, α4,±4 ≡ −√

5/14 o4

α6,0 ≡ o6, α6,±4 ≡ +√

7/2 o6

Octahedral deformations of the second order that are compatible with thetetrahedral deformation in 226Th with two Skyrme parameterisations

Force Q32 Q40

√5/14 Q44 Q60

√7/2 Q64

SkM* 3.4166 0.5582 0.5583 0.1537 0.1538SLy4 3.3353 0.5471 0.5617 0.1306 0.1341

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Shape Coexistence and Negative-Parity Bands

Negative parity rotational bandsare usually associated either withharmonic or an-harmonic oscilla-tions vs. α30 ...

... with the presence of the stableoctupole minima & accompanyingdoublet structures ...

... or with the negative-parityparticle-hole excitations of e.g.quadrupole-deformed nuclei

Stable Octupole Deformation

Anharmonic VibrationsHarmonic Vibrations

OctupoleDeformation

Tot

al E

nerg

y

Doublets

particle−hole

excitation

Rotational FrequencySin

gle

−N

ucl

eon

En

erg

y

π = −1π = +1

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Shape Coexistence and Negative-Parity Bands

Negative parity rotational bandsare usually associated either withharmonic or an-harmonic oscilla-tions vs. α30 ...

... with the presence of the stableoctupole minima & accompanyingdoublet structures ...

... or with the negative-parityparticle-hole excitations of e.g.quadrupole-deformed nuclei

Stable Octupole Deformation

Anharmonic VibrationsHarmonic Vibrations

OctupoleDeformation

Tot

al E

nerg

y

Doublets

particle−hole

excitation

Rotational FrequencySin

gle

−N

ucl

eon

En

erg

y

π = −1π = +1

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Shape Coexistence and Negative-Parity Bands

Negative parity rotational bandsare usually associated either withharmonic or an-harmonic oscilla-tions vs. α30 ...

... with the presence of the stableoctupole minima & accompanyingdoublet structures ...

... or with the negative-parityparticle-hole excitations of e.g.quadrupole-deformed nuclei

Stable Octupole Deformation

Anharmonic VibrationsHarmonic Vibrations

OctupoleDeformation

Tot

al E

nerg

y

Doublets

particle−hole

excitation

Rotational FrequencySin

gle

−N

ucl

eon

En

erg

y

π = −1π = +1

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Do We Know How the Octahedral Nuclei Rotate?

Spin-Orientation Probability

Z=

64,N

=86

,Jx=

31.9

,Jy=

31.9

,Jz=

31.9

;H0P

.150

00H

4P.0

8964

I=3, n=1; Erot=1.00000000000 (A2)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at increasing spins of an octahedral nucleus

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Do We Know How the Octahedral Nuclei Rotate?

Spin-Orientation Probability

Z=

64,N

=86

,Jx=

31.9

,Jy=

31.9

,Jz=

31.9

;H0P

.150

00H

4P.0

8964

I=4, n=1; Erot=1.00000000000 (T2)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at increasing spins of an octahedral nucleus

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Do We Know How the Octahedral Nuclei Rotate?

Spin-Orientation Probability

Z=

64,N

=86

,Jx=

31.9

,Jy=

31.9

,Jz=

31.9

;H0P

.150

00H

4P.0

8964

I=5, n=1; Erot=1.00000000000 (T1)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at increasing spins of an octahedral nucleus

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Do We Know How the Octahedral Nuclei Rotate?

Spin-Orientation Probability

Z=

64,N

=86

,Jx=

31.9

,Jy=

31.9

,Jz=

31.9

;H0P

.150

00H

4P.0

8964

I=6, n=1; Erot=1.00000000000 (A1)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at increasing spins of an octahedral nucleus

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Do We Know How the Octahedral Nuclei Rotate?

Spin-Orientation Probability

Z=

64,N

=86

,Jx=

31.9

,Jy=

31.9

,Jz=

31.9

;H0P

.150

00H

4P.0

8964

I=7, n=1; Erot=1.00000000000 (T1)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at increasing spins of an octahedral nucleus

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Do We Know How the Octahedral Nuclei Rotate?

Spin-Orientation Probability

Z=

64,N

=86

,Jx=

31.9

,Jy=

31.9

,Jz=

31.9

;H0P

.150

00H

4P.0

8964

I=8, n=1; Erot=1.00000000000 (T2)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at increasing spins of an octahedral nucleus

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Do We Know How the Octahedral Nuclei Rotate?

Spin-Orientation Probability

Z=

64,N

=86

,Jx=

31.9

,Jy=

31.9

,Jz=

31.9

;H0P

.150

00H

4P.0

8964

I=9, n=1; Erot=1.00000000000 (A2)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at increasing spins of an octahedral nucleus

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

New Theory: Verification, Relation to ExperimentConstructing a Series of Theoretical PredictionsExperimental Facts: Comparison with Experiment

Do We Know How the Octahedral Nuclei Rotate?

Spin-Orientation Probability

Z=

64,N

=86

,Jx=

31.9

,Jy=

31.9

,Jz=

31.9

;H0P

.150

00H

4P.0

8964

I=10, n=1; Erot=1.00000000000 (T2)

-1

0

1X -1

0

1

Y-1

0

1

Z

Spin-orientation probability at increasing spins of an octahedral nucleus

Jerzy DUDEK, University of Strasbourg, France Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria