Tetrahedral Symmetry in Nuclei: Theory, Experimental Criteria

Transcription

1 Tetrahedral Symmetry in Nuclei: Theory and Experimental Criteria Jerzy DUDEK Department of Subatomic Research, CNRS/IN 2 P 3 and University of Strasbourg I 4th October 28

2 The Most Fundamental Issue In Particle Physics are 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

3 The Most Fundamental Issue In Particle Physics are 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

4 The Most Fundamental Issue In Particle Physics are 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

5 The Most Fundamental Issue In Particle Physics are 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

6 The Most Fundamental Issue In Particle Physics are 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

7 Pathways for Future Facilities and Astrophysics... Superheavy Nuclei with new [tetrahedral] SH magic numbers! Proton Number Nuclei Known Today New Islands of Stability [special astrophysical consequences] Neutron Number

8 COLLABORATORS: Dominique CURIEN Jacek DOBACZEWSKI Andrzej GÓŹDŹ Florent HAAS Daryl HARTLEY Adam MAJ Kasia MAZUREK Hervé MOLIQUE Lee RIEDINGER Nicolas SCHUNCK

9 Group Theory and Geometry for Historians [I] Recall: Platonic Figures = Polyhedra whose faces are identical regular convex polygons Allowed polygons are equilateral triangles, or squares or regular pentagons Plato Tetrahedron, Cube, Octahedron, Icosahedron, Dodecahedron

10 Group Theory and Geometry for Historians [I] Recall: Platonic Figures = Polyhedra whose faces are identical regular convex polygons Allowed polygons are equilateral triangles, or squares or regular pentagons Plato Tetrahedron, Cube, Octahedron, Icosahedron, Dodecahedron

11 Group Theory and Geometry for Historians [I] Recall: Platonic Figures = Polyhedra whose faces are identical regular convex polygons Allowed polygons are equilateral triangles, or squares or regular pentagons Plato Tetrahedron, Cube, Octahedron, Icosahedron, Dodecahedron

12 Group Theory and Geometry for Historians [I] Recall: Platonic Figures = Polyhedra whose faces are identical regular convex polygons Allowed polygons are equilateral triangles, or squares or regular pentagons Plato Tetrahedron, Cube, Octahedron, Icosahedron, Dodecahedron

13 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 the five Platonic solids about -3 years before Plato (stone models in Ashmolean Museum, Oxford)... in religious context In what follows we stick to the aspect of beauty and nuclear reality - similarity to any other context will be purely accidental

14 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 the five Platonic solids about -3 years before Plato (stone models in Ashmolean Museum, Oxford)... in religious context In what follows we stick to the aspect of beauty and nuclear reality - similarity to any other context will be purely accidental

15 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 the five Platonic solids about -3 years before Plato (stone models in Ashmolean Museum, Oxford)... in religious context In what follows we stick to the aspect of beauty and nuclear reality - similarity to any other context will be purely accidental

16 Non-Trivial Discrete Symmetries in Nuclei: T d Let us recall one of the magic forms introduced long time by Plato. The implied symmetry leads to the tetrahedral group denoted T d A tetrahedron has four equal walls. Its shape is invariant with respect to 24 symmetry elements. Tetrahedron is not invariant with respect to the inversion. Of course nuclei cannot be represented by a sharp-edge pyramid... but rather in a form of a regular spherical harmonic expansion: λ max λ R(ϑ, ϕ) = R c({α})[ + α λ,µ Y λ,µ (ϑ, ϕ)] λ µ= λ

17 Non-Trivial Discrete Symmetries in Nuclei: T d Let us recall one of the magic forms introduced long time by Plato. The implied symmetry leads to the tetrahedral group denoted T d A tetrahedron has four equal walls. Its shape is invariant with respect to 24 symmetry elements. Tetrahedron is not invariant with respect to the inversion. Of course nuclei cannot be represented by a sharp-edge pyramid... but rather in a form of a regular spherical harmonic expansion: λ max λ R(ϑ, ϕ) = R c({α})[ + α λ,µ Y λ,µ (ϑ, ϕ)] λ µ= λ

18 Non-Trivial Discrete Symmetries in Nuclei: T d Let us recall one of the magic forms introduced long time by Plato. The implied symmetry leads to the tetrahedral group denoted T d A tetrahedron has four equal walls. Its shape is invariant with respect to 24 symmetry elements. Tetrahedron is not invariant with respect to the inversion. Of course nuclei cannot be represented by a sharp-edge pyramid... but rather in a form of a regular spherical harmonic expansion: R(ϑ, ϕ) = R {+α 3+2 (Y Y 3 2 ) + α 72ˆ(Y7+2 + Y 7 2 ) {z } one parameter 3rd order q 3 (Y Y 7 6 ) } {z } one parameter 7th order

19 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 O h An octahedron has 8 equal walls. Its shape is invariant with respect to 48 symmetry elements that include inversion. However, the nuclear surface cannot be represented in the form of a diamond... but rather in a form of a regular spherical harmonic expansion: λ max λ R(ϑ, ϕ) = R c({α})[ + α λ,µ Y λ,µ (ϑ, ϕ)] λ µ= λ

20 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 O h An octahedron has 8 equal walls. Its shape is invariant with respect to 48 symmetry elements that include inversion. However, the nuclear surface cannot be represented in the form of a diamond... but rather in a form of a regular spherical harmonic expansion: q q 5 R(ϑ, ϕ) = R +α4ˆy4 + 4 (Y Y 4 4 ) + α 6ˆY6 2 (Y Y 6 4 ) {z } {z } one parameter 4th order one parameter 6th order

21 Group Theory and Geometry for Pedestrians [I] Consider a nuclear surface with tetrahedral deformation... This is what we call in jargon: nuclear pyramid = tetrahedron

22 Group Theory and Geometry for Pedestrians [II]... and another surface with octahedral deformation... This is what we call in jargon: nuclear diamond = octahedron

23 Group Theory and Geometry for Pedestrians [III]... or even better, compare them directly... You most likely clearly see the difference between the two shapes: the pyramid [left] and the diamond [right] As one student said: Pyramids are in Egypt and Diamonds Are a Girl s Best Friend ;... but she studied bio-physics...

24 Group Theory and Geometry for Pedestrians [IV] Surprise-Surprise! Difficult to believe! Superposing the tetrahedral- and octahedral-symmetry surfaces gives us a new surface but again of tetrahedral symmetry: new richness

25 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

26 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 octahedral symmetries 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 old spherical harmonic expansions [only very few degrees of freedom]

27 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 of group-theory based families of new nuclear symmetries The approach uses two most powerful tools: group theory and the nuclear mean-field theory the best what we have at the beginning of the XXI st century All other so far known results, criteria of nuclear stability etc. are just a particular case of the new formulation

28 Outline of the New Theory of Nuclear Stability Part I Outline of the New Theory of Nuclear Stability

29 Outline of the New Theory of Nuclear Stability Single-Particle Structure and Global Stability of Nuclei Examples of Most Attractive Point-Group Symmetries Summarising: Global Stability vs. Gaps in SP Spectra Consider a typical outcome of the Mean-Field calculation: the shell structures and the total energies Presence of sufficiently strong gaps correlates with local minima of the total nuclear energy Nucleon Energies Deformation Parameter GAP GAP The Deformation Parameter axis represents several deformations of the mean field e.g. {Q λµ }, {α λµ }, the usual approach in constrained mean-field calculations Nucleus Nucleus 2 Deformation Deformation

30 Outline of the New Theory of Nuclear Stability Single-Particle Structure and Global Stability of Nuclei Examples of Most Attractive Point-Group Symmetries Summarising: Global Stability vs. Gaps in SP Spectra Consider a typical outcome of the Mean-Field calculation: the shell structures and the total energies Presence of sufficiently strong gaps correlates with local minima of the total nuclear energy Nucleon Energies Deformation Parameter GAP GAP The Deformation Parameter axis represents several deformations of the mean field e.g. {Q λµ }, {α λµ }, the usual approach in constrained mean-field calculations Nucleus Nucleus 2 Deformation Deformation

31 Outline of the New Theory of Nuclear Stability Single-Particle Structure and Global Stability of Nuclei Examples of Most Attractive Point-Group Symmetries Summarising: Global Stability vs. Gaps in SP Spectra Consider a typical outcome of the Mean-Field calculation: the shell structures and the total energies Presence of sufficiently strong gaps correlates with local minima of the total nuclear energy Nucleon Energies Deformation Parameter GAP GAP The Deformation Parameter axis represents several deformations of the mean field e.g. {Q λµ }, {α λµ }, the usual approach in constrained mean-field calculations Nucleus Nucleus 2 Deformation Deformation

32 Outline of the New Theory of Nuclear Stability Single-Particle Structure and Global Stability of Nuclei Examples of Most Attractive Point-Group Symmetries Symmetries, Representations and Degeneracies Given Hamiltonian H and a group: G = {O, O 2,... O f } Assume that G is a symmetry group of H i.e. [H, O k ] = with k =, 2,... f Let irreducible representations of G be {R, R 2,... R r } Let their dimensions be {d, d 2,... d r }, respectively Then the eigenvalues {ε ν } of the problem Hψ ν = ε ν ψ ν appear in multiplets d -fold, d 2 -fold... degenerate

33 Outline of the New Theory of Nuclear Stability Single-Particle Structure and Global Stability of Nuclei Examples of Most Attractive Point-Group Symmetries Symmetries, Representations and Degeneracies Given Hamiltonian H and a group: G = {O, O 2,... O f } Assume that G is a symmetry group of H i.e. [H, O k ] = with k =, 2,... f Let irreducible representations of G be {R, R 2,... R r } Let their dimensions be {d, d 2,... d r }, respectively Then the eigenvalues {ε ν } of the problem Hψ ν = ε ν ψ ν appear in multiplets d -fold, d 2 -fold... degenerate

34 Outline of the New Theory of Nuclear Stability Single-Particle Structure and Global Stability of Nuclei Examples of Most Attractive Point-Group Symmetries Symmetries, Representations and Degeneracies Given Hamiltonian H and a group: G = {O, O 2,... O f } Assume that G is a symmetry group of H i.e. [H, O k ] = with k =, 2,... f Let irreducible representations of G be {R, R 2,... R r } Let their dimensions be {d, d 2,... d r }, respectively Then the eigenvalues {ε ν } of the problem Hψ ν = ε ν ψ ν appear in multiplets d -fold, d 2 -fold... degenerate

35 Outline of the New Theory of Nuclear Stability Single-Particle Structure and Global Stability of Nuclei Examples of Most Attractive Point-Group Symmetries Symmetries, Representations and Degeneracies Given Hamiltonian H and a group: G = {O, O 2,... O f } Assume that G is a symmetry group of H i.e. [H, O k ] = with k =, 2,... f Let irreducible representations of G be {R, R 2,... R r } Let their dimensions be {d, d 2,... d r }, respectively Then the eigenvalues {ε ν } of the problem Hψ ν = ε ν ψ ν appear in multiplets d -fold, d 2 -fold... degenerate

36 Outline of the New Theory of Nuclear Stability Single-Particle Structure and Global Stability of Nuclei Examples of Most Attractive Point-Group Symmetries Symmetries, Representations and Degeneracies Given Hamiltonian H and a group: G = {O, O 2,... O f } Assume that G is a symmetry group of H i.e. [H, O k ] = with k =, 2,... f Let irreducible representations of G be {R, R 2,... R r } Let their dimensions be {d, d 2,... d r }, respectively Then the eigenvalues {ε ν } of the problem Hψ ν = ε ν ψ ν appear in multiplets d -fold, d 2 -fold... degenerate

37 Outline of the New Theory of Nuclear Stability Single-Particle Structure and Global Stability of Nuclei Examples of Most Attractive Point-Group Symmetries Symmetries and Gaps in Nuclear Context: Schematic Schematic illustration: Levels of 6 irreps and average spacings/gaps G a p Irrep. Irrep.2 Irrep.3 Irrep.4 Irrep.5 Irrep.6 All Irreps. Roughly: The average level spacings within an irrep increase by a factor of 6. The total spectrum may present big unprecedented gaps.

38 Outline of the New Theory of Nuclear Stability Single-Particle Structure and Global Stability of Nuclei Examples of Most Attractive Point-Group Symmetries Symmetries and Gaps in Nuclear Context: Schematic Schematic illustration: Levels of 6 irreps and average spacings/gaps G a p Irrep. Irrep.2 Irrep.3 Irrep.4 Irrep.5 Irrep.6 All Irreps. Roughly: The average level spacings within an irrep increase by a factor of 6. The total spectrum may present big unprecedented gaps.

39 Outline of the New Theory of Nuclear Stability Single-Particle Structure and Global Stability of Nuclei Examples 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 either look 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 of a selection of point groups when studying nuclear stability It turns out that the above ideas are well confirmed by calculations

40 Outline of the New Theory of Nuclear Stability Single-Particle Structure and Global Stability of Nuclei Examples 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 either look 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 of a selection of point groups when studying nuclear stability It turns out that the above ideas are well confirmed by calculations

41 Outline of the New Theory of Nuclear Stability Single-Particle Structure and Global Stability of Nuclei Examples of Most Attractive Point-Group Symmetries Example: Octahedral Symmetry - Proton Spectra Double group Oh D has four 2-dimensional and two 4-dimensional irreducible representations six distinct families of levels Proton Energies [MeV] Yb 9 {}[5,,5] /2 {8}[5,,3] 7/2 {}[5,,3] 7/2 {}[5,4,] 3/2 {3}[5,,5] /2 {9}[4,4,] /2 {8}[4,,2] 3/2 {}[4,,] /2 {5}[5,,4] 7/2 {7}[5,2,3] 5/2 {}[5,,2] 5/2 {}[5,3,2] 5/2 {23}[5,,4] 9/2 {3}[4,,4] 7/2 {8}[4,2,2] 3/2 {8}[4,3,] /2 {7}[5,2,3] 7/2 {5}[4,3,] /2 {2}[4,,2] 5/2 {}[4,,3] 5/2 {9}[4,2,] /2 {3}[4,,4] 9/2 {9}[4,3,] 3/ Octahedral Deformation {9}[4,,2] 5/2 {6}[5,3,2] 5/2 {8}[5,,4] 9/2 {8}[5,,5] 9/2 {2}[4,,3] 5/2 {7}[4,,3] 7/2 {24}[4,2,2] 3/2 {9}[5,,2] 5/2 {}[5,,5] 9/2 {7}[5,,] /2 {}[5,4,] 3/2 {2}[4,2,] /2 {}[5,,5] /2 {}[4,3,] /2 {9}[4,2,2] 3/2 {8}[3,3,] /2 {3}[3,,] 3/2 {8}[5,,3] 7/2 {25}[4,2,2] 5/2 {6}[4,,3] 7/2 {9}[4,3,] 3/2 {24}[3,,2] 3/2 {}[4,3,] /2 Strasbourg, August 22 Dirac-Woods-Saxon α 4(min)=-.35, α4(max)=.35 α 44(min)=-.29, α44(max)=.29 Figure: Full lines correspond to 4-dimensional irreducible representations - they are marked with double Nilsson labels. Observe huge gap at Z=7.

42 Outline of the New Theory of Nuclear Stability Single-Particle Structure and Global Stability of Nuclei Examples of Most Attractive Point-Group Symmetries Example: Octahedral Symmetry - Neutron Spectra Double group Oh D has four 2-dimensional and two 4-dimensional irreducible representations six distinct families of levels Neutron Energies [MeV] Yb 9 {7}[6,,3] 7/2 {9}[6,4,2] 5/2 {8}[5,,5] 9/2 {8}[6,,5] /2 {2}[5,,5] 9/2 {}[5,4,] 3/2 {9}[6,3,3] 7/2 {2}[6,,5] /2 {7}[5,4,] 3/2 {6}[6,2,4] 9/2 {}[5,,] 3/2 {6}[5,4,] /2 {9}[5,,3] 7/2 {}[5,3,] /2 {7}[5,,5] /2 {3}[5,2,] /2 {2}[5,,5] /2 {3}[5,,5] /2 {2}[5,,4] 7/2 {}[5,2,3] 5/2 {2}[4,,] /2 {22}[4,4,] /2 {2}[4,,2] 3/ Octahedral Deformation {8}[6,,6] /2 {7}[6,4,] /2 {6}[8,,2] 5/2 {}[6,,3] 7/2 {}[6,5,] 3/2 {7}[6,,6] 3/2 {7}[5,,4] 7/2 {2}[5,3,2] 3/2 {}[5,2,] 3/2 {}[5,3,] /2 {7}[8,8,] /2 {8}[4,,2] 5/2 {9}[6,,4] 9/2 {8}[5,3,2] 5/2 {2}[5,2,3] 7/2 {4}[5,3,2] 5/2 {6}[5,,4] 9/2 {8}[5,,5] 9/2 {2}[4,,3] 5/2 {3}[4,,] /2 {23}[4,2,2] 3/2 {8}[5,,5] /2 {8}[5,4,] /2 Strasbourg, August 22 Dirac-Woods-Saxon α 4(min)=-.35, α4(max)=.35 α 44(min)=-.29, α44(max)=.29 Figure: Full lines correspond to 4-dimensional irreducible representations - they are marked with double Nilsson labels. Observe huge gap at N=4.

43 Focus on the Nuclear Tetrahedral Symmetry Part II Focus on Nuclear Tetrahedral Symmetry

44 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry First Goal: Obtain Tetrahedral Magic Numbers... After inspecting many single-particle diagrams in function of tetrahedral deformation we read-out all the magic numbers 2 The tetrahedral symmetric nuclei are predicted to be particularly stable around magic closures: {Z t, N t } = {32, 4, 56, 64, 7, 9, 36} 3... and more precisely around the following nuclei: Ge 32, Ge 4, Ge 56, 8 4 Zr 4, 4 Zr 7, 2 56 Ba 56, Ba 7, Ba 9, Gd 7, Gd 9, 6 7 Yb 9, Th 36

45 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry First Goal: Obtain Tetrahedral Magic Numbers... After inspecting many single-particle diagrams in function of tetrahedral deformation we read-out all the magic numbers 2 The tetrahedral symmetric nuclei are predicted to be particularly stable around magic closures: {Z t, N t } = {32, 4, 56, 64, 7, 9, 36} 3... and more precisely around the following nuclei: Ge 32, Ge 4, Ge 56, 8 4 Zr 4, 4 Zr 7, 2 56 Ba 56, Ba 7, Ba 9, Gd 7, Gd 9, 6 7 Yb 9, Th 36

46 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry First Goal: Obtain Tetrahedral Magic Numbers... After inspecting many single-particle diagrams in function of tetrahedral deformation we read-out all the magic numbers 2 The tetrahedral symmetric nuclei are predicted to be particularly stable around magic closures: {Z t, N t } = {32, 4, 56, 64, 7, 9, 36} 3... and more precisely around the following nuclei: Ge 32, Ge 4, Ge 56, 8 4 Zr 4, 4 Zr 7, 2 56 Ba 56, Ba 7, Ba 9, Gd 7, Gd 9, 6 7 Yb 9, Th 36

47 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Tetrahedral Stability; Tetrahedral Magic Numbers Tetrahedral Symmetry Induced Magic Numbers 2 N=36 N=78 Z=8 Z=2 N=2 Proton Number N=4 N=56 N=7 N=9 Z=56 Z=7 Z=64 Z=9 4 Z= Neutron Number 6 8

48 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Tetrahedral Stability; Tetrahedral Magic Numbers Tetrahedral Symmetry Induced Magic Numbers 2 N=36 N=78 Z=8 Z=2 N=2 Proton Number N=4 N=56 N=7 N=9 Z=56 Z=7 Z=64 Z=9 4 Z= Neutron Number 6 8

49 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Experimental Hints: Where Are Tetrahedra? Total Energy (WS Universal) Total Energy (WS Universal) Energy [MeV] Deformation a[2,] Z=96 IPARAM= 4 Z= N=36 Zpl= 96 Npl=36 ISOSPI= October 22; NPOLYN= 6 HOMFAC=3.2 MeV Energy [MeV] Z= Tetrahedral Deformation IPARAM= 4 Z= N=36 Zpl= 96 Npl=36 ISOSPI= October 22; NPOLYN= 6 HOMFAC=3.2 MeV 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

50 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Experimental Hints: Where Are Tetrahedra? Excitation Energy 4 2 Tetrahedral and octhedral bands Cold Reactions, Coulex,... Yrast Line 2 Spin The phenomenon discussed is a low-spin and relatively high-energy effect

51 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Axial Nuclei Rotate? Spin-Orientation Probability Z I=2, n=; E rot =. (A) - - X - Y Z=4,N=4,J x =24.5,J y =24.5,J z =8.8;No-higher-defs. Spin-orientation probability in the yrast states of the axially symmetric nuclei - observe flattening of the distribution with increasing spin value

52 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Axial Nuclei Rotate? Spin-Orientation Probability Z I=2, n=; E rot =. (A) - - X - Y Z=4,N=4,J x =24.5,J y =24.5,J z =8.8;No-higher-defs. Spin-orientation probability in the yrast states of the axially symmetric nuclei - observe flattening of the distribution with increasing spin value

53 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Axial Nuclei Rotate? Spin-Orientation Probability Z I=6, n=; E rot =. (A) - - X - Y Z=4,N=4,J x =24.5,J y =24.5,J z =8.8;No-higher-defs. Spin-orientation probability in the yrast states of the axially symmetric nuclei - observe flattening of the distribution with increasing spin value

54 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Axial Nuclei Rotate? Spin-Orientation Probability Z I=8, n=; E rot =. (A) - - X - Y Z=4,N=4,J x =24.5,J y =24.5,J z =8.8;No-higher-defs. Spin-orientation probability in the yrast states of the axially symmetric nuclei - observe flattening of the distribution with increasing spin value

55 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Axial Nuclei Rotate? Spin-Orientation Probability Z I=, n=; E rot =. (A) - - X - Y Z=4,N=4,J x =24.5,J y =24.5,J z =8.8;No-higher-defs. Spin-orientation probability in the yrast states of the axially symmetric nuclei - observe flattening of the distribution with increasing spin value

56 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Axial Nuclei Rotate? Spin-Orientation Probability Z I=2, n=; E rot =. (A) - - X - Y Z=4,N=4,J x =24.5,J y =24.5,J z =8.8;No-higher-defs. Spin-orientation probability in the yrast states of the axially symmetric nuclei - observe flattening of the distribution with increasing spin value

57 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Axial Nuclei Rotate? Spin-Orientation Probability Z I=4, n=; E rot =. (A) - - X - Y Z=4,N=4,J x =24.5,J y =24.5,J z =8.8;No-higher-defs. Spin-orientation probability in the yrast states of the axially symmetric nuclei - observe flattening of the distribution with increasing spin value

58 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Axial Nuclei Rotate? Spin-Orientation Probability Z I=4, n=; E rot =. (A) - - X - Y Z=4,N=4,J x =24.5,J y =24.5,J z =8.8;No-higher-defs. Spin-orientation probability at the constant spin I of an axially symmetric nucleus - observe the evolution with nuclear excitation at fixed value I=4

59 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Axial Nuclei Rotate? Spin-Orientation Probability Z I=4, n=5; E rot = (A) - - X - Y Z=4,N=4,J x =24.5,J y =24.5,J z =8.8;No-higher-defs. Spin-orientation probability at the constant spin I of an axially symmetric nucleus - observe the evolution with nuclear excitation at fixed value I=4

60 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Axial Nuclei Rotate? Spin-Orientation Probability Z I=4, n=9; E rot = (A) - - X - Y Z=4,N=4,J x =24.5,J y =24.5,J z =8.8;No-higher-defs. Spin-orientation probability at the constant spin I of an axially symmetric nucleus - observe the evolution with nuclear excitation at fixed value I=4

61 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Axial Nuclei Rotate? Spin-Orientation Probability Z I=4, n=3; E rot = (A) - - X - Y Z=4,N=4,J x =24.5,J y =24.5,J z =8.8;No-higher-defs. Spin-orientation probability at the constant spin I of an axially symmetric nucleus - observe the evolution with nuclear excitation at fixed value I=4

62 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Axial Nuclei Rotate? Spin-Orientation Probability Z I=4, n=7; E rot = (A) - - X - Y Z=4,N=4,J x =24.5,J y =24.5,J z =8.8;No-higher-defs. Spin-orientation probability at the constant spin I of an axially symmetric nucleus - observe the evolution with nuclear excitation at fixed value I=4

63 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Axial Nuclei Rotate? Spin-Orientation Probability Z I=4, n=2; E rot = (A) - - X - Y Z=4,N=4,J x =24.5,J y =24.5,J z =8.8;No-higher-defs. Spin-orientation probability at the constant spin I of an axially symmetric nucleus - observe the evolution with nuclear excitation at fixed value I=4

64 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Axial Nuclei Rotate? Spin-Orientation Probability Z I=4, n=25; E rot = (A2) - - X - Y Z=4,N=4,J x =24.5,J y =24.5,J z =8.8;No-higher-defs. Spin-orientation probability at the constant spin I of an axially symmetric nucleus - observe the evolution with nuclear excitation at fixed value I=4

65 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Axial Nuclei Rotate? Spin-Orientation Probability Z I=4, n=29; E rot = (A) - - X - Y Z=4,N=4,J x =24.5,J y =24.5,J z =8.8;No-higher-defs. Spin-orientation probability at the constant spin I of an axially symmetric nucleus - observe the evolution with nuclear excitation at fixed value I=4

66 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected 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 nuclear physics - although there are entire books discussing the analogous problems in molecular physics!

67 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected 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 nuclear physics - although there are entire books discussing the analogous problems in molecular physics!

68 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected 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 nuclear physics - although there are entire books discussing the analogous problems in molecular physics!

69 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Octahedral Nuclei Rotate? Spin-Orientation Probability Z - - I=6, n=; E rot =. (A) Y X - Z=64,N=86,J x =3.9,J y =3.9,J z =3.9;HP.5H4P.8964 Spin-orientation probability at I = 6 and the increasing excitations of an octahedral nucleus - observe unexpected forms of probability distributions

70 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Octahedral Nuclei Rotate? Spin-Orientation Probability Z - - I=6, n=2; E rot = (T) Y X - Z=64,N=86,J x =3.9,J y =3.9,J z =3.9;HP.5H4P.8964 Spin-orientation probability at I = 6 and the increasing excitations of an octahedral nucleus - observe unexpected forms of probability distributions

71 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Octahedral Nuclei Rotate? Spin-Orientation Probability Z - - I=6, n=3; E rot = (T) Y X - Z=64,N=86,J x =3.9,J y =3.9,J z =3.9;HP.5H4P.8964 Spin-orientation probability at I = 6 and the increasing excitations of an octahedral nucleus - observe unexpected forms of probability distributions

72 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Octahedral Nuclei Rotate? Spin-Orientation Probability Z - - I=6, n=4; E rot = (T) Y X - Z=64,N=86,J x =3.9,J y =3.9,J z =3.9;HP.5H4P.8964 Spin-orientation probability at I = 6 and the increasing excitations of an octahedral nucleus - observe unexpected forms of probability distributions

73 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Octahedral Nuclei Rotate? Spin-Orientation Probability Z - - I=6, n=5; E rot = (T2) Y X - Z=64,N=86,J x =3.9,J y =3.9,J z =3.9;HP.5H4P.8964 Spin-orientation probability at I = 6 and the increasing excitations of an octahedral nucleus - observe unexpected forms of probability distributions

74 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Octahedral Nuclei Rotate? Spin-Orientation Probability Z - - I=6, n=6; E rot = (T2) Y X - Z=64,N=86,J x =3.9,J y =3.9,J z =3.9;HP.5H4P.8964 Spin-orientation probability at I = 6 and the increasing excitations of an octahedral nucleus - observe unexpected forms of probability distributions

75 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Octahedral Nuclei Rotate? Spin-Orientation Probability Z - - I=6, n=7; E rot = (T2) Y X - Z=64,N=86,J x =3.9,J y =3.9,J z =3.9;HP.5H4P.8964 Spin-orientation probability at I = 6 and the increasing excitations of an octahedral nucleus - observe unexpected forms of probability distributions

76 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Octahedral Nuclei Rotate? Spin-Orientation Probability Z - - I=6, n=8; E rot = (A2) Y X - Z=64,N=86,J x =3.9,J y =3.9,J z =3.9;HP.5H4P.8964 Spin-orientation probability at I = 6 and the increasing excitations of an octahedral nucleus - observe unexpected forms of probability distributions

77 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Octahedral Nuclei Rotate? Spin-Orientation Probability Z - - I=6, n=9; E rot = (T2) Y X - Z=64,N=86,J x =3.9,J y =3.9,J z =3.9;HP.5H4P.8964 Spin-orientation probability at I = 6 and the increasing excitations of an octahedral nucleus - observe unexpected forms of probability distributions

78 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Octahedral Nuclei Rotate? Spin-Orientation Probability Z - - I=6, n=; E rot = (T2) Y X - Z=64,N=86,J x =3.9,J y =3.9,J z =3.9;HP.5H4P.8964 Spin-orientation probability at I = 6 and the increasing excitations of an octahedral nucleus - observe unexpected forms of probability distributions

79 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Octahedral Nuclei Rotate? Spin-Orientation Probability Z - - I=6, n=; E rot = (T2) Y X - Z=64,N=86,J x =3.9,J y =3.9,J z =3.9;HP.5H4P.8964 Spin-orientation probability at I = 6 and the increasing excitations of an octahedral nucleus - observe unexpected forms of probability distributions

80 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Octahedral Nuclei Rotate? Spin-Orientation Probability Z - - I=6, n=2; E rot = (E) Y X - Z=64,N=86,J x =3.9,J y =3.9,J z =3.9;HP.5H4P.8964 Spin-orientation probability at I = 6 and the increasing excitations of an octahedral nucleus - observe unexpected forms of probability distributions

81 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Do We Know How the Octahedral Nuclei Rotate? Spin-Orientation Probability Z - - I=6, n=3; E rot = (E) Y X - Z=64,N=86,J x =3.9,J y =3.9,J z =3.9;HP.5H4P.8964 Spin-orientation probability at I = 6 and the increasing excitations of an octahedral nucleus - observe unexpected forms of probability distributions

82 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry What did we learn from these pictures? First message is this: we know the wave functions, we can and we do calculate the transition probabilities that give neccessary and sufficient experimental criteria for the presence of the symmetries These pictures help to understand and interpret geometrically the theoretical predictions for electromagnetic transition probabilities... but this is a looong topic for yet another, separate story!

83 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry What did we learn from these pictures? First message is this: we know the wave functions, we can and we do calculate the transition probabilities that give neccessary and sufficient experimental criteria for the presence of the symmetries These pictures help to understand and interpret geometrically the theoretical predictions for electromagnetic transition probabilities... but this is a looong topic for yet another, separate story!

84 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry What did we learn from these pictures? First message is this: we know the wave functions, we can and we do calculate the transition probabilities that give neccessary and sufficient experimental criteria for the presence of the symmetries These pictures help to understand and interpret geometrically the theoretical predictions for electromagnetic transition probabilities... but this is a looong topic for yet another, separate story!

85 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Survey of Doubly-Magic Tetrahedral Nuclei Deformation t E(fyu)+Shell[e]+Correlation[PNP] Deformation α 4 Zr 2 56 Emin=-3.77, Eo= E [MeV] UNIVERS_COMPACT (D=3, 23) Gp=.96 Gn=.98 N p=35 Nn=45 This stable nucleus is predicted tetrahedral-unstable in the ground-state - and yet will combine the signs of sphericity and tetrahedrality

86 4 4 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Survey of Doubly-Magic Tetrahedral Nuclei Deformation α Deformation α 9 Th 2 36 Emin=-3.55, Eo= E(fyu)+Shell[e]+Correlation[PNP] E [MeV] UNIVERS_COMPACT (D=4, 23) Gp=.96 Gn=.98 N p=35 Nn=45 This nucleus is predicted to manifest tetrahedral minima in competition with the quadrupole ground-state minimum...

87 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Survey of Doubly-Magic Tetrahedral Nuclei Deformation α E(fyu)+Shell[e]+Correlation[PNP] Deformation α 9 Th 2 36 Emin=-6.68, Eo= E [MeV] UNIVERS_COMPACT (D=4, 23) Gp=.96 Gn=.98 N p=35 Nn=45... and in competition with the pear-shape old octupole minimum!!!

88 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected 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) = R + λ µ α λµ Y λµ (ϑ, ϕ) ] Given uniform density ρ Σ ( r ) defined using the surface Σ { ρ : r Σ ρ Σ ( r ) = : r / Σ Express the multipole moments as usual by Q λµ = ρ Σ ( r ) r λ Y λµ d 3 r We will calculate the quadrupole moments as functions of α 3µ

89 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected 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) = R + λ µ α λµ Y λµ (ϑ, ϕ) ] Given uniform density ρ Σ ( r ) defined using the surface Σ { ρ : r Σ ρ Σ ( r ) = : r / Σ Express the multipole moments as usual by Q λµ = ρ Σ ( r ) r λ Y λµ d 3 r We will calculate the quadrupole moments as functions of α 3µ

90 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected 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) = R + λ µ α λµ Y λµ (ϑ, ϕ) ] Given uniform density ρ Σ ( r ) defined using the surface Σ { ρ : r Σ ρ Σ ( r ) = : r / Σ Express the multipole moments as usual by Q λµ = ρ Σ ( r ) r λ Y λµ d 3 r We will calculate the quadrupole moments as functions of α 3µ

91 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected 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) = R + λ µ α λµ Y λµ (ϑ, ϕ) ] Given uniform density ρ Σ ( r ) defined using the surface Σ { ρ : r Σ ρ Σ ( r ) = : r / Σ Express the multipole moments as usual by Q λµ = ρ Σ ( r ) r λ Y λµ d 3 r We will calculate the quadrupole moments as functions of α 3µ

92 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Multipole Moments as Functionals of the Density For small deformations we use Taylor expansion: α= Q λµ (α) Q λµ + Q λµ α + α= 2 Q λµ α α α= We set λ = 2, µ = and λ = λ 2 = 3 and obtain α 3 : Q 2 = 5/(2 5π) α 2 3 ρ R 5 α 3 : Q 2 = 5/(4 5π) α 3+ α 3 ρ R 5 α 32 : Q 2 = α 33 : Q 2 = 25/(2 5π) α 3+3 α 3 3 ρ R 5 Conclusion: Among λ = 3 defs. only α 32 leads to Q 2!!!

93 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Multipole Moments as Functionals of the Density For small deformations we use Taylor expansion: α= Q λµ (α) Q λµ + Q λµ α + α= 2 Q λµ α α α= We set λ = 2, µ = and λ = λ 2 = 3 and obtain α 3 : Q 2 = 5/(2 5π) α 2 3 ρ R 5 α 3 : Q 2 = 5/(4 5π) α 3+ α 3 ρ R 5 α 32 : Q 2 = α 33 : Q 2 = 25/(2 5π) α 3+3 α 3 3 ρ R 5 Conclusion: Among λ = 3 defs. only α 32 leads to Q 2!!!

94 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Multipole Moments as Functionals of the Density For small deformations we use Taylor expansion: α= Q λµ (α) Q λµ + Q λµ α + α= 2 Q λµ α α α= We set λ = 2, µ = and λ = λ 2 = 3 and obtain α 3 : Q 2 = 5/(2 5π) α 2 3 ρ R 5 α 3 : Q 2 = 5/(4 5π) α 3+ α 3 ρ R 5 α 32 : Q 2 = α 33 : Q 2 = 25/(2 5π) α 3+3 α 3 3 ρ R 5 Conclusion: Among λ = 3 defs. only α 32 leads to Q 2!!!

95 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry We Proceed to Suggest a Physical Interpretation When looking for experimental signs of tetrahedral symmetry assume that Q 2 -moments of tetra-configurations nearly vanish Strictly speaking: Exact tetrahedral and octahedral symmetries cause vanishing of quadrupole and dipole moments: the only permitted transitions are octupole - an extreme idea! We know that at reduced transition probabilities B(E) and B(E3) comparable, the E-decay is 2 more probable!! Therefore it will be neccessary to look into problems of partial symmetry breaking in the case of the two high-rank symmetries We begin with the mechanism of the zero-point quadrupole motion about zero-quadrupole deformation following discussion

96 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry We Proceed to Suggest a Physical Interpretation When looking for experimental signs of tetrahedral symmetry assume that Q 2 -moments of tetra-configurations nearly vanish Strictly speaking: Exact tetrahedral and octahedral symmetries cause vanishing of quadrupole and dipole moments: the only permitted transitions are octupole - an extreme idea! We know that at reduced transition probabilities B(E) and B(E3) comparable, the E-decay is 2 more probable!! Therefore it will be neccessary to look into problems of partial symmetry breaking in the case of the two high-rank symmetries We begin with the mechanism of the zero-point quadrupole motion about zero-quadrupole deformation following discussion

97 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry We Proceed to Suggest a Physical Interpretation When looking for experimental signs of tetrahedral symmetry assume that Q 2 -moments of tetra-configurations nearly vanish Strictly speaking: Exact tetrahedral and octahedral symmetries cause vanishing of quadrupole and dipole moments: the only permitted transitions are octupole - an extreme idea! We know that at reduced transition probabilities B(E) and B(E3) comparable, the E-decay is 2 more probable!! Therefore it will be neccessary to look into problems of partial symmetry breaking in the case of the two high-rank symmetries We begin with the mechanism of the zero-point quadrupole motion about zero-quadrupole deformation following discussion

98 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry We Proceed to Suggest a Physical Interpretation When looking for experimental signs of tetrahedral symmetry assume that Q 2 -moments of tetra-configurations nearly vanish Strictly speaking: Exact tetrahedral and octahedral symmetries cause vanishing of quadrupole and dipole moments: the only permitted transitions are octupole - an extreme idea! We know that at reduced transition probabilities B(E) and B(E3) comparable, the E-decay is 2 more probable!! Therefore it will be neccessary to look into problems of partial symmetry breaking in the case of the two high-rank symmetries We begin with the mechanism of the zero-point quadrupole motion about zero-quadrupole deformation following discussion

99 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry We Proceed to Suggest a Physical Interpretation When looking for experimental signs of tetrahedral symmetry assume that Q 2 -moments of tetra-configurations nearly vanish Strictly speaking: Exact tetrahedral and octahedral symmetries cause vanishing of quadrupole and dipole moments: the only permitted transitions are octupole - an extreme idea! We know that at reduced transition probabilities B(E) and B(E3) comparable, the E-decay is 2 more probable!! Therefore it will be neccessary to look into problems of partial symmetry breaking in the case of the two high-rank symmetries We begin with the mechanism of the zero-point quadrupole motion about zero-quadrupole deformation following discussion

100 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Extreme-Symmetry Limit: Q 2 = and Q = Dipole Moment non zero Dipole Moment vanishes Transitions always present The only transitions are the octupole ones Pear shape Tetrahedral

101 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Quadrupole Polarisation through Zero-Point Motion Denoting quadrupole deformations α and classical energy E we have: E(α, α) = 2 B α C α 2 Ĥ = 2B The normalised wave functions are: ϕ n (α) = 2 α C α 2, 2π2 n n! A 2 e α /2σ 2 H n (α/σ), σ df =. 2A, where A df. = [ ϕ n= α 2 ϕ n= ] /2 = [ 2 /(4BC)] /4 We find an effective quadrupole deformation different from zero Most probable deformation: α α ϕn 2 (α) dα

102 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Quadrupole Polarisation through Zero-Point Motion Denoting quadrupole deformations α and classical energy E we have: E(α, α) = 2 B α C α 2 Ĥ = 2B The normalised wave functions are: ϕ n (α) = 2 α C α 2, 2π2 n n! A 2 e α /2σ 2 H n (α/σ), σ df =. 2A, where A df. = [ ϕ n= α 2 ϕ n= ] /2 = [ 2 /(4BC)] /4 We find an effective quadrupole deformation different from zero Most probable deformation: α α ϕn 2 (α) dα

103 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Quadrupole Polarisation through Zero-Point Motion Denoting quadrupole deformations α and classical energy E we have: E(α, α) = 2 B α C α 2 Ĥ = 2B The normalised wave functions are: ϕ n (α) = 2 α C α 2, 2π2 n n! A 2 e α /2σ 2 H n (α/σ), σ df =. 2A, where A df. = [ ϕ n= α 2 ϕ n= ] /2 = [ 2 /(4BC)] /4 We find an effective quadrupole deformation different from zero Most probable deformation: α α ϕn 2 (α) dα

104 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry We Have the Smoking-Gun Signatures Almost There Valence particles cause a certain quadrupole polarisation Additional polarisation caused by Coriolis spin alignments Total Spin Spin-alignment will cause additional quadrupole polarisation

105 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry We Have the Smoking-Gun Signatures Almost There... According to a simplified way of thinking, when all deformations tend to zero (α λµ ) then Q 2 and Q and we are confronted with an ill-defined mathematical problem B(E2) lim α B(E) =??? (undefined symbol )!!! However, because of the residual polarisations in terms of quadrupole deformation and of induced dipole moments at the band-heads we have B(E2) lim α B(E) = Bres(E2) B B res(e)

106 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry We Have the Smoking-Gun Signatures Almost There... According to a simplified way of thinking, when all deformations tend to zero (α λµ ) then Q 2 and Q and we are confronted with an ill-defined mathematical problem B(E2) lim α B(E) =??? (undefined symbol )!!! However, because of the residual polarisations in terms of quadrupole deformation and of induced dipole moments at the band-heads we have B(E2) lim α B(E) = Bres(E2) B B res(e)

107 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry We Have the Smoking-Gun Signatures Almost There... Schematic: Predictions for B(E2)/B(E) B(E2)/B(E) Tetrahedral Band Octupole Band Angular Momentum In other words, we expect a spin dependence: B(E2) B(E) B + B I Conclusion: Tetrahedral symmetry must always be accompanied by static or dynamic quadrupole deformations at α 2 and α 22

108 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry We Have the Smoking-Gun Signatures Almost There... Schematic: Predictions for B(E2)/B(E) B(E2)/B(E) Tetrahedral Band Octupole Band Angular Momentum In other words, we expect a spin dependence: B(E2) B(E) B + B I Conclusion: Tetrahedral symmetry must always be accompanied by static or dynamic quadrupole deformations at α 2 and α 22

109 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Possible El-Magnetic Signs of Tetrahedral Symmetry Table: Experimental ratios B(E2) in /B(E) out 6 Spin 52 Gd 56 Gd 54 Dy 6 Er 64 Er 62 Yb 64 Yb Above: Branching ratios related to the negative parity bands are interpreted as tetrahedral, interband transitions to g.s.band

110 Focus on the Nuclear Tetrahedral Symmetry Tetrahedral Magic Numbers and Nuclear Binding Expected Experimental Signs of Tetrahedral Symmetry Possible El-Magnetic Signs of Tetrahedral Symmetry Table: Experimental ratios B(E2) in /B(E) out 6 Spin 52 Gd 56 Gd 54 Dy 6 Er 64 Er 62 Yb 64 Yb Above: Branching ratios related to the negative parity bands are interpreted as tetrahedral, interband transitions to g.s.band

111 New Theory: Verification, Relation to Experiment Part III New Theory of Nuclear Stability: Its Realisation, Tests and Experimental Background

112 New Theory: Verification, Relation to Experiment Constructing a Series of Theoretical Predictions Experimental 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 T d symmetry is Y 3±2 negative parity bands At the exact symmetry limit Q 2 moments must vanish! Therefore: There must exist negative-parity bands without E2 transitions!!! We suggest looking for the collective negative parity bands without rotational (E2) transitions. The question Where?

113 New Theory: Verification, Relation to Experiment Constructing a Series of Theoretical Predictions Experimental 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 T d symmetry is Y 3±2 negative parity bands At the exact symmetry limit Q 2 moments must vanish! Therefore: There must exist negative-parity bands without E2 transitions!!! We suggest looking for the collective negative parity bands without rotational (E2) transitions. The question Where?

114 New Theory: Verification, Relation to Experiment Constructing a Series of Theoretical Predictions Experimental 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 T d symmetry is Y 3±2 negative parity bands At the exact symmetry limit Q 2 moments must vanish! Therefore: There must exist negative-parity bands without E2 transitions!!! We suggest looking for the collective negative parity bands without rotational (E2) transitions. The question Where?

115 New Theory: Verification, Relation to Experiment Constructing a Series of Theoretical Predictions Experimental 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 T d symmetry is Y 3±2 negative parity bands At the exact symmetry limit Q 2 moments must vanish! Therefore: There must exist negative-parity bands without E2 transitions!!! We suggest looking for the collective negative parity bands without rotational (E2) transitions. The question Where?

116 New Theory: Verification, Relation to Experiment Constructing a Series of Theoretical Predictions Experimental Facts: Comparison with Experiment Where to Look for Experimental Evidence: Summary The Strongest Tetrahedral Islands Predicted by Theory Proton Number N=4 N=56 N=7 N ~ > 9 N~2 N ~ > 36 Z=4 Z=56 Z=7 Z=64 Z ~> 9 In the Actinide region most of the so called octupole bands have never seen their E2 transitions in experiment [detailed discussion] Neutron Number 6

117 New Theory: Verification, Relation to Experiment Constructing a Series of Theoretical Predictions Experimental Facts: Comparison with Experiment Where to Look for Experimental Evidence: Summary The Strongest Tetrahedral Islands Predicted by Theory Proton Number N=4 N=56 N=7 N ~ > 9 N~2 N ~ > 36 Z=4 Z=56 Z=7 Z=64 In the Rare Earth Region the Sm, Gd and Dy nuclei [Z=62,64,66] manifest negative parity bands without E2 transitions [see details] Z ~> Neutron Number 6

118 New Theory: Verification, Relation to Experiment Constructing a Series of Theoretical Predictions Experimental Facts: Comparison with Experiment Where to Look for Experimental Evidence: Summary The Strongest Tetrahedral Islands Predicted by Theory Proton Number N=4 N=56 N=7 N ~ > 9 N~2 N ~ > 36 Z=4 Z=56 Z=7 Z=64 Z ~> 9 In the Zirconium region several nuclei manifest largest ever octupole transitions [B(E3)~(2 6)W.u.] Neutron Number 6

119 New Theory: Verification, Relation to Experiment Constructing a Series of Theoretical Predictions Experimental Facts: Comparison with Experiment Example: This Is Not Our Effect [Pear-Shape] Deformation α E(fyu)+Shell[e]+Correlation[PNP] Th Deformation α E [MeV] UNIVERS_COMPACT (D=4, 23) Gp=.96 Gn=.98 N p=35 Nn=45 Emin=-7.58, Eo= Th

120 New Theory: Verification, Relation to Experiment Constructing a Series of Theoretical Predictions Experimental Facts: Comparison with Experiment Example: This Is Not Our Effect [Pear-Shape] Deformation α Th 32 E(fyu)+Shell[e]+Correlation[PNP] Deformation α E [MeV] UNIVERS_COMPACT (D=4, 23) Gp=.96 Gn=.98 N p=35 Nn=45 Emin=-7.58, Eo=-3.7 B(E2) in /B(E) out 6 e Fm Spin 222 Th Experiment

121 New Theory: Verification, Relation to Experiment Constructing a Series of Theoretical Predictions Experimental Facts: Comparison with Experiment Possible El-Magnetic Signs of Tetrahedral Symmetry Table: Experimental ratios B(E2) in /B(E) out 6 Spin 52 Gd 56 Gd 54 Dy 6 Er 64 Er 62 Yb 64 Yb 222 Th Conclusion: Tetrahedral bands in Rare Earth nuclei behave very differently as compared e.g. to classical octupole 222 Th band!

122 New Theory: Verification, Relation to Experiment Constructing a Series of Theoretical Predictions Experimental Facts: Comparison with Experiment Possible El-Magnetic Signs of Tetrahedral Symmetry Table: Experimental ratios B(E2) in /B(E) out 6 Spin 52 Gd 56 Gd 54 Dy 6 Er 64 Er 62 Yb 64 Yb 222 Th Conclusion: Tetrahedral bands in Rare Earth nuclei behave very differently as compared e.g. to classical octupole 222 Th band!

123 New Theory: Verification, Relation to Experiment Constructing a Series of Theoretical Predictions Experimental Facts: Comparison with Experiment Partial Decay of 56 Gd and Vanishing Q 2 -Moments From I π = 9 down - no E2-transitions are observed despite tries Process Refs Last Exp. 56Eu: β-decay Tb: ec-decay Nd: ( 3 C,A-3n-γ) Sm: (A,2n-γ) 2 54Gd: (t,p) Gd: (n,γ) Gd: (d,p) Gd: (γ,γ ),(e,e ) Gd: (µ,γ) 97 56Gd: (n,n γ) Gd: (p,p ),(d,d ) Gd: (p,d),(3he,a) Gd: (d,t) Gd: (p,t) Coulomb excitation g. s. band 394 (7 - ) According to C. W. Reich, Nucl. Data Sheets (23) a few dozens among those refs have been used to deduce the level scheme on the right n. p. band (5 - ) Gd

124 New Theory: Verification, Relation to Experiment Constructing a Series of Theoretical Predictions Experimental Facts: Comparison with Experiment Tetrahedral/Octahedral Shapes Have No Q 2 -Moments At the exact tetrahedral symmetry the quadrupole moments vanish 394 (7 - ) 335 (5 - ) n. p. band Gd Equilibrium shape t =.5...but, E2-intensities are expected to grow with spin (Coriolis polarisation) g. s. band

125 New Theory: Verification, Relation to Experiment Constructing a Series of Theoretical Predictions Experimental Facts: Comparison with Experiment Comparison Theory - Experiment: Alignments Comparison between three typical hypotheses:. Tetrahedral and Octahedral (defs. from microscopic calculations); 2. Tetrahedral and Octahedral + Zero-Point Motion (α pol. 2 =.7); 3. Prolate ground-state deformation with α 2.25 Nuclear Spin[ ] Gd α 4 α 32 α Experiment ω [MeV]

126 New Theory: Verification, Relation to Experiment Constructing a Series of Theoretical Predictions Experimental Facts: Comparison with Experiment Preliminary Results from our Jyvaskyla Experiment Preliminary 56 Gd level scheme from our experiment g.s. band o.n.p band e.n.p band. Our first-time evidence for the non-stretched E2-transitions from the rotational even-spin negative-parity Q.T. Doan et al., Search bandforto Fingerprints the tetrahedral-suspect of Tetrahedral... / 4 band. 2. These transitions support the interpretation of tetrahedral sequence as a real rotational band, fed and depopulated through I = transitions!

127 New Theory: Verification, Relation to Experiment Constructing a Series of Theoretical Predictions Experimental Facts: Comparison with Experiment Preliminary Results from our Jyvaskyla Experiment Preliminary 56 Gd level scheme from our experiment g.s. band o.n.p band e.n.p band. Our first-time evidence for the non-stretched E2-transitions from the rotational even-spin negative-parity Q.T. Doan et al., Search bandforto Fingerprints the tetrahedral-suspect of Tetrahedral... / 4 band. 2. These transitions support the interpretation of tetrahedral sequence as a real rotational band, fed and depopulated through I = transitions!