first evidence for the presence of nuclear octahedral and

First Evidence∗) for the Presence of NuclearOctahedral and Tetrahedral Symmetries

Jerzy DUDEK

UdS and IPHC, France and UMCS, Poland

∗)Or – The newest news: Recent discoveries – Theory and Experiment

Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries

First Evidence∗) for the Presence of NuclearOctahedral and Tetrahedral Symmetries

Jerzy DUDEK

UdS and IPHC, France and UMCS, Poland

∗)Or – The newest news: Recent discoveries – Theory and Experiment

Dear Jacek: Congratulations!

It was loooong ago...

we met in Warsaw...

became colleagues and friends...

... hmm...

Do you remember?

we met in Warsaw...

... hmm...

Do you remember?

we met in Warsaw...

... hmm...

Do you remember?

we met in Warsaw...

... hmm...

Do you remember?

we met in Warsaw...

... hmm...

Do you remember?

Somewhat later...

we started travelling a lot...

Warsaw began her new era...

Somewhat later...

But since we refresh memories...

... Recall: More than 20 years back ...

you came to Strasbourg...

as Invited Professor of our University

Indeed, one sunny day in Strasbourg...

... looking deeply into the screens of ourHewlett-Packard work stations ...

We said: We have all the FORTRANinfrastructure in the existing WSODD

Let the HFODD come !

And the HFODD came !

In the service of Louis Pasteur, Strasbourg 1

Motto:

Symmetries Are the FactorsDetermining

Stability of Atomic Nuclei

Or – More provocatively:

Do We Owe Our Existenceto the Geometrical Symmetries

on the Sub-atomic Level?

[The Issue of Stability of an Atomic Nucleus]

Before really starting – a short preamble:

In 2017 we will celebrate the 15th anniversaryof the Tetranuc Project

TetraNuc Project opened the way for the studies of theso-called high-rank symmetries in nuclei

– one of the central subjects of this presentation

TetraNuc was contributed by over 110 physicistsfrom over 35 institutions

We begin with a few historical messages...

VOLUME 88, No. 25 P H Y S I C A L R E V I E W L E T T E R S 22 June 2002

Nuclear Tetrahedral Symmetry: Possibly Presentthroughout the Periodic Table

J. Dudek, A. Gozdz, N. Schunck, and M. Miskiewicz

More than half a century after the fundamental, spherical shell structurein nuclei had been established, theoretical predictions indicated that theshell gaps comparable or even stronger than those at spherical shapes mayexist. Group-theoretical analysis supported by realistic mean-field calcula-tions indicate that the corresponding nuclei are characterized by the TD

d(“double-tetrahedral”) symmetry group. Strong shell-gap structure is en-hanced by the existence of its four-dimensional irreducible representations;it can be seen as a geometrical effect that does not depend on a particularrealization of the mean field. Possibilities of discovering the TD

d symmetryin experiment are discussed.

[Follow-up of an earlier pilot projectX. Li and J. Dudek, Phys. Rev. C 94, R1250 (1994).]

VOLUME 88, No. 25 P H Y S I C A L R E V I E W L E T T E R S 22 June 2002

Nuclear Tetrahedral Symmetry: Possibly Presentthroughout the Periodic Table

J. Dudek, A. Gozdz, N. Schunck, and M. Miskiewicz

More than half a century after the fundamental, spherical shell structurein nuclei had been established, theoretical predictions indicated that theshell gaps comparable or even stronger than those at spherical shapes mayexist. Group-theoretical analysis supported by realistic mean-field calcula-tions indicate that the corresponding nuclei are characterized by the TD

d(“double-tetrahedral”) symmetry group. Strong shell-gap structure is en-hanced by the existence of its four-dimensional irreducible representations;it can be seen as a geometrical effect that does not depend on a particularrealization of the mean field. Possibilities of discovering the TD

d symmetryin experiment are discussed.

[Follow-up of an earlier pilot projectX. Li and J. Dudek, Phys. Rev. C 94, R1250 (1994).]

Science Popularisation Articles in the USA [2002]

“Nuclear Pyramids”

SEARCH AIP

Number 593 #1, June 10, 2002 by Phil Schewe, James Riordon, and Ben SteinNuclear Pyramids

Physicists normally think of atomic nuclei as being something like a dropletwith a roughly spherical shape. But if atoms can assemble into tiny pyramidstructures (such as the ammonia molecule, NH4), why not nuclei? It alldepends on how the nuclear forces act in a nucleus.

A group of physicists from the Universite Louis Pasteur (Strasbourg, France)and the Marie Curie University (Lublin, Poland) have, for the first time, triedto imagine how stable nuclei could form with pyramid, or even cubic oroctahedral shapes.

In chemistry many configurations are possible because the interactions (e.g.,Van der Waals, covalent, or hydrogen bonding) can extend over considerabledistances.

The nuclear force, by contrast, is attenuated, and acts not much further thanthe size of nucleons (the protons and neutrons making up the nucleus). Anexcited pyramidal nucleus would turn in space, every now and then throwingout a high-energy photon (gamma ray). This would make for a characteristicspectrum, but one which would most likely require a gamma detectionsensitivity only now being planned for experiments in the US and Europe.

Jerzy Dudek ([email protected], 33-388-10-6498) and his colleagueshave worked out the "magic numbers" for those elements and isotopes mostlikely to be sustainable in tetrahedral form, nuclei with certain numbers ofprotons (e.g., 20, 32, 40, 55/58, 70) and neutrons.

For example, barium-126 (56 protons, 70 neutrons) and barium-146 (56protons, 90 neutrons) have promise, whereas Ba-114 or Ba-168 do not.(Dudek et al. (http://link.aps.org/abstract/PRL/v88/e252502) , PhysicalReview Letters, 24 June 2002.)

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1 sur 1 9/13/13 12:47 PM

Science Popularisation “New Scientist” [2002]

Science Popularisation “Gazeta Wyborcza” [2002]

These are just the first 2 paragraphs of a ∼3 pages article...

In these lines: ”... – because as it is well known – a round body is rolling in totallydifferent direction as compared to the one with corners...”

VOLUME 97, 072501 (2006) P H Y S I C A L R E V I E W L E T T E R S 18 August 2006

Island of Rare Earth Nuclei with Tetrahedral and OctahedralSymmetries: Possible Experimental Evidence

J. Dudek, D. Curien, N. Dubray, J. Dobaczewski, V. Pangon, P. Olbratowski, and N. Schunck

Calculations using realistic mean-field methods suggest the existence of nu-clear shapes with tetrahedral Td and/or octahedral Oh symmetries some-times at only a few hundreds of keV above the ground states in some rareearth nuclei around 156Gd and 160Yb. The underlying single-particle spec-tra manifest exotic fourfold rather than Kramers’s twofold degeneracies.The associated shell gaps are very strong, leading to a new form of shapecoexistence in many rare earth nuclei. We present possible experimen-tal evidence of the new symmetries based on the published experimentalresults–although an unambiguous confirmation will require dedicated ex-periments.

Science Popularisation Articles in the USA [2006]

“Smallest Pyramids in the Universe”

SEARCH AIP

Number 789 #1, August 22, 2006 by Phil Schewe, Ben Stein, and Davide CastelvecchiSmallest Pyramids in the Universe

French physicists believe they can solve the mystery behind dozens ofnuclear experiments conducted years ago. The experiments, conducted witha variety of detectors, energies, and colliding nuclear species, left puzzlingresults, so puzzling and hard to interpret that many of the experimentersturned their attention to the study of highly spinning nuclei, a quitefashionable subject at the time.

Now, Jerzy Dudek of the Université Louis Pasteur in Strasbourg, France, andhis colleagues at Warsaw University and the Universidad Autonoma de Madridclaim that the old results can be explained by arguing that some nuclei, madein the tempestuous conditions of a sufficiently high-energy collision, can existin the form of a tetrahedron or a octahedron.

Like a pyramid-shaped methane (CH4) molecule held together by theelectromagnetic force, a pyramidal nucleus would consist of protons andneutrons held together by the strong nuclear force. Such a nuclear molecule-- in effect the smallest pyramid in the universe -- would be only a fewfemtometers (10-15 meter) on a side and millions of times smaller in volumethan methane molecules.

Just as there are so-called "magic" nuclei with just the right number ofneutrons and protons that readily form stable spherical nuclei, so there areexpected to be such magic numbers for forming pyramid nuclei too. Stable,in this case, means that the state persists for 1012 to 1014 times longer thanthe typical timescale for nuclear reactions, namely 10-21 seconds.

Dudek (contact [email protected], +33-3-88-10-64-98) says thatgadolinium-156 and ytterbium-160 are nuclei very conducive to residing in astable pyramid configuration. Nuclei might exist also in stable octahedral(diamond) forms also. These nuclei would all possess a quantum property notseen before in nuclei: in the process of filling out an energy-level diagram forthe nucleus, four nucleons of the same kind (neutrons or protons) couldshare a single energy level instead of the customary one or two permittednucleons.

This rule-of-four would inhibit the normally observed decay patterns by whichnon-spherical nuclei throw off energy, usually by emitting gamma rays. Infact, in the case of nuclear pyramids it is expected to result in new andunprecedented decay rules. This inhibition would explain the puzzling resultsof earlier experiments.

Dudek and his colleagues plan to test these ideas in upcoming experiments.

Dudek et al. (http://link.aps.org/abstract/PRL/v97/e072501) , PhysicalReview Letters, 18 August 2006Contact Jerzy DudekUniversité Louis [email protected]: +33-3-88-10-64-98

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Point-Group Symmetries

Symmetry-Induced Single-Particle Gaps

Implied Deformed Closed Shells

Reminder: Global Stability vs. Gaps in S.P. Spectra

Qualitatively:

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

Deformation Parameter

Deformation Deformation

Nucleus ’1’ Nucleus ’2’

Qualitatively:

Symmetries, Gaps, Valence Particles

Observe: Particle-hole excitationsmay lower the energy of the systemat increasing deformations thusstrengthening the symmetry effectin the presence of a number of thevalence particles

Importance of P-H configurations

excitations

s Deformation

Deformation

p!h exc. config.

g.s. config.

Energy

Symmetries, Gaps, Valence Particles

Observe: Particle-hole excitationsmay lower the energy of the systemat increasing deformations thusstrengthening the symmetry effectin the presence of a number of thevalence particles

Importance of P-H configurations

excitations

s Deformation

Deformation

p!h exc. config.

g.s. config.

Energy

Gaps, Deformed Shells, Point-Groups and Stability

Posing the problem of global theory of stability:

Construction of Symmetry-Oriented Theory of Nuclear Stabilitywill be equivalent to employing group theory in constructing

a systematic method of looking for big Mean-Field Gaps

Empirical introduction of symmetry groups:

1. Suppose that the Mean-Field potential parameters are fixed

2. We expect that the mean-field calculations will give bigger gapsat shapes with certain symmetries and smaller otherwise

3. The Strategy: Profit from symmetries to maximise the gaps

Symmetries, Group-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, . . . RrLet their dimensions be d1, d2, . . . dr, respectively

Then the eigenvalues εν of the problem

Hψν = ενψν

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

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

Hψν = ενψν

[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

Hψν = ενψν

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

Hψν = ενψν

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

Hψν = ενψν

Symmetries and Gaps in Nuclear Context

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

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

Qualitatively: Average level spacings increase, within an irrep, by a“factor of ∼6”. The full spectrum may present unusually big gaps.

Symmetries and Gaps in Nuclear Context

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

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

Qualitatively: Average level spacings increase, within an irrep, by a“factor of ∼6”. The full spectrum may present unusually big gaps.

Physically Meaningful Point-Groups Contained in SO3

32 Point Groups: Subgroups

C6h 6v

C C C C

i 2 s 3

Figure:Cubic group structure

Dashed lines indicate that the sub-group marked is not invariant

In nuclear structure physics the point-groups used so far, mainly implicitly,are D2 and D2h [‘triaxial nuclei’]. Noother discrete subgroups of SO3 havebeen explicitly used in the past.

The diagram shows possible candidatepoint-groups: How to profit from thisinformation?

C6h 6v

C C C C

i 2 s 3

C6h 6v

C C C C

i 2 s 3

No. Group No. Irr. Dimensions

01. ODh 6 4 x 2D and 2 x 4D

02. OD 3 2 x 2D and 1 x 4D

03. TDd 3 2 x 2D and 1 x 4D

04. CD6h 12→ 6 12 x 1D

05. DD6h 6 6 x 2D

06. TDh 6 6 x 2D

07. DD4h 4 4 x 2D

. . . . . . . . . . . .

17. DD2h 2 2 x 2D (‘reference’)

Point-groups and their Irreducible Representations

Conclusions at This Point

Among the molecular 32 point-groups, subgroups of SO3,there are 16 (!) which satisfy more favourably

the schematic big-gap criteria than the reference D2h group(‘usual’ tri-axial nuclei)

Mathematical Implications:

1. To increase chances of finding mean-field big gaps focus on pointgroups with high-dimension irreps or with many irreps

2. There are only 2 structurally non-equivalent symmetries that givedegeneracies larger than 2: Octahedral OD

h and tetrahedral TDd

3. From now on octahedral and tetrahedral symmetries are the firstcandidates on the list - we call them high-rank symmetries

What Do We Need at This Stage?

• Learn constructing the Mean-Field Hamiltonians invariant underpre-selected point-group → examine new quantum [shell] effects

• Formulate the theoretical predictions [identification criteria]

• Look for the experimental evidence in agreement with the criteria

• Prepare systematic comparison between theory and experiments

Mean-Field Hamiltonians Invariant under Group G

Given group G = O1, O2, . . . Of . How to construct arealistic Hamiltonian invariant under all transformations in G?

Start with Woods-Saxon Hamiltonian; the HF mean-field next

First step: Construct auxiliary invariant surfaces starting with

Σ : R(ϑ, ϕ) = R0c(α)[1 +

λmax∑λ=2

λ∑µ=−λ

α?λµYλµ(ϑ, ϕ)]

The condition of invariance:

ΣO→ Σ ′ ≡ Σ ∀ O

The latter can be written down as

∑λmax

∑λµ=−λ α

∗λµ [OYλµ(ϑ, ϕ)] =

∑λmax

∑λµ=−λ α

∗λµYλµ(ϑ, ϕ)

Σ : R(ϑ, ϕ) = R0c(α)[1 +

λmax∑λ=2

λ∑µ=−λ

ΣO→ Σ ′ ≡ Σ ∀ O

∑λmax

∑λµ=−λ α

∑λmax

∑λµ=−λ α

Σ : R(ϑ, ϕ) = R0c(α)[1 +

λmax∑λ=2

λ∑µ=−λ

ΣO→ Σ ′ ≡ Σ ∀ O

∑λmax

∑λµ=−λ α

∑λmax

∑λµ=−λ α

Σ : R(ϑ, ϕ) = R0c(α)[1 +

λmax∑λ=2

λ∑µ=−λ

ΣO→ Σ ′ ≡ Σ ∀ O

∑λmax

∑λµ=−λ α

∑λmax

∑λµ=−λ α

Σ : R(ϑ, ϕ) = R0c(α)[1 +

λmax∑λ=2

λ∑µ=−λ

ΣO→ Σ ′ ≡ Σ ∀ O

∑λmax

∑λµ=−λ α

∑λmax

∑λµ=−λ α

Invariant Mean-Field Hamiltonian [2]

In what follows we will need a representation of the operatorsO ∈ G adapted to the action on spherical harmonics Yλµ(ϑϕ)

The action of proper rotations can be written down as

O → R(Ω) ≡ exp(iαz + iβy + iγz ′)

Using this notation the invariance condition takes the form

λmax∑λ=2

λ∑µ=−λ

α∗λµ [OYλµ(ϑ, ϕ)] =

λmax∑λ=2

λ∑µ=−λ

α∗λµ

λ∑µ′=−λ

Dλµ′µ(Ω)Yλµ′(ϑ, ϕ)

The latter can be written down ∀ ϑ, ϕ as

∑λµ′=−λ

∑λmax

[∑λµ=−λ α

∗λµD

λµ′µ(Ω)− α∗λµ′

]Yλµ(ϑ, ϕ) = 0

λmax∑λ=2

λ∑µ=−λ

λmax∑λ=2

λ∑µ=−λ

α∗λµ

λ∑µ′=−λ

∑λµ′=−λ

∑λmax

[∑λµ=−λ α

∗λµD

]Yλµ(ϑ, ϕ) = 0

λmax∑λ=2

λ∑µ=−λ

λmax∑λ=2

λ∑µ=−λ

α∗λµ

λ∑µ′=−λ

∑λµ′=−λ

∑λmax

[∑λµ=−λ α

∗λµD

]Yλµ(ϑ, ϕ) = 0

λmax∑λ=2

λ∑µ=−λ

λmax∑λ=2

λ∑µ=−λ

α∗λµ

λ∑µ′=−λ

∑λµ′=−λ

∑λmax

[∑λµ=−λ α

∗λµD

]Yλµ(ϑ, ϕ) = 0

Spherical harmonics are linearly independent → it then follows

∑λµ=−λ

[Dλµ′µ(Ωk)− δµµ′

]α?λµ = 0; k = 1, 2, . . . f .

Above Ωk are fixed sets of Euler angles corresponding to Ok ;for instance for a four-fold Oz -axis→ Ω = π/2, 0, 0 etc. etc.

Solutions can be taken as eigenvectors of the (2λ+1)×(2λ+1)matrix Dλ

µ′µ(Ωk) with the eigenvalue equal +1, cf. ref. [A]→

The above system of equations is uniform → multiplying thecorresponding solutions αλµ by a constant gives again a solution

This allows to select, e.g. αλµ=0 as an independent parameter,which uniquely fixes all the other non-zero components

[A] J. Dudek, J. Dobaczewski, N. Dubray, A. Gozdz, V. Pangon and N. Schunck;Int. J. Mod. Phys. E16, No. 2 (2007) 516-532

∑λµ=−λ

]α?λµ = 0; k = 1, 2, . . . f .

∑λµ=−λ

]α?λµ = 0; k = 1, 2, . . . f .

∑λµ=−λ

]α?λµ = 0; k = 1, 2, . . . f .

∑λµ=−λ

]α?λµ = 0; k = 1, 2, . . . f .

Tetrahedral Symmetry and Spherical Harmonics

Only special combinations of spherical harmonics may form a basisfor surfaces with tetrahedral symmetry and only odd-order Ref. [A]:

Three Lowest Order Solutions: Rank ↔ Multipolarity λ

λ = 3 : α3,±2 ≡ t3

λ = 5 : no solution possible

λ = 7 : α7,±2 ≡ t7; α7,±6 ≡ −√

1113· t7

λ = 9 : α9,±2 ≡ t9; α9,±6 ≡ +√

28198· t9

λ = 3 : α3,±2 ≡ t3

λ = 7 : α7,±2 ≡ t7; α7,±6 ≡ −√

1113· t7

λ = 9 : α9,±2 ≡ t9; α9,±6 ≡ +√

28198· t9

λ = 3 : α3,±2 ≡ t3

λ = 7 : α7,±2 ≡ t7; α7,±6 ≡ −√

1113· t7

λ = 9 : α9,±2 ≡ t9; α9,±6 ≡ +√

28198· t9

λ = 3 : α3,±2 ≡ t3

λ = 7 : α7,±2 ≡ t7; α7,±6 ≡ −√

1113· t7

λ = 9 : α9,±2 ≡ t9; α9,±6 ≡ +√

28198· t9

Nuclear Tetrahedral Symmetry: Td -Group

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)︸︷︷︸one parameter 3rd order

[(Y7+2 + Y7−2)−

√1113

(Y7+6 + Y7−6)]︸︷︷︸

one parameter 7th order

+ higher order odd−λ terms

Nuclear Tetrahedral Shapes - Proton Spectra

Double group TDd has two 2-dimensional - and one 4-dimensional

irreducible representations: Three distinct families of nucleon levels

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

Tetrahedral Deformation

α 32(m

-.200,

α 32(m

100100

09[5,1,4] 9/207[4,3,1] 1/209[4,2,2] 3/215[4,0,4] 7/210[4,0,4] 7/213[4,1,3] 5/216[4,0,2] 3/208[4,4,0] 1/209[4,0,0] 1/211[5,0,5] 11/208[5,2,1] 1/216[5,0,5] 11/209[5,2,3] 7/212[5,1,4] 9/204[4,4,0] 1/206[6,3,3] 7/208[5,0,3] 7/210[6,2,4] 9/206[6,1,5] 11/208[5,2,3] 5/220[5,0,5] 9/207[5,0,5] 9/219[5,1,4] 7/207[5,4,1] 3/2

06[3,1,2] 3/210[3,1,2] 5/2

03[3,1,0] 1/204[3,1,0] 1/206[3,0,1] 1/206[4,0,4] 7/206[4,0,4] 7/2

04[5,0,5] 11/205[4,3,1] 1/2

05[6,1,5] 11/2

10[3,1,2] 3/204[5,0,3] 7/203[4,1,3] 7/204[4,1,3] 7/204[3,0,1] 1/205[4,0,0] 1/206[5,4,1] 1/2

11[5,0,5] 11/207[4,2,2] 5/207[4,1,3] 7/205[6,1,5] 9/206[5,0,5] 9/206[4,2,0] 1/204[4,1,1] 3/2

226Th 136 90

Full lines ↔ 4-dimensional irreducible representations - marked with doubleNilsson labels. Observe huge gaps at N=64, 70, 90-94, 100.

Nuclear Tetrahedral Shapes - Neutron Spectra

Double group TDd has two 2-dimensional - and one 4-dimensional

irreducible representations: Three distinct families of nucleon levels

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

Tetrahedral Deformation

α 32(m

-.200,

α 32(m

112124

136 136

04[5,5,0] 1/207[4,1,1] 1/206[5,1,2] 3/213[5,0,3] 5/207[5,5,0] 1/208[5,0,1] 1/209[6,0,6] 13/219[6,0,6] 13/208[6,3,1] 1/204[6,1,5] 11/205[6,1,5] 11/209[6,2,4] 9/208[6,1,5] 11/204[7,3,4] 9/208[6,0,4] 9/206[7,2,5] 11/205[7,2,5] 11/206[6,0,4] 9/207[6,0,6] 11/211[6,0,6] 11/204[6,4,0] 1/207[6,0,2] 5/203[6,6,0] 1/205[6,5,1] 3/213[6,2,4] 7/2

05[4,1,1] 3/2

03[5,0,3] 5/203[5,0,3] 5/2

06[6,0,6] 13/202[3,0,1] 3/202[4,1,3] 5/2

03[5,4,1] 1/204[5,1,2] 5/2

05[7,2,5] 11/204[5,3,0] 1/204[4,1,3] 5/2

04[7,1,6] 13/206[6,2,4] 9/203[5,0,1] 3/203[5,0,1] 3/208[5,1,4] 9/205[5,3,2] 5/204[5,0,5] 9/206[5,2,3] 7/2

10[6,0,6] 13/206[6,5,1] 1/204[5,3,2] 5/203[5,2,1] 3/202[5,2,1] 3/2

04[6,0,6] 11/2

226Th 136 90

Full lines ↔ 4-dimensional irreducible representations - marked with doubleNilsson labels. Observe huge gaps at N=112, 136.

First Goal: Obtain Tetrahedral Magic Numbers

• After inspecting many single-particle diagrams as functions oftetrahedral deformation we read-out all magic numbers (Zt ,Nt)

• Tetrahedral symmetric (likely) shape-coexisting configurationsare predicted to appear around the tetrahedral magic closures:

Zt ,Nt = 16, 20, 32, 40, 56, 64, 70, 90, 136

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

3216S16, 40

20S20, 6432Ge32, 72

32Ge40, 8832Ge56, 80

40Zr40, 9640Zr56,

11040Zr70, 126

56Ba70, 14656Ba90, 134

64Gd70, 15464Gd90, 160

70Yb90, 22690Th136

Zt ,Nt = 16, 20, 32, 40, 56, 64, 70, 90, 136

3216S16, 40

20S20, 6432Ge32, 72

32Ge40, 8832Ge56, 80

40Zr40, 9640Zr56,

11040Zr70, 126

56Ba70, 14656Ba90, 134

64Gd70, 15464Gd90, 160

70Yb90, 22690Th136

Zt ,Nt = 16, 20, 32, 40, 56, 64, 70, 90, 136

3216S16, 40

20S20, 6432Ge32, 72

32Ge40, 8832Ge56, 80

40Zr40, 9640Zr56,

11040Zr70, 126

56Ba70, 14656Ba90, 134

64Gd70, 15464Gd90, 160

70Yb90, 22690Th136

Zt ,Nt = 16, 20, 32, 40, 56, 64, 70, 90, 136

3216S16, 40

20S20, 6432Ge32, 72

32Ge40, 8832Ge56, 80

40Zr40, 9640Zr56,

11040Zr70, 126

56Ba70, 14656Ba90, 134

64Gd70, 15464Gd90, 160

70Yb90, 22690Th136

A Basis for the Octahedral Symmetry

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

λ = 4 : α40 ≡ o4; α4,±4 ≡ ±√

514· o4

λ = 6 : α60 ≡ o6; α6,±4 ≡ −√

72· o6

λ = 8 : α80 ≡ o8; α8,±4 ≡√

28198· o8; α8,±8 ≡

198· o8

λ = 4 : α40 ≡ o4; α4,±4 ≡ ±√

514· o4

λ = 6 : α60 ≡ o6; α6,±4 ≡ −√

72· o6

λ = 8 : α80 ≡ o8; α8,±4 ≡√

28198· o8; α8,±8 ≡

198· o8

λ = 4 : α40 ≡ o4; α4,±4 ≡ ±√

514· o4

λ = 6 : α60 ≡ o6; α6,±4 ≡ −√

72· o6

λ = 8 : α80 ≡ o8; α8,±4 ≡√

28198· o8; α8,±8 ≡

198· o8

λ = 4 : α40 ≡ o4; α4,±4 ≡ ±√

514· o4

λ = 6 : α60 ≡ o6; α6,±4 ≡ −√

72· o6

λ = 8 : α80 ≡ o8; α8,±4 ≡√

28198· o8; α8,±8 ≡

198· o8

λ = 4 : α40 ≡ o4; α4,±4 ≡ ±√

514· o4

λ = 6 : α60 ≡ o6; α6,±4 ≡ −√

72· o6

λ = 8 : α80 ≡ o8; α8,±4 ≡√

28198· o8; α8,±8 ≡

198· o8

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

[Y40 +

14(Y4+4 + Y4−4)

]︸︷︷︸

one parameter 4th order

[Y60 −

(Y6+4 + Y6−4)]

︸︷︷︸one parameter 6th order

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

Octahedral Deformation

α 40(m

-.350,

α 40(m

α 44(m

-.209,

α 44(m

09[4,3,1] 3/213[4,0,4] 9/209[4,2,0] 1/211[4,1,3] 5/212[4,0,2] 5/215[4,3,1] 1/217[5,2,3] 7/208[4,3,1] 1/208[4,2,2] 3/213[4,0,4] 7/223[5,1,4] 9/210[5,3,2] 5/211[5,1,2] 5/207[5,2,3] 5/215[5,1,4] 7/211[4,0,0] 1/218[4,0,2] 3/219[4,4,0] 1/213[5,0,5] 11/210[5,4,1] 3/210[5,0,3] 7/208[5,0,3] 7/210[5,0,5] 11/2

10[4,3,1] 1/224[3,1,2] 3/2

09[4,3,1] 3/216[4,1,3] 7/225[4,2,2] 5/208[5,0,3] 7/213[3,0,1] 3/208[3,3,0] 1/209[4,2,2] 3/210[4,3,1] 1/2

10[5,0,5] 11/221[4,2,0] 1/211[5,4,1] 3/207[5,1,0] 1/210[5,0,5] 9/209[5,1,2] 5/224[4,2,2] 3/217[4,1,3] 7/221[4,1,3] 5/208[5,0,5] 9/218[5,1,4] 9/216[5,3,2] 5/209[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.

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

Octahedral Deformation

α 40(m

-.350,

α 40(m

α 44(m

-.209,

α 44(m

114 116

118126

21[4,0,2] 3/222[4,4,0] 1/212[4,0,0] 1/210[5,2,3] 5/221[5,1,4] 7/213[5,0,5] 11/212[5,0,5] 11/213[5,2,1] 1/207[5,0,5] 11/210[5,3,0] 1/209[5,0,3] 7/216[5,4,1] 1/211[5,0,1] 3/216[6,2,4] 9/207[5,4,1] 3/220[6,1,5] 11/219[6,3,3] 7/210[5,4,1] 3/221[5,0,5] 9/218[6,1,5] 11/208[5,0,5] 9/209[6,4,2] 5/217[6,1,3] 7/2

08[5,4,1] 1/208[5,0,5] 11/223[4,2,2] 3/213[4,1,1] 1/221[4,1,3] 5/2

08[5,0,5] 9/2

16[5,1,4] 9/214[5,3,2] 5/221[5,2,3] 7/208[5,3,2] 5/209[6,0,4] 9/208[4,0,2] 5/207[8,8,0] 1/211[5,3,0] 1/211[5,2,1] 3/212[5,3,2] 3/207[5,1,4] 7/2

17[6,0,6] 13/211[6,5,1] 3/210[6,1,3] 7/206[8,0,2] 5/207[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.

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 ×√

(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

α4,0 ≡ o4, α4,±4 ≡ −√

5/14 o4

Z N ∆E Q32 Q40 Q44 Q40 ×√

(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

α4,0 ≡ o4, α4,±4 ≡ −√

5/14 o4

Z N ∆E Q32 Q40 Q44 Q40 ×√

(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 −0.130763

62 90 −1.168 1.206017 −0.380334 +0.227293 −0.227293

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

Table: Octahedral deformations of the second order compatible with tetra-hedral 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

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

Table: Octahedral deformations of the second order compatible with tetra-hedral 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

During the Following Years: A Long Preparation

[Various Problems of Theory: Formulated and Solved]

1) Constructing a mean-field Hamiltonian of a predefined symmetry

2) Constructing a quantum rotor of an arbitrary predefined symmetry

3) Relating the Hamiltonian-symmetry groups to the nuclear stability

4) The Schrodinger equation in multi-dimensional curvilinear spaces

5) Between laboratory and rotating frames: Symmetrisation Group

6) Microscopic theories with efficient angular-momentum projection

7) Point groups and quantum numbers: the doublex and the triplex

8) Multi-dimensional deformation spaces: isotropy groups and orbits

9) Theory calculations of transition probabilities & branching ratios

1) Constructing a mean-field Hamiltonian of a predefined symmetryProblem solved in:JD, J. Dobaczewski, N. Dubray, A. Gozdz, V. Pangon and N. Schunck;Int. J. Mod. Phys. E16, No. 2 (2007) 516-532

2) Constructing a quantum rotor of an arbitrary predefined symmetryProblem introduced, presented and solved in:JD, A. Gozdz and D. Ros ly; Acta Phys. Polon. B32, (2001) 2625JD, A. Gozdz, D. Curien, V. Pangon, N. Schunck; Acta Phys. Polon. B38, (2006) 1389M. Miskiewicz, A. Gozdz and JD; Int. J. Mod. Phys. E 13 (2004) 127A. Gozdz, M. Miskiewicz and JD; Int. J. Mod. Phys. E 17 (2008) 272

3) Relating the Hamiltonian-symmetry groups to the nuclear stabilityProblem formulated and solved in:X. Li and JD, Phys. Rev. C49, 1246 (1994)JD, A. Gozdz, N. Schunck and M. Miskiewicz, Phys. Rev. Lett. 88, 252502 (2002)JD, K. Mazurek, D. Curien, A. Dobrowolski, A. Gozdz, D. Hartley, A. Maj, L. Riedingerand N. Schunck; Acta Phys. Polon. B 40, 713 (2009)JD, A. Gozdz, D. Curien, V. Pangon and N. Schunck;Acta Phys. Polon. B 38, 1389 (2007)

4) The Schrodinger equation in multi-dimensional curvilinear spacesThe first solution of a simplified problem in:A. Dobrowolski, A. Gozdz, K. Mazurek and J. Dudek;Int. J. Mod. Phys. E 20, (2011) 500Significant progress towards the full solution in: D. Rouvel, Ph-D thesis, Strasbourg

5) Between laboratory and rotating frames: Symmetrisation GroupProblem solved in:A. Gozdz, M. Miskiewicz, JD, and A. Dobrowolski;Int. J. Mod. Phys. E 18, 1028 (2009)A. Gozdz, A. Szurelecka, A. Dobrowolski and JD; Int. J. Mod. Phys. E 20, 199 (2011)

6) Microscopic theories with efficient angular-momentum projectionProblem significantly advanced in:S. Tagami, Y. R. Shimizu; Prog. Theor. Phys. 127, 79 (2012)c.f. also: S. Tagami, Y. R. Shimizu and JD; Phys. Rev. C 87 (2013) 054306

7) Point groups and quantum numbers: the doublex and the triplexProblem solved in:N. Schunck, JD and S. Frauendorf, Acta Phys. Pol. B 36, 1071 (2005)

8) Multi-dimensional deformation spaces: isotropy groups and orbitsProblem solved in:A. Gozdz, M. Miskiewicz, JD and A. Dobrowolski; Int. J. Mod. Phys. E 18, 1028 (2009)A. Gozdz, A. Szurelecka, A. Dobrowolski and JD; Int. J. Mod. Phys. E 20, 199 (2011)

How decisive these ‘new’ degrees of freedommay become?

See illustrations

Symmetry Concepts Impact Our Ideas about Stability

-3.0-3.0

-1.4-1.4

0.20.2

0.2 0.2 0.2

1.83.4

3.4 3.4

5.05.0

-0.10 0.00 0.10 0.20 0.30 0.40

X196X322126 β2cos(γ+30)

β 2si

• Consider a total energy for a super-heavy nucleus in the form ofthe standard (β, γ)-representation

Symmetry Concepts Impact Our Ideas about Stability

-3.0-3.0

-1.4-1.4

0.20.2

0.2 0.2

1.8 1.8

-0.10 0.00 0.10 0.20 0.30 0.40

X196X322126 β2cos(γ+30)

β 2si

• Consider the similar standard (β, γ)-representation but now let usintroduce an extra minimisation over the tetrahedral deformation

These Concepts Change Our Ideas about Stability

-1.4-1.4

0.20.2

0.2 0.2 0.2

1.83.4

3.4 3.4

3.45.0

5.05.0

-0.10 0.00 0.10 0.20 0.30 0.40

X196X322126 β2cos(γ+30)

β 2si

-1.4-1.4

-1.4-1.4-1.4

0.20.2

0.2 0.2

1.8 1.8

-0.10 0.00 0.10 0.20 0.30 0.40

X196X322126 β2cos(γ+30)

β 2si

• The mechanism discussed may provide new challenges for theexotic nuclei projects: Observe a qualitative change of the landscape

• Totally different fission barriers - thus experimental search criteria

• The ground-state expected to be otherwise quadrupole deformedmay obtain e.g. zero-quadrupole and non-zero-tetrahedral geometry

Further Illustrations of the Effectsof the High-Rank Symmetries

How Powerful the Symmetry Notions Are – See Maps

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.25

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

∆Np=

35 ∆

Nd90 Nd150 60

0.00.51.01.52.02.53.03.54.04.55.05.56.06.57.07.58.08.59.09.510.010.511.011.5

E [MeV]

Deformation α20

n α 3

Emin=-3.04, Eo= 0.45

Observe the presence of well defined tetrahedral minima at N=90

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.25

∆Np=

35 ∆

Sm90 Sm152 62

0.00.51.01.52.02.53.03.54.04.55.05.56.06.57.07.58.08.59.09.510.010.511.011.5

E [MeV]

Deformation α20

n α 3

Emin=-2.60, Eo= 1.09

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.25

∆Np=

35 ∆

Gd90 Gd154 64

0.00.51.01.52.02.53.03.54.04.55.05.56.06.57.07.58.08.59.09.510.010.511.011.5

E [MeV]

Deformation α20

n α 3

Emin=-2.18, Eo= 1.49

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.25

∆Np=

35 ∆

Dy90 Dy156 66

0.00.51.01.52.02.53.03.54.04.55.05.56.06.57.07.58.08.59.09.510.010.511.011.5

E [MeV]

Deformation α20

n α 3

Emin=-2.01, Eo= 1.77

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.25

∆Np=

35 ∆

Er90 Er158 68

0.00.51.01.52.02.53.03.54.04.55.05.56.06.57.07.58.08.59.09.510.010.511.011.5

E [MeV]

Deformation α20

n α 3

Emin=-1.85, Eo= 1.86

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

Yb90 Yb160 70

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 _20

Emin=-1.99, Eo= 1.45Observe the presence of well defined tetrahedral minima at N=90

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

∆Np=

35 ∆

Hf90 Hf162 72

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

Emin=-2.28, Eo= 0.84

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

∆Np=

35 ∆

W90 W164 74

0.00.51.01.52.02.53.03.54.04.55.05.56.06.57.07.58.08.59.09.510.010.511.011.5

E [MeV]

Deformation α20

n α 3

Emin=-2.78, Eo= 0.13

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

∆Np=

35 ∆

Os90 Os166 76

0.00.51.01.52.02.53.03.54.04.55.05.56.06.57.07.58.08.59.09.510.010.511.011.5

E [MeV]

Deformation α20

n α 3

Emin=-3.10, Eo=-1.49

Conclusion:

We may look for the experimental evidenceof the tetrahedral symmetry

focussing on those N = 90 isotonesfor which the best experimental data exist

Interplay Between Tetrahedral and OctahedralSymmetries

Realistic Calculations of a Rare Earth Nucleus

Consider 15262Sm90: Phenomenological WS Approach

Tetrahedral Symmetry Effect

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

Sm90 Sm152 62

0.00.51.01.52.02.53.03.54.04.55.05.56.06.57.07.58.08.59.09.510.010.511.011.5

E [MeV]

Deformation _20

Emin=-2.60, Eo= 1.09

Observe the presence of well defined tetrahedral minima at α32 ≈ ±0.12

Combined Octahedral and Tetrahedral Symmetry Effect

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3-0.2

Sm90 Sm152 62

0.00.51.01.52.02.53.03.54.04.55.05.56.06.57.07.58.08.59.09.510.010.511.011.5

E [MeV]

Deformation _32

Emin=-1.41, Eo= 1.09

Allowing for octahedral deformation lowers the tetrahedral minimum by 2 MeV

Combined Octahedral and Tetrahedral Symmetry Effect

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

Sm90 Sm152 62

0.00.51.01.52.02.53.03.54.04.55.05.56.06.57.07.58.08.59.09.510.010.511.011.5

E [MeV]

Deformation _20

Emin=-2.60, Eo= 1.09

An alternative illustration of the octahedral symmetry effect

Consider 15262Sm90: Microscopic Gogny Approach

Gogny calculations for increasing constraint Q32 = 500, 1000, . . . 2500 fm3

corresponding to tetrahedral deformation shown. Self-consistency impliesan increase of the octahedral deformation

t1 = α32 o1 = α40 α44 −α40 ×√

o2 = α60 α64 α60 ×√

0.0505 −0.0446 0.0266 0.0267 0.0021 0.0039 0.00390.1003 −0.0640 0.0383 0.0383 0.0065 0.0122 0.01220.1483 −0.0850 0.0508 0.0508 0.0152 0.0284 0.02840.1930 −0.1117 0.0668 0.0668 0.0288 0.0539 0.05390.2343 −0.1402 0.0838 0.0838 0.0460 0.0861 0.0861

• Recall: Octahedral deformationsof the two lowest orders are:

o1 ≡ α4,0 and α4,±4 = −

14α4,0,

o2 ≡ α6,0 and α6,±4 = +

2α6,0,

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3-0.2

Sm90 Sm152 62

0.00.51.01.52.02.53.03.54.04.55.05.56.06.57.07.58.08.59.09.510.010.511.011.5

E [MeV]

Deformation _32

Emin=-1.41, Eo= 1.09

Notice: t1 ≈ ±0.12 and o1 ≈ −0.08

Semantical Exercises

Conclusion:

We find simultaneous octahedral and tetrahedral shapes

However:

Tetrahedral symmetry is a subgroup of the octahedral one

We may say:

Octahedral symmetry is broken∗) by the tetrahedral one ...

If the effect of the octahedral symmetry breaking is weak→ We should have detectable traces of both... →

What could be those traces?

∗) ... should we say ‘spontaneously broken’ ?

Microscopic Approach in This Project: Gogny HFB

About our Projected Hartree-Fock-Bogolyubov Approach

• Our collaboration uses the totally new program by Shingo TAGAMI

• It solves the Gogny-Hartree-Fock-Bogoliubov (G-HFB) problemwith angular momentum, parity and particle number projections1−3)

• Calculation techniques adapted from previous works, cf. e.g. [4-6]

•Matrix elements calculated with Cartesian harmonic oscillator basis

1) S. Tagami, Y. R. Shimizu and J. Dudek, Prog. Th. Phys. Supp. 196 (2012), 792) S. Tagami, Y. R. Shimizu and J. Dudek, Phys. Rev. C 87 (2013), 0543063) S. Tagami, Y. R. Shimizu and J. Dudek, J. Phys. G 42 (2015), 015106

4) J. Decharge and D. Gogny, Phys. Rev. C 21 (1980), 15685) M. Girod and B. Grammaticos, Phys. Rev. C 27 (1983), 23176) M. Anguiano, J. L. Egido and L. M. Robledo, Nucl. Phys. A 683 (2001), 227

• We consider the standard two-body Hamiltonian, `i = 1, 2, . . . M:

H =∑`1`2

t`1`2 c+`1c`2

∑`1`2

∑`3`4

v`1`2`3`4 c+`1c+`2c`4

• Bogolyubov transformation (c+, c-particles, β+, β-quasi-particles):

β+k ≡

[U`k c

+` + V`k c`

]• Quasiparticle vacuum |Φ〉 and Thouless theorem

|Φ〉 = N eZ |0〉 and Z ≡ 12

∑`′`

Z`′`c+`′ c

+` with N ≡ 〈0|Φ〉,

Z`′` = (VU−1)∗`′`

Microscopic Approach in This Project: Gogny HFB II

• The overlap of an arbitrary operator is calculated using the generalisedWick theorem:

〈Φ|c+`1c+`2c`4

c`3|Φ′〉

〈Φ|Φ′〉= ρ

(c)`3`1

ρ(c)`4`2− ρ(c)

`4`1ρ

(c)`3`2

+ κ(c)`2`1

κ(c)`3`4

• The basic contractions, or the transition density matrix ρ(c) and thetransition pairing tensors, κ(c) and κ(c) with respect to the original particlebasis (c†, c ) are defined by

ρ(c)`′` ≡

〈Φ|c†` c`′ |Φ′〉〈Φ|Φ′〉

=(Z ′[1 + Z †Z ′

]−1Z †)`′`,

κ(c)`′` ≡

〈Φ|c`c`′ |Φ′〉〈Φ|Φ′〉

=(Z ′[1 + Z †Z ′

]−1)`′`,

κ(c)`′` ≡

〈Φ|c†` c†`′ |Φ′〉

〈Φ|Φ′〉=([

1 + Z †Z ′]−1

Z †)`′`.

Microscopic Approach in This Project: Gogny HFB III

• After obtaining the constrained HFB state |Φ〉, we perform the full quan-tum number projection from it to obtain the the projected wave function:

|ΨINZ(±)M;α 〉 =

gINZ(±)K ,α P I

MK P±PN PZ |Φ〉,

• The amplitude gINZ(±)K ,α and the energy eigenvalue E

INZ(±)α are obtained

by the so-called Hill-Wheeler relation∑K ′

HINZ(±)K ,K g

INZ(±)K ′,α = E INZ(±)

∑K ′

N INZ(±)K ,K ′ g

INZ(±)K ′,α ,

• The kernels are defined by HINZ(±)K ,K ′

N INZ(±)K ,K ′

= 〈Φ|

P IKK ′ PN PZ P±|Φ〉.

Approaching Degeneracies with Increasing Deformation

Tetrahedral Symmetry Effect

Observe the sequence characteristically mixing both parities and degeneracies

Iπ = 6±, Iπ = 9±, Iπ = 10±, ...

It will be instructive at this point

to recall what is known about

“these degeneracies”

from the point of view of group representation theory

Elementary Group -Theory Properties

• Let G be the symmetry group of the quantum rotor Hamiltonian

• Let Di , i = 1, 2, . . . M be the irreducible representations of G

• The representation D(Iπ) of the rotor states with the definite spin-

parity Iπ, can be decomposed in terms of Di with multiplicities a(Iπ)i :

D(Iπ) =∑M

i=1 a(Iπ)i Di

• Multiplicities [M. Hamermesh, Group Theory, 1962] are given by:

a(Iπ)i =

∑R∈G

χIπ(R)χi (R) =1

M∑α=1

gαχIπ(Rα)χi (Rα);

NG : order of the group G ; χIπ(R), χi (R): characters of D(Iπ),DiR: group element; gα: the number of elements in the class α, whoserepresentative element is Rα.

Elementary Td-Group -Theory Properties

• Tetrahedral group has 5 irreducible representations and 5 classes

• The representative elements R are: E , C2 (= S24 ), C3, σd , S4

• The characters of irreducible representation of Td are listed below

Td E C3(8) C2(3) σd (2) S4(6)

A1 1 1 1 1 1A2 1 1 1 −1 −1E 2 −1 2 0 0

F1(T1) 3 0 −1 −1 1F2(T2) 3 0 −1 1 −1

• The characters χIπ(Rα) for the rotor representations are as follows:

χIπ(E) = 2I+1, χIπ(Cn) =I∑

K=−I

e2πKn

i , χIπ(σd ) = π×χIπ(C2), χIπ(S4) = π×χIπ(C4)

• From these relations we obtain ‘employing the pocket calculator’:

a(Iπ)i =

M∑α=1

gαχIπ(Rα)χi (Rα) ↔ a(I±)A1

= a(I∓)A2

, a(I+)E = a

(I−)E , a

(I±)F1

= a(I∓)F2

Elementary Td-Group -Theory Properties

• The number of states a(Iπ)i within five irreducible representations.

If a(Iπ)i = 0 → states not allowed; a

(Iπ)i = 2 → doubly degenerate

I+ 0+ 1+ 2+ 3+ 4+ 5+ 6+ 7+ 8+ 9+ 10+

A1 1 0 0 0 1 0 1 0 1 1 1A2 0 0 0 1 0 0 1 1 0 1 1E 0 0 1 0 1 1 1 1 2 1 2

F1(T1) 0 1 0 1 1 2 1 2 2 3 2F2(T2) 0 0 1 1 1 1 2 2 2 2 3

I− 0− 1− 2− 3− 4− 5− 6− 7− 8− 9− 10−

A1 0 0 0 1 0 0 1 1 0 1 1A2 1 0 0 0 1 0 1 0 1 1 1E 0 0 1 0 1 1 1 1 2 1 2

F1(T1) 0 0 1 1 1 1 2 2 2 2 3F2(T2) 0 1 0 1 1 2 1 2 2 3 2

• In this way we find the spin-parity sequence for A1-representation

A1 : 0+, 3−, 4+, 6+, 6−, 7−, 8+, 9+, 9−, 10+, 10−, 11−, 2× 12+, 12−, · · ·

Quantum Rotors: Tetrahedral vs. Octahedral

• The tetrahedral symmetry group has 5 irreducible representations

• The ground-state Iπ = 0+ belongs to A1 representation given by:

A1 : 0+, 3−, 4+, (6+, 6−)︸︷︷︸doublet

, 7−, 8+, (9+, 9−)︸︷︷︸doublet

, (10+, 10−)︸︷︷︸doublet

, 11−, 2× 12+, 12−︸︷︷︸triplet

, · · ·

︸︷︷︸Forming a common parabola

• There are no states with spins I = 1, 2 and 5. We have paritydoublets: I = 6, 9, 10 . . ., at energies: E6− = E6+ , E9− = E9+ , etc.

• One shows that the analogue structure in the octahedral symmetry

A1g : 0+, 4+, 6+, 8+, 9+, 10+, . . . , Iπ = I+︸︷︷︸Forming a common parabola

A2u : 3−, 6−, 7−, 9−, 10−, 11−, . . . , Iπ = I−︸︷︷︸Forming another (common) parabola

Consequently we should expect two independent parabolic structures

A1 : 0+, 3−, 4+, (6+, 6−)︸︷︷︸doublet

, 7−, 8+, (9+, 9−)︸︷︷︸doublet

, (10+, 10−)︸︷︷︸doublet

, 11−, 2× 12+, 12−︸︷︷︸triplet

, · · ·

A1 : 0+, 3−, 4+, (6+, 6−)︸︷︷︸doublet

, 7−, 8+, (9+, 9−)︸︷︷︸doublet

, (10+, 10−)︸︷︷︸doublet

, 11−, 2× 12+, 12−︸︷︷︸triplet

, · · ·

We Are Accustomed to See This Type of Spectra

Just a small fragment of the decay scheme containing numerous rotational bands

with very strong E2-transitions

Characteristic Spectroscopic Difficulty

Given Spectrum of a Nucleuswith a Td -symmetric structures

• ‘Usually’ one observes the strongE2 transitions of ∼ 200 W.u., ‘easy’to detect in the coincidence spectra

• This is the case of the g. s. band !

• However the tetrahedral band hasno strong neither intra-band E2 norinter-band E1; next strong: E3 & E4

• It is not only difficult to detect butalso populate via el-mag transitions

• A possible feeding: Directly vianucleon-evaporation processes or CE

Tetrahedral states: BLACK HOLES

ergy [

g.s.b.

transtionswith NO E2

Tetrahedral band

Schematic Illustration

Two parabolic structures: One connected with

strong E2’s – another one with no E2 (nor E1)

ergy [

g.s.b.

Tetrahedral band

ergy [

g.s.b.

Tetrahedral band

ergy [

g.s.b.

Tetrahedral band

ergy [

g.s.b.

Tetrahedral band

ergy [

g.s.b.

Tetrahedral band

ergy [

g.s.b.

Tetrahedral band

We Start by Connecting to the NNDC Data Base

Precious experimental information which is ready to use for our Td -project

We Continue Reading from the NNDC Data Base

... many pages ...

... and after 11 pages we have the full list

Ready To Search

How to start? What To Start With?

Trying To Find the Experimental Evidence

How to start finding specific levelssatisfying very specific criteria?

We propose proceeding like this:

• We must try to find the sequence

4+, 6+, 8+, 10+ . . .

which is parabolic, no E2 transitions

• If successful, we will fit coefficientsof the reference seed-band parabola

• Once this parabola is known – westart selection of the other candidateexper. states close to this reference

ergy [

3− R

g.s.b.

Tetrahedral band

We begin by looking for experimental

candidates for the ‘reference seed band’

4+, 6+, 8+, 10+ . . .

ergy [

3− R

g.s.b.

Tetrahedral band

4+, 6+, 8+, 10+ . . .

ergy [

3− R

g.s.b.

Tetrahedral band

4+, 6+, 8+, 10+ . . .

ergy [

3− R

g.s.b.

Tetrahedral band

4+, 6+, 8+, 10+ . . .

• Once this parabola is known – westart selection of the other candidateexper. states close to this reference E2

ergy [

3− R

g.s.b.

Tetrahedral band

Start Looking for the Reference Band with no E2’s

• We must try to find the sequence which is parabolic, no E2 transitions

4+, 6+, 8+, 10+ . . .

245122

213126

563 685963 842

279 153

855515

919675

562 492314

14581213

15111389 1408

1112867

1193853

955756 667

16811559

996870 717639756

12031025

12961026

0 0.02 121.8

4 366.5

6 706.9

8 1125.4

10 1609.3

12 2148.8

14 2736.2

16 3365.0

18 4047.7

20 4749.6

0 684.82 810.5

4 1023.0

6 1310.5

8 1666.4

10 2079.6

12 2525.7

14 3292.8

16 3857.2

1963.431041.151221.6

71505.8

91879.1

112326.9

132833.3

153383.4

173973.2

133080.1

112641.1

92290.4

72004.3

51764.331579.4 11510.8 21529.8

41682.1

61929.9

82201.5

102510.6

122901.4

143378.4

112832.9

92375.5

71945.9

51559.6

31233.9

133390.9

112808.9

92445.9

72176.6

51977.2

31779.111680.6

164524.8

143931.2

123352.3

102905.2

82458.6

Kpi=0+ g.s. band

Kpi=0+ beta-vib. band

Kpi=0- octupole vib. band

Kpi=1- (odd)

Kpi=1- (even)

Kpi=2+ g-vib. band(odd)

Kpi=1-

Source -> ENSDF

Experimental spectrum of 152Sm

4+, 6+, 8+, 10+ . . .

245122

213126

563 685963 842

279 153

855515

919675

562 492314

14581213

15111389 1408

1112867

1193853

955756 667

16811559

996870 717639756

12031025

12961026

0 0.02 121.8

4 366.5

6 706.9

8 1125.4

10 1609.3

12 2148.8

14 2736.2

16 3365.0

18 4047.7

20 4749.6

0 684.82 810.5

4 1023.0

6 1310.5

8 1666.4

10 2079.6

12 2525.7

14 3292.8

16 3857.2

1963.431041.151221.6

71505.8

91879.1

112326.9

132833.3

153383.4

173973.2

133080.1

112641.1

92290.4

72004.3

51764.331579.4 11510.8 21529.8

41682.1

61929.9

82201.5

102510.6

122901.4

143378.4

112832.9

92375.5

71945.9

51559.6

31233.9

133390.9

112808.9

92445.9

72176.6

51977.2

31779.111680.6

164524.8

143931.2

123352.3

102905.2

82458.6

Kpi=0+ g.s. band

Kpi=0+ beta-vib. band

Kpi=0- octupole vib. band

Kpi=1- (odd)

Kpi=1- (even)

Kpi=2+ g-vib. band(odd)

Kpi=1-

Source -> ENSDF

OUUPPPSSSS! Too small a scale!!

4+, 6+, 8+, 10+ . . .

I Could NOT Stop Laughing Seeing It For the First Time

4+, 6+, 8+, 10+ . . .

From NNDC data base: Notice the fantasist nomenclature of the bands

... invented long ago by the NNDC data base evaluator

“OUR BAND” is called ... Band (T) like ...

(T)ransportable or (T)ransatlantic... or (T)etrahedral ... or ...

4+, 6+, 8+, 10+ . . .

“OUR BAND” is called ... Band (T) like ... (T)ransportable

or (T)ransatlantic... or (T)etrahedral ... or ...

4+, 6+, 8+, 10+ . . .

“OUR BAND” is called ... Band (T) like ... (T)ransportable or (T)ransatlantic

... or (T)etrahedral ... or ...

4+, 6+, 8+, 10+ . . .

“OUR BAND” is called ... Band (T) like ... (T)ransportable or (T)ransatlantic... or (T)etrahedral ... or ...

Possible Candidate Reference Band

• The sequence 4+, 6+, 8+, 10+ . . . (very) parabolic with no E2 transitions

245122

0 0.02 121.8

4 366.5

6 706.9

8 1125.4

10 1609.3

12 2148.8

14 2736.2

16 3365.0

18 4047.7

20 4749.6

164524.8

143931.2

123352.3

102905.2

82458.6

62099.8

41796.9

9 2506.39 2445.9

2227.77 2176.6

7 2121.06 1929.96

3 1779.13 1730.2

3 1579.4

3 1041.1

21554.8

Kpi=0+ g.s. band

Source -> ENSDF

(A)(T)

(5-,6-,7-)Kpi=1-

Kpi=1-Kpi=1-

Kpi=7-Kpi=1-Kpi=5-(even)

Kpi=1-Kpi=2-(odd)

Kpi=1-(odd)

Kpi=0- Octupole vib.band

(even) 1920.46

(L)(V)

(S)(H)

(L)(J)

itatio

Nest Steps in the Procedure

We Proceed With the Other Tetrahedral-Candidate States:

Criterion no. 1:Accepted states must neither be populated nor depopulated by anystrong E1 or E2 transitions, preferably populated by nuclear reaction

Criterion No. 2:Their energies should be ‘reasonably’ close to the reference parabola

Observation:Since they do not decay via a single strong transition it is instructiveverifying that they decay into several states – with weak intensities

Next Steps in the Procedure: Part II

A typical diagram among a hundred in this analysis

Decay from the tetrahedral Iπ = 3− candidate (among five others)

Let us note that 3− does not decay to the 0+ ground-states (suggesting that it isnot an octuple vibrational state built on the other) and that there are numerousstates populating it suggesting that its structure is exotic from our point of view.

Decay from the tetrahedral Iπ = 3− candidate (among five others)

Let us observe that this state decays to many others suggesting its ‘exotic’structure as in the previous case

Decay from the tetrahedral Iπ = 4+ candidate level

Let us observe that this state decays to many others via very weak transitionssuggesting no resemblance to quadrupole-deformed rotational states

Proceeding Towards a Summary

Proposed experimental energy levels candidates as members of the tetrahedralband in 152Sm after analysing numerous hypotheses. Columns 3 and 4 give the

numbers of decay-out transitions and feeding transitions, respectively.

Spin E[keV] No. D-out No. Feed Reaction

3− 1579.4 10 none CE & α4+ 1757.0 9 1+(1) CE & α6− 1929.9 2 (1) CE & α6+ 2040.1 7 none CE & α7− 2057.5 6 2+(1) CE & α8+ 2391.7 3 1 CE & α9− 2388.8 4 3 CE & α9+ 2588 2 1 α

10− 2590.7 4 1 α(10+) 2810 2 none α11− 2808.9 2 none CE

Parabolic Relations: R.M.S.-Deviation Analysis

Tetrahedral Symmetry Hypothesis: One Parabolic Branch

A1 : 0+, 3−, 4+, (6+, 6−)︸︷︷︸doublet

, 7−, 8+, (9+, 9−)︸︷︷︸doublet

, (10+, 10−)︸︷︷︸doublet

, 11−, 2× 12+, 12−︸︷︷︸triplet

, · · ·

•We performed the test of the tetrahedral A1-type hypothesis by fitting the parametersof the parabola to the energies in the Table. The obtained root-mean-square deviation:

Td : A1 → r .m.s. ≈ 80.5 keV ↔ 11 levels Iπ = I±

For comparison:

G.s.b. → r .m.s. ≈ 52.4 keV ↔ 7 levels Iπ = I+

A1 : 0+, 3−, 4+, (6+, 6−)︸︷︷︸doublet

, 7−, 8+, (9+, 9−)︸︷︷︸doublet

, (10+, 10−)︸︷︷︸doublet

, 11−, 2× 12+, 12−︸︷︷︸triplet

, · · ·

For comparison:

A1 : 0+, 3−, 4+, (6+, 6−)︸︷︷︸doublet

, 7−, 8+, (9+, 9−)︸︷︷︸doublet

, (10+, 10−)︸︷︷︸doublet

, 11−, 2× 12+, 12−︸︷︷︸triplet

, · · ·

For comparison:

Octahedral Symmetry Hypothesis: Two Parabolic Branches

•We performed the test of the octahedral A1g -A2u hypothesis by fitting the parameters

of the parabolas to the energies in the Table. The obtained root-mean-square deviations:

Oh : A1g → r .m.s. ≈ 1.6 keV ↔ Iπ = I+,

Oh : A2u → r .m.s. ≈ 7.5 keV ↔ Iπ = I−.

For comparison:

Oh : A1g → r .m.s. ≈ 1.6 keV ↔ Iπ = I+,

Oh : A2u → r .m.s. ≈ 7.5 keV ↔ Iπ = I−.

For comparison:

Oh : A1g → r .m.s. ≈ 1.6 keV ↔ Iπ = I+,

Oh : A2u → r .m.s. ≈ 7.5 keV ↔ Iπ = I−.

For comparison:

Oh : A1g → r .m.s. ≈ 1.6 keV ↔ Iπ = I+,

Oh : A2u → r .m.s. ≈ 7.5 keV ↔ Iπ = I−.

For comparison:

Dominating Octahedral-Symmetry Hypothesis

Sm152 62 90

0 1 2 3 4 5 6 7 8 9 10 11 12Spin ~

Experimental Results [Td -vs.-Oh]

Symmetry Hypotheses:

Tetrahedral: Td

Octahedral: Oh

A1 → r.m.s.=80.5 keV

A1g → r.m.s.=1.6 keVA2u → r.m.s.=7.5 keV

9+ 11−

7−6+

Graphical representation of the experimental data from the summary Table.Curves represent the fit and are not meant ‘to guide the eye’. Markedly, point[Iπ = 0+], is a prediction by extrapolation - not an experimental datum.

A Comment About Extrapolation to Iπ → 0+

Sm152 62 90

0 1 2 3 4 5 6 7 8 9 10 11 12Spin ~

Experimental Results [Td -vs.-Oh]

Symmetry Hypotheses:

Tetrahedral: Td

Octahedral: Oh

A1 → r.m.s.=80.5 keV

A1g → r.m.s.=1.6 keVA2u → r.m.s.=7.5 keV

9+ 11−

7−6+

Notice: The negative parity sequence lies entirely below the positive parity one.Extrapolating the parabolas to zero-spin we find E−I=0 = 1.396 8 MeV comparedto E+

I=0 = 1.396 1 MeV, the difference of 0.7 keV at the level 1.4 MeV excitation!

COLLABORATORS:

Dominique CURIEN and Irene DEDES

IPHC and University of Strasbourg, France

Kasia MAZUREK

Institute of Nuclear Physics PAN, Cracow, Poland

Shingo TAGAMI and Yoshifumi R SHIMIZU

Department of Physics, Faculty of Sciences, Kyushu University,Fukuoka 8190359, Japan

Tumpa BHATTACHARJEE

Variable Energy Cyclotron Centre,IN-700 064 Kolkata, India

Nuclear D2d -Group: 3D Examples

The nuclear D2d -symmetric shapes have been predicted to coexistwith the axial super-deformed shapes at high spins (JD and X. Li)

Figure:Perspective 1 Figure:Perspective 2 Figure:Perspective 3

Observations:

Nuclear elongation in the range of α20 ∼ (0.45→ 0.55);

Barriers between the coexisting minima ∼ (1→ 2) MeV

Nuclear D3d -Group: 3D Examples

The nuclear D3d -symmetric shapes are expected at high spins; theycorrespond to superposition of α20 and α43 (inversion symmetric)

Observations:

Moderately elongated nuclei can form D3d -symmetry shapes

Probably seen already (remain mis-interpreted as tri-axiality)

Nuclear C3h-Group (’Octupole’): 3D Examples

The nuclear C3h-symmetric shapes are expected at high spins; theycorrespond to superposition of α20 and α33

Observations:

Nuclei with C3h-symmetry predicted to coexist with octupoles

Probably seen already (and mis-interpreted in terms of Iπ=3−)

first evidence for the presence of nuclear octahedral and

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