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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
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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
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Dear Jacek: Congratulations!
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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It was loooong ago...
we met in Warsaw...
became colleagues and friends...
... hmm...
Do you remember?
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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It was loooong ago...
we met in Warsaw...
became colleagues and friends...
... hmm...
Do you remember?
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 6: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/6.jpg)
It was loooong ago...
we met in Warsaw...
became colleagues and friends...
... hmm...
Do you remember?
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 7: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/7.jpg)
It was loooong ago...
we met in Warsaw...
became colleagues and friends...
... hmm...
Do you remember?
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 8: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/8.jpg)
It was loooong ago...
we met in Warsaw...
became colleagues and friends...
... hmm...
Do you remember?
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 9: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/9.jpg)
Somewhat later...
we started travelling a lot...
Warsaw began her new era...
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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Somewhat later...
we started travelling a lot...
Warsaw began her new era...
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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Somewhat later...
we started travelling a lot...
Warsaw began her new era...
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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Somewhat later...
we started travelling a lot...
Warsaw began her new era...
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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But since we refresh memories...
... Recall: More than 20 years back ...
you came to Strasbourg...
as Invited Professor of our University
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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But since we refresh memories...
... Recall: More than 20 years back ...
you came to Strasbourg...
as Invited Professor of our University
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 15: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/15.jpg)
But since we refresh memories...
... Recall: More than 20 years back ...
you came to Strasbourg...
as Invited Professor of our University
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 16: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/16.jpg)
But since we refresh memories...
... Recall: More than 20 years back ...
you came to Strasbourg...
as Invited Professor of our University
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 17: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/17.jpg)
But since we refresh memories...
... Recall: More than 20 years back ...
you came to Strasbourg...
as Invited Professor of our University
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 18: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/18.jpg)
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 !
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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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 !
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 20: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/20.jpg)
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 !
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 21: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/21.jpg)
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 !
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 22: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/22.jpg)
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 !
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 23: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/23.jpg)
And the HFODD came !
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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In the service of Louis Pasteur, Strasbourg 1
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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In the service of Louis Pasteur, Strasbourg 1
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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In the service of Louis Pasteur, Strasbourg 1
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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Motto:
Symmetries Are the FactorsDetermining
Stability of Atomic Nuclei
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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Or – More provocatively:
Do We Owe Our Existenceto the Geometrical Symmetries
on the Sub-atomic Level?
[The Issue of Stability of an Atomic Nucleus]
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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...
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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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...
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 31: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/31.jpg)
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...
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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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).]
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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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).]
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Science Popularisation Articles in the USA [2002]
“Nuclear Pyramids”
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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.)
Physics News Update http://www.aip.org/pnu/2002/split/593-1.html
1 sur 1 9/13/13 12:47 PM
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Science Popularisation “New Scientist” [2002]
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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...”
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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.
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Science Popularisation Articles in the USA [2006]
“Smallest Pyramids in the Universe”
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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
Back to Physics News Update (/pnu)
Smallest Pyramids in the Universe -- Physics News Update 789 http://www.aip.org/pnu/2006/split/789-1.html
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Point-Group Symmetries
Symmetry-Induced Single-Particle Gaps
and
Implied Deformed Closed Shells
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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
GAP
GAP
Deformation Parameter
Nuc
leon
Ene
rgie
s
Deformation Deformation
Nucleus ’1’ Nucleus ’2’
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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
GAP
GAP
Deformation Parameter
Nuc
leon
Ene
rgie
s
Deformation Deformation
Nucleus ’1’ Nucleus ’2’
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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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
GAP
GAP
Deformation Parameter
Nuc
leon
Ene
rgie
s
Deformation Deformation
Nucleus ’1’ Nucleus ’2’
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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Symmetries, Gaps, Valence Particles
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
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
p−h
Nu
cleo
n E
ener
gie
s Deformation
Deformation
p!h exc. config.
g.s. config.
Energy
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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Symmetries, Gaps, Valence Particles
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
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
p−h
Nu
cleo
n E
ener
gie
s Deformation
Deformation
p!h exc. config.
g.s. config.
Energy
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 45: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/45.jpg)
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 46: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/46.jpg)
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 47: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/47.jpg)
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 48: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/48.jpg)
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 49: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/49.jpg)
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 50: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/50.jpg)
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 51: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/51.jpg)
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 52: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/52.jpg)
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, . . . Rr
Let their dimensions be d1, d2, . . . dr, respectively
Then the eigenvalues εν of the problem
Hψν = ενψν
appear in multiplets d1-fold, d2-fold ... degenerate
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 53: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/53.jpg)
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 54: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/54.jpg)
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 55: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/55.jpg)
Symmetries and Gaps in Nuclear Context
Schematic illustration: Levels of 6 irreps and average spacings/gaps
ap
G
Irrep.1 Irrep.2 Irrep.3 Irrep.4 Irrep.5 Irrep.6 All Irreps.
Qualitatively: Average level spacings increase, within an irrep, by a“factor of ∼6”. The full spectrum may present unusually big gaps.
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 56: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/56.jpg)
Symmetries and Gaps in Nuclear Context
Schematic illustration: Levels of 6 irreps and average spacings/gaps
ap
G
Irrep.1 Irrep.2 Irrep.3 Irrep.4 Irrep.5 Irrep.6 All Irreps.
Qualitatively: Average level spacings increase, within an irrep, by a“factor of ∼6”. The full spectrum may present unusually big gaps.
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 57: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/57.jpg)
Physically Meaningful Point-Groups Contained in SO3
32 Point Groups: Subgroups
C4h 4
D2d
C4v
D2h
T D6
C6h 6v
C
4C
4D
2C
6C
2vC
2hC
3iD
3C
3vC
3h
3hD
3dDD
D4h
Th
O Td
D6h
hO
C C C C
C
i 2 s 3
1
.
.
S
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?
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 58: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/58.jpg)
Physically Meaningful Point-Groups Contained in SO3
32 Point Groups: Subgroups
C4h 4
D2d
C4v
D2h
T D6
C6h 6v
C
4C
4D
2C
6C
2vC
2hC
3iD
3C
3vC
3h
3hD
3dDD
D4h
Th
O Td
D6h
hO
C C C C
C
i 2 s 3
1
.
.
S
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?
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 59: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/59.jpg)
Physically Meaningful Point-Groups Contained in SO3
32 Point Groups: Subgroups
C4h 4
D2d
C4v
D2h
T D6
C6h 6v
C
4C
4D
2C
6C
2vC
2hC
3iD
3C
3vC
3h
3hD
3dDD
D4h
Th
O Td
D6h
hO
C C C C
C
i 2 s 3
1
.
.
S
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?
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 60: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/60.jpg)
Symmetries, Group-Representations and Degeneracies
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 61: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/61.jpg)
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 62: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/62.jpg)
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 63: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/63.jpg)
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 64: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/64.jpg)
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 65: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/65.jpg)
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 66: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/66.jpg)
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 67: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/67.jpg)
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 68: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/68.jpg)
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 69: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/69.jpg)
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 70: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/70.jpg)
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 71: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/71.jpg)
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 72: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/72.jpg)
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
λ=2
∑λµ=−λ α
∗λµ [OYλµ(ϑ, ϕ)] =
∑λmax
λ=2
∑λµ=−λ α
∗λµYλµ(ϑ, ϕ)
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 73: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/73.jpg)
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
λ=2
∑λµ=−λ α
∗λµ [OYλµ(ϑ, ϕ)] =
∑λmax
λ=2
∑λµ=−λ α
∗λµYλµ(ϑ, ϕ)
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 74: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/74.jpg)
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
λ=2
∑λµ=−λ α
∗λµ [OYλµ(ϑ, ϕ)] =
∑λmax
λ=2
∑λµ=−λ α
∗λµYλµ(ϑ, ϕ)
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 75: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/75.jpg)
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
λ=2
∑λµ=−λ α
∗λµ [OYλµ(ϑ, ϕ)] =
∑λmax
λ=2
∑λµ=−λ α
∗λµYλµ(ϑ, ϕ)
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 76: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/76.jpg)
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
λ=2
∑λµ=−λ α
∗λµ [OYλµ(ϑ, ϕ)] =
∑λmax
λ=2
∑λµ=−λ α
∗λµYλµ(ϑ, ϕ)
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 77: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/77.jpg)
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
λ=2
[∑λµ=−λ α
∗λµD
λµ′µ(Ω)− α∗λµ′
]Yλµ(ϑ, ϕ) = 0
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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
λ=2
[∑λµ=−λ α
∗λµD
λµ′µ(Ω)− α∗λµ′
]Yλµ(ϑ, ϕ) = 0
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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
λ=2
[∑λµ=−λ α
∗λµD
λµ′µ(Ω)− α∗λµ′
]Yλµ(ϑ, ϕ) = 0
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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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
λ=2
[∑λµ=−λ α
∗λµD
λµ′µ(Ω)− α∗λµ′
]Yλµ(ϑ, ϕ) = 0
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Invariant Mean-Field Hamiltonian [3]
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
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Invariant Mean-Field Hamiltonian [3]
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
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Invariant Mean-Field Hamiltonian [3]
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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Invariant Mean-Field Hamiltonian [3]
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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Invariant Mean-Field Hamiltonian [3]
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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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
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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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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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
+α72
[(Y7+2 + Y7−2)−
√1113
(Y7+6 + Y7−6)]︸ ︷︷ ︸
one parameter 7th order
+ higher order odd−λ terms
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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
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
Tetrahedral Deformation
Pro
ton
En
erg
ies
[MeV
]
α 32(m
in)=
-.200,
α 32(m
ax)=
.400
Str
asb
ou
rg, A
ugu
st 2
002
Wood
s-S
axon
Un
ivers
al
Para
ms.
58
6464
70
82
90
94
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.
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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
-11
-10
-9
-8
-7
-6
-5
-4
Tetrahedral Deformation
Neu
tron
En
erg
ies
[MeV
]
α 32(m
in)=
-.200,
α 32(m
ax)=
.400
Str
asb
ou
rg, A
ugu
st 2
002
Wood
s-S
axon
Un
ivers
al
Para
ms.
112124
126
136 136
142
148
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.
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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
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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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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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:
Three Lowest Order Solutions: Rank ↔ Multipolarity λ
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 ≡
√65
198· o8
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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:
Three Lowest Order Solutions: Rank ↔ Multipolarity λ
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 ≡
√65
198· o8
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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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:
Three Lowest Order Solutions: Rank ↔ Multipolarity λ
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 ≡
√65
198· o8
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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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:
Three Lowest Order Solutions: Rank ↔ Multipolarity λ
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 ≡
√65
198· o8
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 101: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/101.jpg)
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:
Three Lowest Order Solutions: Rank ↔ Multipolarity λ
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 ≡
√65
198· o8
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 102: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/102.jpg)
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 +
√5
14(Y4+4 + Y4−4)
]︸ ︷︷ ︸
one parameter 4th order
+α60
[Y60 −
√72
(Y6+4 + Y6−4)]
︸ ︷︷ ︸one parameter 6th order
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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Example: Octahedral Symmetry - Proton Spectra
Double group ODh has four 2-dimensional and two 4-dimensional
irreducible representations → six distinct families of levels
-.35 -.25 -.15 -.05 .05 .15 .25 .35
-12
-10
-8
-6
-4
-2
0
2
Octahedral Deformation
Pro
ton
En
erg
ies
[MeV
]
α 40(m
in)=
-.350,
α 40(m
ax)=
.350
α 44(m
in)=
-.209,
α 44(m
ax)=
.209
Str
asb
ou
rg, A
ugu
st 2
002
Dir
ac-
Wood
s-S
axon
5252
5658
64
70
7282
8888
94
09[4,3,1] 3/213[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.
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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Example: Octahedral Symmetry - Neutron Spectra
Double group ODh has four 2-dimensional and two 4-dimensional
irreducible representations → six distinct families of levels
-.35 -.25 -.15 -.05 .05 .15 .25 .35
-12
-10
-8
-6
-4
-2
Octahedral Deformation
Neu
tron
En
erg
ies
[MeV
]
α 40(m
in)=
-.350,
α 40(m
ax)=
.350
α 44(m
in)=
-.209,
α 44(m
ax)=
.209
Str
asb
ou
rg, A
ugu
st 2
002
Dir
ac-
Wood
s-S
axon
82
86
8888
94 94
100
110
114 116
118126
21[4,0,2] 3/222[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.
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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Example: Results with the HFB Solutions in RE
The HFB results for tetrahedral solutions in light Rare-Earth nuclei
α4,0 ≡ o4, α4,±4 ≡ −√
5/14 o4
Z N ∆E Q32 Q40 Q44 Q40 ×√
514
(MeV) (b3/2) (b2) (b2) (b2)
64 86 −1.387 0.941817 −0.227371 +0.135878 −0.13588064 90 −3.413 1.394656 −0.428250 +0.255929 −0.25592864 92 −3.972 0.000000 −0.447215 +0.267263 −0.267262
62 86 −0.125 0.487392 −0.086941 +0.051954 −0.05195762 88 −0.524 0.812103 −0.218809 +0.130760 −0.13076362 90 −1.168 1.206017 −0.380334 +0.227293 −0.227293
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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Example: Results with the HFB Solutions in RE
The HFB results for tetrahedral solutions in light Rare-Earth nuclei
α4,0 ≡ o4, α4,±4 ≡ −√
5/14 o4
Z N ∆E Q32 Q40 Q44 Q40 ×√
514
(MeV) (b3/2) (b2) (b2) (b2)
64 86 −1.387 0.941817 −0.227371 +0.135878 −0.13588064 90 −3.413 1.394656 −0.428250 +0.255929 −0.25592864 92 −3.972 0.000000 −0.447215 +0.267263 −0.267262
62 86 −0.125 0.487392 −0.086941 +0.051954 −0.05195762 88 −0.524 0.812103 −0.218809 +0.130760 −0.13076362 90 −1.168 1.206017 −0.380334 +0.227293 −0.227293
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 107: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/107.jpg)
Example: Results with the HFB Solutions in RE
The HFB results for tetrahedral solutions in light Rare-Earth nuclei
α4,0 ≡ o4, α4,±4 ≡ −√
5/14 o4
Z N ∆E Q32 Q40 Q44 Q40 ×√
514
(MeV) (b3/2) (b2) (b2) (b2)
64 86 −1.387 0.941817 −0.227371 +0.135878 −0.13588064 90 −3.413 1.394656 −0.428250 +0.255929 −0.25592864 92 −3.972 0.000000 −0.447215 +0.267263 −0.267262
62 86 −0.125 0.487392 −0.086941 +0.051954 −0.05195762 88 −0.524 0.812103 −0.218809 +0.130760 −0.130763
62 90 −1.168 1.206017 −0.380334 +0.227293 −0.227293
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 111: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/111.jpg)
During the Following Years: A Long Preparation
[Various Problems of Theory: Formulated and Solved]
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 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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 112: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/112.jpg)
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 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 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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 113: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/113.jpg)
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 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 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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 114: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/114.jpg)
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 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 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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 115: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/115.jpg)
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 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 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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 116: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/116.jpg)
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 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 triplex
8) Multi-dimensional deformation spaces: isotropy groups and orbits
9) Theory calculations of transition probabilities & branching ratios
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 117: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/117.jpg)
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 triplexProblem solved in:N. Schunck, JD and S. Frauendorf, Acta Phys. Pol. B 36, 1071 (2005)
8) Multi-dimensional deformation spaces: isotropy groups and orbits
9) Theory calculations of transition probabilities & branching ratios
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 118: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/118.jpg)
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 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)
9) Theory calculations of transition probabilities & branching ratios
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 119: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/119.jpg)
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 120: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/120.jpg)
How decisive these ‘new’ degrees of freedommay become?
→
See illustrations
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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Symmetry Concepts Impact Our Ideas about Stability
-3.0
-3.0-3.0
-1.4
-1.4
-1.4-1.4
-1.4-1.4
0.2
0.20.2
0.2
0.2 0.2 0.2
0.2
0.2
1.8
1.8
1.8
1.83.4
3.4 3.4
3.43.
45.0
5.05.0
5.0
-0.10 0.00 0.10 0.20 0.30 0.40
-0.10
0.00
0.10
0.20
0.30
0.40
X196X322126 β2cos(γ+30)
β 2si
n(γ+
30)
• Consider a total energy for a super-heavy nucleus in the form ofthe standard (β, γ)-representation
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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Symmetry Concepts Impact Our Ideas about Stability
-3.0
-3.0-3.0
-3.0
-1.4
-1.4-1.4
-1.4
-1.4-1.4
-1.4
-1.4
-1.4
-1.4
-1.4-1.4
0.20.2
0.20.2
0.2
0.2
0.2
0.2 0.2
0.2
0.2
1.8
1.8
1.8 1.8
1.8
3.4
3.4
-0.10 0.00 0.10 0.20 0.30 0.40
-0.10
0.00
0.10
0.20
0.30
0.40
X196X322126 β2cos(γ+30)
β 2si
n(γ+
30)
• Consider the similar standard (β, γ)-representation but now let usintroduce an extra minimisation over the tetrahedral deformation
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 123: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/123.jpg)
These Concepts Change Our Ideas about Stability
-3.0
-3.0
-3.0
-1.4
-1.4
-1.4
-1.4
-1.4-1.4
0.2
0.20.2
0.2
0.2 0.2 0.2
0.2
0.2
1.8
1.8
1.8
1.83.4
3.4 3.4
3.4
3.45.0
5.05.0
5.0
-0.10 0.00 0.10 0.20 0.30 0.40
-0.10
0.00
0.10
0.20
0.30
0.40
X196X322126 β2cos(γ+30)
β 2si
n(γ+
30)
-3.0
-3.0
-3.0
-3.0
-1.4
-1.4-1.4
-1.4
-1.4-1.4
-1.4-1.4
-1.4
-1.4-1.4-1.4
0.20.2
0.20.2
0.2
0.2
0.2
0.2 0.2
0.2
0.2
1.8
1.8
1.8 1.8
1.8
3.4
3.4
-0.10 0.00 0.10 0.20 0.30 0.40
-0.10
0.00
0.10
0.20
0.30
0.40
X196X322126 β2cos(γ+30)
β 2si
n(γ+
30)
• 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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 124: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/124.jpg)
Further Illustrations of the Effectsof the High-Rank Symmetries
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 125: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/125.jpg)
How Powerful the Symmetry Notions Are – See Maps
2
2 2
2
2
2
2
2
2
2
2
4
4 4
4
4
44
4
66
6
6
8
8
8
10
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
E(fyu)+Shell[e]+Correlation[PNP]
Gp=
0.95
0 G
n=0.
960
∆Np=
35 ∆
Nn=
45M
inim
isat
ion
over
α 4
0
; o2
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
Def
orm
atio
n α 3
2
Emin=-3.04, Eo= 0.45
Observe the presence of well defined tetrahedral minima at N=90
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 126: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/126.jpg)
How Powerful the Symmetry Notions Are – See Maps
2
2
2
2
2
2
2
2
22
2
2
2
4
4 4
4
4
4
44
4
6
6
6
6
8
8
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
E(fyu)+Shell[e]+Correlation[PNP]
Gp=
0.95
0 G
n=0.
960
∆Np=
35 ∆
Nn=
45M
inim
isat
ion
over
α 4
0
; o2
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
Def
orm
atio
n α 3
2
Emin=-2.60, Eo= 1.09
Observe the presence of well defined tetrahedral minima at N=90
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 127: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/127.jpg)
How Powerful the Symmetry Notions Are – See Maps
2
2
2 2
2
2
2
22
2
2
2
2
2
4
4 4
4
4
44
6
6
6
8
8
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
E(fyu)+Shell[e]+Correlation[PNP]
Gp=
0.95
0 G
n=0.
960
∆Np=
35 ∆
Nn=
45M
inim
isat
ion
over
α 4
0
; o2
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
Def
orm
atio
n α 3
2
Emin=-2.18, Eo= 1.49
Observe the presence of well defined tetrahedral minima at N=90
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 128: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/128.jpg)
How Powerful the Symmetry Notions Are – See Maps
2
22
2 2
22
2
2
2
2
2
2
2
2
24
44
4
4
4
44
6
6
8
8
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
E(fyu)+Shell[e]+Correlation[PNP]
Gp=
0.95
0 G
n=0.
960
∆Np=
35 ∆
Nn=
45M
inim
isat
ion
over
α 4
0
; o2
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
Def
orm
atio
n α 3
2
Emin=-2.01, Eo= 1.77
Observe the presence of well defined tetrahedral minima at N=90
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 129: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/129.jpg)
How Powerful the Symmetry Notions Are – See Maps
2
2
2
22
2
2
2
2
2
2 2
2
2
4 4
4
4
4
6
6
6
8
8
10
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
E(fyu)+Shell[e]+Correlation[PNP]
Gp=
0.95
0 G
n=0.
960
∆Np=
35 ∆
Nn=
45M
inim
isat
ion
over
α 4
0
; o2
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
Def
orm
atio
n α 3
2
Emin=-1.85, Eo= 1.86
Observe the presence of well defined tetrahedral minima at N=90
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 130: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/130.jpg)
How Powerful the Symmetry Notions Are – See Maps
02
22
2
2
2
2
2
2
2
2
2
24 4
4
4
4
6 6
6
6
66
8
8
10
10
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
E(fyu)+Shell[e]+Correlation[PNP]
UN
IVER
S_CO
MPA
CT (D
=3, 2
3)G
p=0.
960
Gn=
0.98
0 6
Np=
35 6
Nn=
45
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
Def
orm
atio
n t 1
Emin=-1.99, Eo= 1.45Observe the presence of well defined tetrahedral minima at N=90
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 131: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/131.jpg)
How Powerful the Symmetry Notions Are – See Maps
2
2
2
2
2
2
2
2
2
2
2
44
4
4
44
6 6
6
6
66
8
8
10
10
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
E(fyu)+Shell[e]+Correlation[PNP]
UN
IVE
RS_
CO
MPA
CT
(D
=3,
23)
Gp=
0.96
0 G
n=0.
980
∆Np=
35 ∆
Nn=
45
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
Def
orm
atio
n t 1
Emin=-2.28, Eo= 0.84
Observe the presence of well defined tetrahedral minima at N=90
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 132: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/132.jpg)
How Powerful the Symmetry Notions Are – See Maps
0
2
2
2
2
2
2
2
2 2
2
2
2
2
4
4
44
4
4
4
4
44
4
6
6
66
66
6
6
6
68
8
8
8
8
8
10
10
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
E(fyu)+Shell[e]+Correlation[PNP]
UN
IVE
RS_
CO
MPA
CT
(D
=3,
23)
Gp=
0.95
0 G
n=0.
960
∆Np=
35 ∆
Nn=
45M
inim
isat
ion
over
α 4
0
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
Def
orm
atio
n α 3
0
Emin=-2.78, Eo= 0.13
Observe the presence of well defined tetrahedral minima at N=90
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 133: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/133.jpg)
How Powerful the Symmetry Notions Are – See Maps
02
2
2
2
2
2
2
2
2
4
4
4
4
4
4
44
4
4
6
6
6
66
6
66
68
8
88
88
10
10
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
E(fyu)+Shell[e]+Correlation[PNP]
UN
IVE
RS_
CO
MPA
CT
(D
=3,
23)
Gp=
0.95
0 G
n=0.
960
∆Np=
35 ∆
Nn=
45M
inim
isat
ion
over
α 4
0
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
Def
orm
atio
n α 3
0
Emin=-3.10, Eo=-1.49
Observe the presence of well defined tetrahedral minima at N=90
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 134: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/134.jpg)
Conclusion:
We may look for the experimental evidenceof the tetrahedral symmetry
focussing on those N = 90 isotonesfor which the best experimental data exist
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 135: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/135.jpg)
Interplay Between Tetrahedral and OctahedralSymmetries
Realistic Calculations of a Rare Earth Nucleus
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 136: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/136.jpg)
Consider 15262Sm90: Phenomenological WS Approach
Tetrahedral Symmetry Effect
2
2
2
2
2
2
2
22
2
2
4
4
4
4
4
6
6
6
8
8
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.3
-0.2
-0.1
0.0
0.1
0.2
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
Def
orm
atio
n _
32
Emin=-2.60, Eo= 1.09
Observe the presence of well defined tetrahedral minima at α32 ≈ ±0.12
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 137: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/137.jpg)
Consider 15262Sm90: Phenomenological WS Approach
Combined Octahedral and Tetrahedral Symmetry Effect
0
0
0
0
1
1
1
1
1
1
1
12
2
2
2
2
22
2
2
2 33
3
3
3
33
3
3
3 4 4
4
4
44
4
4
5
5
555
5
6
6
6
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3-0.2
-0.1
0.0
0.1
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
Def
orm
atio
n o 1
Emin=-1.41, Eo= 1.09
Allowing for octahedral deformation lowers the tetrahedral minimum by 2 MeV
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 138: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/138.jpg)
Consider 15262Sm90: Phenomenological WS Approach
Combined Octahedral and Tetrahedral Symmetry Effect
2
2
2
2
2
2
2
2
4
4
4
4
4
4
444
4
6
66
6
6
6
66
8
8
8
8
8
1010
10
10
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.2
-0.1
0.0
0.1
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
Def
orm
atio
n o 1
Emin=-2.60, Eo= 1.09
An alternative illustration of the octahedral symmetry effect
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 139: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/139.jpg)
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 ×√
514
o2 = α60 α64 α60 ×√
72
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 = −
√5
14α4,0,
o2 ≡ α6,0 and α6,±4 = +
√7
2α6,0,
0
0
0
0
1
1
1
1
1
1
1
12
2
2
2
2
22
2
2
2 33
3
3
3
33
3
3
3 4 4
4
4
44
4
45
5
555
5
6
6
6
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3-0.2
-0.1
0.0
0.1
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
Def
orm
atio
n o 1
Emin=-1.41, Eo= 1.09
Notice: t1 ≈ ±0.12 and o1 ≈ −0.08
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 140: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/140.jpg)
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’ ?
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 141: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/141.jpg)
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 142: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/142.jpg)
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
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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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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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
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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
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Microscopic Approach in This Project: Gogny HFB
• We consider the standard two-body Hamiltonian, `i = 1, 2, . . . M:
H =∑`1`2
t`1`2 c+`1c`2
+ 12
∑`1`2
∑`3`4
v`1`2`3`4 c+`1c+`2c`4
c`3,
• 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)∗`′`
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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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 †)`′`.
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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 〉 =
∑K
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(±)
α
∑K ′
N INZ(±)K ,K ′ g
INZ(±)K ′,α ,
• The kernels are defined by HINZ(±)K ,K ′
N INZ(±)K ,K ′
= 〈Φ|
H
1
P IKK ′ PN PZ P±|Φ〉.
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Approaching Degeneracies with Increasing Deformation
Tetrahedral Symmetry Effect
Observe the sequence characteristically mixing both parities and degeneracies
Iπ = 6±, Iπ = 9±, Iπ = 10±, ...
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It will be instructive at this point
to recall what is known about
“these degeneracies”
from the point of view of group representation theory
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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It will be instructive at this point
to recall what is known about
“these degeneracies”
from the point of view of group representation theory
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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It will be instructive at this point
to recall what is known about
“these degeneracies”
from the point of view of group representation theory
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 153: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/153.jpg)
It will be instructive at this point
to recall what is known about
“these degeneracies”
from the point of view of group representation theory
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 154: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/154.jpg)
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 =
1
NG
∑R∈G
χIπ(R)χi (R) =1
NG
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.
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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 =
1
NG
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
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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−, · · ·
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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
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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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 160: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/160.jpg)
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
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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
E2
1
2
3
4
0 0
4
6
8 +
+
+
2 ++
12+
10+
5
9−
6
En
ergy [
MeV
]
14+
4 +
6 +
9 +
8 +
10+10
−
6−
7−
.
.
0+
3−
g.s.b.
transtionswith NO E2
Tetrahedral band
E2
E2
E2
E2
E2
Schematic Illustration
E2
Two parabolic structures: One connected with
strong E2’s – another one with no E2 (nor E1)
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 162: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/162.jpg)
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
E2
1
2
3
4
0 0
4
6
8 +
+
+
2 ++
12+
10+
5
9−
6
En
ergy [
MeV
]
14+
4 +
6 +
9 +
8 +
10+10
−
6−
7−
.
.
0+
3−
g.s.b.
transtionswith NO E2
Tetrahedral band
E2
E2
E2
E2
E2
Schematic Illustration
E2
Two parabolic structures: One connected with
strong E2’s – another one with no E2 (nor E1)
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 163: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/163.jpg)
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
E2
1
2
3
4
0 0
4
6
8 +
+
+
2 ++
12+
10+
5
9−
6
En
ergy [
MeV
]
14+
4 +
6 +
9 +
8 +
10+10
−
6−
7−
.
.
0+
3−
g.s.b.
transtionswith NO E2
Tetrahedral band
E2
E2
E2
E2
E2
Schematic Illustration
E2
Two parabolic structures: One connected with
strong E2’s – another one with no E2 (nor E1)
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 164: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/164.jpg)
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
E2
1
2
3
4
0 0
4
6
8 +
+
+
2 ++
12+
10+
5
9−
6
En
ergy [
MeV
]
14+
4 +
6 +
9 +
8 +
10+10
−
6−
7−
.
.
0+
3−
g.s.b.
transtionswith NO E2
Tetrahedral band
E2
E2
E2
E2
E2
Schematic Illustration
E2
Two parabolic structures: One connected with
strong E2’s – another one with no E2 (nor E1)
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 165: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/165.jpg)
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
E2
1
2
3
4
0 0
4
6
8 +
+
+
2 ++
12+
10+
5
9−
6
En
ergy [
MeV
]
14+
4 +
6 +
9 +
8 +
10+10
−
6−
7−
.
.
0+
3−
g.s.b.
transtionswith NO E2
Tetrahedral band
E2
E2
E2
E2
E2
Schematic Illustration
E2
Two parabolic structures: One connected with
strong E2’s – another one with no E2 (nor E1)
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 166: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/166.jpg)
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
E2
1
2
3
4
0 0
4
6
8 +
+
+
2 ++
12+
10+
5
9−
6
En
ergy [
MeV
]
14+
4 +
6 +
9 +
8 +
10+10
−
6−
7−
.
.
0+
3−
g.s.b.
transtionswith NO E2
Tetrahedral band
E2
E2
E2
E2
E2
Schematic Illustration
E2
Two parabolic structures: One connected with
strong E2’s – another one with no E2 (nor E1)
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 167: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/167.jpg)
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
E2
1
2
3
4
0 0
4
6
8 +
+
+
2 ++
12+
10+
5
9−
6
En
ergy [
MeV
]
14+
4 +
6 +
9 +
8 +
10+10
−
6−
7−
.
.
0+
3−
g.s.b.
transtionswith NO E2
Tetrahedral band
E2
E2
E2
E2
E2
Schematic Illustration
E2
Two parabolic structures: One connected with
strong E2’s – another one with no E2 (nor E1)
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 168: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/168.jpg)
We Start by Connecting to the NNDC Data Base
PAGE 1: Precious experimental information which is ready to use for our Td -project
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 169: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/169.jpg)
We Continue Reading from the NNDC Data Base
PAGE 2: Precious experimental information which is ready to use for our Td -project
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 170: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/170.jpg)
... many pages ...
... many pages ...
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 171: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/171.jpg)
... and after 11 pages we have the full list
PAGE 11: Precious experimental information which is ready to use for our Td -project
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 172: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/172.jpg)
Ready To Search
How to start? What To Start With?
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 173: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/173.jpg)
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
E2
1
2
3
4
0 0
4
6
8 +
+
+
2 ++
12+
10+
5
9−
6
En
ergy [
MeV
]
14+
4 +
6 +
9 +
8 +
10+10
−
6−
7−
.
.
0+
3− R
efer
ence
‘S
eed
’ B
an
d (
No E
2 T
ran
siti
on
)
g.s.b.
transtionswith NO E2
Tetrahedral band
E2
E2
E2
E2
E2
Schematic Illustration
E2
We begin by looking for experimental
candidates for the ‘reference seed band’
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 174: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/174.jpg)
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
E2
1
2
3
4
0 0
4
6
8 +
+
+
2 ++
12+
10+
5
9−
6
En
ergy [
MeV
]
14+
4 +
6 +
9 +
8 +
10+10
−
6−
7−
.
.
0+
3− R
efer
ence
‘S
eed
’ B
an
d (
No E
2 T
ran
siti
on
)
g.s.b.
transtionswith NO E2
Tetrahedral band
E2
E2
E2
E2
E2
Schematic Illustration
E2
We begin by looking for experimental
candidates for the ‘reference seed band’
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 175: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/175.jpg)
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
E2
1
2
3
4
0 0
4
6
8 +
+
+
2 ++
12+
10+
5
9−
6
En
ergy [
MeV
]
14+
4 +
6 +
9 +
8 +
10+10
−
6−
7−
.
.
0+
3− R
efer
ence
‘S
eed
’ B
an
d (
No E
2 T
ran
siti
on
)
g.s.b.
transtionswith NO E2
Tetrahedral band
E2
E2
E2
E2
E2
Schematic Illustration
E2
We begin by looking for experimental
candidates for the ‘reference seed band’
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 176: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/176.jpg)
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
E2
1
2
3
4
0 0
4
6
8 +
+
+
2 ++
12+
10+
5
9−
6
En
ergy [
MeV
]
14+
4 +
6 +
9 +
8 +
10+10
−
6−
7−
.
.
0+
3− R
efer
ence
‘S
eed
’ B
an
d (
No E
2 T
ran
siti
on
)
g.s.b.
transtionswith NO E2
Tetrahedral band
E2
E2
E2
E2
E2
Schematic Illustration
E2
We begin by looking for experimental
candidates for the ‘reference seed band’
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 177: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/177.jpg)
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 E2
1
2
3
4
0 0
4
6
8 +
+
+
2 ++
12+
10+
5
9−
6
En
ergy [
MeV
]
14+
4 +
6 +
9 +
8 +
10+10
−
6−
7−
.
.
0+
3− R
efer
ence
‘S
eed
’ B
an
d (
No E
2 T
ran
siti
on
)
g.s.b.
transtionswith NO E2
Tetrahedral band
E2
E2
E2
E2
E2
Schematic Illustration
E2
We begin by looking for experimental
candidates for the ‘reference seed band’
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 178: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/178.jpg)
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+ . . .
702
683
629
587
540
484
418
340
245122
564
767
446
413
356
288
213126
563 685963 842
279 153
590
506
448
373
608
647
684
718
754
799
855515
919675
931
1032
562 492314
1165
681
1297
1398
14581213
15111389 1408
477
391
309
1076
1223
1316
457
386
326
1112867
1193853
1239
821
1250
766
1064
930
940
955756 667
16811559
996870 717639756
1789
1782
1195
1743
12031025
1780
12961026
1752
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-
K=?
Source -> ENSDF
(A)
(B)
(C)
(G)
(H)
(F)
(L)
(T)
152Sm
Experimental spectrum of 152Sm
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 179: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/179.jpg)
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+ . . .
702
683
629
587
540
484
418
340
245122
564
767
446
413
356
288
213126
563 685963 842
279 153
590
506
448
373
608
647
684
718
754
799
855515
919675
931
1032
562 492314
1165
681
1297
1398
14581213
15111389 1408
477
391
309
1076
1223
1316
457
386
326
1112867
1193853
1239
821
1250
766
1064
930
940
955756 667
16811559
996870 717639756
1789
1782
1195
1743
12031025
1780
12961026
1752
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-
K=?
Source -> ENSDF
(A)
(B)
(C)
(G)
(H)
(F)
(L)
(T)
152Sm
OUUPPPSSSS! Too small a scale!!
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 180: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/180.jpg)
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+ . . .
Experimental spectrum of 152Sm
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 181: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/181.jpg)
I Could NOT Stop Laughing Seeing It For the First Time
• We must try to find the sequence which is parabolic, no E2 transitions
4+, 6+, 8+, 10+ . . .
Experimental spectrum of 152Sm
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 ...
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 182: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/182.jpg)
I Could NOT Stop Laughing Seeing It For the First Time
• We must try to find the sequence which is parabolic, no E2 transitions
4+, 6+, 8+, 10+ . . .
Experimental spectrum of 152Sm
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 ...
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 183: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/183.jpg)
I Could NOT Stop Laughing Seeing It For the First Time
• We must try to find the sequence which is parabolic, no E2 transitions
4+, 6+, 8+, 10+ . . .
Experimental spectrum of 152Sm
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 ...
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 184: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/184.jpg)
I Could NOT Stop Laughing Seeing It For the First Time
• We must try to find the sequence which is parabolic, no E2 transitions
4+, 6+, 8+, 10+ . . .
Experimental spectrum of 152Sm
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 ...
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 185: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/185.jpg)
Possible Candidate Reference Band
• The sequence 4+, 6+, 8+, 10+ . . . (very) parabolic with no E2 transitions
702
683
629
587
540
484
418
340
245122
1789
1782
1195
1743
1203
1780
1296
1752
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
K=?
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)
(L)(V)
(S)(H)
(L)(J)
(G)
(C)
152Sm
Exc
itatio
n E
nerg
y (M
eV)
0
2
4
6
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 186: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/186.jpg)
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 187: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/187.jpg)
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.
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 188: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/188.jpg)
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 observe that this state decays to many others suggesting its ‘exotic’structure as in the previous case
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 189: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/189.jpg)
Next Steps in the Procedure: Part II
A typical diagram among a hundred in this analysis
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
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries
![Page 190: First Evidence for the Presence of Nuclear Octahedral and](https://reader036.vdocument.in/reader036/viewer/2022062400/62aa99b9706c902c70196671/html5/thumbnails/190.jpg)
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
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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
, · · ·
︸ ︷︷ ︸Forming a common parabola
•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+
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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
, · · ·
︸ ︷︷ ︸Forming a common parabola
•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+
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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
, · · ·
︸ ︷︷ ︸Forming a common parabola
•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+
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Parabolic Relations: R.M.S.-Deviation Analysis
Octahedral Symmetry Hypothesis: Two Parabolic Branches
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
•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:
Td : A1 → r .m.s. ≈ 80.5 keV ↔ 11 levels Iπ = I±
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Parabolic Relations: R.M.S.-Deviation Analysis
Octahedral Symmetry Hypothesis: Two Parabolic Branches
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
•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:
Td : A1 → r .m.s. ≈ 80.5 keV ↔ 11 levels Iπ = I±
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Parabolic Relations: R.M.S.-Deviation Analysis
Octahedral Symmetry Hypothesis: Two Parabolic Branches
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
•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:
Td : A1 → r .m.s. ≈ 80.5 keV ↔ 11 levels Iπ = I±
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Parabolic Relations: R.M.S.-Deviation Analysis
Octahedral Symmetry Hypothesis: Two Parabolic Branches
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
•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:
Td : A1 → r .m.s. ≈ 80.5 keV ↔ 11 levels Iπ = I±
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Dominating Octahedral-Symmetry Hypothesis
Sm152 62 90
.
.
0 1 2 3 4 5 6 7 8 9 10 11 12Spin ~
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0R
otat
iona
lEne
rgy
[MeV
]
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
8+
9+ 11−
3−
10−
7−6+
4+
9−
[0+]
10+
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.
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A Comment About Extrapolation to Iπ → 0+
Sm152 62 90
.
.
0 1 2 3 4 5 6 7 8 9 10 11 12Spin ~
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0R
otat
iona
lEne
rgy
[MeV
]
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
8+
9+ 11−
3−
10−
7−6+
4+
9−
[0+]
10+
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!
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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
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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
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Nuclear D3d -Group: 3D Examples
The nuclear D3d -symmetric shapes are expected at high spins; theycorrespond to superposition of α20 and α43 (inversion symmetric)
Figure:Perspective 1 Figure:Perspective 2 Figure:Perspective 3
Observations:
Moderately elongated nuclei can form D3d -symmetry shapes
Probably seen already (remain mis-interpreted as tri-axiality)
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Nuclear C3h-Group (’Octupole’): 3D Examples
The nuclear C3h-symmetric shapes are expected at high spins; theycorrespond to superposition of α20 and α33
Figure:Perspective 1 Figure:Perspective 2 Figure:Perspective 3
Observations:
Nuclei with C3h-symmetry predicted to coexist with octupoles
Probably seen already (and mis-interpreted in terms of Iπ=3−)
Jerzy DUDEK, UdS/IPHC CNRS and UMCS Evidence for Octahedral & Tetrahedral Symmetries