1. Workshop Spiral II - Caen 4-6th October 2005
Exotic Nuclear Shapes
I >> 1
T > 0
I = 0
T = 0
Nicolas Schunck
Departamento de Fisica Teorica,
Universidad Autonoma de Madrid
Cantoblanco 28049, Madrid, Spain
Workshop Spiral II - Caen 4-6th October 2005

2. Standard Nuclear Shapes
Types of nuclear shapes:
Spherical
Prolate
Oblate
Tri-axial
Pear-shape
Workshop Spiral II - Caen 4-6th October 2005

3. Point Symmetries in the Microscopic World
Synthetic inorganic-organic compound with ZnO4 tetrahedral clusters linked by C6H4-C-O2 “struts” (Li, Nature, 1999).
1.29 nm spacing between centers of adjacent clusters.
Workshop Spiral II - Caen 4-6th October 2005

4. Point Symmetries and Nuclear Stability
Shell Gaps  Stable configurations
In nuclei: Higher degeneracies  Larger shell gaps
Degeneracies are a direct consequence of the underlying point symmetry of the shape
J. Dudek, A. Góźdź, N. Schunck and M. Miskiewicz
Phys. Rev. Lett (2002)
Workshop Spiral II - Caen 4-6th October 2005

5. Point Groups and Level Degeneracy
Properties
High symmetries
Low symmetries
Type
Spherical
Tetrahedral
Octahedral
Any other…
Number of Symmetry Elements  48 96 …
New
Degeneracies
2j + 1
4, 2, 2
2, 2
Kramers Degeneracy ( = time-reversal symmetry)
Survey of the properties of a few point groups
Workshop Spiral II - Caen 4-6th October 2005

6. Symmetry Groups Td and Oh
Parameterization of the shape symmetries:
Tetrahedral: α32 ≠ 0
Octahedral: α40, α44 ≠ 0
Other possibilities involving higher order multipoles αλμ
J. Dudek, A. Góźdź and N. Schunck
Act. Phys. Pol. B (2003)
Workshop Spiral II - Caen 4-6th October 2005

7. Tetrahedral Shell GapsTd
Spherical
J. Dudek, A. Góźdź and N. Schunck
Act. Phys. Pol. B (2003)
Workshop Spiral II - Caen 4-6th October 2005

8. Tetrahedral Shell Effects
Shell Correction [MeV]
Tetrahedral Deformation (Rank 7)
Workshop Spiral II - Caen 4-6th October 2005

9. Neutron-rich Zr Isotopes
N. Schunck, J. Dudek, A. Góźdź, P. Regan
Phys. Rev. C (R) (2004)
Workshop Spiral II - Caen 4-6th October 2005

10. Tetrahedral Magic Numbers
From a WS potential:
20, 32, 40, 56-58, 64, 70, 90, 100, 112, …
Existence of Td magic numbers independent of the realization of the mean-field = Universality
Yb isotopes
Zr isotopes
Best candidates: proton-rich or neutron-rich nuclei…
J. Dudek, A. Góźdź, N. Schunck and M. Miskiewicz
Phys. Rev. Lett (2002)
Workshop Spiral II - Caen 4-6th October 2005

11. Experimental signatures (1/2)
Stable tetrahedral minimum: Shape isomer
Low-spin physics: don’t expect I > 10 ħ…
Possibly high excitation energy
Static octupole moment Q3
No dipole moment
Schematic illustrations of where to look for tetrahedral states
Workshop Spiral II - Caen 4-6th October 2005

12. Experimental signatures (2/2)
Weak collective rotation
Yrast traps with parity doublet ? (because of parity-breaking)
Bunch of quasi-particle excitations (because of the 4-fold degeneracy of s.p. levels)
Investigations under way:
RPA (Strasbourg)
GCM (Warsaw)
Spectroscopy with point symmetries (Lublin)
Four-fold degenerate single-particle levels should give rise to multiple quasi-particle excitations
Workshop Spiral II - Caen 4-6th October 2005

13. Workshop Spiral II - Caen 4-6th October 2005
Summary (for Part I)
Mean-field Theories predict tetrahedral configurations in islands of nuclei throughout the nuclear chart
Low-lying states (by opposition to cluster states)
New types of shape coexistence: prolate, oblate, spherical, tetrahedral, octahedral, pear-shaped, etc.
The best candidates will be found away from the valley of stability
The nuclear tetrahedral symmetry reflects the quantal nature of the nucleus (always competing with the macroscopic, liquid-drop aspects)
Workshop Spiral II - Caen 4-6th October 2005

14. Deformed nuclei… in motion
Consequence of the breaking of the rotational invariance: deformed nuclei can rotate…
Example of a super-deformed rotational band
Jacobi Shape transition
Workshop Spiral II - Caen 4-6th October 2005

15. The Jacobi Shape Transition
Example of the Jacobi shape transition in 152Dy
Workshop Spiral II - Caen 4-6th October 2005

16. The various forms of the Jacobi transition
Instability of the Jacobi transition leading to fission
Workshop Spiral II - Caen 4-6th October 2005

17. Jacobi transition and Hyper-deformation
Liquid drop = High-temperature Limit
Workshop Spiral II - Caen 4-6th October 2005

18. Hyper-deformed configurations
Neat Jacobi Transition = a prerequisite for populating SD and HD states
Mean-field at the limits:
Single-particle structure
Effective interaction
Jacobi Shapes and many-body problem
N=8 orbital in N=5 shell
Workshop Spiral II - Caen 4-6th October 2005

19. Workshop Spiral II - Caen 4-6th October 2005
Summary (Part II)
Hyper-deformation (and Jacobi shapes) can be observed only at very high angular momentum
Short spin-window in-between the Jacobi transition and fission
Mass A ~ 100 among the best candidates
The Jacobi shape transition reflects the macroscopic, liquid-drop nature of the nucleus (always competing with the quantal aspects)
Workshop Spiral II - Caen 4-6th October 2005

20. Collaboration Network
IReS-ULP, Strasbourg, France
Institute of Theoretical Physics, Warsaw University, Poland
University Marie Curie-Skłodowska, Lublin, Poland
University of Notre Dame, USA
Université Libre de Bruxelles, Bruxelles, Belgium
University of Surrey, Guildford, UK
Niels Bohr Institute, Copenhagen, Denmark
Henrik Niewodniczanski Institute of Nuclear Physics, Kraków, Poland
Workshop Spiral II - Caen 4-6th October 2005

21. Workshop Spiral II - Caen 4-6th October 2005
Bibliography
X. Li, J. Dudek, Phys. Rev. C49 (1994) 1250(R)
S. Takami , K. Yabana, M. Matsuo, Phys. Lett. B431 (1998)
M. Yamagami, K. Matsuyanagi, M. Matsuo, Nucl. Phys. A693 (2001)
J. Dudek, A. Góźdź, N. Schunck and M. Miskiewicz, Phys. Rev. Lett. 88 (2002)
J. Dudek, A. Góźdź and N. Schunck, Act. Phys. Pol. B (2003)
N. Schunck, J. Dudek, A. Góźdź, P. Regan, Phys. Rev. C69 (2004) (R)
Workshop Spiral II - Caen 4-6th October 2005