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Bohr space in six dimensions

Bohr space in six dimensions. P.E. Georgoudis Institute of Nuclear and Particle Physics, N.C.S.R. “ Demokritos ” and Physics Department, National Technical University of Athens. Athens, June 17, 2013. Bohr space in six dimensions. Motivation Nuclear collective modes of motion

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Bohr space in six dimensions

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  1. Bohr space in six dimensions P.E. Georgoudis Institute of Nuclear and Particle Physics, N.C.S.R. “Demokritos” and Physics Department, National Technical University of Athens Athens, June 17, 2013

  2. Bohr space in six dimensions • Motivation Nuclear collective modes of motion Structural Evolution of Atomic Nuclei Bohr Model Interacting Boson Model • The embedding in six dimensions Moments of inertia, Conformal factor and cosmological similarities O(6) Bohr hypersurfaces and the Infinity Contraction to E(5) • Outcomes IBM and Critical Point Symmetries

  3. Collective Motion and Structural Evolution A<150 150<A<190, A>220

  4. Bohr Model: Nuclear Shapes Almost spherical (γ unstable) Axially symmetric Spherical Ground State Quadrupole Deformation

  5. Interacting Boson Model Collective motion and Structural Evolution is mainly characterized by the valence nucleons. F. Iachello, A. Arima, The Interacting Boson Model, Cambridge University Press, 1987. s and d bosons represent paired valence nucleons of angular momentum zero and two. Nuclear states sit in the representations of the U(6) symmetry group. U(6)

  6. Interacting Boson Model O(6) Deformed phase Spherical phase 2nd order U(5) SU(3) 1st order Geometrical limit: Coherent states of U(6) translate the Dynamical symmetries into phases of nuclear structure, expressed as energy surfaces in Bohr coordinates. But the correspondent hamiltonians differ from Bohr’s. A.E.L Dieperink, O. Scholten, F. Iachello, Phys. Rev. Lett. 44, 1747, (1980).

  7. Critical Point Symmetries Phase Transistions in the Bohr Model : E(5) and X(5) symmetries F. Iachello, Phys. Rev. Lett. 85, 3580 (2000) F. Iachello, Phys. Rev. Lett. 87, 052502 (2001) γ-unstable SU(1,1) x SO(5)D.J.Rowe, J.L Wood, Fundamentals of Nuclear Models, World Scientific, 2010. SU(1,1) x SO(5) E(5) spherical X(5) prolate SU(1,1) x SO(5) E(5)

  8. Questions • The standard one: Connection between the symmetries of the IBM and those of the • Bohr Model. O(6) and SU(3) absent in Bohr Model. • The new one: Connection between Critical Point Symmetries of Bohr Model • and the Quantum Phase Transitions of the IBM. E(5) and X(5) absent in the IBM. E(5): As a symmetry characterizing a second order phase transition, is there any relevance with conformal invariance?

  9. Bohr Space SU(1,1) x SO(3) SU(1,1) x SO(5) D.J.Rowe, T.A. Welsh, M.A.Caprio, Phys.Rev.C 79,054304 ,(2009)

  10. Moments of Inertia D. Bonatsos, P. E. Georgoudis, D. Lenis, N. Minkov, and C. Quesne, Phys. Rev. C 83, 044321 (2011).

  11. Formal Correspondence with R-W geometry Formal Correspondence with the cosmological Robertson-Walker line element, which maps a hypersphere onto a tangent plane.

  12. Five Sphere The five sphere lives in a 5+1 Euclidean space. Bohr space is embedded in six dimensions. A six dimensional geometry is also predicted by the geometrical limit of the IBM. Bohr Hypersurface O(6) χ Bohr Space SU(1,1)xSO(5)

  13. O(6) and Contraction to E(5) at infinity O(6) generators at the boundary of the five sphere Contracts to E(5) at infinity

  14. Conclusions Geometric Manifestation of O(6) symmetry in Bohr model. γ-unstable The conformal factor connects the E(5) with the O(6) symmetry.This is a framework where conformal invariance and its relevance with E(5) can be discussed. SU(1,1) x SO(5) E(5) spherical X(5) prolate

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