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Geometric model

Geometric model. A Bohr, 1952, ..................... G Gneuss, W Greiner 1971,. quadrupole tensor of collective coordinates (2 shape param’s, 3 Euler angles ). …corresponding tensor of momenta. neglect higher-order terms. neglect …. Potential energy :. Shape “phases”. B. oblate.

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Geometric model

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  1. Geometric model A Bohr, 1952, ..................... G Gneuss, W Greiner 1971, .... quadrupole tensor of collective coordinates (2 shape param’s, 3 Euler angles ) …corresponding tensor of momenta neglect higher-order terms neglect … Potential energy: Shape “phases” B oblate spherical depends on 2 internal shape variables y prolate Cartesian polar A x Motion in the xy-plane represents evolution of nuclear shape in the principal axis system (PAS). See e.g.: JM Eisenberg, W Greiner: Nuclear Theory, vol.1 (North Holland, Amsterdam, 1970).

  2. Scaling properties 3 independent scales Energy Time Coordinates 4 external parameters => only 1 essential parameter arbitrary choice: (a) A=variable, B,C,K=1 (b) B=variable, A= –1, C,K=1 A=+1, C,K=1 (c) C=variable, A= –1, B,K=1 A=+1, B,K=1 (a) Dynamically equivalent classes determined by the dimensionless parameter (b+) (b–) In the quantum case, energy & time scales connected by the Planck constant =>2 essential parameters

  3. Angular momentum 5D system Principal Axes System Special case: 2D system Dynamics is fully determined by 2 coordinates and the associated 2 momenta

  4. Quantum dynamics Now only the restricted case: J=0 2 physically important quantization options (b) 5D system restricted to 2D (true geometric model of nuclei) (a) 2D system with an additional ansatz on the angular wave function (to avoid quasi-degeneracies due to the three-fold symmetry of V ) Although both options have the same classical limit, they yield different quantum spectra...

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