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II. Spontaneous symmetry breaking

II. Spontaneous symmetry breaking. States of different IM are so dense that the tiniest interaction With the surroundings generates a wave packet that is well oriented. II.1 Weinberg’s chair. Hamiltonian rotational invariant. Why do we see the chair shape?. Spontaneously broken symmetry.

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II. Spontaneous symmetry breaking

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  1. II. Spontaneous symmetry breaking

  2. States of different IM are so dense that the tiniest interaction With the surroundings generates a wave packet that is well oriented. II.1 Weinberg’s chair Hamiltonian rotational invariant Why do we see the chair shape? Spontaneously broken symmetry

  3. Tiniest external fields generate a superposition of the |JM> that is oriented in space, which is stable. Spontaneous symmetry breaking Macroscopic (“infinite”) system

  4. 3 Axial rotor 2 1 The molecular rotor

  5. . . Born-Oppenheimer Approximation Electronic motion Vibrations Rotations CO

  6. Microscopic (“finite system”) Rotational levels become observable. Spontaneous symmetry breaking = Appearance of rotational bands. Energy scale of rotational levels in

  7. HCl Microwave absorption spectrum Rotational bands are the manifestation of spontaneous symmetry breaking.

  8. II.2 The collective model Most nuclei have a deformed axial shape. The nucleus rotates as a whole. (collective degrees of freedom) The nucleons move independently inside the deformed potential (intrinsic degrees of freedom) The nucleonic motion is much faster than the rotation (adiabatic approximation)

  9. Axial symmetry The nucleus does not have an orientation degree of freedom with respect to the symmetry axis. Nucleons are indistinguishable

  10. Limitations: Single particle and collective degrees of freedom become entangled at high spin and low deformation. Rotational bands in Adiabatic regime Collective model

  11. II.3 Microscopic approach: Mean field theory + concept of spontaneous symmetry breaking for interpretation. Retains the simple picture of an anisotropic object going round.

  12. Reaction of the nucleons to the inertial forces must be taken into account Rotating mean field (Cranking model): Start from the Hamiltonian in a rotating frame Mean field approximation: find state |> of (quasi) nucleons moving independently in mean field generated by all nucleons. Selfconsistency : effective interactions, density functionals (Skyrme, Gogny, …), Relativistic mean field, Micro-Macro (Strutinsky method) …….

  13. Rotational response Low spin: simple droplet. High spin: clockwork of gyroscopes. Quantization of single particle motion determines relation J(w). Uniform rotation about an axis that is tilted with respect to the principal axes is quite common. New discrete symmetries Mean field theory: Tilted Axis Cranking TAC S. Frauendorf Nuclear Physics A557, 259c (1993)

  14. Spontaneous symmetry breaking Full two-body Hamiltonian H’ Mean field approximation Mean field Hamiltonian h’ and m.f. state h’|>=e’|>. Symmetry operation S and Spontaneous symmetry breaking Symmetry restoration

  15. Broken by m.f. rotational bands Combinations of discrete operations spin parity sequence Obeyed by m.f. doubling of states broken by m.f. Which symmetries can be broken? is invariant under

  16. nucleons on high-j orbits specify orientation Deformed charge distribution Rotational degree of freedom and rotational bands.

  17. Isotropy conserved Isotropy broken

  18. Current in rotating J. Fleckner et al. Nucl. Phys. A339, 227 (1980) Lab frame Body fixed frame Moments of inertia reflect the complex flow. No simple formula.

  19. Deformed?

  20. Rotor composed of current loops, which specify the orientation. Orientation specified by the magnetic dipole moment. Magnetic rotation.

  21. Combinations of discrete operations II.3 Discrete symmetries

  22. Common bands PAC solutions (Principal Axis Cranking) TAC solutions (planar) (Tilted Axis Cranking) Many cases of strongly broken symmetry, i.e. no signature splitting

  23. Rotational bands in

  24. Chiral bands

  25. Examples for chiral sister bands

  26. Chirality It is impossible to transform one configuration into the other by rotation. mirror

  27. Only left-handed neutrinos: Parity violation in weak interaction mirror mass-less particles

  28. Simplex quantum number Reflection asymmetric shapes,two reflection planes Parity doubling

  29. II.4 Spontaneous breaking of isospin symmetry Form a condensate “isovector pair field”

  30. The relative strengths of pp, nn, and pn pairing are determined by the isospin symmetry

  31. Symmetry restoration –Isorotations (strong symmetry breaking – collective model)

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