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Symmetries of the Cranked Mean Field

Explore how symmetries characterize rotations in nuclei, from Tilted Axis Cranking to Magnetorotational Bands, revealing new chirality types and discrete symmetries. Paradigm for fermionic systems.

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Symmetries of the Cranked Mean Field

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  1. Symmetries of the Cranked Mean Field S. Frauendorf IKH, Forschungszentrum Rossendorf, Dresden Germany Department of Physics University of Notre Dame USA

  2. HCl Moment of inertia of the dumbbell Microwave absorption spectrum

  3. . Upper particles Lower particles . Indistinguishable Particles 2 Restriction of orientation

  4. Nuclei are different Nucleons are not on fixed positions Most particles are identical All particles have the same mass. What is rotating? The nuclear mean field

  5. Rotating mean field: Tilted Axis Cranking model Seek a mean field state |> carrying finite angular momentum, where |> is a Slater determinant (HFB vacuum state) Use the variational principle with the auxiliary condition The state |> is the stationary mean field solution in the frame that rotates uniformly with the angular velocity w about the z axis. S. Frauendorf Nuclear Physics A557, 259c (1993)

  6. NEW: The principal axes of the density distribution need not coincide with the rotational axis (z). Variational principle : Hartree-Fock effective interaction Density functionals (Skyrme, Gogny, …) Relativistic mean field Micro-Macro (Strutinsky method) ……. (Pairing+QQ) X

  7. Tilted rotation The nucleus is not a simple piece of matter, but more like a clockwork of gyroscopes. Uniform rotation about an axis that is tilted with respect to the principal axes is quite common.

  8. Spontaneous symmetry breaking Symmetry operation S

  9. invariant? leave Broken by m.f. rotational bands Combinations of discrete operations spin parity sequence Obeyed by m.f. Which symmetries

  10. Deformed charge distribution Rotational degree of freedom and rotational bands. Microscopic approach to the Unified Model.

  11. Moments of inertia The moment of inertia are determined by the quantal orbits of the nucleons and the pair correlations. A complicated relationship, but the cranking model provides accurate values.

  12. No deformation – no bands?

  13. E2 radiation - electric rotation M1 radiation - magnetic rotation

  14. Rotor composed of current loops, which specify the orientation. Axial vector deformation. Orientation specified by the magnetic dipole moment. Magnetic rotation.

  15. Shears mechanism Most of interaction is due to polarization of the core. TAC calculations describe the phenomenon. Residual interaction between high-j orbitals may play an important role.

  16. TAC Long transverse magnetic dipole vectors, strong B(M1) B(M1) decreases with spin.

  17. Anti-Ferromagnet Ferromagnet Antimagnetic rotation Magnetic rotor Antimagnetic rotor

  18. A. Simons et al. PRL, in press

  19. J Quadrupole deformation Axial vector deformation Orientation is specified by the order parameter Electric quadrupole moment magnetic dipole moment Magnetic rotation is manifest by regular rotational bands in nuclei with near spherical charge distribution. 23/42

  20. invariant? leave Broken by m.f. rotational bands spin parity sequence Obeyed by m.f. Which symmetries Combinations of discrete operations

  21. Principal Axis Cranking PAC solutions Tilted Axis Cranking TAC or planar tilted solutions Chiral or aplanar solutions Doubling of states

  22. TAC PAC Rotational bands in 1 1’ 2 3 4 7

  23. Consequence of chirality: Two identical rotational bands.

  24. band 2 band 1 134Pr ph11/2 nh11/2

  25. The prototype of a chiral rotor Frauendorf, Meng, Nucl. Phys. A617, 131 (1997)

  26. 13 0.18 26 observed 13 0.21 14 observed predicted 13 0.21 40 13 0.21 14 predicted predicted 45 0.32 26 23 0.20 29 observed Chiral sister bands Representative nucleus 31/37

  27. Chirality of molecules mirror The two enantiomers of 2-iodubutene

  28. mirror z Chirality of mass-less particles

  29. New type of chirality Chirality Changed invariant Molecules Massless particles space inversion time reversal Nuclei time reversal space inversion

  30. Combinations of discrete operations

  31. Good simplex Several examples in mass 230 region Other regions?

  32. Parity doubling Only good case. Must be better studied!

  33. Tetrahedral shapes J. Dudek et al. PRL 88 (2002) 252502

  34. Combinations of discrete operations

  35. E3

  36. M3 E3 Parity doubling

  37. Summary Symmetries of the mean field are very useful to characterize nuclear rotational bands. Orientation does not always mean a deformed charge density: Magnetic rotation. Nuclei can rotate about a tilted axis: New discrete symmetries. New type of chirality: Time reversal changes left-handed into right handed system. Paradigm for non-nuclear fermionic systems.

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