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Symmetries of the Cranked Mean Field S. Frauendorf Department of Physics University of Notre Dame USA IKH, Forschungszentrum Rossendorf, Dresden Germany
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HCl Microwave absorption spectrum Moment of inertia of the dumbbell
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Indistinguishable Particles.. 2 Upper particlesLower particles Restriction of orientation
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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
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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 about the z axis. S. Frauendorf Nuclear Physics A557, 259c (1993)
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Variational principle : Hartree-Fock effective interaction Density functionals (Skyrme, Gogny, …) Relativistic mean field Micro-Macro (Strutinsky method) ……. (Pairing+QQ) X NEW: The principal axes of the density distribution need not coincide with the rotational axis (z).
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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. Tilted rotation
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Spontaneous symmetry breaking Symmetry operation S
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Which symmetries Combinations of discrete operations leave invariant? Broken by m.f. rotational bands Obeyed by m.f. spin parity sequence
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Rotational degree of freedom and rotational bands. Microscopic approach to the Unified Model. Deformed charge distribution
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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.
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No deformation – no bands?
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E2 radiation - electric rotation M1 radiation - magnetic rotation
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Rotor composed of current loops, which specify the orientation. Orientation specified by the magnetic dipole moment. Magnetic rotation. Axial vector deformation.
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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.
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TAC Long transverse magnetic dipole vectors, strong B(M1) B(M1) decreases with spin.
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Antimagnetic rotation Magnetic rotorAntimagnetic rotor Anti-Ferromagnet Ferromagnet
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A. Simons et al. PRL, in press
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Magnetic rotation is manifest by regular rotational bands in nuclei with near spherical charge distribution. J Quadrupole deformation Axial vector deformation Orientation is specified by the order parameter Electric quadrupole moment magnetic dipole moment 23/42
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Which symmetries Combinations of discrete operations leave invariant? Broken by m.f. rotational bands Obeyed by m.f. spin parity sequence
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Principal Axis Cranking PAC solutions Tilted Axis Cranking TAC or planar tilted solutions Chiral or aplanar solutions Doubling of states
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Rotational bands in 11’2347 PAC TAC
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Consequence of chirality: Two identical rotational bands.
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band 2 band 1 134 Pr h 11/2 h 11/2
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The prototype of a chiral rotor Frauendorf, Meng, Nucl. Phys. A617, 131 (1997 )
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Chiral sister bands Representative nucleus observed13 0.21 14 13 0.21 40 13 0.21 14 predicted 45 0.32 26 observed 23 0.20 29 observed13 0.18 26 31/37
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Chirality of molecules mirror The two enantiomers of 2-iodubutene
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mirror Chirality of mass-less particles z
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New type of chirality Chirality Changed invariant Molecules Massless particles space inversion time reversal Nuclei time reversal space inversion
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Combinations of discrete operations
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Good simplex Several examples in mass 230 region Other regions?
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Parity doubling Only good case. Must be better studied!
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Tetrahedral shapes J. Dudek et al. PRL 88 (2002) 252502
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Combinations of discrete operations
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E3
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Parity doubling E3M3
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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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