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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Presentation transcript:

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

HCl Microwave absorption spectrum Moment of inertia of the dumbbell

Indistinguishable Particles.. 2 Upper particlesLower particles Restriction of orientation

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

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)

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).

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

Spontaneous symmetry breaking Symmetry operation S

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

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

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.

No deformation – no bands?

E2 radiation - electric rotation M1 radiation - magnetic rotation

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

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.

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

Antimagnetic rotation Magnetic rotorAntimagnetic rotor Anti-Ferromagnet Ferromagnet

A. Simons et al. PRL, in press

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

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

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

Rotational bands in 11’2347 PAC TAC

Consequence of chirality: Two identical rotational bands.

band 2 band Pr  h 11/2 h 11/2

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

Chiral sister bands Representative nucleus observed predicted observed observed /37

Chirality of molecules mirror The two enantiomers of 2-iodubutene

mirror Chirality of mass-less particles z

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

Combinations of discrete operations

Good simplex Several examples in mass 230 region Other regions?

Parity doubling Only good case. Must be better studied!

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

Combinations of discrete operations

E3

Parity doubling E3M3

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.