5. Exotic modes of nuclear rotation Tilted Axis Cranking -TAC

Cranking Model Seek a mean field solution carrying finite angular momentum. Use the variational principle with the auxillary condition The state |> is the stationary mean field solution in the frame that rotates uniformly with the angular velocity  about the z axis. In the laboratory frame it corresponds to a uniformly rotating mean field state

Low spin: simple droplet. High spin: clockwork of gyroscopes. Uniform rotation about an axis that is tilted with respect to the principal axes is quite common. New discrete symmetries Rotational response Quantization of single particle motion determines relation J(  ).

Quadrupole deformation a

Intrinsic frame Principal axes

Symmetries Broken by m.f. rotational bands

Discrete symmetries Combinations of discrete operations

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

Rotational bands in

Chiral bands

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

mirror Chirality of mass-less particles Only left-handed neutrinos: Parity violation in weak interaction

Consequence of static chirality: Two identical rotational bands with the same parity. Best example of chirality so far Chiral Vibration Tunneling Weak symmetry breaking

Reflection asymmetric shapes, two reflection planes Simplex quantum number Parity doubling 20/23

Weak symmetry breaking

Summary The different discrete symmetries of the m.f. are manifest by different level sequences in the rotational bands. For reflection symmetric shapes, a band has fixed parity and one has: Rotation about a principal axis (signature selects every second I) Rotation about an axis in a principal plane (all I) Rotation about an axis not in a principal plane (all I, for each I a pair of states – chiral doubling) For reflection asymmetric shapes, a band contains both parities. If the rotational axis is normal to one of two reflection planes the bands contain all I and the levels have alternating parity. For reflection asymmetric shapes exists 16 different symmetry types.

5. Emergence of bands

Orienteded mean field solutions This is clearly the case for a well deformed nucleus. Deformed nuclei show regular rotational bands. Spherical nuclei have irregular spectra.

Isotropy broken Isotropy conserved

The rotating nucleus: A Spinning clockwork of gyroscopes Nucleonic orbitals – Highly tropic gyroscopes Orbitals with high nodal structure at the Fermi surface generate orientation

How does orientation come about? Orientation of the gyroscopes Deformed density / potential Deformed potential aligns the partially filled orbitals Partially filled orbitals are highly tropic Nucleus is oriented – rotational band Well deformed 5

Angular momentum is generated by alignment of the spin of the orbitals with the rotational axis Gradual – rotational band Abrupt – band crossing, no bands 7

M1 band in the spherical Nucleus Magnetic rotation – orientation specified by few orbitals SD

Magnetic Rotation Weakly deformed 8

TAC Measurements confirmed the length of the parallel component of the magnetic moment.

Soft deformation: Terminating bands A. Afanasjev et al. Phys. Rep. 322, 1 (99) Orientation of the gyroscopes Deformed density / potential

termination

The nature of nuclear rotational bands The experimentalist’s definition of rotational bands: Requirements for the mean field:

deformationsupernormalweak axes ratio (  1:2 (0.6)1:1.5 (0.3)1:1.1 (0.1) mass /21/ D

Transition to the classical limit

Classical periodic orbits in a deformed potential

Summary Breaking of rotational symmetry does not always mean substantial deviation of the density distribution from sphericity. Magnetic rotors have a non-spherical arrangement of current loops. They represent the quantized rotation of a magnetic dipole. The angular momentum is generated by the shears mechanism. Antimagnetic rotors are like magnetic ones, without a net magnetic moment and signature symmetry. Bands terminate when all angular momentum of the valence nucleons is aligned. The current loops of the valence orbits determine the current pattern and the moment of inertia.

Isotropy broken Isotropy conserved

Summary The mean field may spontaneously break symmetries. The non-spherical mean field defines orientation and the rotational degrees of freedom. The rotating mean field (cranking model) describes the response of the nucleonic motion to rotation. The inertial forces align the angular momentum of the orbits with the rotational axis. The bands are classified as single particle configurations in the rotating mean field. The cranked shell model (fixed shape) is a very handy tool. At moderate spin one must take into account pair correlations. The bands are classified as quasiparticle configurations. Band crossings (backbends) are well accounted for. Nuclei may rotate about a tilted axis New types of discrete symmetries of the mean field.