Orbits, shapes and currents S. Frauendorf Department of Physics University of Notre Dame

Cranking  rotational response of nuclei, magnetic response of clusters Mean field  shapes, shell structure

Na clusters Shell correction method (Micro-macro method) Jellium approximation All energy density functionals that generate a leptodermic density profile give similar shapes. Shapes reflect the quantized motion of the fermions in the average potential. Frauendorf, Pashkevich, Ann. Physik 5, 34 (1996)

What is the relation between quantized fermionic motion and shapes? What is the current pattern if one sets a deformed nucleus into rotation or put a metal cluster into a magnetic field?

Two transparent situations Large systems: gross structure, Periodic Orbit theory Measures to avoid echoes in the Crowell concert hall Small systems: geometry of the valence orbitals Chemical regimeAcoustic regime =hybridization

Shapes reflect geometry of the occupied orbitals (s-,p-, d- spherical harmonics). Nuclei have a higher surface energy than alkali clusters  more rounded. Surface tension tries to keep the shape spherical. Nuclei and clusters System tries to keep the density near the equilibrium value.

M. Koskinen, P.O. Lipas, M. Manninen, Nucl. Phys. A591, 421 (1995)

M. Koskinen, P.O. Lipas, M. Manninen, Z. Phys. D35, 285 (1995) Hybridization tries to make part of the system“closed shell like”.

Currents and velocity fields of rotating nuclei

Spherical harmonic oscillator N=Z=4 or N=Z=8-2 Deformed harmonic oscillator N=Z=4 (equilibrium shape)

For the harmonic oscillator at equilibrium, the contributions of the vortices to the total angular momentum cancel exactly. The moment of inertia takes the rigid body value. For more realistic (leptodermic) potentials the contributions of the vortices do not cancel. The moment of inertia differs from the rigid body value.

Magnetic rotation of near-spherical nuclei

The acoustic regime System tries to avoid high level density at the Fermi surfaces, seeks a shape with low level density. Bunches of single particle levels make the shell structure. Periodic orbit theory relates level density and shapes.

Periodic orbit theory L length of orbit, k wave numberdamping factor Gross shell structure given by the shortest orbits.

Classical periodic orbits in a spheroidal cavity with small-moderate deformation Equator plane Meridian plane one fold degenerate two fold degenerate

L equator =const L meridian =const Shell energy of a Woods-Saxon potential

Quadrupole: -Sudden onset, gradual decrease  path along meridian valley Strutinsky et al., Z. Phys. A283, 269 (1977) - preponderance of prolate shapes  meridian valley has steeper slope on prolate side H. Frisk, Nucl. Phys. A511, 309 (1990)

Meridian ridge Equator ridge Experimental shell energy of nuclei M. A. Deleplanque et al. Phys. Rev. C69, (2004)

Shapes of Na clusters S. Frauendorf, V.V. Pashkevich, Ann. Physik 5, 34 (1996)

L equator =const L meridian =const Shell energy of a Woods-Saxon potential

Hexadecapole: -positve at beginning of shell, negative at end  system tries to stay in equator valley

Currents Without shell effects (Fermi gas) the flow pattern is rigid.

Deviations from rigid flow orbit Rotational flux is proportional to the orbit area.

equator meridian sphere Modification by rotation/magnetic field flux through orbit perpendicular to rotational axis

Meridian orbits generate for rotation perpendicular to symmetry axis. Moments of inertia and energies classical angular momentum of the orbit Equator orbits generate for rotation parallel to symmetry axis.

rotational alignment Backbends Meridian ridge right scale K-isomers equator ridge M. A. Deleplanque et al. Phys. Rev. C69, (2004) area of the orbit

Current in rotating Lab frame Body fixed frame J. Fleckner et al. Nucl. Phys. A339, 227 (1980)

Superdeformed nuclei equator meridian +- M. A. Deleplanque et al. Phys. Rev. C69, (2004) Moments of inertia rigid although strong shell energy. Orbits do not carry flux.

Shell energy at high spin equator meridian sphere M. A. Deleplanque et al. Phys. Rev. C69, (2004)

parallel perpendicular N M. A. Deleplanque et al. Phys. Rev. C69, (2004)

Summary For small particle number: Hybridized spherical harmonics determine the pattern Shapes and currents reflect the quantized motion of the particles near the Fermi surface For large particle number: Gross shell structure controlled by the shortest classical orbits. Orbit length plays central role. Constant length of meridian orbits  quadrupole deformation Constant length of equator orbits  hexadecapole deformation At zero pairing: Currents in rotating frame are substantial. Moments of inertia differ from rigid body value. Strong magnetic response. Flux through orbit plays central role.

Shapes of Na clusters S. Frauendorf, V.V. Pashkevich, Annalen der Physik 5, 34 (1996)

spherical quadrupole full Meridian ridgeEquator ridge Na clusters stay in the equator valley. Nuclei cannot completely adjust. S. Frauendorf, V.V. Pashkevich, Ann. Physik 5, 34 (1996)

For each term area of the orbit

Two transparent situations Large systems: gross structure, Periodic Orbit theory Chladni pattern of nodes of standing waves in a violin Measures to avoid echoes in the Crowell concert hall Small systems: geometry of the valence orbitals Chemical regimeAcoustic regime

The chemical regime Molecules: The geometry of s- and p- orbitals determines the geometry of molecules. The shape of the lightest nuclei follows the shape of the Valence s-, p-, d- orbitals or combinations thereof (hybridization).

N=138 N=136 N=134 N=132 N=130 N=128 N=126 N=124 N=122

M. Koskinen, P.O. Lipas, M. Manninen, Nucl. Phys. A591, 421 (1995)