Orbits, shapes and currents

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

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

4. 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. Na clusters Shell correction method (Micro-macro method) Jellium approximation Frauendorf, Pashkevich, Ann. Physik 5, 34 (1996)

6. What is the relation between quantizedfermionic 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?

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

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

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

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

11. Currents and velocity fields of rotating nuclei

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

13. 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.

14. Magnetic rotation of near-spherical nuclei

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

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

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

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

19. 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)

20. Meridian ridge Equator ridge Experimental shell energy of nuclei M. A. Deleplanque et al. Phys. Rev. C69, 044309 (2004)

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

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

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

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

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

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

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

30. area of the orbit rotational alignment Backbends Meridian ridge K-isomers equator ridge right scale M. A. Deleplanque et al. Phys. Rev. C69, 044309 (2004)

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

32. Superdeformed nuclei equator meridian + - Orbits do not carry flux. Moments of inertia rigid although strong shell energy. M. A. Deleplanque et al. Phys. Rev. C69, 044309 (2004)

33. sphere meridian equator Shell energy at high spin M. A. Deleplanque et al. Phys. Rev. C69, 044309 (2004)

34. parallel N perpendicular M. A. Deleplanque et al. Phys. Rev. C69, 044309 (2004)

35. Summary Shapes and currents reflect the quantized motion of the particles near the Fermi surface For small particle number: Hybridized spherical harmonics determine the pattern 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.

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

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

46. For each term area of the orbit

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

48. 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).

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