Orbits, shapes and currents

S. Frauendorf

Department of Physics

University of Notre Dame

Cranking rotational response of nuclei, magnetic response of clusters

)()]}([{0)]([ rrr iiiVtde

)(),(})]([{ rjrr iiizlVt

Mean field shapes, shell structure

Na clusters Shell correction method(Micro-macro method)Jellium approximation

All energy density functionals thatgenerate a leptodermic density profilegive similar shapes.

Shapes reflect the quantized motionof the fermions in the average potential.

Frauendorf, Pashkevich, Ann. Physik 5, 34 (1996)

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?

Two transparent situations

Large systems: gross structure, Periodic Orbit theory

Measures to avoid echoes in theCrowell concert hall

Small systems: geometryof the valence orbitals

Chemical regime Acoustic regime

=hybridization

Shapes reflect geometry of the occupiedorbitals (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.

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M. K

oskinen, P.O

. Lipas, M

. Manninen, N

ucl. Phys. A

591, 421 (1995)

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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

look"curly " a have harmonics spherical

current finite orbitalsempty admixes term zl

details cmicroscopion

depend inertia ofmoment and vorticesofstrength

)/(|| orbitalsempty of admixture ijz jli

)()(

frame fixedbody in

rervrv xLB

Spherical harmonic oscillatorN=Z=4 or N=Z=8-2

Deformed harmonic oscillatorN=Z=4 (equilibrium shape)

)(/)()(

fieldvelocity

rrjrv mL

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 number damping factor

Gross shell structure given by the shortest orbits.

Classical periodic orbits in a spheroidal cavitywith 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 shapesmeridian 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, 044309 (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 endsystem tries to stay in equator valley

Currents

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

)( rerv xrig

Deviations from rigid flow

fdmrot

2

Larmor :flux magnetic

orbit

Rotational fluxis proportional to theorbit area.

equator

meridian

sphere

Modification by rotation/magnetic field

...14

1factor modulation

rotation

rotation

2),(

aedM

EME

gMg

i

flux through orbitperpendicular to rotational axis

Meridian orbits generate for rotation perpendicular to symmetry axis .

Moments of inertia and energies

classical angular momentum of the orbit

shbodyrigid

sh

Equator orbits generate for rotation parallel to symmetry axis .

sh

rotational alignmentBackbendsMeridian ridge

right scale

K-isomersequator ridge

M. A. Deleplanque et al. Phys. Rev. C69, 044309 (2004)

area of the orbit

Current in rotating

162

Yb162

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, 044309 (2004)

Moments of inertia rigidalthough strong shell energy.

Orbits do not carry flux.

Shell energy at high spin

equator

meridian

sphere

M. A. Deleplanque et al. Phys. Rev. C69, 044309 (2004)

-0.2 0 0.2 0.440

60

80

100

120

140

-0.2 0 0.2 0.440

60

80

100

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-0.2 0 0.2 0.440

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80

100

120

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-0.2 0 0.2 0.440

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80

100

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parallel

perpendicular

N

0 MeV3.0 MeV6.0M. A. Deleplanque et al. Phys. Rev. C69, 044309 (2004)

0equatorM0equatorM

0meridianMreduced meridianM

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)

em

m

i

l

xLB

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L

z

current electric

)()( fieldvelocity

fixedbody

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)(

)()( fieldvelocity

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1)( current mass

)()()( density particle

current finite orbitalsempty admixes term

rervrv

rerv

r

rjrv

rprrprrj

rrr

spherical

quadrupole

full

Meridian ridge Equator 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 standingwaves in a violin

Measures to avoid echoes in theCrowell concert hall

Small systems: geometryof the valence orbitals

Chemical regime Acoustic regime

The chemical regimeMolecules: 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

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228 ))12(21(2000

2220 ))12(21(2000

1120 ))12(21(2000 2000

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228 212000

182000

800

181000

281000

121000

221000

122 ))11(11(1000

22 00)0,0( nml

M. K

oskinen, P.O

. Lipas, M

. Manninen, N

ucl. Phys. A

591, 421 (1995)

282000

228 ))12(21(2000

2220 ))12(21(2000

1120 ))12(21(2000 2000

283000

228 212000

182000

800

181000

281000

121000

221000

122 ))11(11(1000

22 00)0,0( nml

M. K

oskinen, P.O

. Lipas, M

. Manninen, N

ucl. Phys. A

591, 421 (1995)