Post on 28-Dec-2015
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.
282000
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2220 ))12(21(2000
1120 ))12(21(2000 2000
2203000
2228 )1(2212000
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)
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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.
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
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)
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
120
140
-0.2 0 0.2 0.440
60
80
100
120
140
-0.2 0 0.2 0.440
60
80
100
120
140
-0.2 0 0.2 0.440
60
80
100
120
140
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.
em
m
i
l
xLB
xrig
L
z
current electric
)()( fieldvelocity
fixedbody
)( rotation rigid
)(
)()( fieldvelocity
))()()()((2
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)
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).
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)