Normal persistent currents and
gross shell structure at high spin
S. Frauendorf
Department of Physics
University of Notre Dame
Interested in 3D system that adjust its shape – quantum droplets:
Nuclei and metal clusters
Collaborators:
M. Deleplanque
V.V. Pashkevich
A. Sanzhur
Relation between rotation and
magnetism
Hamiltonian in the rotating frame
Hamiltonian with external
magnetic field
Larmor frequency
H R  H  lz
m L2 2
H M  H   Ll z 
( x  y2 )
2
B B
L 

For small frequency the quadratic term can be neglected,
and Larmor’s theorem holds.
Will be considered: Magnetic length (cyclotron radius) >> size
Induced currents give the magnetic response of the electron system
Currents in the rotating frame give the deviation from rigid rotation.

jlab  jbody    r
Large system : unmagnetic/rigid rotation
Small systems: shell effects
Strong magnetic response/deviation from rigid rotation
Susceptibility
A  shell    rigid
Normal persistent currents
Moment of inertia
Magnetism and shapes are difficult to measure for clusters.
Moments of inertia and shapes of rotating are nuclei well known.
Look at high spin. At low spin there are pair correlations
Harmonic oscillator is misleading.
Phys. Rev. C 69 044309 (2004)
160
150
Er
Dy
Experimental
moments
of inertia
Imax>15,16,17 for N<50, 50<N<82, N>82 resp.
 rgid
Imax>20
 rgid
Microscopic calculations
Shell correction method, using Woods Saxon potential
Mimimize
Periodic orbit theory
L length of orbit, k wave number
damping factor
Basic and supershell structure
Square and triangle are the dominant orbits.
The difference of their lengths causes the supershell beat.
Basic shell structure determined by one orbit (square).
Eshell  sin[( Lsquare  Ltriangle)k f ] sin[( Lsquare  Ltriangle)k f / 2]
super oscillation
basic oscillation
Equator plane
one fold degenerate
Meridian plane
two fold degenerate
Classical periodic orbits in a spheroidal cavity
with small-moderate deformation
L equator =const
L meridian =const
Shell energy of a spheroidal cavity
L equator =const
L meridian =const
Shell energy of a Woods-Saxon potential
Meridian ridge
Equator ridge
Influence of rotation
H '  H  l
S. C. Craegh, Ann. Phys. (N.Y.) 248, 60 (1996)
1st order perturbation theory:
Change of action: integrate the perturbation over the
unperturbed orbit
rotational flux
Area field
Modulation factor
 g 
rotation   M  g 
 E
rotation   M  E


1
i  ( , )
2
modulation factor M  
d

e

1

a

  ...

4
sphere
meridian
equator
Moments of inertia and energies
classical angular momentum of the orbit
For each term
right scale
spikes
rotational alignment
Backbends
Meridian
ridge
K-isomers
equator
ridge
spikes
Superdeformed nuclei
equator
+
-
meridian
Shell energy at high spin
sphere
meridian
equator
parallel
N

perpendicular
  0
  0.3MeV
  0.6MeV
Summary
Far-reaching analogy between magnetic susceptibility and
deviation from rigid body moment of inertia
Shell structure leads to normal persistent currents.
Basic shell structure qualitatively described by one family
of periodic orbits (triangle or four angle). Supershells are due
to the interference of two (triangle and four angle).
Rotational/magnetic flux through the orbit controls the shell
oscillations as functions of the rotational/Larmor frequency
Length of the orbit controls the shell oscillations as functions
Of the particle number.
Shell structure of ground state energy and moment of inertia
strongly correlated.
Oblate parallel
Prolate parallel
Oblate pependicular
Prolate perpendicular
optimal