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5. Exotic modes of nuclear rotation
Tilted Axis Cranking -TAC
Cranking Model
Seek a mean field solution carrying finite angular momentum.
.0|| zJ
Use the variational principle
with the auxillary condition
0|| HEi
0||' zJHEi
The state |> is the stationary mean field solution in the frame that rotates uniformly with the angular velocity about the z axis. In the laboratory frame it corresponds to a uniformly rotating mean field state
symmetry). rotational (broken 1|||| if ||
zz tJitJi
eet
tency selfconsis mfi V
functions) (wave states particle single
routhians) p. (s. frame rotatingin energies particle single '
ial)(potentent fieldmean energy kinetic
(routhian) frame rotating in then hamiltonia fieldmean '
'' -'
i
i
mf
iiizmf
e
Vt
h
ehJVth
Low spin: simple droplet.High spin: clockwork of gyroscopes.
Uniform rotation about an axis that is tilted with respect to the principal axes is quite common. New discrete symmetries
Rotational response
Quantization of single particlemotion determines relation J().
)ˆ(xR
)/))ˆ(exp((1)(
)ˆ(1)ˆ(
0
2
2 20
axRr
VxV
xYRxR
mf
Quadrupole deformation
0)( VxV
0)( xVa
Intrinsic frame
Principal axes
2/sincos
00
20
2211
q
,ˆ toparallel bemust
,
:tencyselfconsis
cossinsincossin
)('
20
321
332211
JJ
JJJxVth mf
Symmetries
zJvtH 12'
Broken by m.f. rotationalbands
Discrete symmetries
Combinations of discrete operations
rotation withreversal time- )(
inversion space-
angleby axis-zabout rotation - )(
y
z
TR
P
R
Common bands
by axis-zabout rotation - )(
rotation withreversal time- 1 )(
inversion space - 1
z
y
R
TR
P
PAC solutions(Principal Axis Cranking)
nI
e iz
2
signature ||)(
R
TAC solutions (planar)(Tilted Axis Cranking) Many cases of strongly brokensymmetry, i.e. no signature splitting
Rotationalbands in
Er163
Chiral bands
Chirality of molecules
mirror
It is impossible to transform one configurationinto the other by rotation.
mirror
Chirality of mass-less particles
Only left-handed neutrinos:Parity violation in weak interaction
Consequence of static chirality: Two identical rotational bands with the same parity.
Best example of chirality so far 7513560 Nd
Chi
ral V
ibra
tion
T
unne
ling
Weak
symmetr
y breaking
Reflection asymmetric shapes,
two reflection planes
Simplex quantum number
I
i
z
parity
e
)(
||
)(
S
PRS
Parity doubling
20/23
Th223
Weak
symmetr
y breaking
Summary
• The different discrete symmetries of the m.f. are manifest by different level sequences in the rotational bands.
• For reflection symmetric shapes, a band has fixed parity and one has:
• Rotation about a principal axis (signature selects every second I)
• Rotation about an axis in a principal plane (all I)
• Rotation about an axis not in a principal plane (all I, for each I a pair of states – chiral doubling)
• For reflection asymmetric shapes, a band contains both parities.
• If the rotational axis is normal to one of two reflection planes the bands contain all I and the levels have alternating parity.
• For reflection asymmetric shapes exists 16 different symmetry types.
5. Emergence of bands
Orienteded mean field solutions
zJiz e )( axis-z about the Rotation R
This is clearly the case for a well deformed nucleus.Deformed nuclei show regular rotational bands.Spherical nuclei have irregular spectra.
n.orientatiodifferent
of states theseall of ionsuperposita is state rotational The
.energy same thehave )( | states field mean All
peaked.sharply is 1|||
.but
|R
|R
RRRR
z
z
zzzz hhHH
deformed
Er163
Isotropybroken
spherical
Pb200
Isotropyconserved
The rotating nucleus: A Spinning clockwork of gyroscopes
Nucleonicorbitals –Highly tropicgyroscopes
Orbitals with high nodal structureat the Fermi surface generate orientation
How does orientation come about?
Orientation of the gyroscopes
Deformed density / potential
Deformed potential aligns thepartially filled orbitals
Partially filled orbitals are highly tropic
Nucleus is oriented – rotational band
Well deformed Hf174 -90 0 90 180 2700.0
0.2
0.4
0.6
0.8
1.0
over
lap
5
Angular momentum is generated by alignment of the spin of the orbitals with the rotational axisGradual – rotational bandAbrupt – band crossing, no bands
7
M1 band in the sphericalNucleus Magnetic rotation –orientation specified byfew orbitals
SD
Pb199
Magnetic Rotation
-90 0 90 180 2700.0
0.2
0.4
0.6
0.8
1.0
over
lap
Weakly deformed Pb199
8
TAC
Measurements confirmed the length of the parallelcomponent of the magnetic moment.
Soft deformation:Terminating bands
A. Afanasjev et al. Phys. Rep. 322, 1 (99)
Orientation of the gyroscopes
Deformed density / potential
],[],[ 2/112/112/9lki hhglik
termination
5810951 Sb
The nature of nuclear rotational bands
The experimentalist’s definition of rotational bands:
Requirements for the mean field:
norientatio peaked.sharply is 1||| |Rz
smoothness 1. toclose is 1|)1(|)(| DII
deformation super normal weak
axes ratio (
1:2 (0.6) 1:1.5 (0.3) 1:1.1 (0.1)
mass 150 180 200
1 1/2 1/7
2 4 20
4 8 20
60 30 8
D 0.005 0.03 0.05
rig /)2(
irrot /)2(
][o
][J
4,, 2 J
R
Rrigirrot
Transition to the classical limit
Classical periodic orbits in a deformed potential
Summary
• Breaking of rotational symmetry does not always mean substantial deviation of the density distribution from sphericity.
• Magnetic rotors have a non-spherical arrangement of current loops. They represent the quantized rotation of a magnetic dipole.
• The angular momentum is generated by the shears mechanism.
• Antimagnetic rotors are like magnetic ones, without a net magnetic moment and signature symmetry.
• Bands terminate when all angular momentum of the valence nucleons is aligned.
• The current loops of the valence orbits determine the current pattern and the moment of inertia.
deformed
Er163
spherical
Pb200
Isotropybroken
Isotropyconserved
Summary
• The mean field may spontaneously break symmetries. • The non-spherical mean field defines orientation and the rotational
degrees of freedom.• The rotating mean field (cranking model) describes the response of the
nucleonic motion to rotation.• The inertial forces align the angular momentum of the orbits with the
rotational axis. • The bands are classified as single particle configurations in the
rotating mean field. The cranked shell model (fixed shape) is a very handy tool.
• At moderate spin one must take into account pair correlations. The bands are classified as quasiparticle configurations.
• Band crossings (backbends) are well accounted for. • Nuclei may rotate about a tilted axis• New types of discrete symmetries of the mean field.
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