Spontaneous symmetry breaking and rotational bands. S. Frauendorf. Department of Physics University of Notre Dame. x. The collective model. Eveneven nuclei, low spin. Deformed surface breaks rotational
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S. Frauendorf
Department of Physics
University of Notre Dame
x
Eveneven nuclei, low spin
Deformed surface breaks rotational
the spherical symmetry band
Collective and single particle degrees of freedom
On each single particle state
(configuration) a rotational band
is built (like in molecules).
Limitations:
Single particle and collective degrees of freedom
become entangled at high spin and low deformation.
Rotational
bands in
More microscopic approach:
Mean field theory
+
concept of spontaneous symmetry
breaking for interpretation.
Retains the simple picture of an anisotropic object going round.
Reaction of the
nucleons to
the inertial
forces must be
taken into account
Start from the Hamiltonian in a rotating frame
Mean field approximation:
find state > of (quasi) nucleons moving independently in
mean field generated by all nucleons.
Selfconsistency : effective interactions, density functionals (Skyrme, Gogny, …),
Relativistic mean field, MicroMacro (Strutinsky method)
…….
Low spin: simple droplet.
High spin: clockwork of gyroscopes.
Quantization of single particle
motion determines relation J(w).
Uniform rotation about an axis that
is tilted with respect to the principal
axes is quite common.
New discrete symmetries
Mean field theory:
Tilted Axis Cranking TAC
S. Frauendorf Nuclear Physics A557, 259c (1993)
Full twobody Hamiltonian H’
Mean field approximation
Mean field Hamiltonian h’ and m.f. state h’>=e’>.
Symmetry operation S and
Spontaneous symmetry breaking
Symmetry restoration
Broken by m.f.
rotational
bands
Combinations of discrete operations
spin
parity
sequence
Obeyed by m.f.
doubling
of
states
broken by m.f.
is invariant under
nucleons on
highj orbits
specify orientation
Deformed charge distribution
Rotational degree of freedom and rotational bands.
Principal Axis Cranking
PAC solutions
TAC or planar tilted solutions
Many cases of strongly broken
symmetry, i.e.
no signature splitting
Rotational
bands in
E2 radiation  electric rotation
I1/2
23
24
25
22
26
27
21
M1 radiation  magnetic rotation
28
20
19
No deformation – no bands?
10’
Baldsiefen et al. PLB 275, 252 (1992)
2 proton particles
2 neutron holes
Magnetic rotor composed of two current loops
The nice rotor
consists of
four highj
orbitals only!
repulsive looploop
interaction
E
J
Shears mechanism
Staggering in Multiplets!
Why so regular?
Most of the ll interaction due to a slight
quadrupole polarization of the nucleus.
Keeps two highj holes/particles
in the blades well aligned.
The 4 highj orbitals contribute incoherently
to staggering.
First clear experimental evidence: Clark et al. PRL 78 , 1868 (1997)
TAC
Long transverse magnetic dipole vectors, strong B(M1)
B(M1) decreases with spin: band termination
Experimental magnetic moment confirms picture.
Experimental B(E2) values and spectroscopic quadrupole
moments give the calculated small deformation.
AntiFerromagnet
Ferromagnet
Magnetic rotor
Antimagnetic rotor
24
24
23
22
22
21
20
20
19
18
18
weak
electric
quadrupole
transitions
strong
magnetic
dipole
transitions
Band termination
A. Simons et al. PRL 91, 162501 (2003)
Ordinary rotor
Magnetic rotor
J
Terminating bands
Degree of orientation (A=180, width of
Many particles
2 particles, 2 holes
Deformation:
20’
Chiral or aplanar solutions:
The rotational axis is out of all principal planes.
Consequence of chirality: Two identical rotational bands.
Frauendorf, Meng,
Nucl. Phys. A617, 131 (1997)
20 0.22 29
23 0.20 29
Composite chiral bands
Demonstration of the symmetry concept:
It does not matter how the three components
of angular momentum are generated.
Best candidates
Composite chiral band in
S. Zhu et al.
Phys. Rev. Lett.
91, 132501 (2003)
chiral regime
chiral
regime
chiral
regime
Chiral sister states:
Tunneling between the left and
righthanded configurations
causes splitting.
Rotationalfrequency
Energy difference between chiral sister bands

+
B(out)
B(in)
Sensitive to details of the system
Branching B(out)/B(in) sensitive to details.
Robust: B(in)+B(out)=B(+in)+B(+out)=B(lh)=B(rh)
Chiral
regime
J. Timar et al.
Phys Lett. B 598
178 (2004)
Chirality
Oddodd: 1p1h
Evenodd: 2p1h, 1p2h
Eveneven: 2p2h
Best
13 0.18 26
observed
13 0.21 14
observed
predicted
13 0.21 40
13 0.21 14
predicted
predicted
45 0.32 26
Chiral
sister bands
Representative
nucleus
nucleus
massless particle
molecule
New type of
chirality
29’
Two mirror planes
Combinations of discrete operations
Good simplex
Several examples in mass 230 region
Parity doubling
Only good case.
J. Dudek et al. PRL 88 (2002) 252502
minimum
maximum
Which orientation has the rotational axis?
Classical no preference
E3 M2
E3 M2
Predicted as best case (so far):
Prolate ground state
Tetrahedral isomer at 2 MeV
Comes down by particle alignment
34’
Orientation is generated by the asymmetric distribution
quantal orbits near the Fermi surface
Orientation does not always mean a deformed charge density:
Magnetic rotation – axial vector deformation.
Nuclei can rotate about a tilted axis: New discrete symmetries.
New type of chirality in rotating triaxial nuclei:
Time reversal changes lefthanded into right handed system.
Bands in nuclei with tetrahedral symmetry predicted
Thanks to my collaborators!
V. Dimitrov, S. Chmel, F. Doenau, N. Schunck,
Y. Zhang, S. Zhu
Microscopic (“finite system”)
Rotational levels become observable.
Spontaneous symmetry breaking
=
Appearance of rotational bands.
Energy scale of rotational levels in
Tiniest external fields generate a superposition of the JM>
that is oriented in space, which is stable.
Spontaneous symmetry breaking
Macroscopic (“infinite”) system
Hamiltonian rotational invariant
Why do we see the chair shape?
3
2
1
Symmetry broken state:
approximation, superposition of IM> states:
calculate electronic state for given position of nuclei.
Quadrupole deformation
Axial vector deformation
J
Degree of orientation (width of
Orientation is specified by the order parameter
Electric quadrupole moment magnetic dipole moment
Ordinary “electric” rotor
Magnetic rotor
Robust:

+
out
in
Branching sensitive to details.