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II. Spontaneous symmetry breaking

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II. Spontaneous symmetry breaking II.1 Weinberg s chair The molecular rotor Born-Oppenheimer Approximation II.2 The collective model Rotating mean field (Cranking ... – PowerPoint PPT presentation

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Title: II. Spontaneous symmetry breaking


1
II. Spontaneous symmetry breaking
2
II.1 Weinbergs chair
Hamiltonian rotational invariant
Why do we see the chair shape?
Spontaneously broken symmetry
3
Tiniest external fields generate a superposition
of the JMgt that is oriented in space, which is
stable.
Spontaneous symmetry breaking
Macroscopic (infinite) system
4
The molecular rotor
5
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6
Born-Oppenheimer Approximation
Electronic motion
Vibrations
Rotations
CO
7
Microscopic (finite system)
Rotational levels become observable.
Spontaneous symmetry breaking Appearance of
rotational bands.
Energy scale of rotational levels in
8
Microwave absorption spectrum
Rotational bands are the manifestation of
spontaneous symmetry breaking.
9
II.2 The collective model
Most nuclei have a deformed axial shape.
The nucleus rotates as a whole. (collective
degrees of freedom)
The nucleons move independently inside the
deformed potential (intrinsic degrees of freedom)
The nucleonic motion is much faster than the
rotation (adiabatic approximation)
10
Nucleons are indistinguishable
11
Adiabatic regime Collective model
12
II.3 Microscopic approach
Mean field theory concept of spontaneous
symmetry breaking for interpretation.
Retains the simple picture of an anisotropic
object going round.
13
Rotating mean field (Cranking model)
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 gt of
(quasi) nucleons moving independently in mean
field generated by all nucleons.
Selfconsistency effective interactions,
density functionals (Skyrme, Gogny, ),
Relativistic mean field, Micro-Macro
(Strutinsky method)
.
14
Rotational response
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)
15
Spontaneous symmetry breaking
Full two-body Hamiltonian H
Mean field approximation
Mean field Hamiltonian h and m.f. state
hgtegt.
Symmetry operation S and
Spontaneous symmetry breaking
Symmetry restoration
16
Which symmetries can be broken?
is invariant under
17
Deformed charge distribution
Rotational degree of freedom and rotational bands.
18
Isotropy conserved
Isotropy broken
19
Current in rotating
J. Fleckner et al. Nucl. Phys. A339, 227 (1980)
Lab frame
Body fixed frame
Moments of inertia reflect the complex flow. No
simple formula.
20
Deformed?
21
Rotor composed of current loops, which specify
the orientation.
Orientation specified by the magnetic dipole
moment.
Magnetic rotation.
22
II.3 Discrete symmetries
23
Common bands
PAC solutions (Principal Axis Cranking)
TAC solutions (planar) (Tilted Axis Cranking)
Many cases of strongly broken symmetry, i.e. no
signature splitting
24
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25
Chiral bands
26
Examples for chiral sister bands
27
Chirality
It is impossible to transform one
configuration into the other by rotation.
mirror
28
Only left-handed neutrinos Parity violation in
weak interaction
mirror
mass-less particles
29
Reflection asymmetric shapes,two reflection
planes
Parity doubling
30
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31
II.4 Spontaneous breaking of isospin symmetry
Form a condensate isovector pair field
32
The relative strengths of pp, nn, and pn pairing
are determined by the isospin symmetry
33
Symmetry restoration Isorotations (strong
symmetry breaking collective model)
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