II. Spontaneous symmetry breaking
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Transcript of II. Spontaneous symmetry breaking
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II. Spontaneous symmetry breaking
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II.1 Weinberg’s chairHamiltonian rotational invariant
)( weight the
withnsorientatio allover averagean ison that distributidensity a have
IM| :momentumangular good of seigenstate
IMKD
Why do we see the chair shape?
States of different IM are so dense that the tiniest interactionWith the surroundings generates a wave packet that is well oriented.
IM
IM IMaca ||Spontaneously broken symmetry
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momentumangular
]s m kg[eV10~eV10~ levels rotational of distanceenergy
eV10~ levels rotational of scaleenergy
1-215-49-2
49-2
J
JJJ
Tiniest external fields generate a superposition of the |JM>that is oriented in space, which is stable.
Spontaneous symmetry breakingMacroscopic (“infinite”) system
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The molecular rotor
3NH
1
2
3 21 Axial rotor
3
23
1
23
2
21 JJJ
H
3
2
1
2)1(21 KKIIE
0],[0],[0],[ 23 JHJHJH z
3
23
1
22
21
21 JJJH
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aKMI |,,| :seigenstate
function Wigner D iKIMK
iMIMK edeD )(),,(
),,(8
12,,|,, :rotor ofn orientatiofor
amplitudey probabilit2/1
2
IMKDIKMI
symmetry. rotational breaksly spontanousthat structure intrinsic"" thedescibes | a
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.
.
Born-Oppenheimer Approximation
Electronic motion
Vibrations
Rotations eVrot410~
eVel 1~
CO
eVvib110~
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Microscopic (“finite system”)
Rotational levels become observable.
eV 10 :scale intrinsiceV10~ :molecules 1-6-2
Spontaneous symmetry breaking=
Appearance of rotational bands.
Energy scale of rotational levels in
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HCl
)1()()1()1()(
IBIEIEJIIBIIE
Microwave absorptionspectrum
Rotational bands are the manifestation of spontaneous symmetry breaking.
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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 independentlyinside the deformed potential (intrinsic degrees of freedom)
The nucleonic motion is much fasterthan the rotation (adiabatic approximation)
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Nucleons are indistinguishable
),,(),,()( rotKrotin
rotin
xEEE
2
)1( 2KIIEE in
The nucleus does not have an orientation degree of freedomwith respect to the symmetry axis.
03
2
K
Axial symmetryin
iKin e )(3R
K
2/1
2 ),,(8
12
IMKDI
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Single particle and collective degrees of freedom become entangled at high spin and low deformation.
Limitations:scale intrinsic~MeV10~ :scaleenergy rotational 1-
2
Rotationalbands in
Er163
Adiabatic regimeCollective model
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II.3 Microscopic approach:
Retains the simple picture of an anisotropic object going round.
Mean field theory + concept of spontaneous symmetry breaking for interpretation.
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Rotating mean field (Cranking model):
Start from the Hamiltonian in a rotating frame
zjvtH 12'momentumangular
ninteractiobody - twoeffective energy kinetic
12
zjv
t
Mean field approximation:find state |> of (quasi) nucleons moving independently inmean field generated by all nucleons.mfV
(routhian) frame rotating thein nhamiltonia field mean '
},| { :tencyselfconsis , -' , |'|' 12
h
VvJVtheh mfzmf
Selfconsistency : effective interactions, density functionals (Skyrme, Gogny, …), Relativistic mean field, Micro-Macro (Strutinsky method) …….
Reaction of thenucleons to the inertial forces must be taken into account
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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
Mean field theory:Tilted Axis Cranking TACS. Frauendorf Nuclear Physics A557, 259c (1993)
Quantization of single particlemotion determines relation J().
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Spontaneous symmetry breaking
Symmetry operation S and
.|'|'|'
energy same the withsolutions field mean are states All1||| and ,''
HHE
hh
|SS|S
|SSS
Full two-body Hamiltonian H’ Mean field approximation
Mean field Hamiltonian h’ and m.f. state h’|>=e’|>.
Symmetry restoration |Siic
'' HH SS
Spontaneous symmetry breaking
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Which symmetries can be broken?
Combinations of discrete operations
rotation withreversal time- )(inversion space-
angleby axis-zabout rotation - )(
y
z
TR P
R
zJHH ' is invariant under
axis-zabout rotation - )(zR Broken by m.f. rotationalbands
Obeyed by m.f.spinparitysequence
broken by m.f.doublingofstates
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zmf jVth '
zJiz e )( axis-z about the Rotation R
peaked.sharply is 1||| .''but ''
|RRRRR
z
zzzz hhHH
Rotational degree of freedom and rotational bands.
Deformed charge distribution
nucleons on high-j orbitsspecify orientation
.|21II|momentumangular good of State
.energy same thehave )(| nsorientatio All
deiI
z |R
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deformedEr163
sphericalPb200
Isotropybroken
Isotropyconserved
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Current in rotating Yb162
Lab frame Body fixed frame
J. Fleckner et al. Nucl. Phys. A339, 227 (1980)
Moments of inertia reflect the complex flow. No simple formula.
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Deformed?
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Rotor composed of current loops, which specify the orientation.
Orientation specified by the magnetic dipole moment.
Magnetic rotation.
.energy same thehave )(| nsorientatio Allpeaked.sharply is 1|||
.''but ''
|R|R
RRRR
z
z
zzzz hhHH
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II.3 Discrete symmetries
Combinations of discrete operations
rotation withreversal time- )(inversion space-
angleby axis-zabout rotation - )(
y
z
TR P
R
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Common bands
by axis-zabout rotation - )(
rotation withreversal time- 1 )(inversion space - 1
z
y
RTRP
PAC solutions(Principal Axis Cranking)
nIe i
z
2 signature ||)(
R
TAC solutions (planar)(Tilted Axis Cranking) Many cases of strongly brokensymmetry, i.e. no signature splitting
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Rotationalbands in
Er163
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Chiral bands
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Examples for chiral sister bands
7513459 Pr 1
2/112/11hh
5910445 Rh 2/11
12/9 hg
7513560 Nd 1
2/112
2/11hh
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Chirality
mirror
It is impossible to transform one configurationinto the other by rotation.
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mirror
mass-less particles
Only left-handed neutrinos:Parity violation in weak interaction
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Reflection asymmetric shapes,
two reflection planes
Simplex quantum number
I
i
z
parity
e
)(
||
)(
SPRS
Parity doubling
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Th226
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0110
TSTS
II.4 Spontaneous breaking of isospin symmetry
Form a condensate“isovector pair field”
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02
ˆ
0ˆ
np
ppnn
np
ppnn
y
z
The relative strengths of pp, nn, and pn
pairing are determined by the isospin symmetry
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Symmetry restoration –Isorotations (strong symmetry breaking – collective model)
2)1( :energy nalisorotatio
|)0,,( :state nalisorotatio
| :state intrinsic
intrinsic
0
TTH)E(T,T
D
z
TTz
AMeVTTTTE z
7521,
2)1()(
:alExperiment
exp