ii. spontaneous symmetry breaking. ii.1 weinberg’s chair hamiltonian rotational invariant why do...
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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
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
The molecular rotor
3NH
1
2
3 21 Axial rotor
3
23
1
23
2
2
1 JJJH
3
2
1
2)1(
2
1 KKIIE
0],[0],[0],[ 23 JHJHJH z
3
23
1
22
21
2
1 JJJH
aKMI |,,| :seigenstate
function Wigner D iKIMK
iMIMK edeD )(),,(
),,(8
12,,|,, :rotor ofn orientatiofor
amplitudey probabilit2/1
2
I
MKDI
KMI
symmetry. rotational breaksly spontanousthat
structure intrinsic"" thedescibes | a
.
.
Born-Oppenheimer Approximation
Electronic motion
Vibrations
Rotations eVrot410~
eVel 1~
CO
eVvib110~
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
HCl
)1()()1(
)1()(
IBIEIE
JIIBIIE
Microwave absorptionspectrum
Rotational bands are the manifestation of spontaneous symmetry breaking.
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)
Nucleons are indistinguishable
),,(),,()( rotKrotin
rotin
x
EEE
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
IMKD
I
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
II.3 Microscopic approach:
Retains the simple picture of an anisotropic object going round.
Mean field theory + concept of spontaneous symmetry breaking for interpretation.
Rotating mean field (Cranking model):
Start from the Hamiltonian in a rotating frame
zjvtH 12'momentumangular
ninteractiobody - twoeffective
energy kinetic
12
zj
v
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
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().
Spontaneous symmetry breaking
Symmetry operation S and
.|'|'|'
energy same the withsolutions field mean are states All
1||| 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
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 - )(zRBroken by m.f. rotational
bands
Obeyed by m.f.spinparitysequence
broken by m.f.doublingofstates
zmf jVth '
zJiz e )( axis-z about the Rotation R
peaked.sharply is 1|||
.''but ''
|R
RRRR
z
zzzz hhHH
Rotational degree of freedom and rotational bands.
Deformed charge distribution
nucleons on high-j orbitsspecify orientation
.|2
1II|momentumangular good of State
.energy same thehave )(| nsorientatio All
deiI
z |R
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.
Rotor composed of current loops, which specify the orientation.
Orientation specified by the magnetic dipole moment.
Magnetic rotation.
.energy same thehave )(| nsorientatio All
peaked.sharply is 1|||
.''but ''
|R
|R
RRRR
z
z
zzzz hhHH
II.3 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
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
Reflection asymmetric shapes,
two reflection planes
Simplex quantum number
I
i
z
parity
e
)(
||
)(
S
PRS
Parity doubling
02
ˆ
0
ˆ
np
ppnn
np
ppnn
y
z
The relative strengths of pp, nn, and pn
pairing are determined by the isospin symmetry