Transcript How do nuclei rotate?

II. Spontaneous symmetry breaking
II.1 Weinberg’s chair
Hamiltonian rotational invariant  eigenst at es of good
angular momentum: | IM 
havea densitydistribution thatis
an averageover all orientations wit h
I
theweight DMK
( )
Why do we see the chair shape?
States of different IM are so dense
that the tiniest interaction
With the surroundings generates a
wave packet that is well oriented.
Spontaneously broken symmetry
| a   cIM | IMa 
2
energy scale of rotationallevels ~ 10-49 eV

2 J
J
energy distanceof rotationallevels
~ 10-49 eV ~ 10-15 eVJ [kg m 2 s-1 ]
 

J angular momentum
Tiniest external fields generate a superposition of the |JM>
that is oriented in space, which is stable.
Spontaneous symmetry breaking
Macroscopic (“infinite”) system
The molecular rotor
Axial rotor
3
2
NH3
1
1  2
1  J12  J 22 J 32 

H  

2  1
3 
2
2

J

J
J 32 
1
3
H


2  1
3 
[H , J z ]  0 [H , J3 ]  0 [H , J 2 ]  0
1  I ( I  1)  K 2 K 2 

E  

2
1
3 
eigenstates : | I , M , K | a 
probability amplitude
 2I  1 
I
for orientation of rotor:   , ,  | I , M , K  
D

MK ( , ,  )
2
 8 
1/ 2
Wigner D  function
I
I
DMK
( , ,  )  eiM d MK
( )eiK
| a  descibes the" intrinsic"structure
thatspontanously breaks rotationalsymmetry.
Born-Oppenheimer Approximation
.
.
Electronic motion el ~ 1eV
Vibrations
vib ~ 101 eV
Rotations
rot ~ 104 eV
CO
Microscopic (“finite system”)
Rotational levels become observable.
Spontaneous symmetry breaking
=
Appearance of rotational bands.
Energy scale of rotational levels in
2
molecules:
~ 10-6 eV  intrinsicscale : 10-1 eV

E ( I )  BI ( I  1) I  J
  E ( I  1)  E ( I )  B( I  1)
HCl
Microwave absorption
spectrum
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 independently
inside the deformed potential (intrinsic degrees of freedom)
The nucleonic motion is much faster
than the rotation (adiabatic approximation)
E  Ein  Erot
  in ( x )rot ( , ,  )   K rot ( , ,  )
Nucleons are indistinguishable
Axial symmetry
R3 ( )in  eiK in
The nucleus does not have an orientation degree of freedom
with respect to the symmetry axis.
K2
0
3
I ( I  1)  K 2
E  Ein 
2
 2I  1 
I
 
D

MK ( , ,  ) K
2
 8 
1/ 2
Rotational
bands in
Limitations:
163
2
rotationalenergy scale :

~ 10-1 MeV ~ intrinsicscale

Er
Single particle and collective degrees of freedom
become entangled at high spin and low deformation.
Adiabatic regime
Collective model
II.3 Microscopic approach:
Mean field theory
+
concept of spontaneous symmetry
breaking for interpretation.
Retains the simple picture of an anisotropic object going round.
Rotating mean field
(Cranking model):
Start from the Hamiltonian in a rotating frame
H '  t  v12  jz
t
v12
kineticenergy
effectivetwo - body interaction
jz
angular momentum
Reaction of the
nucleons to
the inertial
forces must be
taken into account
Mean field approximation:
find state |> of (quasi) nucleons moving independently in
mean field Vmf generated by all nucleons.
h' | e ' | ,
h'
h'  t  Vmf - J z , selfconsistency: { |, v12 }  Vmf
mean field hamiltonian in therotatingframe(routhian)
Selfconsistency :
effective interactions, density functionals (Skyrme, Gogny, …),
Relativistic mean field, Micro-Macro (Strutinsky method)
…….
Rotational response
Low spin: simple droplet.
High spin: clockwork of gyroscopes.
Quantization of single particle
motion determines relation J().
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)
Spontaneous symmetry breaking
Full two-body Hamiltonian H’
Mean field approximation
Mean field Hamiltonian h’ and m.f. state h’|>=e’|>.
Symmetry operation S and
SH ' S   H '
Spontaneous symmetry breaking
Sh' S   h' , and
|| S | | 1
All states S |  are mean field solut ionswith thesame energy
E ' | S  H ' S | | H ' | .
Symmetry restoration
c
i
Si | 
Which symmetries can be broken?
H '  H  J z
Rz ( )
is invariant under
- rotationabout z - axis
Broken by m.f.
rotational
bands
Combinations of discrete operations
Rz ( )
- rotationabout z - axis by angle 
P
- space inversion
TRy ( ) - timereversal with rotation
Obeyed by m.f.
broken by m.f.
spin
parity
sequence
doubling
of
states
Deformed charge distribution
h'  t  Vmf  jz
Rotationabout thez - axis
Rz ( )  eiJ z
nucleons on
high-j orbits
specify orientation
Rz H ' Rz   H ' but Rz h' Rz   h'.
|| Rz | | 1
is sharplypeaked.
Rotational degree of freedom and rotational bands.
All orientations |   Rz ( )|  have thesame energy .
1
iI
State of good angular momentum| II 
e
|   d .

2
200
163
Er
deformed
Pb
spherical
Isotropy
broken
Isotropy
conserved
Current in rotating
162
Lab frame
Yb
J. Fleckner et al. Nucl. Phys. A339, 227 (1980)
Body fixed frame
Moments of inertia reflect the complex flow. No simple formula.
Deformed?
Rz H ' Rz   H ' but Rz h' Rz   h'.
|| Rz | | 1
is sharplypeaked.
All orientations |  Rz ( )|  have thesame energy .
Rotor composed of current loops, which specify the orientation.
Orientation specified by the magnetic dipole moment.
Magnetic rotation.
II.3 Discrete symmetries
Combinations of discrete operations
Rz ( )
- rotationabout z - axis by angle 
P
- space inversion
TRy ( ) - timereversal with rotation
Common bands
P 1
- space inversion
TRy ( )  1 - timereversal with rot at ion
Rz ( )
- rot at ionabout z - axis by 
PAC solutions
(Principal Axis Cranking)
Rz ( ) | e i | signature
I    2n
TAC solutions (planar)
(Tilted Axis Cranking)
Many cases of strongly broken
symmetry, i.e.
no signature splitting
Rotational
bands in
163
Er
Chiral bands
Examples for chiral sister bands
134
59
104
45
135
60
Pr75 h11 / 2h111/ 2
Rh59 g h11/ 2
Nd75
1
9/2
h h
2
11 / 2
1
11 / 2
Chirality
It is impossible to transform one configuration
into the other by rotation.
mirror
Only left-handed neutrinos:
Parity violation in weak interaction
mirror
mass-less particles
Reflection asymmetric
shapes,
two reflection planes
Simplex quantum number
S  Rz ( ) P
S | e i |
parity   (  ) I 
Parity doubling
226
Th
II.4 Spontaneous breaking of isospin
symmetry
S  0 T 1
Form
S a1condensate
T 0
“isovector pair field”

  zˆ
 nn   pp  0
 np  

  yˆ
 nn   pp
 np  0
The relative strengths of pp, nn, and pn
pairing are determined by the isospin symmetry


2
Symmetry restoration –Isorotations
(strong symmetry breaking – collective model)
intrinsicstate:
|
isorotational state: DTTz 0 ( ,  ,0) |
T (T  1)
isorotational energy: E(T,Tz )  H intrinsic  
2
Experimental :
T (T  1)
1 75MeV
Eexp (T  Tz ) 
,

2
2
A