Spontaneous symmetry breaking

Laboratory frame and intrinsic frame
(ammonia molecule)
H
H
H
|R› = |L› =
" #
r
D " #
= 0
N
)
P L = R " #
= 1 2 1+ # P
)
but
( ) L
L r
D L " 0
!
!
The height of the barrier: 0.254 eV at 0 K
Minima separation of about 0.75 A
Parity splitting (inversion doubling): 0.0001 eV
Application: ammonia maser based on the inversion doubling 1954
(Nobel Prize for Charles Townes in 1964)

The transition matrix element between the members of the doublet
carries important information about the collectivity of the system.
The normalized signature-projected LM wave functions are:
Since K x is signature-odd, one has
Consequently, the transition matrix element is:
For large values of χ, this matrix element becomes
In the limit of large χ, the transition matrix element carries
information about the deformation of the system!

Laboratory frame and intrinsic frame
(molecular or nuclear quadrupole moment)
Q =
2K 2 -I(I+1)
Q
(I+1)(2I+3) 0
MEASURED
INTRINSIC
RELATIVE
MOTION
Q=0 for I=K=0, independently of Q 0 !

Symmetry breaking
q = !
= 0 but
0 0 0
Q ˆ
! crit
E
ω RPA
=0
ω RPA
>0 ω RPA
-
imag.
Vibrator
(weak
coupling)
Soft
(interm.
coupling)
Rotor
(strong
coupling)
No phase
Band structures
transition
labelled by
in the finite
different q.n.
system!
[ H ˆ S ˆ ] h ˆ , S ˆ
deformed
intrinsic
system!
Q 2 !
“exact”
mean field
of the internally
broken symmetries
[ ] ! 0

Nuclear deformation:
spontaneous symmetry breaking
Molecular physics: Jahn-Teller effect 1937
Any configuration of atoms or ions (except a
linear chain) can develop a stable symmetrybreaking
deformation provided the coupling
between degenerate electronic excitations and
collective molecular vibrations is strong.
Nuclear physics: Bohr-Mottelson 1952-53
Any nuclear configuration can develop a stable
symmetry-breaking deformation provided the
coupling between degenerate single-nucleonic
excitations and collective nuclear modes is
strong.
The unified model. Particle vibration coupling
V int
= !" r
( ) # $µ
$µ
( )
% Y $µ
&

Intrinsic symmetries
5 - ν=1
3 -
1 -
(parallel band)
D •h (C 2 , P)
O
C
O
6 +
4 +
2 +
0 +
ν=0
(ground band)
CO 2
- an axially symmetric molecule with no static dipole moment.
O
C
O
C •ν (S=C 2 P)
3 -
2 +
O N
4 +
1 -
0 +
N 2
O - an axially symmetric molecule with static dipole moment.