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Symmetries in Nuclei
Symmetry and its mathematical description
The role of symmetry in physics
Symmetries of the nuclear shell model
Symmetries of the interacting boson model
Symmetries in Nuclei, Tokyo, 2008
Interacting boson approximation
Dominant interaction between nucleons has pairing
character  two nucleons form a pair with
angular momentum J=0 (S pair).
Next important interaction between nucleons with
angular momentum J=2 (D pair).
Approximation: Replace S and D fermion pairs by s
and d bosons. Argument:
ˆS, Sˆ  1 nˆ 1 while s,s 1

 
Symmetries in Nuclei, Tokyo, 2008

Microscopy of IBM
In a boson mapping, fermion pairs are represented
as bosons:



 0



 2
ˆ
ˆ
s  S    j a j  a j 0 , d  D    jj' a j  a j' 
j
jj'
Mapping of operators (such as hamiltonian) should
take account of Pauli effects.
Two different methods by
requiring same commutation relations;
associating state vectors.
T. Otsuka et al., Nucl. Phys. A 309 (1978) 1
Symmetries in Nuclei, Tokyo, 2008

The interacting boson model
Describe the nucleus as a system of N interacting s
and d bosons. Hamiltonian:
6
Hˆ IBM  ibibi 
i1
6
 

b
 ijkl i b j bkbl
ijkl1
Justification from
Shell model: s and d bosons are associated with S and D
fermion (Cooper) pairs.
Geometric model: for large boson number the IBM
reduces to a liquid-drop hamiltonian.
A. Arima & F. Iachello, Ann. Phys. (NY) 99 (1976) 253; 111 (1978) 201; 123 (1979) 468
Symmetries in Nuclei, Tokyo, 2008
Dimensions
Assume  available 1-fermion states. Number of Nfermion states is

!
 
N  N!  N !
Assume  available 1-boson states. Number of Nboson states is   N 1   N 1!


 N  N! 1!

Example: 162Dy96 with 14 neutrons (=44) and 16
protons (=32) (132Sn82 inert core).
SM dimension: 7·1019

IBM dimension:
15504
Symmetries in Nuclei, Tokyo, 2008



U(6) algebra and symmetry
Introduce 6 creation & annihilation operators:
b ,i  1,

i
,6 s ,d ,d ,d ,d ,d


2

1

0

1

2
,
bi  b

 
i
The hamiltonian (and other operators) can be
written in terms of generators of U(6):




b
b
,b
b

b
b


b
 i j k l  i l jk k b jil
The harmonic hamiltonian has U(6) symmetry
6


Hˆ U(6)  E 0   bibi  Hˆ U(6),bib j  0
i1
Additional terms break U(6) symmetry.
Symmetries in Nuclei, Tokyo, 2008

The IBM hamiltonian
Rotational invariant hamiltonian with up to N-body
interactions (usually up to 2):
L 
L

 L 
ˆ
˜
˜
HIBM  E 0  snˆ s  d nˆ d   l l l l  bl  bl   bl   bl  
l1 l 2 l1l 2,L
1 2 1 2
1
2
1
2
For what choice of single-boson energies  and
boson-boson interactions  is the IBM hamiltonian
solvable?
This problem is equivalent to the enumeration of all
algebras G satisfying


U6  G  SO3  Lˆ   10 d   d˜

1

Symmetries in Nuclei, Tokyo, 2008
Dynamical symmetries of the IBM
U(6) has the following subalgebras:
          
SU3  
d  d˜  ,s  d˜  d  s˜  d  d˜  
SO6  
d  d˜  ,s  d˜  d  s˜ ,d  d˜  
SO5  
d  d˜  ,d  d˜  
Three solvable limits are found:
U5  d  d˜


0

, d  d˜
1

1

, d  d˜





1
1



, d  d˜

2
7
4





2
2



3

, d  d˜

2


3

3

U5  SO5 


U6   SU3
 SO3
SO 6  SO 5 
 
  
Symmetries in Nuclei, Tokyo, 2008
4 


Dynamical symmetries of the IBM
The general IBM hamiltonian is
L 
L

 L 
ˆ
˜
˜
HIBM  E 0  snˆ s  d nˆ d   l l l l  bl  bl   bl   bl  
l1 l 2 l1l 2,L
1 2 1 2
1
2
1
2
An entirely equivalent form of HIBM is
Hˆ IBM  E 0  0Cˆ1 U6 1Cˆ1 U5 0Cˆ1 U6Cˆ1 U5
  0Cˆ 2 U6 1Cˆ 2 U5  2Cˆ 2 SU3
  3Cˆ 2 SO6  4 Cˆ 2 SO5  5Cˆ 2 SO3
The coefficients  and  are certain combinations of
the coefficients  and .

Symmetries in Nuclei, Tokyo, 2008

The solvable IBM hamiltonians
Excitation spectrum of HIBM is determined by
Hˆ IBM  1Cˆ1 U5 1Cˆ 2 U5  2Cˆ 2 SU3
  3Cˆ 2 SO6  4 Cˆ 2 SO5  5Cˆ 2 SO3
If certain coefficients are zero, HIBM can be written
as a sum of commuting operators:
Hˆ U5  1Cˆ1 U5 1Cˆ 2 U5  4 Cˆ 2 SO5  5Cˆ 2 SO3
Hˆ SU3   2Cˆ 2 SU3  5Cˆ 2 SO3
Hˆ SO6   3Cˆ 2 SO6  4 Cˆ 2 SO5  5Cˆ 2 SO3
Symmetries in Nuclei, Tokyo, 2008
The U(5) vibrational limit
U(5) Hamiltonian:
Hˆ U5   nˆ d 
 c d
1
L 2

d

 
L 
 d˜  d˜

L 
L 0,2,4
Energy eigenvalues:

E n d , ,L   n d  1n d n d  4    4   3   5 LL  1
with
1  121 c 0
 4   101 c 0  17 c 2  703 c 4
 5   141 c 2  141 c 4
Symmetries in Nuclei, Tokyo, 2008
The U(5) vibrational limit
Anharmonic vibration spectrum associated with the
quadrupole oscillations of a spherical surface.
Conserved quantum numbers: nd, , L.
A. Arima & F. Iachello, Ann. Phys. (NY) 99 (1976) 253
D. Brink et al., Phys. Lett. 19 (1965) 413
Symmetries in Nuclei, Tokyo, 2008

The SU(3) rotational limit
SU(3) Hamiltonian:
ˆ Q
ˆ  bLˆ  Lˆ
Hˆ SU3  aQ


Energy eigenvalues:
E , ,L   2 2   2  3  3    5 LL  1
with
 2  12 a
 5  b  38 a
Symmetries in Nuclei, Tokyo, 2008
The SU(3) rotational limit
Rotation-vibration spectrum of quadrupole
oscillations of a spheroidal surface.
Conserved quantum numbers: (,), L.
A. Arima & F. Iachello,
Ann. Phys. (NY) 111 (1978) 201
A. Bohr & B.R. Mottelson, Dan. Vid.
Selsk. Mat.-Fys. Medd. 27 (1953) No 16
Symmetries in Nuclei, Tokyo, 2008

The SO(6) -unstable limit
SO(6) Hamiltonian:
Hˆ SO6  aPˆ   Pˆ  bTˆ3  Tˆ3  cLˆ  Lˆ
Energy eigenvalues:
E  , ,L   3 N N  4      4  4   3   5 LL  1
with
 3  14 a
 4  12 b
 5   101 b  c
Symmetries in Nuclei, Tokyo, 2008
The SO(6) -unstable limit
Rotation-vibration spectrum of quadrupole
oscillations of a -unstable spheroidal surface.
Conserved quantum numbers: , , L.
A. Arima & F. Iachello, Ann. Phys. (NY) 123 (1979) 468
L. Wilets & M. Jean, Phys. Rev. 102 (1956) 788
Symmetries in Nuclei, Tokyo, 2008
The IBM symmetries
Three analytic solutions: U(5), SU(3) & SO(6).
Symmetries in Nuclei, Tokyo, 2008
Synopsis of IBM symmetries
Three standard solutions: U(5), SU(3), SO(6).
Solution for the entire U(5)  SO(6) transition via
the SU(1,1) Richardson-Gaudin algebra.
Hidden symmetries because of parameter
transformations: SU±(3) and SO±(6).
Partial dynamical symmetries.
Symmetries in Nuclei, Tokyo, 2008
Applications of IBM
Symmetries in Nuclei, Tokyo, 2008
The ratio R42
Symmetries in Nuclei, Tokyo, 2008

Partial dynamical symmetries
Solvable models with dynamical symmetry:
G  G  G 






Dynamical symmetries can be partial
Type 1: All labels , ’… remain good quantum numbers
for some eigenstates.
Type 2: Some of the labels , ’… remain good quantum
numbers for all eigenstates.
Y. Alhassid and A. Leviatan, J. Phys. A 25 (1995) L1285
A. Leviatan et al., Phys. Lett. B 172 (1986) 144
Symmetries in Nuclei, Tokyo, 2008
Example of type-1 PDS
A. Leviatan, Phys. Rev. Lett. 77 (1996) 818
Symmetries in Nuclei, Tokyo, 2008
Example of type-2 PDS
P. Van Isacker, Phys. Rev. Lett. 83 (1999) 4269
Symmetries in Nuclei, Tokyo, 2008
Modes of nuclear vibration
Nucleus is considered as a droplet of nuclear matter
with an equilibrium shape. Vibrations are modes
of excitation around that shape.
Character of vibrations depends on symmetry of
equilibrium shape. Two important cases in nuclei:
Spherical equilibrium shape
Spheroidal equilibrium shape
Symmetries in Nuclei, Tokyo, 2008
Vibration about a spherical shape
Vibrations are characterized by a multipole quantum
number  in surface parametrization:



*
R ,    R0 1     Y  , 


   
=0: compression (high energy)
=1: translation (not an intrinsic excitation)
=2: quadrupole vibration


Symmetries in Nuclei, Tokyo, 2008
Vibration about a spheroidal shape
The vibration of a shape with
axial symmetry is
characterized by a.
Quadrupolar oscillations:
=0: along the axis of
symmetry ()
=1: spurious rotation
=2: perpendicular to axis of
symmetry ()



Symmetries in Nuclei, Tokyo, 2008





Classical limit of IBM
For large boson number N, a coherent (or intrinsic)
state is an approximate eigenstate,
Hˆ IBM N;   E N;  ,

N;   s     d

o
N
The real parameters  are related to the three
Euler angles and shape variables  and .
Any IBM hamiltonian yields energy surface:
N;  Hˆ IBM N;   N; Hˆ IBM N;  V , 
J.N. Ginocchio & M.W. Kirson, Phys. Rev. Lett. 44 (1980) 1744.
A.E.L. Dieperink et al., Phys. Rev. Lett. 44 (1980) 1747.
A. Bohr & B.R. Mottelson, Phys. Scripta 22 (1980) 468.
Symmetries in Nuclei, Tokyo, 2008
Classical limit of IBM
For large boson number N the minimum of
V()=N;H approaches the exact
ground-state energy:


U(5) :


4  4
V ,    SU(3) :


SO(6) :


2
1  2
2 3 cos 3  8 2
81 

2 2
1  2 

2 
1  
2
Symmetries in Nuclei, Tokyo, 2008

Geometry of IBM
A simplified, much used IBM hamiltonian:
2
 ˆ

˜


ˆ
ˆ
ˆ
˜
HCQF  d nˆ d  Q  Q ,
Q  s d  d s   d  d

HCQF can acquire the three IBM symmetries.
HCQF has the following classical limit:
VCQF ,    N; Hˆ CQF N;
 d N

2
1 
2
 N
5  1  2  2
1  2

N N 1 2 2 4
2 3
2

    4  cos 3  4 
2 2 7
7

1   
Symmetries in Nuclei, Tokyo, 2008
Phase diagram of IBM
J. Jolie et al. , Phys. Rev. Lett. 87 (2001) 162501.
Symmetries in Nuclei, Tokyo, 2008
Extensions of the IBM
Neutron and proton degrees freedom (IBM-2):
F-spin multiplets (N+N=constant).
Scissors excitations.
Fermion degrees of freedom (IBFM):
Odd-mass nuclei.
Supersymmetry (doublets & quartets).
Other boson degrees of freedom:
Isospin T=0 & T=1 pairs (IBM-3 & IBM-4).
Higher multipole (g,…) pairs.
Symmetries in Nuclei, Tokyo, 2008
Scissors excitations
Collective displacement
modes between
neutrons and protons:
Linear displacement
(giant dipole resonance):
R-R  E1 excitation.
Angular displacement
(scissors resonance):
L-L  M1 excitation.
D. Bohle et al., Phys. Lett. B 137 (1984) 27
Symmetries in Nuclei, Tokyo, 2008
SO(6) (mixed-)symmetry in 94Mo
Analytic calculation in
SO(6) limit of IBM-2.
Complex spectrum with
mixed-symmetry states.
E2 and M1 transition
rates reproduced with
two effective boson
charges e and e.
N. Pietralla et al., Phys. Rev. Lett. 83 (1999) 1303
Symmetries in Nuclei, Tokyo, 2008

Bosons + fermions
Odd-mass nuclei are fermions.
Describe an odd-mass nucleus as N bosons + 1
fermion mutually interacting. Hamiltonian:

Hˆ IBFM  Hˆ IBM   j a j a j 
Algebra:
j1
6


 i
i1 i2 1 j1 j 2 1
 
b
a j1 bi2 a j 2
1 j1 i2 j 2 i1
bibi

U6  U   1 2


a j1 a j 2 

Many-body problem is solved analytically for certain
energies  and interactions .

Symmetries in Nuclei, Tokyo, 2008
Example: 195Pt117
Symmetries in Nuclei, Tokyo, 2008
Example: 195Pt117 (new data)
Symmetries in Nuclei, Tokyo, 2008

Nuclear supersymmetry
Up to now: separate description of even-even and
odd-mass nuclei with the algebra
bibi

U6  U  


a j a j 

Simultaneous description of even-even and oddmass nuclei with the superalgebra
1
2
1
bibi
U6 /   1 2
a j1 bi2
2
bi1 a j2 


a j1 a j2 
F. Iachello, Phys. Rev. Lett. 44 (1980) 777
Symmetries in Nuclei, Tokyo, 2008
U(6/12) supermultiplet
Symmetries in Nuclei, Tokyo, 2008
Example: 194Pt116 &195Pt117
Symmetries in Nuclei, Tokyo, 2008
Quartet supersymmetry
A simultaneous description of even-even, even-odd,
odd-even and odd-odd nuclei (quartets).
Example of 194Pt, 195Pt, 195Au & 196Au:
195
78
ab 
194
78
a b

Pt117
 b a
Pt116



b a
a b



b a
196
79
Au117
ab 
195
79
 b a
Au116
P. Van Isacker et al., Phys. Rev. Lett. 54 (1985) 653
Symmetries in Nuclei, Tokyo, 2008
Quartet supersymmetry
A. Metz et al., Phys. Rev. Lett. 83 (1999) 1542
Symmetries in Nuclei, Tokyo, 2008
Example of 196Au
Symmetries in Nuclei, Tokyo, 2008
Isospin invariant boson models
Several versions of IBM depending on the fermion
pairs that correspond to the bosons:
IBM-1: single type of pair.
IBM-2: T=1 nn (MT=-1) and pp (MT=+1) pairs.
IBM-3: full isospin T=1 triplet of nn (MT=-1), np (MT=0)
and pp (MT=+1) pairs.
IBM-4: full isospin T=1 triplet and T=0 np pair (with S=1).
Schematic IBM-k has only s (L=0) bosons, full IBM-k
has s (L=0) and d (L=2) bosons.
Symmetries in Nuclei, Tokyo, 2008
Symmetry rules
Symmetry is a universal concept relevant in
mathematics, physics, chemistry, biology, art…
In science in particular it enables
To describe invariance properties in a rigorous manner.
To predict properties before any detailed calculation.
To simplify the solution of many problems and to clarify
their results.
To classify physical systems and establish analogies.
To unify knowledge.
Symmetries in Nuclei, Tokyo, 2008
Symmetry in physics
Fundamental symmetries: Laws of physics are
invariant under a translation in time or space, a
rotation, a change of inertial frame and under a
CPT transformation.
Noether’s theorem (1918): Every (continuous)
global symmetry gives rise to a conservation law.
Approximate symmetries: Countless physical
systems obey approximate invariances which
enable an understanding of their spectral
properties.
Symmetries in Nuclei, Tokyo, 2008
Symmetry in nuclear physics
The integrability of quantum many-body (bosons
and/or fermions) systems can be analyzed with
algebraic methods.
Two nuclear examples:
Pairing vs. quadrupole interaction in the nuclear shell
model.
Spherical, deformed and -unstable nuclei with s,d-boson
IBM.
Symmetries in Nuclei, Tokyo, 2008