Symmetry Approach to
Nuclear Collective Motion II
P. Van Isacker, GANIL, France
Symmetry and dynamical symmetry
Symmetry in nuclear physics:
Nuclear shell model
Interacting boson model
Nuclear Collective Dynamics II, Istanbul, July 2004
The three faces of the shell model
Nuclear Collective Dynamics II, Istanbul, July 2004
Symmetries of the shell model
• Three bench-mark solutions:
– No residual interaction  IP shell model.
– Pairing (in jj coupling)  Racah’s SU(2).
– Quadrupole (in LS coupling)  Elliott’s SU(3).
• Symmetry triangle:
 p 2

1
2 2
2
k
H   
 m  rk   ls l k  s k   ll l k 
2m 2

k  1 
A
A

 V r ,r 
RI
k
l
k l
Nuclear Collective Dynamics II, Istanbul, July 2004
Evidence for shell structure
• Evidence for nuclear shell structure from
–
–
–
–
2+ in even-even nuclei [Ex, B(E2)].
Nucleon-separation energies & nuclear masses.
Nuclear level densities.
Reaction cross sections.
• Is nuclear shell structure
modified away from the
line of stability?
Nuclear Collective Dynamics II, Istanbul, July 2004
Shell structure from Ex(21)
• High Ex(21) indicates stable shell structure:
Nuclear Collective Dynamics II, Istanbul, July 2004

Weizsäcker mass formula
• Total nuclear binding energy:
B W N , Z   a V A  a S A
2/3
 aC

Tz Tz  1
1  a I A 2 


Z Z  1
A
1/3
 aP
1
A
1/2
1
  0
1
pair  pair
impair
impair  impair
• For 2149 nuclei (N,Z≥8) in AME03: aV16,
aS18, aI7.3, aC0.71, aP13  rms2.5
MeV.
Nuclear Collective Dynamics II, Istanbul, July 2004
Shell structure from masses
• Deviations from Weizsäcker mass formula:
Nuclear Collective Dynamics II, Istanbul, July 2004
Racah’s SU(2) pairing model
• Assume large spin-orbit splitting ls which
implies a jj coupling scheme.
• Assume pairing interaction in a single-j shell:
j JM J V pairing r1 ,r2  j JM
2
2
J

 

1
2
2 j  1g,
0,
• Spectrum of 210Pb:

Nuclear Collective Dynamics II, Istanbul, July 2004
J0
J0

Solution of pairing hamiltonian
• Analytic solution of pairing hamiltonian for
identical nucleons in a single-j shell:
n
j J
n
 V pairing rk ,rl  j  J   4 G n   2 j  n    3
n
1
k l
• Seniority  (number of nucleons not in pairs
coupled to J=0) is a good quantum number.
• Correlated ground-state solution (cfr. superfluidity in solid-state physics).
G. Racah, Phys. Rev. 63 (1943) 367
Nuclear Collective Dynamics II, Istanbul, July 2004
Pairing and superfluidity
• Ground states of a pairing hamiltonian have a
superfluid character:
– Even-even nucleus (=0):
– Odd-mass nucleus (=1):
S  
n/2
a j S  

o
n/ 2
o
• Nuclear superfluidity leads to
– Constant energy of first 2+ in even-even nuclei.
– Odd-even staggering in masses.
– Smooth variation of two-nucleon separation
energies with nucleon number.
– Two-particle (2n or 2p) transfer enhancement.
Nuclear Collective Dynamics II, Istanbul, July 2004
Superfluidity in semi-magic nuclei
• Even-even nuclei:
– Ground state has
=0.
– First-excited state has
=2.
– Pairing produces
constant energy gap:

1
E x 21   2 2 j  1g
• Example of Sn
nuclei:
Nuclear Collective Dynamics II, Istanbul, July 2004
Two-nucleon separation energies
• Two-nucleon separation energies S2n:
(a) Shell splitting dominates over interaction.
(b) Interaction dominates over shell splitting.
(c) S2n in tin isotopes.
Nuclear Collective Dynamics II, Istanbul, July 2004
Pairing with neutrons and protons
• For neutrons and protons two pairs and hence
two pairing interactions are possible:
– Isoscalar (S=1,T=0):
 S  S  ,
10
10
S 
10
– Isovector (S=0,T=1):
 S  S ,
01
01
S 
01
 010 
1  

l  2 a l 1 1  al 1 1 

2 2
2 2
S  S 

,
S  S

 001 
1  

l  2 a l 1 1  a l 1 1 

2 2
2 2
Nuclear Collective Dynamics II, Istanbul, July 2004

,
10
01
10
01

Neutron-proton pairing hamiltonian
• A hamiltonian with two pairing terms,
H   g 0 S   S  g 1 S  S 
10
10
01
01
• …has an SO(8) algebraic structure.
• H is solvable (or has dynamical symmetries)
for g0=0, g1=0 and g0=g1.
Nuclear Collective Dynamics II, Istanbul, July 2004
SO(8) ‘quasi-spin’ formalism
• A closed algebra is obtained with the pair
operators S± with in addition

000
 

n  2 2l  1 al 1 1  al 1 1 
,
 2 2

2 2
000
Y  
 010 
S 
 

2l  1 al 1 1  a l 1 1 
,

2 2
2 2
0 0
 011 
 

2l  1 a l 1 1  al 1 1 
 2 2
0 
2 2
 001 
T 
 

2l  1 a l 1 1  a l 1 1 
00 
2 2
2 2
• This set of 28 operators forms the Lie algebra
SO(8) with subalgebras
SO 6   SU  4   S, T , Y , SO 5   n, S , S ,
10

S
SO T  5   n, T , S
01
,
SO S  3   S ,
SO T 3   T 
B.H. Flowers & S. Szpikowski, Proc. Phys. Soc. 84 (1964) 673
Nuclear Collective Dynamics II, Istanbul, July 2004

Solvable limits of SO(8) model
• Pairing interactions can expressed as follows:
S 01  S 01 
C 2 SO T  5  
2
1
S   S   S  S  
01
01
S  S 
10
10
10
10
2
C2 SO T 3  
C 2 SO  8  
2
1
C 2 SO S  5  
2
1
1
1
2
1
2
1
8
( 2l  n  1)( 2l  n  7 )
C2 SO 6  
C 2 SO S  3 
1
8
1
8
( 2l  n  1)( 2l  n  13 )
(2l  n  1)( 2l  n  7 )
• Symmetry lattice of the SO(8) model:
SO S 5   SO T 3 


SO (8)   SO (6 )  SU 4    SO


SO
5

SO
3




 T

S
S
3   SO T 3 
• Analytic solutions for g0=0, g1=0 and g0=g1.
Nuclear Collective Dynamics II, Istanbul, July 2004
Superfluidity of N=Z nuclei
• T=0 & T=1 pairing has quartet superfluid
character with SO(8) symmetry. Pairing
ground state of an N=Z nucleus:
cos  S  S  sin  S  S  o
•  Condensate of ’s ( depends on g01/g10).
• Observations:
10
10
01
01




n /4
– Isoscalar component in condensate survives only
in N~Z nuclei, if anywhere at all.
– Spin-orbit term reduces isoscalar component.
Nuclear Collective Dynamics II, Istanbul, July 2004
Deuteron transfer in N=Z nuclei
• Deuteron intensity cT2
calculated in schematic
model based on SO(8).
• Parameter ratio b/a
fixed from masses.
• In lower half of 28-50
shell: b/a5.
Nuclear Collective Dynamics II, Istanbul, July 2004
Symmetries of the shell model
• Three bench-mark solutions:
– No residual interaction  IP shell model.
– Pairing (in jj coupling)  Racah’s SU(2).
– Quadrupole (in LS coupling)  Elliott’s SU(3).
• Symmetry triangle:
 p 2

1
2 2
2
k
H   
 m  rk   ls l k  s k   ll l k 
2m 2

k  1 
A
A

 V r ,r 
RI
k
l
k l
Nuclear Collective Dynamics II, Istanbul, July 2004
Wigner’s SU(4) symmetry
• Assume the nuclear hamiltonian is invariant
under spin and isospin rotations:
H
nucl


A
S 

, S   H nucl , T   H nucl , Y    0
 s  k ,
k1
A
T 
 t k ,
k 1
A
Y  
 s   k t  k 
k1
• Since {S,T,Y} form an SU(4) algebra:
– Hnucl has SU(4) symmetry.
– Total spin S, total orbital angular momentum L,
total isospin T and SU(4) labels () are
conserved quantum numbers.
E.P. Wigner, Phys. Rev. 51 (1937) 106
F. Hund, Z. Phys. 105 (1937) 202
Nuclear Collective Dynamics II, Istanbul, July 2004
Physical origin of SU(4) symmetry
• SU(4) labels specify the separate spatial and
spin-isospin symmetry of the wave function:
• Nuclear interaction is short-range attractive
and hence favours maximal spatial symmetry.
Nuclear Collective Dynamics II, Istanbul, July 2004
Breaking of SU(4) symmetry
• Non-dynamical breaking of SU(4) symmetry
as a consequence of
– Spin-orbit term in nuclear mean field.
– Coulomb interaction.
– Spin-dependence of residual interaction.
• Evidence for SU(4) symmetry breaking from
– Masses: rough estimate of nuclear BE from
B  N , Z   a  bg     a  b    C 2SU 4    
–  decay: Gamow-Teller operator Y,1 is a
generator of SU(4)  selection rule in ().
Nuclear Collective Dynamics II, Istanbul, July 2004
SU(4) breaking from masses
• Double binding energy difference Vnp
 Vnp  N , Z  
1
4
B  N , Z   B  N  2, Z   B  N , Z  2   B  N  2, Z  2 
• Vnp in sd-shell nuclei:
P. Van Isacker et al., Phys. Rev. Lett. 74 (1995) 4607
Nuclear Collective Dynamics II, Istanbul, July 2004
SU(4) breaking from  decay
• Gamow-Teller decay into odd-odd or eveneven N=Z nuclei:
P. Halse & B.R. Barrett, Ann. Phys. (NY) 192 (1989) 204
Nuclear Collective Dynamics II, Istanbul, July 2004
Elliott’s SU(3) model of rotation
• Harmonic oscillator mean field (no spin-orbit)
with residual interaction of quadrupole type:
 pk

1
2 2
H  
 m  rk   Q  Q ,


2m
2


k 1
A
Q 
2
A
4 
 r 2 Y ˆr  

k 2
k

5 k 1
A

k1

p k Y 2   ˆpk 

2
• State labelling in LS coupling:
U  4 

1 
M
U    SU 3   SO 3
L
L
L




  
 ˜

˜
˜

L 
 
   
SU ST 4   SU S 2   SU T 2 



 


    
S
T

J.P. Elliott, Proc. Roy. Soc. A 245 (1958) 128; 562
Nuclear Collective Dynamics II, Istanbul, July 2004
Importance/limitations of SU(3)
• Historical importance:
– Bridge between the spherical shell model and the
liquid droplet model through mixing of orbits.
– Spectrum generating algebra of Wigner’s SU(4)
supermultiplet.
• Limitations:
– LS (Russell-Saunders) coupling, not jj coupling
(zero spin-orbit splitting)  beginning of sd shell.
– Q is the algebraic quadrupole operator  no
major-shell mixing.
Nuclear Collective Dynamics II, Istanbul, July 2004
Tripartite classification of nuclei
• Evidence for seniority-type, vibrational- and
rotational-like nuclei:
• Need for model of vibrational nuclei.
N.V. Zamfir et al., Phys. Rev. Lett. 72 (1994) 3480
Nuclear Collective Dynamics II, Istanbul, July 2004
The interacting boson model
• Spectrum generating algebra for the nucleus is
U(6). All physical observables (hamiltonian,
transition operators,…) are expressed in terms
of s and d bosons.
• 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
Nuclear Collective Dynamics II, Istanbul, July 2004
Algebraic structure of the IBM
• The U(6) algebra consists of the generators
U 6   s s , s d , d s , d d , m , m'  2, , 2


m


m
m
m'
• The harmonic oscillator in 6 dimensions,
H = n  n  s s   d d  C U 6   N
2

s
d

m
m
1
m  2
• …has U(6) symmetry since
 g  U 6  : H, g = 0
• Can the U(6) symmetry be lifted while
preserving the rotational SO(3) symmetry?
i
i
Nuclear Collective Dynamics II, Istanbul, July 2004
The IBM hamiltonian
• Rotational invariant hamiltonian with up to Nbody interactions (usually up to 2):
 
 
H
  n   n    b  b   b˜  b˜  
• For what choice of single-boson energies s
and d and boson-boson interactions Lijkl is the
IBM hamiltonian solvable?
• This problem is equivalent to the enumeration
of all algebras G that satisfy
 
IBM
s
s
d
d
L


ijkl
i
j
L
L
k
l
ijklJ

U 6   G  SO 3   L  


10 d  d˜


1

Nuclear Collective Dynamics II, Istanbul, July 2004



Dynamical symmetries of the IBM
• The general IBM hamiltonian is
 
H
  n   n    b  b   b˜
IBM
s
s
d
d
L


ijkl
i
j
L
k
 b˜ l
 L

ijklJ
• An entirely equivalent form of HIBM is
H IBM   1C 1 U 6    2 C 1 U  5   0C 1 U 6 C 1 U  5 
  1 C 2 U 6    2 C 2 U 5    3 C 2 SU  3 
  4C 2 SO  6    5 C 2 SO  5   6 C 2 SO  3 
• The coefficients i and j are certain
combinations of the coefficients i and Lijkl.
Nuclear Collective Dynamics II, Istanbul, July 2004
The solvable IBM hamiltonians
• Without N-dependent terms in the hamiltonian
(which are always diagonal)
H IBM   1C 1 U 5    1 C2 U  5    2 C2 SU 3 
  3C 2 SO  6    4 C 2 SO 5    5C 2 SO  3 
• If certain coefficients are zero, HIBM can be
written as a sum of commuting operators:
H     C U  5    C U 5    C SO  5    C SO 3 
U 5
1
1
1
2
4
2
5
H SU  3    2C 2 SU  3    5 C 2SO 3 
H SO  6    3 C2 SO  6    4 C 2 SO  5   5C 2 SO  3 
Nuclear Collective Dynamics II, Istanbul, July 2004
2
The U(5) vibrational limit
• Spectrum of an anharmonic oscillator in 5
dimensions associated with the quadrupole
oscillations of a droplet’s 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
Nuclear Collective Dynamics II, Istanbul, July 2004
The SU(3) rotational limit
• Rotation-vibration spectrum with - and vibrational bands.
• 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
Nuclear Collective Dynamics II, Istanbul, July 2004
The SO(6) -unstable limit
• Rotation-vibration spectrum of a -unstable
body.
• Conserved quantum numbers: , , L.
A. Arima & F. Iachello, Ann. Phys. (NY) 123 (1979) 468
L. Wilets & M. Jean, Phys. Rev. 102 (1956) 788
Nuclear Collective Dynamics II, Istanbul, July 2004
Synopsis of IBM symmetries
• Symmetry triangle of the IBM:
–
–
–
–
–
–
Three standard solutions: U(5), SU(3), SO(6).
SU(1,1) analytic solution for U(5) SO(6).
Hidden symmetries (parameter transformations).
Deformed-spherical coexistent phase.
Partial dynamical symmetries.
Critical-point symmetries?
Nuclear Collective Dynamics II, Istanbul, July 2004
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.
Nuclear Collective Dynamics II, Istanbul, July 2004
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.
N. Lo Iudice & F. Palumbo, Phys. Rev. Lett. 41 (1978) 1532
F. Iachello, Phys. Rev. Lett. 53 (1984) 1427
D. Bohle et al., Phys. Lett. B 137 (1984) 27
Nuclear Collective Dynamics II, Istanbul, July 2004
Supersymmetry
• A simultaneous description of even- and oddmass nuclei (doublets) or of even-even, evenodd, odd-even and odd-odd nuclei (quartets).
• Example of 194Pt, 195Pt, 195Au & 196Au:
F. Iachello, Phys. Rev. Lett. 44 (1980) 772
P. Van Isacker et al., Phys. Rev. Lett. 54 (1985) 653
A. Metz et al., Phys. Rev. Lett. 83 (1999) 1542
Nuclear Collective Dynamics II, Istanbul, July 2004
Example of
195Pt
Nuclear Collective Dynamics II, Istanbul, July 2004
Example of
196Au
Nuclear Collective Dynamics II, Istanbul, July 2004
Algebraic many-body models
• The integrability of any quantum many-body
(bosons and/or fermions) system 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.
 U 5   SO  5  


U 6   
SU 3 
  SO  3 
SO 6   SO  5 


Nuclear Collective Dynamics II, Istanbul, July 2004
Other fields of physics
• Molecular physics:
– U(4) vibron model with s,p-bosons.
 U 3  
U  4   
  SO  3 
SO 4 
– Coupling of many SU(2) algebras for polyatomic
molecules.
• Similar applications in hadronic, atomic, solidstate, polymer physics, quantum dots…
• Use of non-compact groups and algebras for
scattering problems.
F. Iachello, 1975 to now
Nuclear Collective Dynamics II, Istanbul, July 2004