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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

The nuclear many-body problem
The nucleus is a system of A interacting nucleons
described by the hamiltonian
A
Hˆ  
k1 1
pk21
2mk1
  r , sˆ
ki
ki
ki

A
Vˆ 
2
1k1 k2
k1 , k2 
, tˆki , p ki  i  ki
A
Vˆ 
3
1k1 k2 k3
k1
, k2 , k3 

Separation in mean field + residual interaction:
 p 2
  A
k
Hˆ   1  Vˆ  k1   Vˆ2  k1 , k2 

2mk1
 
k1 1
1k1 k2
A
mean field

 Vˆ  k1 

k1 1

residual interaction
Symmetries in Nuclei, Tokyo, 2008
A


The nuclear shell model
Hamiltonian with one-body term (mean field) and
two-body interactions:
A
Hˆ SM  Uˆ  k1 
k1 1
A
Wˆ 
2
1k1 k2
k1
, k2 
Entirely equivalent form of the same hamiltonian in
second quantization:
Hˆ SM  i ai ai   ijklai aj ak al
i
ijkl
, : single-particle energies & interactions
ijkl: single-particle quantum numbers
Symmetries in Nuclei, Tokyo, 2008


Boson and fermion statistics
Fermions have half-integer spin and obey FermiDirac statistics:





a
,a

a
a

a
a


,
a
,a

a
,a
 i j  i j j i ij  i j   i j  0
Bosons have integer spin and obey Bose-Einstein
statistics:



 
b
,b

b
b

b
b


,
b
,b

b
 i j  i j j i ij  i j   i ,b j  0
Matter is carried by fermions. Interactions are
carried by bosons. Composite matter particles can
be fermions or bosons.
Symmetries in Nuclei, Tokyo, 2008


Boson-fermion systems
Denote particle (boson or fermion) creation and
annihilation operators by c+ and c.
Introduce (q=0 for bosons; q=1 for fermions):
q
uˆ,vˆq  uˆ vˆ   vˆuˆ
Boson/fermion statistics is summarized by

c i ,c j   ij ,
q

Many-particle state:

i
c 
 ni
i
n i!

c i ,c j   c i ,c j   0
q
q
o , ci o  0
Symmetries in Nuclei, Tokyo, 2008


Algebraic structure
Introduce bilinear operators
uˆ ij  cic j  uˆ ij , uˆ kl  uˆ il jk  uˆ kjil  U
Rewrite many-body hamiltonian
Hˆ  E 0  ic ic i   ijklc ic j c kc l
i
ijkl


q
q
ˆ


 E 0  iil    ijkl uil    ijkl uˆ ik uˆ jl
il 
j

ijkl
Symmetries in Nuclei, Tokyo, 2008
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:
pk2 1

2 2
2
ˆ
ˆ
ˆ
H    m rk   ls lk  sˆk   ll lk 
2m 2

k1 
A

A
Wˆ 
2
1k1 k2
k1
, k2 
Symmetries in Nuclei, Tokyo, 2008
Racah’s SU(2) pairing model
Assume pairing interaction in a single-j shell:
j 2 JM J Vˆpairing j 2 JM J
 1 2 j  1g0, J  0
  2
0,
J 0

Spectrum 210Pb:

Symmetries in Nuclei, Tokyo, 2008

Pairing between identical nucleons
Analytic solution of the pairing hamiltonian based on
SU(2) symmetry. E.g., energies:
j nJ
n
1
n
ˆ
V
k,l
j

J

g
 pairing 
0 4 n   2 j  n    3
1kl
Seniority  (number of nucleons not in pairs coupled
to J=0) is a good quantum number.
Correlated ground-state solution (cf. BCS).
G. Racah, Phys. Rev. 63 (1943) 367
G. Racah, L. Farkas Memorial Volume (1952) p. 294
B.H. Flowers, Proc. Roy. Soc. (London) A 212 (1952) 248
A.K. Kerman, Ann. Phys. (NY) 12 (1961) 300
Symmetries in Nuclei, Tokyo, 2008
Nuclear superfluidity
Ground states of pairing hamiltonian have the
following correlated character:

Even-even nucleus (=0):   o , Sˆ    am
amÝÝÝ
m
Odd-mass nucleus (=1): a Sˆ n / 2 o

m
Sˆ 
n /2
Nuclear superfluidity leads to
Constant energy offirst 2+ in even-even nuclei.
 in masses.
Odd-even staggering
Smooth variation of two-nucleon separation energies with
nucleon number.
Two-particle (2n or 2p) transfer enhancement.
Symmetries in Nuclei, Tokyo, 2008
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.
Symmetries in Nuclei, Tokyo, 2008

Wigner’s SU(4) symmetry
Assume the nuclear hamiltonian is invariant under
spin and isospin rotations:
Hˆ
 
 

ˆ  Hˆ , Tˆ  Hˆ ,Yˆ  0
,
S
nucl 
nucl 
nucl 
A
A
A
k1
k1
k1
Sˆ    sˆ k , Tˆ   tˆ k , Yˆ   sˆ k tˆ k 
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
Symmetries in Nuclei, Tokyo, 2008
Physical origin of SU(4) symmetry
SU(4) labels specify the separate spatial and spinisospin symmetry of the wave function.
Nuclear interaction is short-range attractive and
hence favours maximal spatial symmetry.
Symmetries in Nuclei, Tokyo, 2008
Pairing with neutrons and protons
For neutrons and protons two pairs and hence two
pairing interactions are possible:
1S
0

isovector or spin singlet (S=0,T=1): Sˆ    am
am 
m0

3S
1
isoscalar or spin triplet (S=1,T=0):

Pˆ   am
am 
m0

Symmetries in Nuclei, Tokyo, 2008
Neutron-proton pairing hamiltonian
The nuclear hamiltonian has two pairing interactions
Vˆpairingalgebraic
 g0 Sˆ   Sˆstructure.
 g1Pˆ  Pˆ
SO(8)
Integrable and solvable for g0=0, g1=0 and g0=g1.

B.H. Flowers & S. Szpikowski, Proc. Phys. Soc. 84 (1964) 673
Symmetries in Nuclei, Tokyo, 2008

Quartetting in N=Z nuclei
Pairing ground state of an N=Z nucleus:

cos Sˆ   Sˆ   sin  Pˆ  Pˆ

n /4
o
 Condensate of “-like” objects.
Observations:
Isoscalar component in condensate survives only in NZ
nuclei, if anywhere at all.
Spin-orbit term reduces isoscalar component.
Symmetries in Nuclei, Tokyo, 2008
(d,) and (p,3He) transfer
Symmetries in Nuclei, Tokyo, 2008

Wigner energy
Extra binding energy of
N=Z nuclei (cusp).
Wigner energy BW is
decomposed in two
parts:
BW  W A N  Z
 dAN ,Z  np
W(A) and d(A) can be
fixed empirically from
binding energies.
P. Möller & R. Nix, Nucl. Phys. A 536 (1992) 20
J.-Y. Zhang et al., Phys. Lett. B 227 (1989) 1
W. Satula et al., Phys. Lett. B 407 (1997) 103
Symmetries in Nuclei, Tokyo, 2008
Connection with SU(4) model
Wigner’s explanation of the ‘kinks in the mass
defect curve’ was based on SU(4) symmetry.
Symmetry contribution to the nuclear binding energy
is



K Ag, ,   K A N  Z   8 N  Z  8N,Z  np  6pairing
2
SU(4) symmetry is broken by spin-orbit term. Effects
of SU(4) mixing must be included.
D.D. Warner et al., Nature Physics 2 (2006) 311
Symmetries in Nuclei, Tokyo, 2008

Elliott’s SU(3) model of rotation
Harmonic oscillator mean field (no spin-orbit) with
residual interaction of quadrupole type:
pk2 1

2 2
ˆ
H    m rk  g2Qˆ  Qˆ ,
2

k1 2m
A
A
Qˆ    rk2Y2 rˆk 
k1
A
  pk2Y2 pˆ k 
k1
J.P. Elliott, Proc. Roy. Soc. A 245 (1958) 128; 562
Symmetries in Nuclei, Tokyo, 2008
Importance & limitations of SU(3)
Historical importance:
Bridge between the spherical shell model and the liquiddrop model through mixing of orbits.
Spectrum generating algebra of Wigner’s SU(4) model.
Limitations:
LS (Russell-Saunders) coupling, not jj coupling (no spinorbit splitting)  (beginning of) sd shell.
Q is the algebraic quadrupole operator  no major-shell
mixing.
Symmetries in Nuclei, Tokyo, 2008
Breaking of SU(4) symmetry
SU(4) symmetry breaking as a consequence of
Spin-orbit term in nuclear mean field.
Coulomb interaction.
Spin-dependence of the nuclear interaction.
Evidence for SU(4) symmetry breaking from masses
and from Gamow-Teller  decay.
Symmetries in Nuclei, Tokyo, 2008
SU(4) breaking from masses
Double binding energy difference Vnp
Vnp N,Z   14 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
Symmetries in Nuclei, Tokyo, 2008
SU(4) breaking from  decay
Gamow-Teller decay into odd-odd or even-even
N=Z nuclei.
P. Halse & B.R. Barrett, Ann. Phys. (NY) 192 (1989) 204
Symmetries in Nuclei, Tokyo, 2008
Pseudo-spin symmetry
Apply a helicity transformation to the spin-orbit +
orbit-orbit nuclear mean field:


ˆ  sˆ   lˆ  lˆ uˆ  4   lˆ  sˆ   lˆ  lˆ  c te
uˆ 1

l
k
k
k
k
k
k
k
k
k
k
sˆk  rk
uˆ k  2i
rk
Degeneracies
for 4=.

K.T. Hecht & A. Adler, Nucl. Phys. A 137 (1969) 129
A. Arima et al., Phys. Lett. B 30 (1969) 517
R.D. Ratna et al., Nucl. Phys. A 202 (1973) 433
J.N. Ginocchio, Phys. Rev. Lett. 78 (1998) 436
Symmetries in Nuclei, Tokyo, 2008
occur

Pseudo-SU(4) symmetry
Assume the nuclear hamiltonian is invariant under
pseudo-spin and isospin rotations:





Hˆ nucl, Sˆ˜   Hˆ nucl, Tˆ  Hˆ nucl,Yˆ˜  0
A
A
A
k1
Consequences:
k1
k1
Sˆ˜    sˆ˜ k , Tˆ   tˆ k , Yˆ˜   sˆ˜ k tˆ k 
Hamiltonian has pseudo-SU(4) symmetry.
Total pseudo-spin, total pseudo-orbital angular
momentum, total isospin and pseudo-SU(4) labels are
conserved quantum numbers.
D. Strottman, Nucl. Phys. A 188 (1971) 488
Symmetries in Nuclei, Tokyo, 2008
Test of pseudo-SU(4) symmetry
Shell-model test of
pseudo-SU(4).
Realistic interaction in
pf5/2g9/2 space.
Example: 58Cu.
P. Van Isacker et al., Phys. Rev. Lett. 82 (1999) 2060
Symmetries in Nuclei, Tokyo, 2008
Pseudo-SU(4) and  decay
Pseudo-spin transformed Gamow-Teller operator is
deformation dependent:

ˆs˜ tˆ  uˆ sˆ tˆ uˆ   1 sˆ tˆ  20 1 r  r 2  sˆ
 
 
 
3
3 r2
1
Test:  decay of
58Zn.

A. Jokinen et al., Eur. Phys. A. 3 (1998) 271
Symmetries in Nuclei, Tokyo, 2008
 tˆ
1


18Ne
vs.
Three faces of the shell model
Symmetries in Nuclei, Tokyo, 2008