Nuclear Structure
(II) Collective models
P. Van Isacker, GANIL, France
NSDD Workshop, Trieste, February 2006
Overview of collective models
•
•
•
•
•
•
(Rigid) rotor model
(Harmonic quadrupole) vibrator model
Liquid-drop model of vibrations and rotations
Interacting boson model
Particle-core coupling model
Nilsson model
NSDD Workshop, Trieste, February 2006
+
(2 )
Evolution of Ex
J.L. Wood, private communication
NSDD Workshop, Trieste, February 2006


Quantum-mechanical symmetric top
• Energy spectrum:
E rot I  
2
I I  1
2
 A II  1, I  0,2,4,
• Large deformation 
large   low Ex(2+).
• R42 energy ratio:
Erot 4  / Erot 2  3.333
NSDD Workshop, Trieste, February 2006

Rigid rotor model
• Hamiltonian of quantum-mechanical rotor in
terms of ‘rotational’ angular momentum R:
R12 R22 R32  2 3 Ri2
Hˆ rot   
  
2 1  2  3  2 i1  i
2
• Nuclei have an additional intrinsic part Hintr
with ‘intrinsic’ angular momentum J.
• The total angular momentum is I=R+J.
NSDD Workshop, Trieste, February 2006


Rigid axially symmetric rotor
• For 1=2= ≠ 3 the rotor hamiltonian is
3
Hˆ rot  
i1
2
2 i
3
Ii  Ji   
2
i1
• Eigenvalues of H´rot:
2
3
Ii  
2
2
2 i
i1

Hˆ rot
i
Ii J i  
Coriolis
 1 1  2
 
E KI
II 1    K
2
2  3  
2
2
• Eigenvectors KIM of H´rot satisfy:
I 2 KIM  I I  1 KIM ,
Iz KIM  M KIM , I3 KIM  K KIM
NSDD Workshop, Trieste, February 2006
3
i1
2
2 i
Ji
intrinsic
2
Ground-state band of an axial rotor
• The ground-state spin of
even-even nuclei is I=0.
Hence K=0 for groundstate band:
EI 
2
2
I I  1

NSDD Workshop, Trieste, February 2006
The ratio R42
NSDD Workshop, Trieste, February 2006
Electric (quadrupole) properties
• Partial -ray half-life:
1
2  1


E 
8

 1

T1/ 2 E  ln2
 BE
2 


 2 1!!  c 

• Electric quadrupole transitions:

1
BE2;Ii  If  
If M f


2Ii  1 M i M f 
A
2
e
r
 k k Y2 k , k  Ii M i
k1
• Electric quadrupole moments:
16
2
eQI   IM  I
e
r

k k Y20  k , k  IM  I
5 k1
A

NSDD Workshop, Trieste, February 2006
2


Magnetic (dipole) properties
• Partial -ray half-life:
1
2  1


E 
8

 1

T1/ 2 M  ln2
 BM
2 


 2 1!!  c 

• Magnetic dipole transitions:
1
BM1;Ii  If  
If M f


2Ii  1 M i M f 
A
l
s
g
l

g
  k k, k sk,  Ii M i
k1
• Magnetic dipole moments:
A
I  IM  I gkl lk,z  gks sk,z  IM  I
k1
NSDD Workshop, Trieste, February 2006
2


E2 properties of rotational nuclei
• Intra-band E2 transitions:
5
2 2
2
BE2;KIi  KIf  
IiK 20 If K e Q0 K
16
• E2 moments:
3K 2  I I  1
QKI  
Q0 K 
I  12I  3
• Q0(K) is the ‘intrinsic’ quadrupole moment:
ˆ 
eQ
0
2
2


r
r
3cos
    1dr, Q0K  K Qˆ 0 K
NSDD Workshop, Trieste, February 2006

E2 properties of ground-state bands
• For the ground state (usually K=I):
I2I 1
QK  I 
Q0 K 
I 12I  3
• For the gsb in even-even nuclei (K=0):
II 1
15
BE2;I  I  2 
e 2Q02
32 2I 12I  1
I
QI  
Q0
2I  3
2

 eQ21  
16  BE2;21  01 
7
NSDD Workshop, Trieste, February 2006
Generalized intensity relations
• Mixing of K arises from
– Dependence of Q0 on I (stretching)
– Coriolis interaction
– Triaxiality
• Generalized intra- and inter-band matrix
elements (eg E2):
BE2;K i Ii  K f If 
2
IiK i 2K f  K i If K f
 M 0  M1  M 2 
with   If If  1  Ii Ii  1
NSDD Workshop, Trieste, February 2006
Inter-band E2 transitions
• Example of g
transitions in 166Er:
BE2;I  Ig 
I 2 2  2 Ig 0
 M 0  M1  M 22 
  Ig Ig  1 I I  1
W.D. Kulp et al., Phys. Rev. C 73 (2006) 014308
NSDD Workshop, Trieste, February 2006
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
NSDD Workshop, Trieste, February 2006

Vibrations 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


NSDD Workshop, Trieste, February 2006



Properties of spherical vibrations
• Energy spectrum:
Evib n  n  52  , n  0,1
• R42 energy ratio:
Evib 4  / Evib 2  2
• E2 transitions:
BE2;21  01   2
BE2;2 2  01  0
BE2;n  2  n  1  2 2
NSDD Workshop, Trieste, February 2006
Example of
112Cd
NSDD Workshop, Trieste, February 2006
Possible vibrational nuclei from R42
NSDD Workshop, Trieste, February 2006
Vibrations about a spheroidal shape
• The vibration of a shape
with axial symmetry is
characterized by a.
• Quadrupole oscillations:
– =0: along the axis of
symmetry ()
– =1: spurious rotation
– =2: perpendicular to
axis of symmetry ()
NSDD Workshop, Trieste, February 2006






Spectrum of spheroidal vibrations
NSDD Workshop, Trieste, February 2006
Example of
166Er
NSDD Workshop, Trieste, February 2006

Rigid triaxial rotor
• Triaxial rotor hamiltonian 1 ≠ 2 ≠ 3 :
3
  
Hˆ rot
i1
2
2 i
Ii2 
2
2
I2 
2
2 f
I32 
2
2 g

Hˆ axial
2
2
I

I
  

Hˆ mix
1 1  1
1  1
1 1
1 1  1
1 
   ,

 ,
   
 2 1  2   f  3   g 4 1  2 
• H´mix non-diagonal in axial basis KIM  K
is not a conserved quantum number
NSDD Workshop, Trieste, February 2006
Rigid triaxial rotor spectra
  15
  30


NSDD Workshop, Trieste, February 2006
Tri-partite classification of nuclei
• Empirical 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
NSDD Workshop, Trieste, February 2006

Interacting boson model
• Describe the nucleus as a system of N
interacting s and d bosons. Hamiltonian:
6
Hˆ IBM  ibˆibˆi 
i1
6
 ˆ ˆ ˆ
ˆ

b
 i i i i i bi bi bi
i1 i2 i3 i4 1
1 2 3 4
1
2
3
4
• 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.
NSDD Workshop, Trieste, February 2006
Dimensions
• Assume  available 1-fermion states. Number
of n-fermion states is  !
 
n  n!  n!
• Assume  available 1-boson states. Number of
  n 1   n 1!
n-boson states is



n

 n! 1!
• Example: 162Dy96 with 14 neutrons (=44) and
16 protons (=32) (132Sn82 inert core).
 ~7·1019
– SM dimension:
– IBM dimension: 15504
NSDD Workshop, Trieste, February 2006


Dynamical symmetries
• Boson hamiltonian is of the form
6
Hˆ IBM  ibˆibˆi 
i1
6
 ˆ ˆ ˆ
ˆ

b
 i i i i i bi bi bi
i1 i2 i3 i4 1
1 2 3 4
1
2
3
4
• In general not solvable analytically.
• Three solvable cases with SO(3) symmetry:
U6  U 5  SO 5  SO 3
U6  SU 3  SO 3
U6  SO 6  SO 5  SO 3
NSDD Workshop, Trieste, February 2006
U(5) vibrational limit:
110Cd
NSDD Workshop, Trieste, February 2006
62
SU(3) rotational limit:
156Gd
NSDD Workshop, Trieste, February 2006
92
SO(6) -unstable limit:
196Pt
118
NSDD Workshop, Trieste, February 2006
Applications of IBM
NSDD Workshop, Trieste, February 2006
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 cos3  8 2
81 

2 2
1  2 

2 
1  
2
NSDD Workshop, Trieste, February 2006
Phase diagram of IBM
J. Jolie et al. , Phys. Rev. Lett. 87 (2001) 162501.
NSDD Workshop, Trieste, February 2006
The ratio R42
NSDD Workshop, Trieste, February 2006
Extensions of 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
NSDD Workshop, Trieste, February 2006
Scissors mode
• Collective displacement
modes between neutrons
and protons:
– Linear displacement
(giant dipole resonance):
R-R  E1 excitation.
– Angular displacement
(scissors resonance):
L-L  M1 excitation.
NSDD Workshop, Trieste, February 2006
Supersymmetry
• A simultaneous description of even- and odd-mass
nuclei (doublets) or of even-even, even-odd, oddeven and odd-odd nuclei (quartets).
• Example of 194Pt, 195Pt, 195Au & 196Au:
NSDD Workshop, Trieste, February 2006

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 aˆ j aˆ j 
j1
• Algebra:
6


  ˆ
ˆ

b
 i j i j i aˆ j bi aˆ j
i1 i2 1 j1 j 2 1

bˆi1 bˆi2
U6  U  


1 1 2 2
1
1
2
2




aˆ j1 aˆ j2 

• Many-body problem is solved analytically for
certain energies  and interactions .

NSDD Workshop, Trieste, February 2006
Example:
195Pt
117
NSDD Workshop, Trieste, February 2006
Example:
195Pt
117 (new
data)
NSDD Workshop, Trieste, February 2006

Nuclear supersymmetry
• Up to now: separate description of even-even
and odd-mass nuclei with the algebra

bˆi1 bˆi2
U6  U  






aˆ j1 aˆ j2 

• Simultaneous description of even-even and
odd-mass nuclei with the superalgebra

bˆi1 bˆi2
U6 /    ˆ

aˆ j1 bi2

bˆi1 aˆ j 2 


aˆ j1 aˆ j2 

NSDD Workshop, Trieste, February 2006
U(6/12) supermultiplet
NSDD Workshop, Trieste, February 2006
Example:
194Pt
116
195
& Pt117
NSDD Workshop, Trieste, February 2006
Example:
196Au
117
NSDD Workshop, Trieste, February 2006
Bibliography
• A. Bohr and B.R. Mottelson, Nuclear Structure. I
Single-Particle Motion (Benjamin, 1969).
• A. Bohr and B.R. Mottelson, Nuclear Structure. II
Nuclear Deformations (Benjamin, 1975).
• R.D. Lawson, Theory of the Nuclear Shell Model
(Oxford UP, 1980).
• K.L.G. Heyde, The Nuclear Shell Model (SpringerVerlag, 1990).
• I. Talmi, Simple Models of Complex Nuclei (Harwood,
1993).
NSDD Workshop, Trieste, February 2006
Bibliography (continued)
• P. Ring and P. Schuck, The Nuclear Many-Body
Problem (Springer, 1980).
• D.J. Rowe, Nuclear Collective Motion (Methuen,
1970).
• D.J. Rowe and J.L. Wood, Fundamentals of Nuclear
Collective Models, to appear.
• F. Iachello and A. Arima, The Interacting Boson Model
(Cambridge UP, 1987).
NSDD Workshop, Trieste, February 2006