Nuclear Structure Models
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Transcript Nuclear Structure Models
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 II 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 i1 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
i1
2
2 i
3
Ii Ji
2
i1
• Eigenvalues of H´rot:
2
3
Ii
2
2
2 i
i1
Hˆ rot
i
Ii J i
Coriolis
1 1 2
E KI
II 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
i1
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
BE
2
2 1!! c
• Electric quadrupole transitions:
1
BE2;Ii If
If M f
2Ii 1 M i M f
A
2
e
r
k k Y2 k , k Ii M i
k1
• Electric quadrupole moments:
16
2
eQI IM I
e
r
k k Y20 k , k IM I
5 k1
A
NSDD Workshop, Trieste, February 2006
2
Magnetic (dipole) properties
• Partial -ray half-life:
1
2 1
E
8
1
T1/ 2 M ln2
BM
2
2 1!! c
• Magnetic dipole transitions:
1
BM1;Ii If
If M f
2Ii 1 M i M f
A
l
s
g
l
g
k k, k sk, Ii M i
k1
• Magnetic dipole moments:
A
I IM I gkl lk,z gks sk,z IM I
k1
NSDD Workshop, Trieste, February 2006
2
E2 properties of rotational nuclei
• Intra-band E2 transitions:
5
2 2
2
BE2;KIi KIf
IiK 20 If K e Q0 K
16
• E2 moments:
3K 2 I I 1
QKI
Q0 K
I 12I 3
• Q0(K) is the ‘intrinsic’ quadrupole moment:
ˆ
eQ
0
2
2
r
r
3cos
1dr, Q0K K Qˆ 0 K
NSDD Workshop, Trieste, February 2006
E2 properties of ground-state bands
• For the ground state (usually K=I):
I2I 1
QK I
Q0 K
I 12I 3
• For the gsb in even-even nuclei (K=0):
II 1
15
BE2;I I 2
e 2Q02
32 2I 12I 1
I
QI
Q0
2I 3
2
eQ21
16 BE2;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):
BE2;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:
BE2;I Ig
I 2 2 2 Ig 0
M 0 M1 M 22
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:
BE2;21 01 2
BE2;2 2 01 0
BE2;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
i1
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 ibˆibˆi
i1
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 ibˆibˆi
i1
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:
U6 U 5 SO 5 SO 3
U6 SU 3 SO 3
U6 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
81
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
j1
• Algebra:
6
ˆ
ˆ
b
i j i j i aˆ j bi aˆ j
i1 i2 1 j1 j 2 1
bˆi1 bˆi2
U6 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
bˆi1 bˆi2
U6 U
aˆ j1 aˆ j2
• Simultaneous description of even-even and
odd-mass nuclei with the superalgebra
bˆi1 bˆi2
U6 / ˆ
aˆ j1 bi2
bˆi1 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