nsdd workshop, trieste, february 2006 nuclear structure (ii) collective models p. van isacker,...
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NSDD Workshop, Trieste, February 2006
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
Evolution of Ex(2+)
J.L. Wood, private communication
NSDD Workshop, Trieste, February 2006
Quantum-mechanical symmetric top• Energy spectrum:
• Large deformation large low Ex(2+).
• R42 energy ratio:
€
E rot I( ) =h2
2ℑI I +1( )
≡ A I I +1( ), I = 0,2,4,K
€
E rot 4+( ) / E rot 2+
( ) = 3.333K
NSDD Workshop, Trieste, February 2006
Rigid rotor model
• Hamiltonian of quantum-mechanical rotor in terms of ‘rotational’ angular momentum R:
• Nuclei have an additional intrinsic part Hintr with ‘intrinsic’ angular momentum J.
• The total angular momentum is I=R+J.
€
ˆ H rot =h2
2
R12
ℑ 1
+R2
2
ℑ 2
+R3
2
ℑ 3
⎡
⎣ ⎢
⎤
⎦ ⎥=
h2
2
Ri2
ℑ ii=1
3
∑
NSDD Workshop, Trieste, February 2006
Rigid axially symmetric rotor
• For 1=2= ≠ 3 the rotor hamiltonian is
• Eigenvalues of H´rot:
• Eigenvectors KIM of H´rot satisfy:
€
ˆ H rot =h2
2ℑ i
Ii − Ji( )2
i=1
3
∑ =h2
2ℑ i
Ii2
i=1
3
∑ˆ ′ H rot
1 2 4 3 4 −
h2
ℑ i
IiJi
i=1
3
∑Coriolis
1 2 4 3 4 +
h2
2ℑ i
Ji2
i=1
3
∑intrinsic
1 2 4 3 4
€
′ E KI =h2
2ℑI I +1( ) +
h2
2
1
ℑ 3
−1
ℑ
⎛
⎝ ⎜
⎞
⎠ ⎟K
2
€
I2 KIM = I I +1( ) KIM ,
Iz KIM = M KIM , I3 KIM = K KIM
NSDD Workshop, Trieste, February 2006
Ground-state band of an axial rotor• The ground-state spin of
even-even nuclei is I=0. Hence K=0 for ground-state band:
€
E I =h2
2ℑI I +1( )
NSDD Workshop, Trieste, February 2006
The ratio R42
NSDD Workshop, Trieste, February 2006
Electric (quadrupole) properties
• Partial -ray half-life:
• Electric quadrupole transitions:
• Electric quadrupole moments:
€
B E2;Ii → If( ) =1
2Ii +1If M f ekrk
2Y2μ θk,ϕ k( )k=1
A
∑ IiM i
2
M f μ
∑M i
∑
€
eQ I( ) = IM = I16π
5ekrk
2
k=1
A
∑ Y20 θk,ϕ k( ) IM = I
€
T1/ 2γ Eλ( ) = ln2
8π
h
λ +1
λ 2λ +1( )!![ ]2
Eγ
hc
⎛
⎝ ⎜
⎞
⎠ ⎟
2λ +1
B Eλ( ) ⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
−1
NSDD Workshop, Trieste, February 2006
Magnetic (dipole) properties
• Partial -ray half-life:
• Magnetic dipole transitions:
• Magnetic dipole moments:
€
B M1;Ii → If( ) =1
2Ii +1If M f gk
l lk,μ + gkssk,μ( ) IiM i
k=1
A
∑2
M f μ
∑M i
∑
€
μ I( ) = IM = I gkl lk,z + gk
ssk,z( )k=1
A
∑ IM = I
€
T1/ 2γ Mλ( ) = ln2
8π
h
λ +1
λ 2λ +1( )!![ ]2
Eγ
hc
⎛
⎝ ⎜
⎞
⎠ ⎟
2λ +1
B Mλ( ) ⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
−1
NSDD Workshop, Trieste, February 2006
E2 properties of rotational nuclei
• Intra-band E2 transitions:
• E2 moments:
• Q0(K) is the ‘intrinsic’ quadrupole moment:
€
B E2;KIi → KIf( ) =5
16πIiK 20 IfK
2e2Q0 K( )
2
€
Q KI( ) =3K 2 − I I +1( )I +1( ) 2I + 3( )
Q0 K( )
€
e ˆ Q 0 ≡ ρ ′ r ( )∫ r2 3cos2 ′ θ −1( )d ′ r , Q0 K( ) = K ˆ Q 0 K
NSDD Workshop, Trieste, February 2006
E2 properties of ground-state bands
• For the ground state (usually K=I):
• For the gsb in even-even nuclei (K=0):
€
Q K = I( ) =I 2I −1( )
I +1( ) 2I + 3( )Q0 K( )
€
B E2;I → I − 2( ) =15
32π
I I −1( )2I −1( ) 2I +1( )
e2Q02
Q I( ) = −I
2I + 3Q0
⇒ eQ 21+
( ) =2
716π ⋅B E2;21
+ → 01+
( )
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 iIi → K fIf( )
IiK i 2K f − K i IfK f
= M0 + M1Δ + M2Δ2 +L
with Δ = If If +1( ) − Ii Ii +1( )
NSDD Workshop, Trieste, February 2006
Inter-band E2 transitions• Example of g
transitions in 166Er:
€
B E2;Iγ → Ig( )
Iγ 2 2 − 2 Ig0
= M0 + M1Δ + M2Δ2 +L
Δ = Ig Ig +1( ) − Iγ Iγ +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:
=0: compression (high energy) =1: translation (not an intrinsic excitation) =2: quadrupole vibration
€
R θ,ϕ( ) = R0 1+ α λμYλμ* θ,ϕ( )
μ =−λ
+λ
∑λ
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
⇔ ⇔
NSDD Workshop, Trieste, February 2006
Properties of spherical vibrations• Energy spectrum:
• R42 energy ratio:
• E2 transitions:
€
Evib n( ) = n + 5
2( )hω, n = 0,1K
€
Evib 4+( ) / Evib 2+
( ) = 2
€
B E2;21+ → 01
+( ) = α 2
B E2;22+ → 01
+( ) = 0
B E2;n = 2 → n =1( ) = 2α 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 ()
c βγ
⇔γ⇔
c β
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 :
• H´mix non-diagonal in axial basis KIM K is not a conserved quantum number
€
ˆ ′ H rot =h2
2ℑ i
Ii2
i=1
3
∑ =h2
2ℑI2 +
h2
2ℑ f
I32
ˆ ′ H axial
1 2 4 4 3 4 4
+h2
2ℑ g
I+2 + I−
2( )
ˆ ′ H mix
1 2 4 4 3 4 4
1
ℑ=
1
2
1
ℑ 1
+1
ℑ 2
⎛
⎝ ⎜
⎞
⎠ ⎟,
1
ℑ f
=1
ℑ 3
−1
ℑ,
1
ℑ g
=1
4
1
ℑ 1
−1
ℑ 2
⎛
⎝ ⎜
⎞
⎠ ⎟
NSDD Workshop, Trieste, February 2006
Rigid triaxial rotor spectra
€
=30o
€
=15o
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:
• 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.
€
ˆ H IBM = ε iˆ b i
+ ˆ b ii=1
6
∑ + υ i1i2i3i4ˆ b i1
+ ˆ b i2+ ˆ b i3
ˆ b i4i1i2i3i4 =1
6
∑
NSDD Workshop, Trieste, February 2006
Dimensions• Assume available 1-fermion states. Number
of n-fermion states is
• Assume available 1-boson states. Number of n-boson states is
• Example: 162Dy96 with 14 neutrons (=44) and 16 protons (=32) (132Sn82 inert core).– SM dimension: ~7·1019
– IBM dimension: 15504
€
n
⎛
⎝ ⎜
⎞
⎠ ⎟=
Ω!
n! Ω − n( )!
€
+ n −1
n
⎛
⎝ ⎜
⎞
⎠ ⎟=
Ω + n −1( )!
n! Ω −1( )!
NSDD Workshop, Trieste, February 2006
Dynamical symmetries
• Boson hamiltonian is of the form
• In general not solvable analytically.
• Three solvable cases with SO(3) symmetry:€
ˆ H IBM = ε iˆ b i
+ ˆ b ii=1
6
∑ + υ i1i2i3i4ˆ b i1
+ ˆ b i2+ ˆ b i3
ˆ b i4i1i2i3i4 =1
6
∑
€
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: 110Cd62
NSDD Workshop, Trieste, February 2006
SU(3) rotational limit: 156Gd92
NSDD Workshop, Trieste, February 2006
SO(6) -unstable limit: 196Pt118
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:
€
V β,γ( )∝
U(5) :β 2
1+ β 2
SU(3) :β 4 − 4 2β 3 cos3γ + 8β 2
8 1+ β 2( )
2
SO(6) :1− β 2
1+ β 2
⎛
⎝ ⎜
⎞
⎠ ⎟
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, odd-even 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:
• Algebra:
• Many-body problem is solved analytically for certain energies and interactions .
€
ˆ H IBFM = ˆ H IBM + ε j ˆ a j+ ˆ a j
j=1
Ω
∑ + υ i1 j1i2 j2
ˆ b i1+ ˆ a j1
+ ˆ b i2 ˆ a j2
j1 j2 =1
Ω
∑i1i2 =1
6
∑
€
U 6( )⊕U Ω( ) =ˆ b i1
+ ˆ b i2ˆ a j1
+ ˆ a j2
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
NSDD Workshop, Trieste, February 2006
Example: 195Pt117
NSDD Workshop, Trieste, February 2006
Example: 195Pt117 (new data)
NSDD Workshop, Trieste, February 2006
Nuclear supersymmetry
• Up to now: separate description of even-even and odd-mass nuclei with the algebra
• Simultaneous description of even-even and odd-mass nuclei with the superalgebra
€
U 6( )⊕U Ω( ) =ˆ b i1
+ ˆ b i2ˆ a j1
+ ˆ a j2
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
€
U 6/Ω( ) =ˆ b i1
+ ˆ b i2ˆ b i1
+ ˆ a j2
ˆ a j1+ ˆ b i2 ˆ a j1
+ ˆ a j2
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
NSDD Workshop, Trieste, February 2006
U(6/12) supermultiplet
NSDD Workshop, Trieste, February 2006
Example: 194Pt116 &195Pt117
NSDD Workshop, Trieste, February 2006
Example: 196Au117
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 (Springer-
Verlag, 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).