2358-20 Joint ICTP-IAEA Workshop on Nuclear Structure Decay Data: Theory and Evaluation 6-17 August 2012 Introduction to Nuclear Physics - 2 P. Van Isacker GANIL, Grand Accelerateur National d'ions Lourds France
Nuclear Structure (II) Collective models P. Van Isacker, GANIL, France
Some collective models (Rigid) rotor model (Harmonic quadrupole) vibrator model Liquid-drop model of vibrations and rotations Interacting boson model
Evolution of E x (2 + ) J.L. Wood, private communication
Symmetric top Energy spectrum: E rot ( I) = 2 ( ) 2I I I +1 A I( I +1), I = 0,2,4, Large deformation large I low E x (2 + ). R 42 energy ratio: E ( rot 4 + ) / E ( rot 2 + ) = 3.333
Rigid rotor model Hamiltonian of quantum-mechanical rotor in terms of rotational angular momentum R: ˆ H rot = 2 2 2 R 1 + R 2 2 + R 2 3 = 2 I 1 I 2 2 I 3 3 i=1 Nuclei have an additional intrinsic part H intr with intrinsic angular momentum J. 2 R i I i The total angular momentum is I=R+J.
Rigid axially symmetric rotor For I 1 =I 2 =I I 3 the rotor hamiltonian is ˆ H rot = 3 i=1 2 2 I 2I i i = 2 2I Eigenvalues of H rot : E KI = 2 2I I( I +1) + 2 2 3 i=1 I 2 i + 2 2 1 1 K 2 I 3 I 1 1 2 I I 3 I 3 Eigenvectors KIM of H rot satisfy: I 2 KIM = I( I +1) KIM, I z KIM = M KIM, I 3 KIM = K KIM
Ground-state band of axial rotor The ground-state spin of even-even nuclei is I=0. Hence K=0 for ground-state band: E I = 2 2I I ( I +1 )
The ratio R 42
Electric (quadrupole) properties Partial γ-ray half-life: γ T 1/ 2 ( Eλ) = ln2 8π 2λ+1 λ +1 E γ λ[ ( 2λ +1)!! ] 2 B( Eλ) c Electric quadrupole transitions: B( E2;I i I f ) = 1 2I i +1 M i M f µ I f M f Electric quadrupole moments: eq( I) = IM = I 16π 5 A k=1 A k=1 2 e k r k Y 20 θ k,ϕ k e k r 2 k Y 2µ ( θ k,ϕ k ) ( ) IM = I 1 I i M i 2
Magnetic (dipole) properties Partial γ-ray half-life: γ T 1/ 2 ( Mλ) = ln2 8π λ +1 λ ( 2λ +1)!! Magnetic dipole transitions: B( M1;I i I f ) = 1 2I i +1 M i M f µ Magnetic dipole moments: ( ) = IM = I g k l l k,z + g k s s k,z µ I A k=1 E γ [ ] 2 c ( ) I f M f A ( g l k l k,µ + g s k s ) k,µ I i M i k=1 IM = I 2λ+1 B Mλ ( ) 1 2
E2 properties of rotational nuclei Intra-band E2 transitions: B( E2;KI i KI f ) = 5 16π I ik 20 I f K 2 e 2 Q 0 ( K) 2 E2 moments: ( ) = 3K 2 I( I +1) ( I +1) ( 2I + 3) Q 0( K) Q KI Q 0 (K) is the intrinsic quadrupole moment: eq ˆ 0 ρ( r ʹ ) r 2 ( 3cos 2 θ ʹ 1)dr ʹ, Q 0 ( K) = K Q ˆ 0 K
E2 properties of gs bands For the ground state (K=I): Q( K = I) = I( 2I 1) ( I +1) ( 2I + 3) Q 0( K) For the gsb in even-even nuclei (K=0): B( E2;I I 2) = 15 I( I 1) 32π ( 2I 1) ( 2I +1) e2 2 Q 0 Q( I) = eq 2 1 + I 2I + 3 Q 0 ( ) = 2 7 16π B ( E2;2 + + 1 0 ) 1
Generalized intensity relations Mixing of K arises from Dependence of Q 0 on I (stretching) Coriolis interaction Triaxiality Generalized intra- and inter-band matrix elements (eg E2): B( E2;K i I i K f I ) f = M 0 + M 1 Δ + M 2 Δ 2 + I i K i 2K f K i I f K f with Δ = I ( f I f +1) I ( i I i +1)
Inter-band E2 transitions Example of γ g transitions in 166 Er: B( E2;I γ I ) g I γ 2 2 2 I g 0 = M 0 + M 1 Δ + M 2 Δ 2 + Δ = I ( g I g +1) I ( γ I γ +1) W.D. Kulp et al., Phys. Rev. C 73 (2006) 014308
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
Vibrations about a spherical shape Vibrations are characterized by a multipole quantum number λ in surface parametrization: * ( ) = R 0 1+ α λµ Y λµ ( θ,ϕ) R θ,ϕ λ +λ µ= λ λ=0: compression (high energy) λ=1: translation (not an intrinsic excitation) λ=2: quadrupole vibration
Properties of spherical vibrations Energy spectrum: E vib ( n) = ( n + 5 2) ω, n = 0,1 R 42 energy ratio: E vib 4 + ( ) / E ( vib 2 + ) = 2 E2 transitions: ( ) = α 2 B E2;2 1 + 0 1 + B( E2;2 + + 2 0 ) 1 = 0 ( ) = 2α 2 B E2;n = 2 n =1
Example of 112 Cd
Possible vibrational nuclei from R 42
Vibrations of a spheroidal shape The vibration of a shape with axial symmetry is characterized by a λν. Quadrupole oscillations: ν=0: along the axis of symmetry (β) ν=±1: spurious rotation ν=±2: perpendicular to axis of symmetry (γ) γ c β c β γ
Spectrum of spheroidal vibrations
Example of 166 Er
Rigid triaxial rotor Triaxial rotor hamiltonian I 1 I 2 I 3 : ˆ ʹ H rot = 1 I = 1 2 3 i=1 2 2 I i 2I i 1 + 1, I 1 I 2 = 2 2I I 2 + 2 2 I 3 2I f ˆ ʹ H axial + 2 ( ) I 2 2 + + I 2I g H' mix non-diagonal in axial basis KIM K is not a conserved quantum number ˆ ʹ H mix 1 = 1 1 I f I 3 I, 1 = 1 I g 4 1 1 I 1 I 2
Rigid triaxial rotor spectra γ = 30 γ =15
Tri-partite classification of nuclei Empirical evidence for seniority-type, vibrationaland rotational-like nuclei. N.V. Zamfir et al., Phys. Rev. Lett. 72 (1994) 3480
Interacting boson model Describe the nucleus as a system of N interacting s and d bosons. Hamiltonian: ˆ H IBM = ε ˆ + i b ˆ i b i + υ ˆ + i1 i 2 i 3 i 4 b ˆ + i1 b ˆ i2 b ˆ i3 i=1 Justification from 6 6 i 1 i 2 i 3 i 4 =1 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. b i4
Dimensions Assume Ω available 1-fermion states. Number of n-fermion states is Ω Ω! = n n! ( Ω n)! Assume Ω available 1-boson states. Number of n- boson states is Ω + n 1 n = ( Ω + n 1 )! n! ( Ω 1)! Example: 162 Dy 96 with 14 neutrons (Ω=44) and 16 protons (Ω=32) ( 132 Sn 82 inert core). SM dimension: 7 10 19 IBM dimension: 15504
Dynamical symmetries Boson hamiltonian is of the form ˆ H IBM = 6 ε ˆ + i b ˆ i b i + υ ˆ + i1 i 2 i 3 i 4 b ˆ + i1 b ˆ i2 b ˆ i3 i=1 6 i 1 i 2 i 3 i 4 =1 In general not solvable analytically. Three solvable cases with SO(3) symmetry: ( ) U( 5) SO( 5) SO( 3) ( ) SU( 3) SO( 3) ( ) SO( 6) SO( 5) SO( 3) U 6 U 6 U 6 b i4
U(5) vibrational limit: 110 Cd 62
SU(3) rotational limit: 156 Gd 92
SO(6) γ-unstable limit: 196 Pt 118
Applications of IBM
Classical limit of IBM For large boson number N, a coherent (or intrinsic) state is an approximate eigenstate, ( ) N o µ H ˆ IBM N;α µ E N;α µ, N;α µ s + + + α µ d µ The real parameters α µ are related to the three Euler angles and shape variables β and γ. Any IBM hamiltonian yields energy surface: N;α µ ˆ H IBM N;α µ = N;βγ H ˆ IBM N;βγ V ( β,γ)
Phase diagram of IBM J. Jolie et al., Phys. Rev. Lett. 87 (2001) 162501
The ratio R 42
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).
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).
Extensions of IBM Neutron and proton degrees freedom (IBM-2): F-spin multiplets (N ν +N π =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
Scissors mode Collective displacement modes between neutrons and protons: Linear displacement (giant dipole resonance): R ν -R π E1 excitation. Angular displacement (scissors resonance): L ν -L π M1 excitation.
Supersymmetry A simultaneous description of even- and oddmass nuclei (doublets) or of even-even, even-odd, odd-even and odd-odd nuclei (quartets). Example of 194 Pt, 195 Pt, 195 Au & 196 Au:
Bosons + fermions Odd-mass nuclei are fermions. Describe an odd-mass nucleus as N bosons + 1 fermion mutually interacting. Hamiltonian: H ˆ IBFM = H ˆ IBM + Algebra: Ω ε j a ˆ j+ a ˆ j + υ ˆ i1 j 1 i 2 j 2 b i1+ ˆ j=1 U( 6) U( Ω) = b ˆ + ˆ i1 6 i 1 i 2 =1 Ω j 1 j 2 =1 a ˆ + j1 a ˆ j2 Many-body problem is solved analytically for certain energies ε and interactions υ. b i2 a j1 + ˆ b i2 ˆ a j2
Example: 195 Pt 117
Example: 195 Pt 117 (new data)
Nuclear supersymmetry Up to now: separate description of even-even and odd-mass nuclei with the algebra U( 6) U( Ω) = b ˆ + ˆ i1 b i2 a ˆ + j1 a ˆ j2 Simultaneous description of even-even and oddmass nuclei with the superalgebra U( 6 /Ω) = b ˆ + ˆ i1 + a ˆ ˆ j1 b i2 b i2 ˆ b i1+ ˆ ˆ a j1 a j2 + ˆ a j2
U(6/12) supermultiplet
Example: 194 Pt 116 & 195 Pt 117
Example: 196 Au 117