Nuclear Structure (II) Collective models P. Van Isacker, GANIL, France NSDD Workshop, Trieste, March 2014
TALENT school TALENT (Training in Advanced Low-Energy Nuclear Theory, see http://www.nucleartalent.org). Course 5: Theory for exploring nuclear structure experiments. Lecturers: M.A. Caprio, R.F. Casten, J. Jolie, A.O. Macchiavelli, N. Pietralla and P. Van Isacker. Audience: Master of Science and PhD students, and early post-doctoral students. Place: GANIL. Dates: 11-29 August 2014.
Collective nuclear models Rotational models Vibrational models Liquid-drop model of vibrations and rotations Interacting boson model
Rotation of a 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
Evolution of E x (2 + ) J.L. Wood, private communication
The ratio R42
Reflection Asymmetric Rotation of an asymmetric top At "high-spins" the spectrum for 222 Ra shows a I=1 sequence with alternating parity; consistent with E ( rot a stable I π ) = 2 octupole deformation, and similar to the reflectionasymmetric rotor, HCl. Shapes in Nuclei Energy spectrum: Reflection symmetry only allows even I with positive with parity alternating π. parity; Reflection Asymmetric ( ) 2I I I +1 I π = 0 +,1, 2 +,3, 4 +, At "high-spins" the spectrum for 222 Ra shows a I=1 sequence consistent with a stable octupole deformation, and similar to the reflectionasymmetric rotor, HCl. Shapes in Nuclei
Nuclear shapes Shapes are characterized by the variables α λµ in the surface parameterization: ( +λ + * R( θ,ϕ) = R 0 * 1+ α λµ Y λµ ( θ,ϕ) - ) λ µ= λ, λ=0: compression (high energy) λ=1: translation (not an intrinsic deformation) λ=2: quadrupole deformation λ=3: octupole deformation
Quadrupole shapes Since the surface R(θ,φ) is real: ( α ) * λµ = 1 ( ) µ α λ µ è Five independent quadrupole variables (λ=2). Equivalent to three Euler angles and two intrinsic variables β and γ: ( ) α 2µ = 2 a 2ν D µν Ω, a 21 = a 2 1 = 0, a 22 = a 2 2 ν a 20 = β cosγ, a 22 = 1 2 β sinγ
The (β,γ) plane
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 λ in the surface parameterization: λ=0: compression (high energy) λ=1: translation (not an intrinsic excitation) λ=2: quadrupole vibration λ=3: octupole vibration
Spherical quadrupole 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: B( E2;2 + + 1 0 ) 1 = α 2 B( E2;2 + + 2 0 ) 1 = 0 ( ) = 2α 2 B E2;n = 2 n =1
Example of 112 Cd
Possible vibrational nuclei from R42
Spheroidal quadrupole vibrations 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 β γ β
Spectrum of spheroidal vibrations
Example of 166 Er
Quadrupole-octupole shapes It is difficult to define an intrinsic frame for a pure octupole shape. Quadrupole-octupole: use quadrupole frame è two quadrupole and seven octupole intrinsic variables α 3µ. Most important case: β 3 =α 30 (axial symmetry).
Quadrupole-octupole vibrations
s in Nuclei Shapes in N e Octupole rotation-vibrations At "high-spins" the spectrum for 222 Ra shows a I=1 sequence with alternating parity; consistent with a stable octupole deformation, and similar to the reflectionasymmetric rotor, HCl. le vibrational spectrum; the I+3 negative parity than the even parity states with spin I. 3 At low spins 222 Ra has an octupole vibratio states are one h higher in energy than the e 3 3
s in Nuclei e Octupole rotation-vibrations le vibrational spectrum; the I+3 negative parity than the even parity states with spin I. 3 3 3
s in Nuclei e Octupole rotation-vibrations le vibrational spectrum; the I+3 negative parity than the even parity states with spin I. 3 3 3
Example: 222 Ra
Example: 220 Rn and 224 Ra a 220 Rn 8 + 1244 b 224 Ra 1128 7 2 + 938 937.8 696.9 370.4 6 + 874 340.2 4 + 534 292.7 2 + 241 241.0 0 + 0 254.3 276.2 318.3 852 663 645 188.8 422.0 404.2 645.4 5 3 1 2 + 966 965.5 881.1 10 + 1067 312.6 8 + 755 113.8 275.7 6 + 479 228.4 182.3 4 + 251 166.4 2 + 84 0 + 84.4 0 151.5 161.7 906 641 433 290 265.3 207.7 142.6 216 74.4 205.9 131.6 216.0 9 7 5 3 1 L.P. Gaffney et al., Nature 497 (2013) 199
Discrete nuclear symmetries Tetrahedral symmetry: α 3±2 0 Octahedral symmetry: α 40 = 14 5 α 4±2 0 Experimental evidence? J. Dudek et al., Phys. Rev. Lett. 88 (2002) 252502
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. The total angular momentum is I=R+J. 2 R i I i
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 )
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 $ ) r 2 ( 3cos 2 θ $ 1)d r $, 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
Rigid triaxial rotor Triaxial rotor hamiltonian I 1 I 2 I 3 : ˆ " 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 " 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 " 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: 110Cd 62
SU(3) rotational limit: 156Gd 92
SO(6) γ-unstable limit: 196Pt 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 R42
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).
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) 0.- & c ) 1 - 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 ( ) / - 0 1-1 2
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: 195Pt 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: 194Pt 116 &195Pt117
Example: 196Au 117