1. NSDD Workshop, Trieste, February 2006 Nuclear Structure (II) Collective models P. Van Isacker, GANIL, France

2. 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

3. NSDD Workshop, Trieste, February 2006 Evolution of E x (2 + ) J.L. Wood, private communication

4. NSDD Workshop, Trieste, February 2006 Quantum-mechanical symmetric top Energy spectrum: Large deformation  large   low E x (2 + ). R 42 energy ratio:

5. 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 H intr with ‘intrinsic’ angular momentum J. The total angular momentum is I=R+J.

6. 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:

7. 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:

8. NSDD Workshop, Trieste, February 2006 The ratio R 42

9. NSDD Workshop, Trieste, February 2006 Electric (quadrupole) properties Partial  -ray half-life: Electric quadrupole transitions: Electric quadrupole moments:

10. NSDD Workshop, Trieste, February 2006 Magnetic (dipole) properties Partial  -ray half-life: Magnetic dipole transitions: Magnetic dipole moments:

11. NSDD Workshop, Trieste, February 2006 E2 properties of rotational nuclei Intra-band E2 transitions: E2 moments: Q 0 (K) is the ‘intrinsic’ quadrupole moment:

12. 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):

13. NSDD Workshop, Trieste, February 2006 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):

14. NSDD Workshop, Trieste, February 2006 Inter-band E2 transitions Example of  g transitions in 166 Er: W.D. Kulp et al., Phys. Rev. C 73 (2006) 014308

15. 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

16. 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

17. NSDD Workshop, Trieste, February 2006 Properties of spherical vibrations Energy spectrum: R 42 energy ratio: E2 transitions:

18. NSDD Workshop, Trieste, February 2006 Example of 112 Cd

19. NSDD Workshop, Trieste, February 2006 Possible vibrational nuclei from R 42

20. 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 (  )

21. NSDD Workshop, Trieste, February 2006 Spectrum of spheroidal vibrations

22. NSDD Workshop, Trieste, February 2006 Example of 166 Er

23. 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

24. NSDD Workshop, Trieste, February 2006 Rigid triaxial rotor spectra

25. 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

26. 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.

27. 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: 162 Dy 96 with 14 neutrons (  =44) and 16 protons (  =32) ( 132 Sn 82 inert core). –SM dimension: ~7·10 19 –IBM dimension: 15504

28. 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:

29. NSDD Workshop, Trieste, February 2006 U(5) vibrational limit: 110 Cd 62

30. NSDD Workshop, Trieste, February 2006 SU(3) rotational limit: 156 Gd 92

31. NSDD Workshop, Trieste, February 2006 SO(6)  -unstable limit: 196 Pt 118

32. NSDD Workshop, Trieste, February 2006 Applications of IBM

33. 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:

34. NSDD Workshop, Trieste, February 2006 Phase diagram of IBM J. Jolie et al., Phys. Rev. Lett. 87 (2001) 162501.

35. NSDD Workshop, Trieste, February 2006 The ratio R 42

36. 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

37. 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.

38. 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 194 Pt, 195 Pt, 195 Au & 196 Au:

39. 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 .

40. NSDD Workshop, Trieste, February 2006 Example: 195 Pt 117

41. NSDD Workshop, Trieste, February 2006 Example: 195 Pt 117 (new data)

42. 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

43. NSDD Workshop, Trieste, February 2006 U(6/12) supermultiplet

44. NSDD Workshop, Trieste, February 2006 Example: 194 Pt 116 & 195 Pt 117

45. NSDD Workshop, Trieste, February 2006 Example: 196 Au 117

46. 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).

47. 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).