1. Nuclear Collective Dynamics II, Istanbul, July 2004 Symmetry Approach to Nuclear Collective Motion II P. Van Isacker, GANIL, France Symmetry and dynamical symmetry Symmetry in nuclear physics: Nuclear shell model Interacting boson model

2. Nuclear Collective Dynamics II, Istanbul, July 2004 The three faces of the shell model

3. Nuclear Collective Dynamics II, Istanbul, July 2004 Symmetries of the shell model Three bench-mark solutions: –No residual interaction  IP shell model. –Pairing (in jj coupling)  Racah’s SU(2). –Quadrupole (in LS coupling)  Elliott’s SU(3). Symmetry triangle:

4. Nuclear Collective Dynamics II, Istanbul, July 2004 Evidence for shell structure Evidence for nuclear shell structure from –2 + in even-even nuclei [E x, B(E2)]. –Nucleon-separation energies & nuclear masses. –Nuclear level densities. –Reaction cross sections. Is nuclear shell structure modified away from the line of stability?

5. Nuclear Collective Dynamics II, Istanbul, July 2004 Shell structure from E x (2 1 ) High E x (2 1 ) indicates stable shell structure:

6. Nuclear Collective Dynamics II, Istanbul, July 2004 Weizsäcker mass formula Total nuclear binding energy: For 2149 nuclei (N,Z≥8) in AME03: a V  16, a S  18, a I  7.3, a C  0.71, a P  13   rms  2.5 MeV.

7. Nuclear Collective Dynamics II, Istanbul, July 2004 Shell structure from masses Deviations from Weizsäcker mass formula:

8. Nuclear Collective Dynamics II, Istanbul, July 2004 Racah’s SU(2) pairing model Assume large spin-orbit splitting  ls which implies a jj coupling scheme. Assume pairing interaction in a single-j shell: Spectrum of 210 Pb:

9. Nuclear Collective Dynamics II, Istanbul, July 2004 Solution of pairing hamiltonian Analytic solution of pairing hamiltonian for identical nucleons in a single-j shell: Seniority  (number of nucleons not in pairs coupled to J=0) is a good quantum number. Correlated ground-state solution (cfr. super- fluidity in solid-state physics). G. Racah, Phys. Rev. 63 (1943) 367

10. Nuclear Collective Dynamics II, Istanbul, July 2004 Pairing and superfluidity Ground states of a pairing hamiltonian have a superfluid character: –Even-even nucleus (  =0): –Odd-mass nucleus (  =1): Nuclear superfluidity leads to –Constant energy of first 2 + in even-even nuclei. –Odd-even staggering in masses. –Smooth variation of two-nucleon separation energies with nucleon number. –Two-particle (2n or 2p) transfer enhancement.

11. Nuclear Collective Dynamics II, Istanbul, July 2004 Superfluidity in semi-magic nuclei Even-even nuclei: –Ground state has  =0. –First-excited state has  =2. –Pairing produces constant energy gap: Example of Sn nuclei:

12. Nuclear Collective Dynamics II, Istanbul, July 2004 Two-nucleon separation energies Two-nucleon separation energies S 2n : (a) Shell splitting dominates over interaction. (b) Interaction dominates over shell splitting. (c) S 2n in tin isotopes.

13. Nuclear Collective Dynamics II, Istanbul, July 2004 Pairing with neutrons and protons For neutrons and protons two pairs and hence two pairing interactions are possible: –Isoscalar (S=1,T=0): –Isovector (S=0,T=1):

14. Nuclear Collective Dynamics II, Istanbul, July 2004 Neutron-proton pairing hamiltonian A hamiltonian with two pairing terms, …has an SO(8) algebraic structure. H is solvable (or has dynamical symmetries) for g 0 =0, g 1 =0 and g 0 =g 1.

15. Nuclear Collective Dynamics II, Istanbul, July 2004 SO(8) ‘quasi-spin’ formalism A closed algebra is obtained with the pair operators S ± with in addition This set of 28 operators forms the Lie algebra SO(8) with subalgebras B.H. Flowers & S. Szpikowski, Proc. Phys. Soc. 84 (1964) 673

16. Nuclear Collective Dynamics II, Istanbul, July 2004 Solvable limits of SO(8) model Pairing interactions can expressed as follows: Symmetry lattice of the SO(8) model:  Analytic solutions for g 0 =0, g 1 =0 and g 0 =g 1.

17. Nuclear Collective Dynamics II, Istanbul, July 2004 Superfluidity of N=Z nuclei T=0 & T=1 pairing has quartet superfluid character with SO(8) symmetry. Pairing ground state of an N=Z nucleus:  Condensate of  ’s (  depends on g 01 /g 10 ). Observations: –Isoscalar component in condensate survives only in N~Z nuclei, if anywhere at all. –Spin-orbit term reduces isoscalar component.

18. Nuclear Collective Dynamics II, Istanbul, July 2004 Deuteron transfer in N=Z nuclei Deuteron intensity c T 2 calculated in schematic model based on SO(8). Parameter ratio b/a fixed from masses. In lower half of 28-50 shell: b/a  5.

19. Nuclear Collective Dynamics II, Istanbul, July 2004 Symmetries of the shell model Three bench-mark solutions: –No residual interaction  IP shell model. –Pairing (in jj coupling)  Racah’s SU(2). –Quadrupole (in LS coupling)  Elliott’s SU(3). Symmetry triangle:

20. Nuclear Collective Dynamics II, Istanbul, July 2004 Wigner’s SU(4) symmetry Assume the nuclear hamiltonian is invariant under spin and isospin rotations: Since {S ,T,Y  } form an SU(4) algebra: –H nucl has SU(4) symmetry. –Total spin S, total orbital angular momentum L, total isospin T and SU(4) labels (  ) are conserved quantum numbers. E.P. Wigner, Phys. Rev. 51 (1937) 106 F. Hund, Z. Phys. 105 (1937) 202

21. Nuclear Collective Dynamics II, Istanbul, July 2004 Physical origin of SU(4) symmetry SU(4) labels specify the separate spatial and spin-isospin symmetry of the wave function: Nuclear interaction is short-range attractive and hence favours maximal spatial symmetry.

22. Nuclear Collective Dynamics II, Istanbul, July 2004 Breaking of SU(4) symmetry Non-dynamical breaking of SU(4) symmetry as a consequence of –Spin-orbit term in nuclear mean field. –Coulomb interaction. –Spin-dependence of residual interaction. Evidence for SU(4) symmetry breaking from –Masses: rough estimate of nuclear BE from –  decay: Gamow-Teller operator Y ,  1 is a generator of SU(4)  selection rule in (  ).

23. Nuclear Collective Dynamics II, Istanbul, July 2004 SU(4) breaking from masses Double binding energy difference  V np  V np in sd-shell nuclei: P. Van Isacker et al., Phys. Rev. Lett. 74 (1995) 4607

24. Nuclear Collective Dynamics II, Istanbul, July 2004 SU(4) breaking from  decay Gamow-Teller decay into odd-odd or even- even N=Z nuclei: P. Halse & B.R. Barrett, Ann. Phys. (NY) 192 (1989) 204

25. Nuclear Collective Dynamics II, Istanbul, July 2004 Elliott’s SU(3) model of rotation Harmonic oscillator mean field (no spin-orbit) with residual interaction of quadrupole type: State labelling in LS coupling: J.P. Elliott, Proc. Roy. Soc. A 245 (1958) 128; 562

26. Nuclear Collective Dynamics II, Istanbul, July 2004 Importance/limitations of SU(3) Historical importance: –Bridge between the spherical shell model and the liquid droplet model through mixing of orbits. –Spectrum generating algebra of Wigner’s SU(4) supermultiplet. Limitations: –LS (Russell-Saunders) coupling, not jj coupling (zero spin-orbit splitting)  beginning of sd shell. –Q is the algebraic quadrupole operator  no major-shell mixing.

27. Nuclear Collective Dynamics II, Istanbul, July 2004 Tripartite classification of nuclei 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

28. Nuclear Collective Dynamics II, Istanbul, July 2004 The interacting boson model Spectrum generating algebra for the nucleus is U(6). All physical observables (hamiltonian, transition operators,…) are expressed in terms of s and d bosons. 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. A. Arima & F. Iachello, Ann. Phys. (NY) 99 (1976) 253; 111 (1978) 201; 123 (1979) 468

29. Nuclear Collective Dynamics II, Istanbul, July 2004 Algebraic structure of the IBM The U(6) algebra consists of the generators The harmonic oscillator in 6 dimensions, …has U(6) symmetry since Can the U(6) symmetry be lifted while preserving the rotational SO(3) symmetry?

30. Nuclear Collective Dynamics II, Istanbul, July 2004 The IBM hamiltonian Rotational invariant hamiltonian with up to N- body interactions (usually up to 2): For what choice of single-boson energies  s and  d and boson-boson interactions  L ijkl is the IBM hamiltonian solvable? This problem is equivalent to the enumeration of all algebras G that satisfy

31. Nuclear Collective Dynamics II, Istanbul, July 2004 Dynamical symmetries of the IBM The general IBM hamiltonian is An entirely equivalent form of H IBM is The coefficients  i and  j are certain combinations of the coefficients  i and  L ijkl.

32. Nuclear Collective Dynamics II, Istanbul, July 2004 The solvable IBM hamiltonians Without N-dependent terms in the hamiltonian (which are always diagonal) If certain coefficients are zero, H IBM can be written as a sum of commuting operators:

33. Nuclear Collective Dynamics II, Istanbul, July 2004 The U(5) vibrational limit Spectrum of an anharmonic oscillator in 5 dimensions associated with the quadrupole oscillations of a droplet’s surface. Conserved quantum numbers: n d, , L. A. Arima & F. Iachello, Ann. Phys. (NY) 99 (1976) 253 D. Brink et al., Phys. Lett. 19 (1965) 413

34. Nuclear Collective Dynamics II, Istanbul, July 2004 The SU(3) rotational limit Rotation-vibration spectrum with  - and  - vibrational bands. Conserved quantum numbers: (,  ), L. A. Arima & F. Iachello, Ann. Phys. (NY) 111 (1978) 201 A. Bohr & B.R. Mottelson, Dan. Vid. Selsk. Mat.-Fys. Medd. 27 (1953) No 16

35. Nuclear Collective Dynamics II, Istanbul, July 2004 The SO(6)  -unstable limit Rotation-vibration spectrum of a  -unstable body. Conserved quantum numbers: , , L. A. Arima & F. Iachello, Ann. Phys. (NY) 123 (1979) 468 L. Wilets & M. Jean, Phys. Rev. 102 (1956) 788

36. Nuclear Collective Dynamics II, Istanbul, July 2004 Synopsis of IBM symmetries Symmetry triangle of the IBM: –Three standard solutions: U(5), SU(3), SO(6). –SU(1,1) analytic solution for U(5)  SO(6). –Hidden symmetries (parameter transformations). –Deformed-spherical coexistent phase. –Partial dynamical symmetries. –Critical-point symmetries?

37. Nuclear Collective Dynamics II, Istanbul, July 2004 Extensions of the 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.

38. Nuclear Collective Dynamics II, Istanbul, July 2004 Scissors excitations Collective displacement modes between neutrons and protons: –Linear displacement (giant dipole resonance): R -R   E1 excitation. –Angular displacement (scissors resonance): L -L   M1 excitation. N. Lo Iudice & F. Palumbo, Phys. Rev. Lett. 41 (1978) 1532 F. Iachello, Phys. Rev. Lett. 53 (1984) 1427 D. Bohle et al., Phys. Lett. B 137 (1984) 27

39. Nuclear Collective Dynamics II, Istanbul, July 2004 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: F. Iachello, Phys. Rev. Lett. 44 (1980) 772 P. Van Isacker et al., Phys. Rev. Lett. 54 (1985) 653 A. Metz et al., Phys. Rev. Lett. 83 (1999) 1542

40. Nuclear Collective Dynamics II, Istanbul, July 2004 Example of 195 Pt

41. Nuclear Collective Dynamics II, Istanbul, July 2004 Example of 196 Au

42. Nuclear Collective Dynamics II, Istanbul, July 2004 Algebraic many-body models The integrability of any quantum many-body (bosons and/or fermions) system can be analyzed with algebraic methods. Two nuclear examples: –Pairing vs. quadrupole interaction in the nuclear shell model. –Spherical, deformed and  -unstable nuclei with s,d-boson IBM.

43. Nuclear Collective Dynamics II, Istanbul, July 2004 Other fields of physics Molecular physics: –U(4) vibron model with s,p-bosons. –Coupling of many SU(2) algebras for polyatomic molecules. Similar applications in hadronic, atomic, solid- state, polymer physics, quantum dots… Use of non-compact groups and algebras for scattering problems. F. Iachello, 1975 to now