1. Generalized pairing models, Saclay, June 2005 Generalized models of pairing in non-degenerate orbits J. Dukelsky, IEM, Madrid, Spain D.D. Warner, Daresbury, United Kingdom A. Frank, UNAM, Mexico P. Van Isacker, GANIL, France Symmetries of pairing models Generalized pairing models Deuteron transfer

2. Generalized pairing models, Saclay, June 2005 The nuclear shell model Mean field plus residual interaction (between valence nucleons). Assume a simple mean-field potential: Contains –Harmonic-oscillator potential with constant . –Spin-orbit term with strength  ls. –Orbit-orbit term with strength  ll.

3. Generalized pairing models, Saclay, June 2005 Shell model for complex nuclei Solve the eigenvalue problem associated with the matrix (n active nucleons): Methods of solution: –Diagonalization (Lanczos): d~10 9. –Monte-Carlo shell model: d~10 15. –Density Matrix Renormalization Group: d~10 120 ?

4. Generalized pairing models, Saclay, June 2005 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:

5. Generalized pairing models, Saclay, June 2005 Racah’s SU(2) pairing model Assume pairing interaction in a single-j shell: Spectrum 210 Pb:

6. Generalized pairing models, Saclay, June 2005 Solution of the 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 (cf. BCS). G. Racah, Phys. Rev. 63 (1943) 367

7. Generalized pairing models, Saclay, June 2005 Nuclear “superfluidity” Ground states of pairing hamiltonian have the following correlated 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.

8. Generalized pairing models, Saclay, June 2005 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.

9. Generalized pairing models, Saclay, June 2005 Integrability of pairing hamiltonian A.K. Kerman, Ann. Phys. (NY) 12 (1961) 300 Pair operators (several shells): The pairing hamiltonian for degenerate shells … is solvable by virtue of an underlying SU(2) “quasi-spin” symmetry:

10. Generalized pairing models, Saclay, June 2005 Generalized pairing model Hamiltonian for pairing interaction in non- degenerate shells: Is the pairing model with non-degenerate orbits integrable?

11. Generalized pairing models, Saclay, June 2005 Richardson-Gaudin models R.W. Richardson, Phys. Lett. 5 (1963) 82 M. Gaudin, J. Phys. (Paris) 37 (1976) 1087. Algebraic structure: The Gaudin operators …commute if X ij and Y ij are antisymmetric and satisfy the equations  Any combination of R i is integrable.

12. Generalized pairing models, Saclay, June 2005 Pairing with non-degenerate orbits J. Dukelsky et al., Phys. Rev. Lett. 87 (2001) 066403 If we choose  A hamiltonian for pairing in non-degenerate shells is integrable! Solution:

13. Generalized pairing models, Saclay, June 2005 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. Generalized pairing models, Saclay, June 2005 Neutron-proton pairing hamiltonian A hamiltonian with two pairing interactions …has an SO(8) algebraic structure. V pairing is integrable and solvable (dynamical symmetries) for g 0 =0, g 0 =0 and g 0 =g 0.

15. Generalized pairing models, Saclay, June 2005 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. Generalized pairing models, Saclay, June 2005 Solvable limits of the SO(8) model Pairing interactions can expressed as follows: Symmetry lattice of the SO(8) model:  Analytic solutions for g 0 =0, g 0 =0 & g 0 =g 0.

17. Generalized pairing models, Saclay, June 2005 Quartetting in N=Z nuclei T=0 and T=1 pairing has a quartet structure with SO(8) symmetry. Pairing ground state of an N=Z nucleus:  Condensate of “  -like” objects. Observations: –Isoscalar component in condensate survives only in N~Z nuclei, if anywhere at all. –Spin-orbit term reduces isoscalar component.

18. Generalized pairing models, Saclay, June 2005 Generalized neutron-proton pairing Hamiltonian for pairing interactions in non- degenerate shells: Solution techniques: –Richardson-Gaudin for SO(8) model. –Boson mappings: requiring same commutation relations; associating state vectors.

19. Generalized pairing models, Saclay, June 2005 Generalized pairing models J. Dukelsky et al., to be published Pairing in degenerate orbits between identical particles has SU(2) symmetry. Richardson-Gaudin models can be generalized to higher-rank algebras:

20. Generalized pairing models, Saclay, June 2005 Example: SO(5) pairing Hamiltonian: “Quasi-spin” algebra is SO(5) (rank 2). Example: 64 Ge in pfg 9/2 shell (d~9  10 14 ). S. Dimitrova, unpublished

21. Generalized pairing models, Saclay, June 2005 Model with L=0 vector bosons Correspondence: Algebraic structure is U(6). Symmetry lattice of U(6): Boson mapping is exact in the symmetry limits [for fully paired states of the SO(8)]. P. Van Isacker et al., J. Phys. G 24 (1998) 1261

22. Generalized pairing models, Saclay, June 2005 Masses of N~Z nuclei Neutron-proton pairing hamiltonian in non- degenerate shells: H F maps into the boson hamiltonian: H B describes masses of N~Z nuclei. E. Baldini-Neto et al., Phys. Rev. C 65 (2002) 064303

23. Generalized pairing models, Saclay, June 2005 Two-nucleon transfer Amplitude for two-nucleon transfer in the reaction  A+a  B+b: Nuclear-structure information contained in G N (L,S,J) which for L=0 transfer reduces to N.K. Glendenning, Direct Nuclear Reactions

24. Generalized pairing models, Saclay, June 2005 Deuteron transfer Overlap of uncorrelated pair: Bosons correspond to correlated pairs: Scale property: P. Van Isacker et al., Phys. Rev. Lett. 94 (2005) 162502

25. Generalized pairing models, Saclay, June 2005 Deuteron transfer with bosons Correspondence does not take account of Pauli principle. The following correspondence is shown to be exact [in the Wigner limit]: –Even-even  odd-odd –Odd-odd  even-even

26. Generalized pairing models, Saclay, June 2005 Masses of pf-shell nuclei Boson hamiltonian: Rms deviation is 306 (or 254) keV. Parameter ratio: b/a  5.

27. Generalized pairing models, Saclay, June 2005 Deuteron transfer in N=Z nuclei Deuteron-transfer intensity c T 2 calculated in sp-boson IBM based on SO(8). Ratio b/a fixed from masses in lower half of 28-50 shell.