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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

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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.

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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 ?

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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:

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

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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

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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.

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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.

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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:

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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?

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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.

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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:

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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):

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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.

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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

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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.

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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.

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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.

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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:

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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

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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

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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

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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

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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

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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

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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.

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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.

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