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The nuclear shell model

The nuclear shell model. Context and assumptions of the model Symmetries of the shell model: Racah’s SU(2) pairing model Wigner’s SU(4) symmetry Elliott’s SU(3) model of rotation. P. Van Isacker, GANIL, France. Overview of nuclear models.

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The nuclear shell model

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  1. The nuclear shell model • Context and assumptions of the model • Symmetries of the shell model: • Racah’s SU(2) pairing model • Wigner’s SU(4) symmetry • Elliott’s SU(3) model of rotation P. Van Isacker, GANIL, France IAEA Workshop on NSDD, Trieste, November 2003

  2. Overview of nuclear models • Ab initio methods: Description of nuclei starting from the bare nn & nnn interactions. • Nuclear shell model: Nuclear average potential + (residual) interaction between nucleons. • Mean-field methods: Nuclear average potential with global parametrization (+ correlations). • Phenomenological models: Specific nuclei or properties with local parametrization. IAEA Workshop on NSDD, Trieste, November 2003

  3. Ab initio methods • Faddeev-Yakubovsky: A≤4 • Coupled-rearrangement-channel Gaussian-basis variational: A≤4 (higher with clusters) • Stochastic variational: A≤7 • Hyperspherical harmonic variational: • Green’s function Monte Carlo: A≤7 • No-core shell model: A≤12 • Effective interaction hyperspherical: A≤6 IAEA Workshop on NSDD, Trieste, November 2003

  4. Benchmark calculation for A=4 • Test calculation with realistic interaction: all methods agree. • But Eexpt=-28.296 MeV  need for three-nucleon interaction. H. Kamada et al., Phys. Rev. C 63 (2001) 034006 IAEA Workshop on NSDD, Trieste, November 2003

  5. Basic symmetries • Non-relativistic Schrödinger equation: • Symmetry or invariance under: • Translations  linear momentum P • Rotations  angular momentum J=L+S • Space reflection  parity  • Time reversal IAEA Workshop on NSDD, Trieste, November 2003

  6. Nuclear shell model • Separation in mean field + residual interaction: • Independent-particle assumption. Choose V and neglect residual interaction: IAEA Workshop on NSDD, Trieste, November 2003

  7. Independent-particle shell model • Solution for one particle: • Solution for many particles: IAEA Workshop on NSDD, Trieste, November 2003

  8. Independent-particle shell model • Antisymmetric solution for many particles (Slater determinant): • Example for A=2 particles: IAEA Workshop on NSDD, Trieste, November 2003

  9. Hartree-Fock approximation • Vary i (ieV) to minize the expectation value of H in a Slater determinant: • Application requires choice of H. Many global parametrizations (Skyrme, Gogny,…) have been developed. IAEA Workshop on NSDD, Trieste, November 2003

  10. Poor man’s Hartree-Fock • Choose a simple, analytically solvable V that approximates the microscopic HF potential: • Contains • Harmonic oscillator potential with constant . • Spin-orbit term with strength ls. • Orbit-orbit term with strength ll. • Adjust , ls and ll to best reproduce HF. IAEA Workshop on NSDD, Trieste, November 2003

  11. Energy levels of harmonic oscillator • Typical parameter values: • ‘Magic’ numbers at 2, 8, 20, 28, 50, 82, 126, 184,… IAEA Workshop on NSDD, Trieste, November 2003

  12. Evidence for shell structure • Evidence for nuclear shell structure from • Excitation energies in even-even nuclei • Nucleon-separation energies • Nuclear masses • Nuclear level densities • Reaction cross sections • Is nuclear shell structure modified away from the line of stability? IAEA Workshop on NSDD, Trieste, November 2003

  13. Shell structure from Ex(21) • High Ex(21) indicates stable shell structure: IAEA Workshop on NSDD, Trieste, November 2003

  14. Shell structure from Sn or Sp • Change in slope of Sn (Sp) indicates neutron (proton) shell closure (constant N-Z plots): A. Ozawa et al., Phys. Rev. Lett. 84 (2000) 5493 IAEA Workshop on NSDD, Trieste, November 2003

  15. Shell structure from masses • Deviations from Weizsäcker mass formula: IAEA Workshop on NSDD, Trieste, November 2003

  16. Shell structure from masses • Deviations from improved Weizsäcker mass formula that includes nn and n+n terms: IAEA Workshop on NSDD, Trieste, November 2003

  17. Validity of SM wave functions • Example: Elastic electron scattering on 206Pb and 205Tl, differing by a 3s proton. • Measured ratio agrees with shell-model prediction for 3s orbit with modified occupation. J.M. Cavedon et al., Phys. Rev. Lett. 49 (1982) 978 IAEA Workshop on NSDD, Trieste, November 2003

  18. Nuclear shell model • The full shell-model hamiltonian: • Valence nucleons: Neutrons or protons that are in excess of the last, completely filled shell. • Usual approximation: Consider the residual interaction VRI among valence nucleons only. • Sometimes: Include selected core excitations (‘intruder’ states). IAEA Workshop on NSDD, Trieste, November 2003

  19. The shell-model problem • Solve the eigenvalue problem associated with the matrix (n active nucleons): • Methods of solution: • Diagonalization (Strasbourg-Madrid): 109 • Monte-Carlo (Pasadena-Oak Ridge): • Quantum Monte-Carlo (Tokyo): • Group renormalization (Madrid-Newark): 10120 IAEA Workshop on NSDD, Trieste, November 2003

  20. Residual shell-model interaction • Four approaches: • Effective: Derive from free nn interaction taking account of the nuclear medium. • Empirical: Adjust matrix elements of residual interaction to data. Examples: p, sd and pf shells. • Effective-empirical: Effective interaction with some adjusted (monopole) matrix elements. • Schematic: Assume a simple spatial form and calculate its matrix elements in a harmonic-oscillator basis. Example:  interaction. IAEA Workshop on NSDD, Trieste, November 2003

  21. Schematic short-range interaction • Delta interaction in harmonic-oscillator basis. • Example of 42Sc21 (1 active neutron + 1 active proton): IAEA Workshop on NSDD, Trieste, November 2003

  22. 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: IAEA Workshop on NSDD, Trieste, November 2003

  23. 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 210Pb: IAEA Workshop on NSDD, Trieste, November 2003

  24. 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 IAEA Workshop on NSDD, Trieste, November 2003

  25. Pairing and superfluidity • Ground states of a pairing hamiltonian have 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. • Two-particle (2n or 2p) transfer enhancement. IAEA Workshop on NSDD, Trieste, November 2003

  26. 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: IAEA Workshop on NSDD, Trieste, November 2003

  27. Two-nucleon separation energies • Two-nucleon separation energies S2n: (a) Shell splitting dominates over interaction. (b) Interaction dominates over shell splitting. (c) S2n in tin isotopes. IAEA Workshop on NSDD, Trieste, November 2003

  28. Generalized pairing models • Trivial generalization from a single-j shell to several degenerate j shells: • Pairing with neutrons and protons: • T=1 pairing: SO(5). • T=0 & T=1 pairing: SO(8). • Non-degenerate shells: • Talmi’s generalized seniority. • Richardson’s integrable pairing model. IAEA Workshop on NSDD, Trieste, November 2003

  29. 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): IAEA Workshop on NSDD, Trieste, November 2003

  30. Superfluidity of N=Z nuclei • Ground state of a T=1 pairing hamiltonian for identical nucleons is superfluid, (S+)n/2o. • Ground state of a T=0 & T=1 pairing hamiltonian with equal number of neutrons and protons has different superfluid character: •  Condensate of ’s ( depends on g0/g1). • Observations: • Isoscalar component in condensate survives only in N~Z nuclei, if anywhere at all. • Spin-orbit term reduces isoscalar component. IAEA Workshop on NSDD, Trieste, November 2003

  31. 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: • Hnucl 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 IAEA Workshop on NSDD, Trieste, November 2003

  32. Physical origin of SU(4) symmetry • SU(4) labels specify the separate spatial and spin-isospin symmetry of the wavefunction: • Nuclear interaction is short-range attractive and hence favours maximal spatial symmetry. IAEA Workshop on NSDD, Trieste, November 2003

  33. Breaking of SU(4) symmetry • 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 (). IAEA Workshop on NSDD, Trieste, November 2003

  34. SU(4) breaking from masses • Double binding energy difference Vnp • Vnp in sd-shell nuclei: P. Van Isacker et al., Phys. Rev. Lett. 74 (1995) 4607 IAEA Workshop on NSDD, Trieste, November 2003

  35. 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 IAEA Workshop on NSDD, Trieste, November 2003

  36. Elliott’s SU(3) model of rotation • Harmonic oscillator mean field (no spin-orbit) with residual interaction of quadrupole type: J.P. Elliott, Proc. Roy. Soc. A 245 (1958) 128; 562 IAEA Workshop on NSDD, Trieste, November 2003

  37. Importance and 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. IAEA Workshop on NSDD, Trieste, November 2003

  38. Generalized SU(3) models • How to obtain rotational features in a jj-coupling limit of the nuclear shell model? • Several efforts since Elliott: • Pseudo-spin symmetry. • Quasi-SU(3) symmetry (Zuker). • Effective symmetries (Rowe). • FDSM: fermion dynamical symmetry model. • ... IAEA Workshop on NSDD, Trieste, November 2003

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