Generalized models of pairing in non-degenerate orbits

1
Generalized models of pairingin 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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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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Shell model for complex nuclei

• Solve the eigenvalue problem associated with the
  matrix (n active nucleons)
• Methods of solution
• Diagonalization (Lanczos) d109.
• Monte-Carlo shell model d1015.
• Density Matrix Renormalization Group d10120?

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Symmetries of the shell model

• Three bench-mark solutions
• No residual interaction ? IP shell model.
• Pairing (in jj coupling) ? Racahs SU(2).
• Quadrupole (in LS coupling) ? Elliotts SU(3).
• Symmetry triangle

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Racahs SU(2) pairing model

• Assume pairing interaction in a single-j shell
• Spectrum 210Pb

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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 J0) is a good quantum number.
• Correlated ground-state solution (cf. BCS).

G. Racah, Phys. Rev. 63 (1943) 367
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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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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.

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Integrability of pairing hamiltonian

• Pair operators (several shells)
• The pairing hamiltonian for degenerate shells
• is solvable by virtue of an underlying SU(2)
  quasi-spin symmetry

A.K. Kerman, Ann. Phys. (NY) 12 (1961) 300
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Generalized pairing model

• Hamiltonian for pairing interaction in
  non-degenerate shells
• Is the pairing model with non-degenerate orbits
  integrable?

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Richardson-Gaudin models

• Algebraic structure
• The Gaudin operators
• commute if Xij and Yij are antisymmetric and
  satisfy the equations
• ?Any combination of Ri is integrable.

R.W. Richardson, Phys. Lett. 5 (1963) 82 M.
Gaudin, J. Phys. (Paris) 37 (1976) 1087.
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Pairing with non-degenerate orbits

• If we choose
• ? A hamiltonian for pairing in non-degenerate
  shells is integrable! Solution

J. Dukelsky et al., Phys. Rev. Lett. 87 (2001)
066403
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Pairing with neutrons and protons

• For neutrons and protons two pairs and hence two
  pairing interactions are possible
• Isoscalar (S1,T0)
• Isovector (S0,T1)

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Neutron-proton pairing hamiltonian

• A hamiltonian with two pairing interactions
• has an SO(8) algebraic structure.
• Vpairing is integrable and solvable (dynamical
  symmetries) for g00, g?00 and g0g?0.

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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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Solvable limits of the SO(8) model

• Pairing interactions can expressed as follows
• Symmetry lattice of the SO(8) model
• ?Analytic solutions for g00, g?00 g0g?0.

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Quartetting in NZ nuclei

• T0 and T1 pairing has a quartet structure with
  SO(8) symmetry. Pairing ground state of an NZ
  nucleus
• ? Condensate of ?-like objects.
• Observations
• Isoscalar component in condensate survives only
  in NZ nuclei, if anywhere at all.
• Spin-orbit term reduces isoscalar component.

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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

• Pairing in degenerate orbits between identical
  particles has SU(2) symmetry.
• Richardson-Gaudin models can be generalized to
  higher-rank algebras

J. Dukelsky et al., to be published
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Example SO(5) pairing

• Hamiltonian
• Quasi-spin algebra is SO(5) (rank 2).
• Example 64Ge in pfg9/2 shell (d9?1014).

S. Dimitrova, unpublished
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Model with L0 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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Masses of NZ nuclei

• Neutron-proton pairing hamiltonian in
  non-degenerate shells
• HF maps into the boson hamiltonian
• HB describes masses of NZ nuclei.

E. Baldini-Neto et al., Phys. Rev. C 65 (2002)
064303
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Two-nucleon transfer

• Amplitude for two-nucleon transfer in the
  reaction ??Aa???Bb
• Nuclear-structure information contained in
  GN(L,S,J) which for L0 transfer reduces to

N.K. Glendenning, Direct Nuclear Reactions
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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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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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Masses of pf-shell nuclei

• Boson hamiltonian
• Rms deviation is 306 (or 254) keV.
• Parameter ratio b/a?5.

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Deuteron transfer in NZ nuclei

• Deuteron-transfer intensity cT2 calculated in
  sp-boson IBM based on SO(8).
• Ratio b/a fixed from masses in lower half of
  28-50 shell.