Symmetry Approach to Nuclear Collective Motion II

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Symmetry Approach toNuclear Collective Motion II

• P. Van Isacker, GANIL, France

Symmetry and dynamical symmetry Symmetry in
nuclear physics Nuclear shell model Interacting
boson model
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The three faces of the shell model
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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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Evidence for shell structure

• Evidence for nuclear shell structure from
• 2 in even-even nuclei Ex, B(E2).
• Nucleon-separation energies nuclear masses.
• Nuclear level densities.
• Reaction cross sections.
• Is nuclear shell structure
  modified away from the
  line of stability?

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Shell structure from Ex(21)

• High Ex(21) indicates stable shell structure

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Weizsäcker mass formula

• Total nuclear binding energy
• For 2149 nuclei (N,Z8) in AME03 aV?16, aS?18,
  aI?7.3, aC?0.71, aP?13 ? ?rms?2.5 MeV.

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Shell structure from masses

• Deviations from Weizsäcker mass formula

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Racahs 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

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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 J0) is a good quantum number.
• Correlated ground-state solution (cfr.
  super-fluidity in solid-state physics).

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

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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

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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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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 terms,
• has an SO(8) algebraic structure.
• H is solvable (or has dynamical symmetries) for
  g00, g10 and g0g1.

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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 SO(8) model

• Pairing interactions can expressed as follows
• Symmetry lattice of the SO(8) model
• ?Analytic solutions for g00, g10 and g0g1.

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Superfluidity of NZ nuclei

• T0 T1 pairing has quartet superfluid
  character with SO(8) symmetry. Pairing ground
  state of an NZ nucleus
• ? Condensate of ?s (? depends on g01/g10).
• Observations
• Isoscalar component in condensate survives only
  in NZ nuclei, if anywhere at all.
• Spin-orbit term reduces isoscalar component.

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Deuteron transfer in NZ nuclei

• Deuteron intensity cT2 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.

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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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Wigners 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
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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.

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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 (???).

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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
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SU(4) breaking from ? decay

• Gamow-Teller decay into odd-odd or even-even NZ
  nuclei

P. Halse B.R. Barrett, Ann. Phys. (NY) 192
(1989) 204
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Elliotts 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
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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 Wigners 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.

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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
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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
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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?

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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 ?Lijkl is the IBM
  hamiltonian solvable?
• This problem is equivalent to the enumeration of
  all algebras G that satisfy

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Dynamical symmetries of the IBM

• The general IBM hamiltonian is
• An entirely equivalent form of HIBM is
• The coefficients ?i and ?j are certain
  combinations of the coefficients ?i and ?Lijkl.

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The solvable IBM hamiltonians

• Without N-dependent terms in the hamiltonian
  (which are always diagonal)
• If certain coefficients are zero, HIBM can be
  written as a sum of commuting operators

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The U(5) vibrational limit

• Spectrum of an anharmonic oscillator in 5
  dimensions associated with the quadrupole
  oscillations of a droplets surface.
• Conserved quantum numbers nd, ?, L.

A. Arima F. Iachello, Ann. Phys. (NY) 99 (1976)
253 D. Brink et al., Phys. Lett. 19 (1965) 413
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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
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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
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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?

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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 T0 T1 pairs (IBM-3 IBM-4).
• Higher multipole (g,) pairs.

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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
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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 194Pt, 195Pt, 195Au 196Au

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
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Example of 195Pt
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Example of 196Au
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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.

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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