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Nuclear Collective Dynamics II, Istanbul, July 2004 Symmetry Approach to Nuclear 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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Nuclear Collective Dynamics II, Istanbul, July 2004 The three faces of the shell model

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Nuclear Collective Dynamics II, Istanbul, July 2004 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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Nuclear Collective Dynamics II, Istanbul, July 2004 Evidence for shell structure Evidence for nuclear shell structure from –2 + in even-even nuclei [E x, 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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Nuclear Collective Dynamics II, Istanbul, July 2004 Shell structure from E x (2 1 ) High E x (2 1 ) indicates stable shell structure:

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Nuclear Collective Dynamics II, Istanbul, July 2004 Weizsäcker mass formula Total nuclear binding energy: For 2149 nuclei (N,Z≥8) in AME03: a V 16, a S 18, a I 7.3, a C 0.71, a P 13 rms 2.5 MeV.

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Nuclear Collective Dynamics II, Istanbul, July 2004 Shell structure from masses Deviations from Weizsäcker mass formula:

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Nuclear Collective Dynamics II, Istanbul, July 2004 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 210 Pb:

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Nuclear Collective Dynamics II, Istanbul, July 2004 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

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Nuclear Collective Dynamics II, Istanbul, July 2004 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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Nuclear Collective Dynamics II, Istanbul, July 2004 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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Nuclear Collective Dynamics II, Istanbul, July 2004 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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Nuclear Collective Dynamics II, Istanbul, July 2004 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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Nuclear Collective Dynamics II, Istanbul, July 2004 Neutron-proton pairing hamiltonian A hamiltonian with two pairing terms, …has an SO(8) algebraic structure. H is solvable (or has dynamical symmetries) for g 0 =0, g 1 =0 and g 0 =g 1.

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Nuclear Collective Dynamics II, Istanbul, July 2004 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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Nuclear Collective Dynamics II, Istanbul, July 2004 Solvable limits of SO(8) model Pairing interactions can expressed as follows: Symmetry lattice of the SO(8) model: Analytic solutions for g 0 =0, g 1 =0 and g 0 =g 1.

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Nuclear Collective Dynamics II, Istanbul, July 2004 Superfluidity of N=Z nuclei T=0 & T=1 pairing has quartet superfluid character with SO(8) symmetry. Pairing ground state of an N=Z nucleus: Condensate of ’s ( depends on g 01 /g 10 ). 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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Nuclear Collective Dynamics II, Istanbul, July 2004 Deuteron transfer in N=Z nuclei Deuteron intensity c T 2 calculated in schematic model based on SO(8). Parameter ratio b/a fixed from masses. In lower half of shell: b/a 5.

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Nuclear Collective Dynamics II, Istanbul, July 2004 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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Nuclear Collective Dynamics II, Istanbul, July 2004 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: –H nucl 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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Nuclear Collective Dynamics II, Istanbul, July 2004 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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Nuclear Collective Dynamics II, Istanbul, July 2004 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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Nuclear Collective Dynamics II, Istanbul, July 2004 SU(4) breaking from masses Double binding energy difference V np V np in sd-shell nuclei: P. Van Isacker et al., Phys. Rev. Lett. 74 (1995) 4607

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Nuclear Collective Dynamics II, Istanbul, July 2004 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

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Nuclear Collective Dynamics II, Istanbul, July 2004 Elliott’s 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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Nuclear Collective Dynamics II, Istanbul, July 2004 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 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.

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Nuclear Collective Dynamics II, Istanbul, July 2004 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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Nuclear Collective Dynamics II, Istanbul, July 2004 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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Nuclear Collective Dynamics II, Istanbul, July 2004 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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Nuclear Collective Dynamics II, Istanbul, July 2004 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 L ijkl is the IBM hamiltonian solvable? This problem is equivalent to the enumeration of all algebras G that satisfy

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Nuclear Collective Dynamics II, Istanbul, July 2004 Dynamical symmetries of the IBM The general IBM hamiltonian is An entirely equivalent form of H IBM is The coefficients i and j are certain combinations of the coefficients i and L ijkl.

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Nuclear Collective Dynamics II, Istanbul, July 2004 The solvable IBM hamiltonians Without N-dependent terms in the hamiltonian (which are always diagonal) If certain coefficients are zero, H IBM can be written as a sum of commuting operators:

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Nuclear Collective Dynamics II, Istanbul, July 2004 The U(5) vibrational limit Spectrum of an anharmonic oscillator in 5 dimensions associated with the quadrupole oscillations of a droplet’s surface. Conserved quantum numbers: n d, , L. A. Arima & F. Iachello, Ann. Phys. (NY) 99 (1976) 253 D. Brink et al., Phys. Lett. 19 (1965) 413

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Nuclear Collective Dynamics II, Istanbul, July 2004 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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Nuclear Collective Dynamics II, Istanbul, July 2004 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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Nuclear Collective Dynamics II, Istanbul, July 2004 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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Nuclear Collective Dynamics II, Istanbul, July 2004 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 T=0 & T=1 pairs (IBM-3 & IBM-4). –Higher multipole (g,…) pairs.

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Nuclear Collective Dynamics II, Istanbul, July 2004 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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Nuclear Collective Dynamics II, Istanbul, July 2004 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 194 Pt, 195 Pt, 195 Au & 196 Au: 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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Nuclear Collective Dynamics II, Istanbul, July 2004 Example of 195 Pt

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Nuclear Collective Dynamics II, Istanbul, July 2004 Example of 196 Au

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Nuclear Collective Dynamics II, Istanbul, July 2004 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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Nuclear Collective Dynamics II, Istanbul, July 2004 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

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