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Symmetries in Nuclei

Symmetries in Nuclei. Symmetry and its mathematical description The role of symmetry in physics Symmetries of the nuclear shell model Symmetries of the interacting boson model. Symmetry in physics. Geometrical symmetries Example: C 3 symmetry of O 3 , nucleus 12 C, nucleon (3q).

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Symmetries in Nuclei

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  1. Symmetries in Nuclei • Symmetry and its mathematical description • The role of symmetry in physics • Symmetries of the nuclear shell model • Symmetries of the interacting boson model Symmetries in Nuclei, Tokyo, 2008

  2. Symmetry in physics • Geometrical symmetries • Example: C3 symmetry of O3, nucleus 12C, nucleon (3q). • Permutational symmetries • Example: SA symmetry of an A-body hamiltonian: Symmetries in Nuclei, Tokyo, 2008

  3. Symmetry in physics • Space-time (kinematical) symmetries • Rotational invariance, SO(3): • Lorentz invariance, SO(3,1): • Parity: • Time reversal: • Euclidian invariance, E(3): • Poincaré invariance, E(3,1): • Dilatation symmetry: Symmetries in Nuclei, Tokyo, 2008

  4. Symmetry in physics • Quantum-mechanical symmetries, U(n) • Internal (global gauge) symmetries. Examples: • pn, isospin SU(2) • uds, flavour SU(3) • (Local) gauge symmetries. Example: Maxwell equations, U(1). • Dynamical symmetries. Example: the Coulomb problem, SO(4). Symmetries in Nuclei, Tokyo, 2008

  5. Symmetry in quantum mechanics • Assume a hamiltonian H which commutes with operators gi that form a Lie algebra G: •  H has symmetry G or is invariant under G. • Lie algebra: a set of (infinitesimal) operators that close under commutation. Symmetries in Nuclei, Tokyo, 2008

  6. Consequences of symmetry • Degeneracy structure: If  is an eigenstate of H with energy E, so is gi: • Degeneracy structure and labels of eigenstates of H are determined by algebra G: • Casimir operators of G commute with all gi: Symmetries in Nuclei, Tokyo, 2008

  7. Hydrogen atom • Hamiltonian of the hydrogen atom is • Standard wave quantum-mechanics gives • Degeneracy in m originates from rotational symmetry. What is the origin of l-degeneracy? E. Schrödinger, Ann. Phys. 79 (1926) 361 Symmetries in Nuclei, Tokyo, 2008

  8. Energy spectrum Symmetries in Nuclei, Tokyo, 2008

  9. Classical Kepler problem • Conserved quantities: • Energy (a=semi-major axis): • Angular momentum (=eccentricity): • Runge-Lentz vector: • Newtonian potential gives rise to closed orbits with constant direction of major axis. Symmetries in Nuclei, Tokyo, 2008

  10. Classical Kepler problem Symmetries in Nuclei, Tokyo, 2008

  11. Quantization of operators • From p-ih: • Some useful commutators & relations: Symmetries in Nuclei, Tokyo, 2008

  12. Conservation of angular momentum • The angular momentum operators L commute with the hydrogen hamiltonian: • L operators generate SO(3) algebra: • H has SO(3) symmetry m-degeneracy. Symmetries in Nuclei, Tokyo, 2008

  13. Conserved Runge-Lentz vector • The Runge-Lentz vector R commutes with H: • R does not commute with the kinetic and potential parts of Hseparately: • Hydrogen atom has a dynamical symmetry. Symmetries in Nuclei, Tokyo, 2008

  14. SO(4) symmetry • L and R (almost) close under commutation: • H is time-independent and commutes with L and R choose a subspace with given E. • L and R'(-M/2E)1/2R form an algebra SO(4) corresponding to rotations in four dimensions. Symmetries in Nuclei, Tokyo, 2008

  15. Energy spectrum • Isomorphism of SO(4) and SO(3)SO(3): • Since L and A' are orthogonal: • The quadratic Casimir operator of SO(4) and H are related: W. Pauli, Z. Phys. 36 (1926) 336 Symmetries in Nuclei, Tokyo, 2008

  16. Dynamical symmetry • Two algebras G1 G2 and a hamiltonian •  H has symmetry G2 but notG1! • Eigenstates are independent of parameters m and n in H. • Dynamical symmetry breaking “splits but does not admix eigenstates”. Symmetries in Nuclei, Tokyo, 2008

  17. Isospin symmetry in nuclei • Empirical observations: • About equal masses of n(eutron) and p(roton). • n and p have spin 1/2. • Equal (to about 1%) nn, np, pp strong forces. • This suggests an isospin SU(2) symmetry of the nuclear hamiltonian: W. Heisenberg, Z. Phys. 77 (1932) 1 E.P. Wigner, Phys. Rev. 51 (1937) 106 Symmetries in Nuclei, Tokyo, 2008

  18. Isospin SU(2) symmetry • Isospin operators form an SU(2) algebra: • Assume the nuclear hamiltonian satisfies •  Hnucl has SU(2) symmetry with degenerate states belonging to isobaric multiplets: Symmetries in Nuclei, Tokyo, 2008

  19. Isospin symmetry breaking: A=49 • Empirical evidence from isobaric multiplets. • Example: T=1/2 doublet of A=49 nuclei. O’Leary et al., Phys. Rev. Lett. 79 (1997) 4349 Symmetries in Nuclei, Tokyo, 2008

  20. Isospin symmetry breaking: A=51 Symmetries in Nuclei, Tokyo, 2008

  21. Isospin SU(2) dynamical symmetry • Coulomb interaction can be approximated as •  HCoul has SU(2) dynamical symmetry and SO(2) symmetry. • MT-degeneracy is lifted according to • Summary of labelling: Symmetries in Nuclei, Tokyo, 2008

  22. Isobaric multiplet mass equation • Isobaric multiplet mass equation: • Example: T=3/2 multiplet for A=13 nuclei. E.P. Wigner, Proc. Welch Found. Conf. (1958) 88 Symmetries in Nuclei, Tokyo, 2008

  23. Isospin selection rules • Internal E1 transition operator is isovector: • Selection rule for N=Z (MT=0) nuclei: No E1 transitions are allowed between states with the same isospin. L.E.H. Trainor, Phys. Rev.85 (1952) 962 L.A. Radicati, Phys. Rev.87 (1952) 521 Symmetries in Nuclei, Tokyo, 2008

  24. E1 transitions and isospin mixing • B(E1;5-4+) in 64Ge from: • lifetime of 5- level; • (E1/M2) mixing ratio of 5-4+ transition; • relative intensities of transitions from 5-. • Estimate of minimum isospin mixing: E.Farnea et al., Phys. Lett. B 551 (2003) 56 Symmetries in Nuclei, Tokyo, 2008

  25. Pairing SU(2) dynamical symmetry • The pairing hamiltonian, • …has a quasi-spin SU(2) algebraic structure: • H has SU(2)  SO(2) dynamical symmetry: • Eigensolutions of pairing hamiltonian: A. Kerman, Ann. Phys. (NY) 12 (1961) 300 Symmetries in Nuclei, Tokyo, 2008

  26. Interpretation of pairing solution • Quasi-spin labels S and MS are related to nucleon number n and seniority : • Energy eigenvalues in terms of n, j and : • Eigenstates have an S-pair character: • Seniority  is the number of nucleons not in S pairs (pairs coupled to J=0). G. Racah, Phys. Rev. 63 (1943) 367 Symmetries in Nuclei, Tokyo, 2008

  27. Quantal many-body systems • Take a generic many-body hamiltonian: • Rewrite H as (bosons: q=0; fermions: q=1) • Operators uij generate U(n) for bosons and for fermions (q=0,1): Symmetries in Nuclei, Tokyo, 2008

  28. Quantal many-body systems • With each chain of nested algebras • …is associated a particular class of many-body hamiltonian • Since H is a sum of commuting operators • …it can be solved analytically! Symmetries in Nuclei, Tokyo, 2008

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