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  1. Sizes Gross Properties of Nuclei Nuclear Deformations and Electrostatic Moments

  2. z e |e|Z q Coulomb Fields of Finite Charge Distributions arbitrary nuclear charge distribution with normalization Coulomb interaction Expansion of for |x|«1: «1 Nuclear Deformations

  3. z e |e|Z q Monopoleℓ = 0 Dipole ℓ = 1 Quadrupole ℓ =2 Multipole Expansion of Coulomb Interaction Point Charges Nuclear Deformations Nuclear distribution

  4. A Quantal Symmetry symmetric nuclear shape  symmetric invariance of Hamiltonian against space inversion both even or odd Nuclear Deformations n even  p = +1n odd  p = -1 If strong nuclear interactions parity conserving

  5. Restrictions on Nuclear Field Expt: No nucleus with non-zero electrostatic dipole moment Consequences for nuclear Hamiltonian (assume some average mean field Uifor each nucleon i): Nuclear Deformations Average mean field for nucleons conserves p inversion invariant, e.g., central potential

  6. Neutron Electric Dipole Moment ? qn =0, possible small dn ≠0.CP and P violation  could explain matter/antimatter asymmetry Measure NMR HF splitting for Transition energiesDw=4dnE B=0.1mG, tune with Bosc B. E = 1MV/m w= 30Hz spin-flipof ultra-cold (kT~mK)Ekin=10-7eV, l =670Åneutrons in mgn.bottleguided in reflecting Ni tubes Nuclear Deformations

  7. Experimental Results for dn dn experimental sensitivity From size of neutron (r0≈ 1.2fm): dn 10-15 e·m.So far, only upper limits for dn PNPI (1996): dn < (2.6 ± 4.0 ±1.6)·10-26 e·cm ILL-Sussex-RAL (1999): dn < (-1.0 ± 3.6)·10-26 e·cm Nuclear Deformations

  8. z q’ Intrinsic Quadrupole Moment Consider axially symmetric nuclei (for simplicity), body-fixed system (’), z =z’ symmetry axis Sphere: Nuclear Deformations  Q0 measures deviation from spherical shape.

  9. z z z z z b a Collective and s.p. Deformations collectivedeformationdisc collectivedeformationcigar single particlearound core singleholearound core Q0>0 “prolate” Q0<0 “oblate” Planar single-particle orbit: Nuclear Deformations Ellipsoidprincipal axes a, b Deformation parameter d

  10. z’ y’ q x’ Spectroscopic Quadrupole Moment z body-fixed {x’, y’, z’}, Lab {x, y, z}Symmetry axis z defined by the experiment intrinsic What is measured in Lab system? finite rotation through q Measured Q depends on orientation of deformed nucleus w/r to Lab symmetry axis.  define Qz as the largest Q measurable. How to control or determine orientation of nuclear Q?  Nuclear spin  to symmetry axis, no quantal rotation about z’ Nuclear Deformations

  11. Angular Momentum and Q Qz =maximum measurable  maximum spin (I) alignment Legendre polyn. complete basis set  z I Nuclear Deformations I couples with I to L =2 I Spins too small to effect alignment of Q in the lab.

  12. z Vector Coupling of Spins I≠0: mI=I q Any orientation quadratic dependence of Qz on mI Nuclear Deformations “The” quadrupole moment

  13. z F+DF E+DE F E Electric Multipole Interactions Inhomogeneous external electric field exerts a torque on deformed nucleus. orientation-dependent energy WQExamples: crystal lattice, fly-by of heavy ions Axial symmetry of field assumed: Taylor expansion of scalar potential U: no mixed dervs. Nuclear Deformations monopole WIS dipole  0 quadrupole WQ WIS: isomer shift, WQ: quadrupole hyper-fine splitting

  14. Maxwell equs.  No external charge axial symm Electric Quadrupole Interactions =Uzz Nuclear Deformations Field gradient x spectroscopic quadrupole moment  mI2

  15. mI=±2 excitedstate I=2 mI=±1 mI=0 E2 isomer shift I=0 mI=0 ground stateUzz=0 Uzz≠0 dN/dEg Uzz=0 Eg Quadrupole Hyper-Fine Splitting Use external electrostatic field, align Q by aligning nuclear spin I,Measure interaction energies WQ (I >1/2 ) Quadrupole hyper-fine splitting of nuclear or atomic energy levels • Slight “hf” splitting of nuclear and atomic levels in Uzz≠0 • splitting of g emission/absorption lines • Estimates: atomic energies ~ eVatomic size ~ 10-8cmpotential gradient Uz ~ 108V/cmfield gradient Uzz ~ 1016V/cm2Q0 ~ 10-24 e cm2 • WQ~ 10-8 eV small ! Nuclear Deformations

  16. . .. . I=8 I=6 I=4 I=2 Experimental Methods for Quadrupole Moments • Small “hf” splitting WQ of nuclear and atomic levels in Uzz≠0 • splitting of X-ray/ g emission/absorption lines Measurable for atomic transitions with laser excitations nuclear transitions with Mössbauer spectroscopy muonic atoms: 107 times larger hf splittings WQ with X-ray and g spectroscopy scattering experiments Uzz(t) Nuclear spectroscopy of collective rotations model for moment of inertia Nuclear Deformations I=0

  17. z b a Collective Rotations b : deformation parameter Nuclei with large Q0 consistent w. collective rotations  lanthanides, actinides Nuclear Deformations Wood et al.,Heyde

  18. Systematics of Electric Quadrupole Moments Mostly prolate (Q>0) heavy nuclei Q(167Er) =30R2 odd-Nodd-Z Q>0 : e.g., hole in spherical core  pattern not obvious. If such nuclei exist, weak effect of hole for Q Prolate Tightly bound nuclei are spherical: “Magic” N or Z = 8, 20, 28, 50, 82, 126, … Nuclear Deformations Q<0 : e.g., extra particle around spherical core. pattern recognizable Oblate 8 20 28 50 82 126

  19. Q0 Systematics Q0 large between magic N, Z numbersQ0≈0 close to magic numbers Nuclear Deformations Møller, Nix, Myers, Swiatecki, LBL 1993

  20. No more deformations! Nuclear Deformations

  21. Nuclear Deformations