Gross Properties of Nuclei Sizes Gross Properties of Nuclei Nuclear Spins and Magnetic Moments

Intrinsic Nuclear Spin Nuclei can be deformed  can rotate quantum mech.  collective spin and magnetic effects (moving charges) Intrinsic spin? Nucleons have spin-1/2 Demonstrate via interaction with external odd-A : I= half-integer multiple of ħ even-A: I= integer multiple of ħ even-Z & even-N: I = 0 “good” quantum numbers: I, mI expt fact mIħ f z x y quantization axis Quantum mechanical spin: Nuclear Spins Interactions via magnetic moment Nuclear Magneton W. Udo Schröder, 2004

Extreme coupling schemes of A-body system Spin Coupling Schemes Nuclear spin built up of nucleonic angular momenta spin and orbital, z Extreme coupling schemes of A-body system mL mS L and S good qu. #s Correspond to different strength of interactions between nucleons and atomic electrons  different symmetry of A-body wave function Nuclear Spins Always  conserved: F, no external torque W. Udo Schröder, 2004

Magnetic Dipole Moments Moving charge e  current density j  vector potential A, influences particles at via magnetic field =0 Nuclear Spins current loop: mLoop = j x A= current x Area W. Udo Schröder, 2004

Magnetic Moments: Units and Scaling Nuclear Spins g factors g<0  m I W. Udo Schröder, 2004

Total Nucleon Magnetic Moment z Superposition of orbital and spin m: below use these single-particle states Precession of m around z-axis slaved by precession of j  all m components perp. to j vanish on average. Nuclear Spins maximum alignment of j W. Udo Schröder, 2004

Effective g Factor gj: effective g-factor Magnetic moment for entire nucleus: analogous definition for maximum alignment, slaved by nuclear spin I precession Nuclear Spins W. Udo Schröder, 2004

Simple s.p. Model: Schmidt-Lines (Odd-A) odd-A: All but one nucleon paired,. Paired nucleons make spinless, spherical core  central potential for extra nucleon  even N even Z (Z, N) = (odd, even) unpaired p, gℓ = 1 units: mN (Z, N) = (even, odd) unpaired n, gℓ = 0 Nuclear Spins units: mN W. Udo Schröder, 2004

Experimental m for Odd-A Nuclei m/mN odd-Z I m/mN odd-N I 7Li: j=3/2 j≈ℓ+1/2  ℓ = 1 Nuclear Spins Reproduction of overall trends Almost all m lie between Schmidt lines=extreme values for m. Quenching of gs factors due to interactions with other nucleons W. Udo Schröder, 2004

Magnetic e-Nucleus Interactions z Energy in homogeneous B-field || z axis Force in inhomogeneous B-field || z axis Atomic electrons (currents) produce B-field at nucleus, aligned with total electronic spin Nuclear Spins Total spin W. Udo Schröder, 2004

Magnetic Hyper-Fine Interactions HF pattern depends on strength Bext weak Bext strong FS HFS Strong Bext breaks [J,I]F coupling. F import for weak Bext, independent for strong Bext 1s2p X-Ray Transition mJ2 Nuclear Spins E1, DmJ=0 electronic splitting 2 separated groups @ 2I+1=4 lines. (F not good qu. #) mJ -2 W. Udo Schröder, 2004

Rabi Atomic/Molecular Beam Experiment (1938) Force on magnetic moment in inhomogeneous B-field ||z axis I. Rabi 1984 Alternating B gradients RF coil  DmI A B homogeneous B Aperture Nuclear Spins Magnet B compensates for effect of magnet A for a given mI Transition induced W. Udo Schröder, 2004

Summary: Gross Properties of Nuclei Nuclear sizes: Finite size, R = r0.A1/3, r0 = 1.2 fm  approximately constant density in interior  saturation of nuclear forces, must have repulsive core Diffuse surface, b ~1fm, weak dependence on A Fermi-type charge and mass distributions Nuclei with magic N or/and Z numbers slightly smaller than average B. Nuclear masses and binding energies: Approximately constant B/A≈ 8 MeV, weakly dependent of A  saturation of nuclear forces, nucleon experiences average interaction with “nearest neighbors” Nuclear liquid drop model describes average A-dependence of B/A  b stable valley, but: paired nucleons are more tightly bound dB ≈12 -1/2A MeV. Structure effects: # of isotopes for odd or even A Nuclei with magic N and or Z are more tightly bound than neighbors, 64Ni, 56Fe most tightly bound nuclei Nuclear Spins W. Udo Schröder, 2004

Summary: Gross Properties of Nuclei C. Nuclear deformations and electrostatic moments: Only even electrostatic moments  monopole, quadrupole Q most important  Spin I=0, ½ nuclei have no measurable Q N-Z regions with large Q (Lanthanides, Actinides), domains defined by magic numbers  magic N, Z have Q = 0 D. Nuclear spins and magnetic moments: Most nuclear spins are small, a few ħ, integer multiple of ħ for e-A  I = 0 for e-e nuclei, half-integer ħ for o-A Nuclear spins =combination of nucleonic orbital and spin angular momenta Only odd magnetostatic moments, dipole is first important moment Magnetic moments of o-A nuclei related to unpaired nucleon  Schmidt Lines (quenching in medium, g factors always smaller than s.p. values) Magic nuclei have I =0, m = 0 Nuclear Spins W. Udo Schröder, 2004

All Grossed Out Nuclear Spins W. Udo Schröder, 2004

Nuclear Magnetic Resonance B0 B(t) Nuclear Spins W. Udo Schröder, 2004

2p (ℓ=1) mℓ=-1 mℓ=+1 mℓ= 0 W B≠0 B=0 m Nuclear Spins W. Udo Schröder, 2004

Coulomb Fields of Finite Charge Distributions |e|Z e q z arbitrary nuclear charge distribution with normalization Coulomb interaction Expansion of for |x|«1: «1 Nuclear Spins W. Udo Schröder, 2004