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

2. 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

3. 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

4. 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

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

6. Total Nucleon Magnetic Momentz
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

7. 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

8. 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

9. 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

10. Magnetic e-Nucleus Interactionsz
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

11. 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 2I+1=4 lines. (F not good qu. #)
mJ -2
W. Udo Schröder, 2004

12. 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

13. 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

14. 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

15. All
Grossed
Out
Nuclear Spins
W. Udo Schröder, 2004

16. Nuclear Magnetic ResonanceB0
B(t)
Nuclear Spins
W. Udo Schröder, 2004

17. 2p (ℓ=1) mℓ=-1 mℓ=+1 mℓ= 0 W B≠0 B=0 m Nuclear Spins
W. Udo Schröder, 2004

18. 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