1. Symmetry Concept: Multipolar Electric and Magnetic Fields
Electric multipole moment
Magnetic multipole moment
Partial integration 
El. multipole moment unit = e×(length)ℓ , magn. multipole moment unit = µB×(length)ℓ−1,
e = elementary charge, µB = Bohr magneton. ∇ · B = 0  there are no magn. monopoles. Concept of magn. monopole useful in describing magn. features of composite objects
Int Elm Rad
Useful representation of spherical harmonics:
zero or positive integers p, q, and s, with p + q + s = ℓ and p − q = m.
W. Udo Schröder, 20018

2. Coulomb Fields of Finite Charge Distributionsq z
symmetry axis e q z
System states are typically not spherical in any state (different types of deformation). Excitations open further types of deformation (see later)
arbitrary nuclear charge distribution with normalization
|e|Z
Coulomb interaction system - e
Details of shape are not “visible” at large distances. From distance, everything looks like a point charge.
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Expansion of «1
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3. Coulomb Fields of Finite Charge Distributions
|e|Z
Test Particle e q z
symmetry axis
Expansion of
for |x|«1:
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Recovered “by accident” (?) expansion of symmetric angular shape in terms of
W. Udo Schröder, 20018

4. Multipole Expansion of Coulomb Interaction
Total charge rd3r =|e|Z e q z
symmetry axis
Different multipole shapes/ distributions have different spatial symmetries and ranges
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W. Udo Schröder, 20018

5. Interactions of El Multipoles with Electric Fieldsq+ q- z d x
Consider example of hydrogen atom in homogeneous external electro-static field E  “Stark” perturbation (electronic charge e)
En,ℓ (eV)
Stark Effect in Hydrogen
2nd order effect
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W. Udo Schröder, 20018

6. Static Magnetic Fields
Examples from solid state and nuclear physics Here: The cubic fluorite crystal structure of the AnO2 compounds. Green spheres: actinide An ion, blue spheres:oxygen atoms.
M.-T. Suzuki, N. Magnani, and P. M. OppeneerJournal of the Physical Society of Japan (2018)
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Schematics of the splitting of the 14 one-electron f orbitals. Spin-orbit interactions
splits the orbitals in j = 5/2 and j = 7/2 orbitals, which are further split by the cubic crystal field. The number in the brackets denotes the degeneracy of the orbitals (M.-T. Suzuki, N. Magnani, and P. M. Oppeneer, Phys. Rev. B 88, (2013)).
W. Udo Schröder, 20018

7. Multipole Magnetization Distributions
Spatial distributions of the magnetic moment densities of (a) UO2 and (b) NpO2, for two different viewing directions, [100] and [111], computed with U = 4 eV and J = 0.5 eV. The magnetic moment distributions are depicted on the isosurfaces of the charge densities for the [111] component, with magnitudes as given by the color bars with µB unit. The thin lines show the contour map of the charge density on a spherical surface. M.-T. Suzuki, N. Magnani, and P. M. Oppeneer, Phys. Rev. B 88, (2013)
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W. Udo Schröder, 20018

8. Magnetic Moment Interaction with Elm Field
Particles with intrinsic spin angular momentum have magnetic dipole moment, always coaxial with spin  (2s+1) possible energy states.
msħ f z x y
quantization axis  
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 Distill component of interaction Hamiltonian from part within parentheses.
W. Udo Schröder, 20018

9. Magnetic Dipole Moments
Moving charge e  current density j  vector potential A, influences particles at via magnetic field =0
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current loop:
mLoop = j x A= current x Area
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10. Magnetic Moments: Units and Scaling
Nuclear Spins
g factors
g<0  m I
W. Udo Schröder, 2011

11. 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, 2011

12. 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
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13. 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
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14. 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, 2011

15. 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, 2011

16. Parity Conservation and Central Potentials
Expt: There are no atoms or nuclei with non-zero electrostatic dipole moment
 Consequences for Hamiltonian with some average mean field Ui for particles i (electrons, nucleons,..):
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Average mean field for particles conserves p  U= inversion invariant, e.g., central potential
W. Udo Schröder, 20018