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

Coulomb Fields of Finite Charge Distributions q 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. Int Elm Rad Expansion of «1 W. Udo Schröder, 20018

Coulomb Fields of Finite Charge Distributions |e|Z Test Particle e q z symmetry axis Expansion of for |x|«1: Int Elm Rad Recovered “by accident” (?) expansion of symmetric angular shape in terms of W. Udo Schröder, 20018

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 Int Elm Rad W. Udo Schröder, 20018

Interactions of El Multipoles with Electric Fields q+ 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 Int Elm Rad W. Udo Schröder, 20018

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) Int Elm Rad 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, 195146 (2013)). W. Udo Schröder, 20018

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, 195146 (2013) Int Elm Rad W. Udo Schröder, 20018

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   Int Elm Rad  Distill component of interaction Hamiltonian from part within parentheses. W. Udo Schröder, 20018

Magnetic Dipole Moments Moving charge e  current density j  vector potential A, influences particles at via magnetic field =0 Int Elm Rad current loop: mLoop = j x A= current x Area W. Udo Schröder, 20018

Magnetic Moments: Units and Scaling Nuclear Spins g factors g<0  m I W. Udo Schröder, 2011

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, 2011

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, 2011

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, 2011

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, 2011

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

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,..): Int Elm Rad Average mean field for particles conserves p  U= inversion invariant, e.g., central potential W. Udo Schröder, 20018