Symmetry Concept Multipolar Electric and Magnetic Fields 1

Symmetry Concept: Multipolar Electric and Magnetic Fields 1 Electric multipole moment Magnetic multipole moment Int Elm Rad 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 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 System states are typically not spherical in any state (different types of deformation). Excitations open further types of deformation (see later) z e arbitrary nuclear charge distribution with normalization |e|Z 2 q xis a try e m m sy Coulomb interaction system - e Int Elm Rad Details of shape are not “visible” at large distances. From distance, everything looks like a point charge. Expansion of « 1 W. Udo Schröder, 20018

Coulomb Fields of Finite Charge Distributions z Test Particle e Expansion of |e|Z 3 q x y a r t me Int Elm Rad m sy is for |x| « 1: Recovered “by accident” (? ) expansion of symmetric angular shape in terms of W. Udo Schröder, 20018

Multipole Expansion of Coulomb Interaction z e Different multipole shapes/ distributions have different spatial symmetries and ranges 4 q x y a r t me m sy Int Elm Rad W. Udo Schröder, 20018 is

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

Static Magnetic Fields Examples from solid state and nuclear physics Here: The cubic fluorite crystal structure of the An. O 2 compounds. Green spheres: actinide An ion, blue spheres: oxygen atoms. Int Elm Rad 6 M. -T. Suzuki, N. Magnani, and P. M. Oppeneer. Journal of the Physical Society of Japan (2018) 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

Int Elm Rad 7 Multipole Magnetization Distributions W. Udo Schröder, 20018 Spatial distributions of the magnetic moment densities of (a) UO 2 and (b) Np. O 2, for two different viewing directions, [100] and [111], computed with U = 4 e. V and J = 0. 5 e. V. 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)

Magnetic Moment Interaction with Elm Field Particles with intrinsic spin angular momentum have magnetic dipole moment, always coaxial with spin (2 s+1) possible energy states. quantization axis ms ħ 8 z y f Int Elm Rad x 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 9 e, m current loop: W. Udo Schröder, 20018 m. Loop = j x A= current x Area

Magnetic Moments: Units and Scaling Magnetic moments: 10 m Nuclear Spins Units g factors g<0 m I W. Udo Schröder, 2011

Total Nucleon Magnetic Moment z Superposition of orbital and spin m: Nuclear Spins 11 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. W. Udo Schröder, 2011 maximum alignment of j

Effective g Factor Nuclear Spins 12 gj: effective g-factor Magnetic moment for entire nucleus: analogous definition for maximum alignment, slaved by nuclear spin I precession W. Udo Schröder, 2011

Magnetic e-Nucleus Interactions z Energy in homogeneous B-field || z axis 13 Force in inhomogeneous B-field || z axis Nuclear Spins Atomic electrons (currents) produce B-field at nucleus, aligned with total electronic spin Total spin W. Udo Schröder, 2011

Magnetic Hyper-Fine Interactions HF pattern depends on strength Bext 14 FS HFS Strong Bext breaks [J, I]F coupling. F import for weak Bext, independent for strong Bext weak Bext strong 1 s 2 p X-Ray Transition m. J electronic splitting Nuclear Spins 2 m. J -2 W. Udo Schröder, 2011 2 separated groups @ 2 I+1=4 lines. (F not good qu. #) E 1, Dm. J=0

Rabi Atomic/Molecular Beam Experiment (1938) Force on magnetic moment in inhomogeneous B-field ||z axis I. Rabi 1984 Alternating B gradients 15 homogeneous B Nuclear Spins A B RF coil Dm. I Magnet B compensates for effect of magnet A for a given m. I W. Udo Schröder, 2011 Aperture Transition induced

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