This article reviews the physics of multipolar interactions and multipolar order in $f$-electron systems, using the actinide dioxides as a paradigm. In the past few years, these apparently simple cubic compounds have been studied intensively, and many new phenomena have been discovered. Here the experimental results are discussed together with the current theoretical understanding of multipolar interactions.

Neutron spectroscopy has been used to observe transitions between the energy levels for the 21 different orientations of the $S=10\mathrm{}$ spin ground state of an octanuclear iron molecular cluster $\left({\mathrm{Fe}}_8\right)$ exhibiting quantum tunneling of the magnetization. The results obtained on nondeuterated samples provide the first direct measurement of the zero-field splitting in a large cluster nanomagnet. A reliable and accurate determination of the composition of the cluster's spin wave functions and the spin-Hamiltonian parameters are deduced from the experimental observations, giving crucial information for the study of the macroscopic quantum tunneling process.

We report the results of resonant x-ray scattering experiments performed at the Np $M_{4,5}$ edges in $\mathrm{N}\mathrm{p}{\mathrm{O}}_{\mathrm{2}}$. Below $T_0=25\mathrm{K}$, the development of long-range order of Np electric quadrupoles is revealed by the growth of superlattice Bragg peaks. The polarization and azimuthal dependence of the intensity of the resonant peaks are well reproduced assuming anisotropic tensor susceptibility scattering from a triple-$\overrightarrow{q}$ longitudinal antiferroquadrupolar structure. Electric-quadrupole order in $\mathrm{N}\mathrm{p}{\mathrm{O}}_{\mathrm{2}}$ could be driven by the ordering at $T_0$ of magnetic octupoles of ${\Gamma }_5$ symmetry, splitting the Np ground state quartet and leading to a singlet ground state with zero dipole-magnetic moment.

High-resolution, high-energy-transfer, inelastic neutron scattering has been used to explore the crystal-field (CF) excitations in ${\mathrm{UO}}_2$. As all the dipole-allowed transitions within the free-ion ground manifold have been identified, the observations provide a complete determination of the crystal-field potential and 5f-electron eigenstates. The fourth- and sixth-degree CF parameters are $V_4$=-123 meV and $V_6$=26.5 meV. In spite of the strength of the CF, the ground state is accurately given by the intermediate-coupling approximation with little modification by J-mixing effects. In the antiferromagnetic phase below $T_N$=30.8 K, a splitting of the cubic CF levels, due to the combined effects of the molecular field and the distortion of the oxygen-ligand cage surrounding the ${\mathrm{U}}^{4+}$ ions, has been observed. Detailed CF calculations are presented both for the case of a double-k magnetic structure with a monoclinic distortion of the oxygen sublattice, and for a combined triple-k distortion and magnetic order. The observed splittings are shown to be more consistent with the triple-k model.