Figure 1

(Color) Schematic view of the free-ion energy levels for the $f^3$ electronic configuration. The total Hamiltonian is the sum of $H_{\mathrm{SO}}$ multiplied by a factor $\chi$ and $H_C$ multiplied by a factor proportional to $1-\chi$. On the left side of the panel, the Coulomb interaction dominates, and ${}^{2S+1}L$ terms are formed; on the right side, the spin-orbit interaction is dominant and $j\text{−}j$ coupling is appropriate. The dashed vertical lines indicate the approximate position of ${\mathrm{Np}}^{4+}$ in the diagram. Arbitrary units are used on the vertical scale.Reuse & Permissions

Figure 2

(Color online) Distribution of the ratio $E(K,K)∕E(1,1)$, with $E(K_i,K_j)$ the ground-state energy of a dimer ($C_{2v}$ bond) of ${\mathrm{Pr}}^{4+}$ and ${\mathrm{Np}}^{4+}$ ions coupled with the part of Eq. (35) associated with a specific pair of ranks $K_i$, $K_j$. The $t_{i,j}^{{\Gamma }_l{\Gamma }_l}$ parameters are varied on a cubic grid.Reuse & Permissions

Figure 3

Results of the multipolar model in $j\text{−}j$ coupling proposed for $\mathrm{Np}{\mathrm{O}}_2$ by 167. (a) Correlation functions for multipolar operators on a fcc lattice, calculated by exact diagonalization on an eight-site cluster, for $\mathbf{q}=(0,0,0)$ (circles), $\mathbf{q}=(0,0,1)$ (squares), and $\mathbf{q}=(1∕2,1∕2,1∕2)$ (triangles). The $T_x^{\alpha ,\beta }$ and $T_y^{\alpha ,\beta }$ operators have the same correlation function as $O_{yz}$ and $O_{zx}$. (b) Magnetic susceptibility and (c) specific heat calculated (lines) for $\mathrm{Np}{\mathrm{O}}_2$ in mean-field approximation, compared with experimental data (triangles).Reuse & Permissions

Figure 4

Specific heat (upper panel) and magnetic susceptibility (lower panel) of $\mathrm{Dy}{\mathrm{B}}_2{\mathrm{C}}_2$. Data from 345.Reuse & Permissions

Figure 5

Temperature variation of the third-order magnetic susceptibility along the [0 0 1] crystallographic direction of $\mathrm{Tm}{\mathrm{Ag}}_2$. Data for the [1 0 0], and [1 1 0] directions are shown in the inset. The nonmagnetic transition at $T_0=4.74\mathrm{K}$ is analyzed assuming ferroquadrupolar order within the orthorhombic ${\Gamma }_3$ symmetry-lowering mode ($G^{\alpha }=0$; $G^{\gamma }=17\mathrm{mK}$). Data from 213.Reuse & Permissions

Figure 6

Temperature dependence of the $C_{\gamma }$ elastic constant in $\mathrm{Tm}{\mathrm{Ag}}_2$. The dashed line indicates the value of $C_{\gamma }^0$ measured for the nonmagnetic isostructural compound $\mathrm{Y}{\mathrm{Ag}}_2$. The solid line is a fit to Eq. (57) with $K^{\gamma }=10\mathrm{mK}$ and $\mid B^{\gamma }\mid =32\mathrm{K}$. Inset: Temperature dependence of the parastriction for an external magnetic field applied along the [1 0 0] direction. Relative length changes are measured along [1 0 0] and [0 1 0]. The solid line is a fit leading to the determination of ME and total quadrupolar coefficients. Data from 213.Reuse & Permissions

Figure 7

(Color online) Time dependence of the zero-field backward-forward positron count rate asymmetry for $\mathrm{Np}{\mathrm{O}}_2$ at different temperatures. A vertical offset has been applied to the different data sets. Data from 163.Reuse & Permissions

Figure 8

(Color online) Schematic of the experimental setup for RXS studies. A photon beam polarized perpendicular to the scattering plane impinges on the sample $S$ and is scattered toward the analyzer crystal $A$. The scattered beam components with polarization parallel $\left(\pi \right)$ or perpendicular $\left(\sigma \right)$ to the scattering plane can be selected by an appropriate orientation of the analyzer. $D$ is a standard scintillation detector. Integrated intensities of Bragg peaks are measured as a function of photon energy for different values of the azimuth angle $\Psi$, defining the crystal orientation about the scattering vector. The unit vectors ${\mathbf{u}}_i$ define the reference frame.Reuse & Permissions

Figure 9

Type I, $3\text{−}\mathbf{k}$ magnetic structure in an fcc lattice. The dipole magnetic moment direction is shown as an arrow; $R_1=000$, $R_2=\frac{1}{2}\frac{1}{2}0$, $R_3=\frac{1}{2}0\frac{1}{2}$, $R_4=0\frac{1}{2}\frac{1}{2}$. Left, the longitudinal (L) structure with ${\mathbf{\mu }}_1={\mu }_0(1,1,1)∕\sqrt{3}$, ${\mathbf{\mu }}_2={\mu }_0(\overline{1},\overline{1},1)∕\sqrt{3}$, ${\mathbf{\mu }}_3={\mu }_0(\overline{1},1,\overline{1})∕\sqrt{3}$, ${\mathbf{\mu }}_4={\mu }_0(1,\overline{1},\overline{1})∕\sqrt{3}$; middle, TA transverse $S$-domain, with ${\mathbf{\mu }}_1={\mu }_0(1,1,1)\sqrt{3}$, ${\mathbf{\mu }}_2={\mu }_0(1,\overline{1},\overline{1})∕\sqrt{3}$, ${\mathbf{\mu }}_3={\mu }_0(\overline{1},\overline{1},1)∕\sqrt{3}$, ${\mathbf{\mu }}_4={\mu }_0(\overline{1},1,\overline{1})∕\sqrt{3}$; right, TB transverse $S$-domain, with ${\mathbf{\mu }}_1={\mu }_0(1,1,1)∕\sqrt{3}$, ${\mathbf{\mu }}_2={\mu }_0(\overline{1},1,\overline{1})∕\sqrt{3}$, ${\mathbf{\mu }}_3={\mu }_0(1,\overline{1},\overline{1})∕\sqrt{3}$, ${\mathbf{\mu }}_4={\mu }_0(\overline{1},\overline{1},1)∕\sqrt{3}$.Reuse & Permissions

Figure 10

Bessel transform of the magnetization distribution for Ho (open circles) and ${\mathrm{Ho}}_{0.9}{\mathrm{Sc}}_{0.1}$ (full circles), derived from the intensity of third-order neutron diffraction satellites in the magnetic spiral phase. The solid line is the theoretical prediction for a single ion model. Data from 94.Reuse & Permissions

Figure 11

(Color online) Splitting of the ${}^{11}\mathrm{B}$ NMR resonance lines observed in the phase II of $\mathrm{Ce}{\mathrm{B}}_6$ at $T=2.78\mathrm{K}$ and $10.7\mathrm{MHz}$, as a function of the angle $\theta$ between the direction of the applied magnetic field and the [1 0 0] axis. Triangles (circles) refer to B sites with axial symmetry along [0 0 1] $\left(B_3\right)$ and [1 0 0] $\left(B_1\right)$, respectively. Dashed lines are theoretical calculations including magnetic dipole and octupole contributions (280). The crystallographic structure of $\mathrm{Ce}{\mathrm{B}}_6$ is shown in the right panel.Reuse & Permissions

Figure 12

(Color online) Energy dependence of the $(3∕23∕23∕2)$ superlattice reflection in ${\mathrm{Ce}}_{0.7}{\mathrm{La}}_{0.3}{\mathrm{B}}_6$, around the Ce $L_2$ absorption edge, at (a) $T=1.0\mathrm{K}$ and (b) $T=1.7\mathrm{K}$. Squares represent the intensity in the $\sigma \text{−}\pi$ channel and circles refer to $\sigma \text{−}\sigma$ polarization. The $E1\text{−}E1$ and $E2\text{−}E2$ energy thresholds are shown by vertical dashed lines. (c) Temperature dependence of the intensity for (triangles) $E1\text{−}E1$ $\sigma \text{−}\pi$ intensity, (squares) $E2\text{−}E2$ $\sigma \text{−}\pi$, and (circles) $E2\text{−}E2$ $\sigma \text{−}\sigma$ signals. Data from 198.Reuse & Permissions

Figure 13

Difference between neutron diffraction rocking curves measured at $T=0.25$ and $2\mathrm{K}$ at several $(h∕2,h∕2,h∕2)$ positions in ${\mathrm{Ce}}_{0.7}{\mathrm{La}}_{0.3}{\mathrm{B}}_6$. The temperature dependence of the integrated intensity and the $Q$ dependence of the magnetic structure factor are shown in the insets. Data from 173.Reuse & Permissions

Figure 14

Nonresonant x-ray diffraction intensity in $\mathrm{Dy}{\mathrm{B}}_2{\mathrm{C}}_2$, obtained from $\mathcal{l}$ scans around three chosen $(hk9∕2)$ superlattice reflections. Data were collected at $T=18\mathrm{K}$, below the AFQ ordering temperature $T_Q=24.7\mathrm{K}$ but above the Néel temperature $T_N=15.3\mathrm{K}$. Data from 4. Right panel: Schematic of the AFQ order in $\mathrm{Dy}{\mathrm{B}}_2{\mathrm{C}}_2$. For the ions Dy1, located at 000, the principal axis of the quadrupole moment forms an angle $\varphi$ with the crystallographic axis $\mathbf{a}$, whereas it is directed at angle $\pi ∕2-\varphi$ from $\mathbf{a}$ in the case of the Dy2 ions at $\frac{1}{2}\frac{1}{2}0$. In the $z=1$ plane, the angle between the $\mathbf{a}$ direction and the quadrupole principal axis is $\pi ∕2+\varphi$ for Dy3 at 001, and $\pi \text{−}\varphi$ for Dy4 at $\frac{1}{2}\frac{1}{2}1$.Reuse & Permissions

Figure 15

(Color online) Azimuth dependence of (upper panel) the $\sigma \text{−}\pi$ and (lower panel) the $\sigma \text{−}\sigma$ scattering intensities of the (1 0 3) superlattice peak in $\mathrm{U}{\mathrm{Pd}}_3$ at the U $M_4$ edge, at $T=7.1\mathrm{K}$. The experimental data are compared with calculations assuming ordering of the $O_{zx}$ (solid line) and $O_{x^2-y^2}$ (dashed line) electric quadrupoles. The right panel is a representation of the $O_{zx}$ AFQ structure. Data from 332.Reuse & Permissions

Figure 16

(Color online) Crystallographic structure of actinide dioxides $A{\mathrm{O}}_2$. (a) The cubic ${\mathrm{CaF}}_2$-type structure with (large red circles) $A^{4+}$ and (small green circles) ${\mathrm{O}}^{2-}$. (b) $3\text{−}\mathbf{k}$ internal distortion of the oxygen sublattice in the AF phase of $\mathrm{U}{\mathrm{O}}_2$. The oxygen displacement along the ⟨1 1 1⟩ directions is of $0.014\mathrm{\AA }$ (not to scale in the figure).Reuse & Permissions

Figure 17

(Color online) Temperature dependence of the reduction factor of the susceptibility of ${\mathrm{U}}_{0.09}{\mathrm{Th}}_{0.91}{\mathrm{O}}_2$ (285; 60). The reduction factor is the ratio of the measured susceptibility to the Curie susceptibility expected from the ${\Gamma }_5$ CF ground state. The solid line is the result of a theoretical model based on the dynamical Jahn-Teller effect (266).Reuse & Permissions

Figure 18

(Color online) CF levels of $\mathrm{U}{\mathrm{O}}_2$ in the paramagnetic phase as deduced from inelastic neutron scattering (12). The splitting of the ${\Gamma }_5$ ground triplet in the magnetically ordered phase is shown in the right-hand side. The shading illustrates the continuum of spin-wave excitations.Reuse & Permissions

Figure 19

Azimuth angle dependence of the (1 1 2) $\sigma \text{−}\sigma$ intensity in $\mathrm{U}{\mathrm{O}}_2$ at $12\mathrm{K}$. Data collected with incident energy $E=3.728\mathrm{keV}$ $\left({\mathrm{U}}_{M_4}\right)$. The solid (dashed) line shows the expected azimuth dependence for a $3\text{−}\mathbf{k}$ transverse (longitudinal) structure of ${\Gamma }_5$ quadrupoles. From 336.Reuse & Permissions

Figure 20

The integrated intensity of the (1 1 2) reflection in $\mathrm{U}{\mathrm{O}}_2$ and of the (0 0 3) superlattice peak in $\mathrm{Np}{\mathrm{O}}_2$ as a function of the incident photon energy around the $M_4$ absorption edge of Np and U, respectively. Data were collected at $T=12\mathrm{K}$. The solid lines are fit to Lorentzian squared line shapes.Reuse & Permissions

Figure 21

The most intense spin-wave modes measured at $16.5\mathrm{K}$ along the principal crystallographic directions. The dashed lines correspond to acoustic phonon branches measured at $270\mathrm{K}$. Inelastic neutron scattering data have been collected in the constant-$Q$ mode, at the reciprocal-lattice points $(0,0,2-\xi )$ and $(1-\xi ,1-\xi ,1)$. Open symbols indicate significantly smaller intensity than the filled points. From 44.Reuse & Permissions

Figure 22

(Color online) Multipolar coupling in a ${\mathrm{U}}^{4+}$ dimer. Left: Distribution of the ratio $E(K,K)∕E(1,1)$, with $E(K_i,K_j)$ the ground-state energy of a dimer ($C_{2v}$ bond) of ${\mathrm{U}}^{4+}$ ions coupled with the part of Eq. (35) associated with a specific pair of ranks $K_i,K_j$. The $t_{i,j}^{{\Gamma }_l{\Gamma }_l}$ parameters are varied on a cubic grid. Right: Distribution of the anisotropy of dipole-dipole interactions for NN U ions, i.e., the part of Eq. (35) with ($K_i=1$, $K_j=1$). The $C_{2v}$ symmetry of the bond allows for three independent exchange constants. In the reference frame bringing the coupling to diagonal form, $H_{ij}=I_{xx}{J}_x\left(i\right)J_x\left(j\right)+I_{yy}{J}_y\left(i\right)J_y\left(j\right)+I_{zz}{J}_z\left(i\right)J_z\left(j\right)$. We have defined $I_{\mathrm{iso}}=(1∕\sqrt{3})(I_{xx}+I_{yy}+I_{zz})$, $I_{\text{axial}}=(1∕\sqrt{6})(I_{xx}+I_{yy}-2I_{zz})$, and $I_{\text{rhombic}}=(1∕\sqrt{2})(I_{yy}-I_{xx})$. The histograms show that the two anisotropic components $I_{\text{axial}}$ and $I_{\text{rhombic}}$ are generally sizable. Note that in the $3\text{−}\mathbf{k}$ phase the lattice distortion decreases the bond symmetry, so that additional anisotropic exchange parameters are switched on, even if these are most likely small.Reuse & Permissions

Figure 23

Temperature dependence of the $\mathrm{Np}{\mathrm{O}}_2$ specific heat in zero magnetic field (circles) and in a magnetic field of $9\mathrm{T}$ (triangles). The specific heat of $\mathrm{Th}{\mathrm{O}}_2$ is shown by diamonds. Inset: The magnetic contribution to the $\mathrm{Np}{\mathrm{O}}_2$ specific heat obtained by taking the difference between the $\mathrm{Np}{\mathrm{O}}_2$ and the $\mathrm{Th}{\mathrm{O}}_2$ curves. Data from 196.Reuse & Permissions

Figure 24

(Color online) Magnetic inelastic neutron scattering cross section obtained for $\mathrm{Np}{\mathrm{O}}_2$ at $T=5\mathrm{K}$ (99). The incident energy was $180\mathrm{meV}$ and the average scattering angle was $\varphi =5°$. The solid line is a fit to the data of two Gaussian peaks (dashed lines) and a background.Reuse & Permissions

Figure 25

(Color online) The energy spectrum of $\mathrm{Np}{\mathrm{O}}_2$ in the paramagnetic phase. Left: Phonon density of states of $\mathrm{U}{\mathrm{O}}_2$ (78), showing the presence of the $M_5$ optical bands near $55\mathrm{meV}$. Center: Bare CF energy levels of $\mathrm{Np}{\mathrm{O}}_2$. Right: Structure of the mixed CF-phonon states resulting from the model of Eq. (93).Reuse & Permissions

Figure 26

Evolution of the inelastic neutron scattering intensity from $\mathrm{Np}{\mathrm{O}}_2$ as the temperature is increased through the phase transition point $T_0=25\mathrm{K}$. Data are collected with uniaxial polarization analysis in the spin-flip channel, with initial polarization parallel to the momentum transfer $\mathbf{Q}$. With this configuration, the signal is entirely magnetic and indicates a splitting of the ground-state quartet below $T_0$. Data from 11.Reuse & Permissions

Figure 27

(Color online) Intensity plot of the magnetic field produced by the ${\Gamma }_2$ OP of $\mathrm{Np}{\mathrm{O}}_2$ at $T=0$ (261), assuming uniform ordering within the (0 0 1) plane. The field is along [0 0 1] with the sign indicated. A zoom on a single ion is shown in the lower part. Arrows indicate the magnetic field within a (1 1 0) plane.Reuse & Permissions

Figure 28

Temperature dependence of the normalized intensity of the (0 0 1) and (0 0 3) superlattice reflections observed at the $M_4$ edge $\left(3.845\mathrm{keV}\right)$ in $\mathrm{Np}{\mathrm{O}}_2$. Data from 197.Reuse & Permissions

Figure 29

(Color online) Azimuth angle dependence of the (0 0 3) intensity, at $T=12\mathrm{K}$, with $\pi$ and $\sigma$ final polarization and specular reflection geometry. Lines are calculations based on Eq. (76) for a longitudinal $3\text{−}\mathbf{k}$ structure of ${\Gamma }_5$ quadrupoles. Data from 239.Reuse & Permissions

Figure 30

Azimuth angle dependence of the Stokes parameters $P_1$ (upper panel) and $P_2$ (lower panel) of $\mathrm{Np}{\mathrm{O}}_2$ superlattice reflections measured at $10\mathrm{K}$ with $E=3.846\mathrm{keV}$. Lines are calculations for a longitudinal $3\text{−}\mathbf{k}$ structure of ${\Gamma }_5$ quadrupoles. $P_1=(I_{\sigma \sigma }-I_{\sigma \pi })∕(I_{\sigma \sigma }+I_{\sigma \pi })$; $P_2$ is defined by the intensities measured with the analyzer oriented at $\pm \pi ∕4$ with respect to the scattering plane, and is determined by the phase relation between the two polarization channels. From 239.Reuse & Permissions

Figure 31

(Color) Arrangement of the ${\Gamma }_5$ MMPs (cones) and induced ${\Gamma }_5$ quadrupoles (ellipsoids) in the four-sublattice $3\text{−}\mathbf{k}$ ground state of $\mathrm{Np}{\mathrm{O}}_2$. The slice displays the direction of the local magnetic field around one of the Np ions. The cones are along the local $D_{3d}$ axes, and the direction where they point illustrates the sign of the local multipole moment.Reuse & Permissions

Figure 32

(Color online) MF splitting of the ground ${\Gamma }_8$ quartet in the ordered phase resulting from Eq. (94) with the “strong” set of CF parameters and rank-5 MMPs. Numbers in parentheses are degeneracies.Reuse & Permissions

Figure 33

CF reduction factor for the ${\Gamma }_8$ ground quartet, $r={\mu }_{\mathrm{CF}}^2∕{\mu }_0^2$, as a function of the CF parameter $x$ $(W<0)$. The arrow indicates the most likely value of $x$. We have neglected $J$ mixing here since the associated small correction to the ${\Gamma }_8$ wave functions does not affect the conclusions. Then, these wave functions depend on the $x$ CF parameter only. From 264.Reuse & Permissions

Figure 34

(Color) Intensity plot illustrating the magnetic field produced by the rank-5 component of the OP of $\mathrm{Np}{\mathrm{O}}_2$ at $T=0$ in two representative planes. The darkest area corresponds to a field intensity larger than $5\mathrm{T}$. The Np ion at the center belongs to the sublattice characterized by a MMP moment having [1 1 1] as $C_3$ axis and its charge distribution has a quadrupole moment (illustrated by the central ellipsoid). Arrows show the field calculated at distances of 1 and $1.2\mathrm{\AA }$ from the Np nucleus. The maximum intensity of the field at $1\mathrm{\AA }$ is close to $1.2\mathrm{T}$.Reuse & Permissions

Figure 35

(Color online) Arrows indicate the magnetic field produced by the rank-3 component of the OP of $\mathrm{Np}{\mathrm{O}}_2$ at $T=0$. As in Fig. 34, the Np ion at the center belongs to the sublattice characterized by a MMP moment having [1 1 1] as $C_3$ axis and the field has been calculated at distances of 0.7 and $0.9\mathrm{\AA }$ from the Np nucleus. The field direction in the plane perpendicular to [1 1 1] closely resembles that of Fig. 34 and is not shown.Reuse & Permissions

Figure 36

Results of NMR experiments on $\mathrm{Np}{\mathrm{O}}_2$. (Upper panel) Angular dependence of the ${}^{17}\mathrm{O}$-NMR line splitting ${\mu }_0\Delta H=(f_{\mathrm{res}}-f_0)∕\gamma$ in the ordered phase of $\mathrm{Np}{\mathrm{O}}_2$; $f_{\mathrm{res}}$ is the center of gravity of one of the three observed peaks, $f_0={\mu }_0\gamma H$, and $\gamma =5.7719\mathrm{MHz}∕\mathrm{T}$. The applied field $\mathbf{H}$ rotates in the $\left(1\overline{1}0\right)$ plane, and $\theta$ is the angle between $\mathbf{H}$ and the [0 0 1] direction. Data have been recorded with ${\mu }_{0}H=10.17\mathrm{T}$, with the sample kept at $17\mathrm{K}$. Solid lines are splitting calculated from hyperfine fields derived from field-induced magnetic dipoles and octupoles. (Lower panel) Principal oscillation frequency of the spin-echo decay at the ${\mathrm{O}}_{\alpha }^{\left(3\right)}$ sites of $\mathrm{Np}{\mathrm{O}}_2$ as a function of $\theta$. The dashed lines are calculated assuming the principal axis of the EFG tensor along [1 0 0], [0 1 0], and [0 0 1] for $\alpha =x$, $y$, and $z$, respectively. Data from 320.Reuse & Permissions

Figure 37

(Color online) Measured powder spin-flip inelastic neutron scattering spectrum for $Q=1.9{\mathrm{\AA }}^{-1}$ (black circles, $T=5\mathrm{K}$; triangles, $T=35\mathrm{K}$) (194). Line: Calculation convoluted with the $1.5\mathrm{meV}$ Gaussian resolution function. The left-hand peak involves excited states where typically one Np ion (indicated by an arrow) has its quadrupole reverted and vanishing ${\Gamma }_5$ MMP moment (i.e., the ${\Gamma }_4$ MF excited state in Fig. 32). The other peak(s) involves excited states where typically one Np ion has its ${\Gamma }_5$ MMP moment reverted.Reuse & Permissions

Figure 38

(Color online) Crystallographic and magnetic structure of $\mathrm{Pr}{\mathrm{O}}_2$. (a) Schematic representation of the distortion of the oxygen sublattice in the ordered phase of $\mathrm{Pr}{\mathrm{O}}_2$ (334). Arrows indicate the direction of the displacements. Pr ions at the vertices and at the face centers of the dashed cube are not shown. (b) Schematic representation of the magnetic dipolar order in $\mathrm{Pr}{\mathrm{O}}_2$ (107). The magnetic moment at each site is represented by its components along the cubic cell axes.Reuse & Permissions

Figure 39

(Color online) $\mathrm{Pr}{\mathrm{O}}_2$ inelastic neutron scattering spectra with $\mathrm{Ce}{\mathrm{O}}_2$ background subtracted, showing the variation of the broad band of vibronic scattering with temperature. The centers of the transition determined from the fits at the two temperatures are indicated by arrows. Data from 334.Reuse & Permissions

Figure 40

Phase diagram of ${\mathrm{U}}_{1-x}{\mathrm{Np}}_x{\mathrm{O}}_2$ solid solutions, showing the evolution of the low-$T$ phase from a transversal type-I magnetic-quadrupolar state to a longitudinal type-I octupolar-quadrupolar state. For intermediate values of $x$, incommensurate short-range magnetic ordering has been detected.Reuse & Permissions